Research ArticleChaos Time Series Prediction Based onMembrane Optimization Algorithms
Meng Li1 Liangzhong Yi2 Zheng Pei1 Zhisheng Gao1 and Hong Peng1
1School of Radio Management Technology Research Center Xihua University Chengdu 610039 China2School of Computer Science and Technology Sichuan Police College Luzhou 646000 China
Correspondence should be addressed to Zheng Pei pqyz263net
Received 26 June 2014 Revised 27 August 2014 Accepted 27 August 2014
Academic Editor Shifei Ding
Copyright copy 2015 Meng Li et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper puts forward a prediction model based on membrane computing optimization algorithm for chaos time series themodel optimizes simultaneously the parameters of phase space reconstruction (120591119898) and least squares support vectormachine (LS-SVM) (120574 120590) by using membrane computing optimization algorithm It is an important basis for spectrum management to predictaccurately the change trend of parameters in the electromagnetic environment which can help decisionmakers to adopt an optimalactionThen themodel presented in this paper is used to forecast band occupancy rate of frequencymodulation (FM) broadcastingband and interphone band To show the applicability and superiority of the proposed model this paper will compare the forecastmodel presented in it with conventional similar models The experimental results show that whether single-step prediction ormultistep prediction the proposed model performs best based on three error measures namely normalized mean square error(NMSE) root mean square error (RMSE) and mean absolute percentage error (MAPE)
1 Introduction
Chaotic time series is a kind of nonlinear dynamic phe-nomenon between certainty and randomness in which Lya-punov exponent is adopted to decide whether a time series ischaos or not that is the time series is chaotic if its Lyapunovexponent is greater than zero [1] Because it can be widelyapplied in real life such as in the network traffic earthquakeprediction and weather forecasting [2ndash5] chaotic time seriesprediction has become a hot spot and many interestingresults have been provided by a lot of researchers in recentyears [6 7]
Initially the traditional statistical fitting methods suchas autoregressive (AR) moving average (MA) and autore-gressive moving average (ARMA) models have been used inchaotic time series prediction However due to the inherentlinearity assumptions the above conventional mathematicaltools are not well suited for dealing with ill-defined anduncertain systems With the recent development in chaostheory numerous nonlinear systems have been identified tobe chaotic despite their random behaviors in which the local
model is an important method for chaotic time series themethod projected chaotic time series into amultidimensionalphase space which is then divided into several subspaceswhere the mapping function is approximated by meansof local approximation [8ndash10] Chaotic time series predic-tion based on nonlinear systems shows in general superiorperformance over the traditional statistical fitting methodsAs another alternative in dealing with nonlinear systemssupport vectormachine (SVM)was proposed in [11 12] basedon the principles of the statistical VC (Vapnik Chervonenkis)dimensional theory and structural risk minimization SVMcan better solve problems such as nonlinear dimensiondisaster and good performance for the small sample It willbe widely used in face recognition speech recognition [13ndash15] and so forth Because of its universal approximation capa-bilities recently least squares support vector machine (LS-SVM) [16] is applied to predict chaotic time series [17 18] Inthe model firstly the phase space reconstruction techniqueof chaotic theory is used to reconstruct the nonlinear datathen the least squares support vector machine regression isapplied in multidimensional phase space
Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 589093 14 pageshttpdxdoiorg1011552015589093
2 The Scientific World Journal
Chaos time Phase space Chaotic time series prediction and outputseries reconstruction (120591 m) LS-SVM (120574 120590)
Figure 1 The flow chart of chaotic time series prediction
Formally phase space reconstruction method is suc-ceeded by delay time and embedding dimension that is fora given time series 119909
1 119909
119899minus1 119909119899(119899 is the number of the
data) by using delay time and embedding dimension thephase points after reconstruction of the time series are 119883
119894=
[119909119894minus(119898minus1)120591
119909119894minus120591 119909119894] (119894 = 1 119872minus1119872) where 120591 is delay
time 119898 is embedding dimension and 119872 is the number ofphase space points [19] Accordingly the prediction value ofnext time 119905 + 1 based on LS-SVM can be expressed as
119909119905+1= 119891 (119883
119905) (1)
where 119891(sdot) is regression estimates functionIn applications there are two key problems in the pre-
diction model based on LS-SVM One is the choice of delaytime (120591) and embedding dimension (119898) in the process ofphase space reconstruction Another is the selection of kernelfunction and its relevant parameters [20] The phase spacereconstruction is used to express out the trace of the evolutionof chaotic time series without singular namely chaotic timeseries is projected into a multidimensional phase spaceKernel function is associated with learning and modelingfor the data set of phase space reconstruction to forecastaccurately the future value A large number of studies haveshown that the selection of delay time (120591) and embeddingdimension (119898) in phase space reconstruction has a directimpact on prediction results of chaotic time series [21] If 120591is too small in the delay neighbor element of the phase spacethere will be information redundancy If it is too big 120591 leadsto loss of information the track of signals will occur foldingphenomenon Similarly if 119898 is too small it is not enough toshow the detailed structure of chaotic systems If119898 is too bigthe calculationwill become complicated and cause the impactof noise
LS-SVM learning performance is largely dependent onthe choice of kernel function A large number of studies haveshown that with the lack of a priori knowledge of specificissues the overall performance of the radial basis kernelfunction model is better than other kernel function modelsand hence this paper selects the radial basis kernel functionas the kernel function of LS-SVM So in the model there aretwo parameters (cost factor (120574) and kernel parameter (120590)) thatneed to be identified cost factor 120574 is generally used to controlthe model complexity and compromise of approximationerror which is commonly in [1 1000] Kernel parameter120590 reflects the structure of high-dimensional feature spaceand affects the generalization ability of the system whenthe value of 120590 is too small it will occur over-learningphenomenon and poor generalization while the value of 120590 istoo large it will emerge less learning phenomenon the range
of 120590 is in [01 10000] [22] Currently there are mainly twoideas for optimization of the parameters of the phase spacereconstruction (120591 119898) and LS-SVM (120574 120590) One is that theparameters were optimized separately as shown in Figure 1in which firstly optimal delay time (120591) and embeddingdimension (119898) in the phase space are selected independently[19 23ndash28] or at the same time [27 29 30] then parameters120574 and 120590 of the LS-SVM are selected by gradient descentmethod [31] genetic algorithm (GA) [32] or particle swarmoptimization (PSO) [33] and so forth Another idea is tooptimize jointly the parameters that is the parameters (120591119898120574 120590) as a whole to carry on the optimization [34]
Membrane systems presented in [35] also called 119875
systems are bioinspired computing models belonging to abroader family of so-called biological or natural computing[36 37] which is a distributed and parallel computing modelwith hierarchy Recently membrane systems are widely usedin many fields such as in gasoline blending scheduling radaremitter signals analyzing and images skeletonizing [38ndash40]This paper uses a membrane computing (cell-like membranecomputing optimization algorithm) to optimize simultane-ously the parameters of the phase space reconstruction andLS-SVM (namely 120591119898 120574 and 120590) It is an important basis forspectrummanagement to predict accurately the change trendof parameters in the electromagnetic environment which canhelp decision makers to develop an optimal action programThen using themodel presented in this paper to predict bandoccupancy rate of frequency modulation (FM) broadcastingband and interphone band
The rest of this paper is organized as follows Section 2briefly reviews phase space reconstruction LS-SVM regres-sion and membrane computing Section 3 introduces specif-ically the algorithm of parameters joint optimization aboutprediction model In Section 4 the prediction model pre-sented in this paper will be used to predict the parametersof electromagnetic environment Conclusions are given inSection 5
2 Preliminaries
21 Phase Space Reconstruction and LS-SVM Regression Letthe time series be 119909
1 119909
119899minus1 119909119899 after the phase space
reconstruction the points in phase space can be expressed as[41 42]
where Φ(sdot) is a nonlinear mapping from the input space tothe feature space 120596 is a vector of weight coefficients and 119887 isa bias constant
The optimal hyperplane will be determined by the maxi-mum geometry interval Hence the LS-SVR problem can betransformed as follows [43]
min 119869 (120596 120577) =1
2120596119879120596 +
1
2120574
119897
sum119894=1
120577119894
st 119910119894= 120596119879Φ(119883119894) + 120577119894+ 119887 119894 = 1 119897 minus 1 119897
(4)
where 120577119894are the error variables and 120574 is hyperparameter
The process of finding the optimal decision function is todetermine the process parameters 120596 and 119887
Introducing Lagrange multipliers one can establishLagrange functions as follows
119871 =1
2120596119879120596 +
1
2120574
119897
sum119894=1
120577119894minus
119897
sum119894=1
120572119894(120596119879Φ(119883119894) + 120577119894+ 119887 minus 119910
119894) (5)
where 120572119894(119894 = 1 119897 minus 1 119897) are the Lagrange multiplier The
conditions for optimality are given by
120597119871
120597120596= 0
120597119871
120597119887= 0
120597119871
120597120577119894
= 0120597119871
120597120572119894
= 0 (6)
After elimination of the variables 120596 and 120577 a set of linearequations can be obtained
(0 1198681015840
119879
1198681015840 Ω + 120574minus1119868)(
119887
120572) = (
0
119884) (7)
where 1198681015840 = (1 1 1)119879 isin 119877119897 119868 isin 119877119897times119897 denotes a unitmatrix 120572 = (120572
1 120572
119897minus1 120572119897)119879 119884 = (119910
1 119910
119897minus1 119910119897)119879 and
Ω119894119895= Φ(119883
119894)119879Φ(119883
119895) (119894 119895 = 1 119897 minus 1 119897)
Then LS-SVM regression model is expressed as
119891 (119883) =
119897
sum119894=1
120572119894Φ(119883119894)119879
Φ(119883119895) + 119887 (8)
The mapping function Φ(sdot) can be paraphrased by a kernelfunction 119870(sdot sdot) because of the application of Mercerrsquos theo-rem whichmeans that119870(sdot sdot) (119894 = 1 119897minus1 119897) are any kernelfunctions satisfying the Mercer condition and the Mercerscondition has been applied
119870(119883119894 119883119895) = Φ (119883
119894)119879
Φ(119883119895) (119894 119895 = 1 119897 minus 1 119897) (9)
This finally results in the following LS-SVM model forfunction regression
119891 (119883) =
119899
sum119894=1
120572119894119870(119883119883
119894) + 119887 (10)
As shown in Figure 1 the prediction model of phasespace reconstruction and LS-SVM regression mainly has twosteps First select the delay time (120591) embedding dimension(119898) and LS-SVM parameters (120574 and 120590) The phase spacereconstruction technique is used to determine the trainingsample pairs based on the parameters 120591 and 119898 which aredetermined Assuming the time series is 119909
1 1199092 119909
119873+1
the training sample set of attributes is as follows
119894in the future Select the attribute sample of the
previous time as input in the phase space and use the trainedLS-SVMmodel to obtain the predicted value of the moment
22 Membrane Computing Membrane computing (namely119901 systems) arises as a newmodel of computation inspired bythe way that cells are structured into vesicles and abstractingthe chemical reactions taking place inside them [44] It is abranch of molecular computing that aims to develop modelsandparadigms that are biologicallymotivatedThere has beena flurry of research activities in this area in recent years[45] Because of the built-in nature of maximal parallelisminherent on the models 119901 systems have a great potential forimplementing massively concurrent systems in an efficientway that would allow us to solve currently intractable prob-lems
A membrane system with degree 119889 (119889 gt 0) can beexpressed as
prod = (119881 119879 119862 1205831198821 119882
119889 (1198771 1205881) (119877
119889 120588119889)) (12)
where 119881 is an alphabet whose elements are called objects119879 denotes the output alphabet 119862 is a catalyst which doesnot exhibit any change in the course of evolution but somereaction must have its participation 120583 is the membranestructure which can be shown by []119882
119894denotes multiple sets
of objects in the membrane structure and (119877119894 120588119894) are the set
of rules in which 119877119894and 120588119894denote rule and the priority of the
rule respectivelyIn general 119901 system contains three core elements mem-
brane structure object multiple sets and evolution rules Amembrane system with givenmembrane structure evolutionrules and decided objects will be performed in the form ofnondeterministic and maximum parallel for the evolutionrules When all the objects are exhausted the rules are nolonger executed the system downtime A typical membranesystem consists of cell-likemembranes placed inside a uniqueldquoskinrdquo membrane Multisets of objectsmdashusually strings ofsymbolsmdashand a set of evolution rules are placed inside theregions delimited by the membranes Each object can betransformed into other objects can pass through a mem-brane or can dissolve or create membranes The evolution
4 The Scientific World Journal
1
234
Envi
ronm
ent
Skin
h
d
Regions
Elementary membrane
Envi
ronm
ent
Rules
ObjectsElementary membrane
Membrane
h rarr hh
d rarr dy
Figure 2 Simple membrane structure diagram
between system configurations is done nondeterministicallyby applying the rules in parallel for all objects able to evolve[46] As shown in Figure 2 a simple membrane structurediagram can be shown by [[[]
3]2[]4]1 The skin membrane
which is the outermost membrane of this structure separatesthe system from its environment Several membranes each ofwhich defines a region are placed inside the skin membraneElementary membranes do not contain any membrane Eachregion forms a different compartment of the membranestructure and contains a multiset of objects or membranesWhere ℎ and 119889 denote objects ℎ rarr ℎℎ and 119889 rarr 119889119910 arerules
3 Parameters Joint Optimization AlgorithmBased on Membrane Computing
The optimization algorithm based on cell-like membranecomputing is an important branch of membrane computingIt is an intelligent optimization algorithm inspired by themechanism and the function of biological cells and basedon the existing framework of membrane computing Thesteps generally are membrane structure establishment theobjects generation and evolution and so forth Shown inFigure 3 is the structure of P-LSSVM prediction modelwith the initial objects as initial parameters of predictionmodel these parameters are substituted into the phase spacereconstruction and LS-SVM model Then parameters jointoptimization algorithm based on membrane computing isused to decide the best combination of parameters Algo-rithm specific process is as shown in Figure 3
31The Establishment of the Cellular Membrane Structure andthe Generation of Objects As shown in Figure 4 this paperadopts two layers structure for membrane a skin contains 119861basic membrane generate initial objects in each membraneGenerally 119901 system uses character or character string toencode real number encoding are adopted in here which canreduce the trouble of decode For instance119874 = (119900
1 1199002 1199003 1199004)
where 119874 is an object and 1199001 1199002 1199003 and 119900
4denote 120591 119898
120574 and 120590 respectively We see each object as a solutionof the optimization problem Evolution of each membraneaccording to its own rules all the membrane are executed in
parallelThe final optimal results are output through the skinthat is the optimal solution
32 Construct the Fitness Function The goal of cell-likemembrane computing optimization algorithm is to find themost suitable combination of parameters (120591 119898 120574 and 120590)in order to establish the optimal forecasting model In thispaper we used the rootmean square prediction error (RMSE)to construct the fitness function That is 119891 = 1RMSERMSE = radic(1119885)sum119885
119894=1(119910119894minus 119910119894)2 where119885 denotes the number
of prediction points and 119910119894 119910119894represent the real values and
predicted values respectively
33 Operation Rules The basic rules of cellular membranecomputing optimization method are selection crossovermutation and communication [47] The specific form is asfollows
(1) Selection rule the rule of selection copies the objectsto the next generation according to the size of the string Thesize of the string is not the three-dimensional size of particlesin biological cells but the value of the fitness function Herewheel disk method is used to select objects to the nextgeneration
(2) Crossover rule for any two objects 119874119894
=
(1199001198941 1199001198942 1199001198943 1199001198944) and 119874
119895= (119900
1198951 1199001198952 1199001198953 1199001198954) use cross
rule to obtain new object 119874119896= (1199001198961 1199001198962 1199001198963 1199001198964)
1199001198961= 119903 times 119900
1198941+ (1 minus 119903) times 119900
1198951
1199001198962= 119903 times 119900
1198942+ (1 minus 119903) times 119900
1198952
1199001198963= 119903 times 119900
1198943+ (1 minus 119903) times 119900
1198953
1199001198964= 119903 times 119900
1198944+ (1 minus 119903) times 119900
1198954
(13)
where 119903 is a random number in (0 1)(3) Mutation rule in evolution according to a certain
mutation probability replace the worst 119905 objects with ran-domly generated 119905 objects Mutating rule is described asfollows
where []119894denotes membrane 119894 119902min 1 119902min 2 119902min 119905 are 119905
objects where fitness is the smallest in membrane 119894 and119902init1 119902init2 119902init119905 are randomly generated 119905 objects
(4)Communication rule each membrane 119894will transportthe best 119871 objects out of the membrane while the best 119871objects of foreign membrane are brought into the membrane119894 [48] This rule can be expressed as follows
119877119894communication = 119877119894communication1 cup 119877119894communication2
Parameters joint optimization based on membrane computing
Phase space reconstruction LS-SVM Chaotic time series
prediction and output
120591 m 120574 120590
Figure 3 The structure of P-LSSVM prediction model
0
1 2 B
Elementary membrane
Skin
O11 O
12 O
1G O2
1 O22 O
2G OB
1 OB2 O
BG
O01 O
02 O
0G
middot middot middot
Figure 4 Membrane structure
119877119894communication2 []119894 119902
1015840
max 1 1199021015840
max 2 1199021015840
max119871
997888rarr [1199021015840
max 1 1199021015840
max 2 1199021015840
max119871]119894
(15)
where []119894denotesmembrane 119894 119902max 1 119902max 2 119902max119871 are the
best 119871 objects in membrane 119894 and 1199021015840max 1 1199021015840
max 2 1199021015840
max119871 arethe best 119871 objects out of membrane 119894
34 Parameters Joint Optimization Algorithm Specific Stepsin P-LSSVM Model First of all generate the initial objectsas initial parameters of prediction model then apply evo-lutionary rules to evolve until the stop conditions are metall membranes are operating in parallel Finally output thefitness of the best object by the skin membrane that is theoptimal solution Specifically consider the following
Step 1 Initialize parameters and build cellular membranestructure
(1) Initialization the number of elementary membranesis 119861 the number of objects in eachmembrane is119866 the largestnumber of iterations is Max119879 crossover probability is 119875
119888
mutation probability is 119875119898 and the current iteration number
is 119896 and so forth(2) Create membrane structure as shown in Figure 4
generating randomly 119866 objects in each membrane eachobject represents a set of parametersrsquo combination expressedin decimal coding
Step 2 Optimize each membrane in turn(1) Every object in the membrane as a set of parameters
(120591 119898 120574 and 120590) of P-LSSVM model calculate the fitness ofeach object by training data and save the optimal object andits fitness
(2) Use the reproduction crossover and mutation rulesto evolve
Step 3 Make use of communication rules each membranewill transport the best 119871 objects out of the membrane at thesame time the best 119871 objects outside the membrane will beshipped into the membrane
Step 4 Determine whether the termination condition issatisfied that is whether it reaches the maximum numberof iterations when the number of iterations is less than themaximum number of iterations to continue iteration or stopiteration
Step 5 Theoptimal object is output from the skinmembrane
4 Electromagnetic Environment ParametersPredictions Based on P-LSSVM Model
Electromagnetic spectrum is a fundamental strategicresource to support the national economy and nationaldefense construction along with the rapid development ofinformation technology and it is widely used in various fieldssuch as economic development national defense constru-ction and social life [49] Strategic value and basic role
6 The Scientific World Journal
increasingly highlight in the electromagnetic spectrum withfrequency contradictions increasingly prominent betweencountries departments and military and space businesses[50] It is an important basis for spectrum management tocontrol comprehensively the change trend of parametersin the electromagnetic environment of country or region[51] It is the basis to master the frequency informationfor the frequency planning frequency allocation andsharing service frequency recovery work The situationof electromagnetic environment can be reflected by theelectromagnetic environment indicator parameters theseparameters mainly include band occupancy rate channeloccupancy rate large-signal ratio frequency offset and thefield strength A large number of experiment shown that timeseries data with chaotic in the electromagnetic environmentHence we used the proposed predictionmodel to predict theindicator parameters of the electromagnetic environmentThe experimental results show that the prediction modelproposed in this paper is reasonable and effective
Here we chose the band occupancy rate to do the testBand occupancy rate is calculated as follows extracting allthe signal points in the spectrum data the signals pointare merged with distance less than bandwidth by the belowformula to calculate the band occupancy rate (OccupyFreband)
OccupyFreband =119878119899lowast 119865119908
119865end minus 119865begin (16)
where 119878119899denotes the total number of signals judged 119865
119908is
necessary bandwidth in this band for the type of specifiedbusiness 119865begin is the start frequency point and 119865end is thecutoff frequency point
41 Experimental Data Sources In this paper we adoptdigital receiver EM100whichwas provided byGermanRohdeamp Schwarz Company and fixed radio monitoring stationof Xihua University to collect data for the experiment Wecollected data including frequency modulation (FM) broad-casting band and interphone band As shown in Figures 5 and6 in which the vertical axis denotes band occupancy rate thehorizontal axis represents the collection time and left pictureshows the data of band occupancy rate in FM broadcastingband we collected for 680 hours that is obtaining 680pieces of data Right figure indicates acquisition data ofband occupancy rate in interphone band we continuouslycollected for 187 hours that is gaining 187 pieces of data Inorder to facilitate narration here we put the band data of FMbroadcasting band and interphone band denoted by ldquodataset 1rdquo and ldquodata set 2rdquo respectively Use the method of smallamount of data to calculate the maximum Lyapunov indexof two groups of data which are 120582
1= 0126 and 120582
2= 014
respectively which show the time series with chaos
42 Data Preprocessing This paper mainly uses the Grubbscriteria to deal with the abnormal data the method is asfollows let 119902(ℎ 119889) be the sequence of the collected datawith the time interval between two data collections 119905 = 1
hour where ℎ = 0 22 23 denote 24 hours of a day119889 = 1 119867 minus 1119867 represents date code in total days of data
025035045055065075
1 71 141 211 281 351 421 491 561 631
Occ
upan
cy ra
te
t (hour)
Figure 5 Band occupancy rate data of FM radio band
000500070009001100130015001700190021
1 21 41 61 81 101 121 141 161 181
Occ
upan
cy ra
te
t (hour)
Figure 6 Band occupancy rate data of interphone band
collection119867 and 119902 denotes the collected data Using data setdenoted by 119876 = 119902
1 1199022 119902
119905 for each time point ℎ we
can get the expectation and variance of data sequence 119902(ℎ 119889)the formula is as follows
119864 (ℎ) =1
119878
119878
sum119896=1
119902 (ℎ 119889)
119863 (ℎ) = 1205902
119894=1
119878
119878
sum119896=1
[119902 (ℎ 119889) minus 119864 (ℎ)]2
(17)
where 119878 denotes the length of a unitAccording to the above two formulas combined with
Grubbs criteria if the sample point meet to1003816100381610038161003816119902 (ℎ 119889) minus 119864 (ℎ)
1003816100381610038161003816 ge 119866 (119899 120576) 120590119894 (18)
The sample point should be removed where 119866(119899 120576) is thecritical value of Grubbs criteria it can be obtained by lookingat Grubbs table 120576 denotes the significance level usuallysignificance level 120576 = 005
The Grubbs criteria are used to deal with ldquodata set 1rdquo andldquodata set 2rdquo respectively For the ldquodata set 1rdquo after processingwith Grubbs criteria the remaining 653 pieces of data we usethe front 600 pieces of data as the training data determiningthe best parameters combination and the surplus 53 pieces ofdata as test data testing the prediction accuracy of themodelFor the ldquodata set 2rdquo after processing with Grubbs criteria theremaining 180 pieces of data we use the front 150 pieces ofdata as the training data determining the best parameterscombination and the surplus 30 pieces of data as test datatesting the prediction accuracy of the model
43 Reference Model and Evaluation Criteria In order toverify the validity of the model this paper will comparethe prediction model (P-LSSVM) proposed in this paperwith conventional similar prediction model The first ref-erence model is the parameters joint optimization based
The Scientific World Journal 7
on genetic algorithm for chaos time series prediction (GA-LSSVM) [34] The second reference model uses the mutualinformation method and Cao method to get the best delaytime 120591 and embedding dimension 119898 respectively And thenuse grid search method to obtain LS-SVM parameters (120574and 120590) (denoted as M-C-LSSVM) [19] The third referencemodel uses the mutual information method and false nearestneighbor method to calculate the optimal delay time 120591 andembedding dimension119898 respectively And then use geneticalgorithm to get the optimal combination parameters of LS-SVM (120574 and 120590) (denoted as M-F-LSSVM) [27] The fourthreference model uses C-Cmethod to seek simultaneously thebest delay time 120591 and embedding dimension 119898 Then theoptimal parameters of LS-SVM (120574 and 120590) by using geneticalgorithm (denoted as C-C-LSSVM) [27 52]
Meanwhile this paper uses three evaluation criterianormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)NMSE RMSE and MAPE are defined respectively as fol-lows
where 119885 is the number of prediction points 119910 is the averagevalue and 119910
119894and 119910
119894denote the real value and the predicted
value of 119894th point respectively
44 Experimental Results In this paper the scope of parame-ters 120591119898 120574 and 120590 is [1 8] [3 17] [1 1000] and [01 10000]respectively In the process of evolution the other parametersare set as follows the number of elementary membranes119861 = 20 the number of objects in each membrane 119866 =
100 evolution algebra Max119879 = 1000 crossover probability119875119888= 085 and mutation probability 119875
119898= 005 The optimal
parameters combinations of eachmodel are shown in Tables 1and 2
441 Single-Step Prediction Selecting the first point as inputto obtain first predicted value then the real value of the firstpoint is added to the historical data predicting the next pointAnd so obtain the predicted value of all points Predictionresults of five models are shown in Tables 3 4 5 6 7 and 8and Figures 7 8 9 and 10
442 Multistep Forecast Selecting a point as input to obtainpredicted value then the prediction value of the first point is
Table 1The optimal parameters combination of fivemodels for FMbroadcasting band
added to the historical data predicting next point And soobtain the predicted value of all points Predicted results offive models are shown in Tables 9 10 11 12 13 and 14 andFigures 11 12 13 and 14
45 Analysis of Experimental Results The optimal parame-ters combinations of five models for FM broadcasting bandand interphone band are shown inTables 1 and 2 respectivelyAs seen from experimental results we can find that theparameters 120591 119898 120574 and 120590 are very sensitive to predictionaccuracy the optimal parameters combination is P-LSSVMmodel FM broadcasting bands are 7 14 1631 and 66475Interphone bands are 3 8 1709 and 21621 It can be seenfrom predicted results diagram (Figures 7 to 14) that whethersingle-step prediction or multistep prediction five models getvery good results However the P-LSSVM model predictscurve best fit to real data and other curves relative deviationfrom far away For five prediction models respectively run10 times computing the maximum minimum mean andvariance of error As can be seen from predicted results inTables 3 to 14 three kinds of models evaluation standard areRMSE NMSE andMAPE the model proposed in this paperis the minimum This shows that not only is the P-LSSVMmodel reasonable and correct but prediction accuracy is alsoenhanced
Comparing single-step prediction with multistep predic-tion it can be found that the error of multistep predictionis larger than the single-step prediction indicating that theeffect of single-step prediction is better than multistep pre-diction The reason is that errors exist in every step andthe accumulation of error will lead to decline in the overallprediction accuracy
5 Conclusion
Modeling and prediction of chaotic time series has become ahot spot in the research field of the chaotic signal processing
8 The Scientific World Journal
Table 3 Five models predicted error based on RMSE for FM broadcasting band
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
Chaos time Phase space Chaotic time series prediction and outputseries reconstruction (120591 m) LS-SVM (120574 120590)
Figure 1 The flow chart of chaotic time series prediction
Formally phase space reconstruction method is suc-ceeded by delay time and embedding dimension that is fora given time series 119909
1 119909
119899minus1 119909119899(119899 is the number of the
data) by using delay time and embedding dimension thephase points after reconstruction of the time series are 119883
119894=
[119909119894minus(119898minus1)120591
119909119894minus120591 119909119894] (119894 = 1 119872minus1119872) where 120591 is delay
time 119898 is embedding dimension and 119872 is the number ofphase space points [19] Accordingly the prediction value ofnext time 119905 + 1 based on LS-SVM can be expressed as
119909119905+1= 119891 (119883
119905) (1)
where 119891(sdot) is regression estimates functionIn applications there are two key problems in the pre-
diction model based on LS-SVM One is the choice of delaytime (120591) and embedding dimension (119898) in the process ofphase space reconstruction Another is the selection of kernelfunction and its relevant parameters [20] The phase spacereconstruction is used to express out the trace of the evolutionof chaotic time series without singular namely chaotic timeseries is projected into a multidimensional phase spaceKernel function is associated with learning and modelingfor the data set of phase space reconstruction to forecastaccurately the future value A large number of studies haveshown that the selection of delay time (120591) and embeddingdimension (119898) in phase space reconstruction has a directimpact on prediction results of chaotic time series [21] If 120591is too small in the delay neighbor element of the phase spacethere will be information redundancy If it is too big 120591 leadsto loss of information the track of signals will occur foldingphenomenon Similarly if 119898 is too small it is not enough toshow the detailed structure of chaotic systems If119898 is too bigthe calculationwill become complicated and cause the impactof noise
LS-SVM learning performance is largely dependent onthe choice of kernel function A large number of studies haveshown that with the lack of a priori knowledge of specificissues the overall performance of the radial basis kernelfunction model is better than other kernel function modelsand hence this paper selects the radial basis kernel functionas the kernel function of LS-SVM So in the model there aretwo parameters (cost factor (120574) and kernel parameter (120590)) thatneed to be identified cost factor 120574 is generally used to controlthe model complexity and compromise of approximationerror which is commonly in [1 1000] Kernel parameter120590 reflects the structure of high-dimensional feature spaceand affects the generalization ability of the system whenthe value of 120590 is too small it will occur over-learningphenomenon and poor generalization while the value of 120590 istoo large it will emerge less learning phenomenon the range
of 120590 is in [01 10000] [22] Currently there are mainly twoideas for optimization of the parameters of the phase spacereconstruction (120591 119898) and LS-SVM (120574 120590) One is that theparameters were optimized separately as shown in Figure 1in which firstly optimal delay time (120591) and embeddingdimension (119898) in the phase space are selected independently[19 23ndash28] or at the same time [27 29 30] then parameters120574 and 120590 of the LS-SVM are selected by gradient descentmethod [31] genetic algorithm (GA) [32] or particle swarmoptimization (PSO) [33] and so forth Another idea is tooptimize jointly the parameters that is the parameters (120591119898120574 120590) as a whole to carry on the optimization [34]
Membrane systems presented in [35] also called 119875
systems are bioinspired computing models belonging to abroader family of so-called biological or natural computing[36 37] which is a distributed and parallel computing modelwith hierarchy Recently membrane systems are widely usedin many fields such as in gasoline blending scheduling radaremitter signals analyzing and images skeletonizing [38ndash40]This paper uses a membrane computing (cell-like membranecomputing optimization algorithm) to optimize simultane-ously the parameters of the phase space reconstruction andLS-SVM (namely 120591119898 120574 and 120590) It is an important basis forspectrummanagement to predict accurately the change trendof parameters in the electromagnetic environment which canhelp decision makers to develop an optimal action programThen using themodel presented in this paper to predict bandoccupancy rate of frequency modulation (FM) broadcastingband and interphone band
The rest of this paper is organized as follows Section 2briefly reviews phase space reconstruction LS-SVM regres-sion and membrane computing Section 3 introduces specif-ically the algorithm of parameters joint optimization aboutprediction model In Section 4 the prediction model pre-sented in this paper will be used to predict the parametersof electromagnetic environment Conclusions are given inSection 5
2 Preliminaries
21 Phase Space Reconstruction and LS-SVM Regression Letthe time series be 119909
1 119909
119899minus1 119909119899 after the phase space
reconstruction the points in phase space can be expressed as[41 42]
where Φ(sdot) is a nonlinear mapping from the input space tothe feature space 120596 is a vector of weight coefficients and 119887 isa bias constant
The optimal hyperplane will be determined by the maxi-mum geometry interval Hence the LS-SVR problem can betransformed as follows [43]
min 119869 (120596 120577) =1
2120596119879120596 +
1
2120574
119897
sum119894=1
120577119894
st 119910119894= 120596119879Φ(119883119894) + 120577119894+ 119887 119894 = 1 119897 minus 1 119897
(4)
where 120577119894are the error variables and 120574 is hyperparameter
The process of finding the optimal decision function is todetermine the process parameters 120596 and 119887
Introducing Lagrange multipliers one can establishLagrange functions as follows
119871 =1
2120596119879120596 +
1
2120574
119897
sum119894=1
120577119894minus
119897
sum119894=1
120572119894(120596119879Φ(119883119894) + 120577119894+ 119887 minus 119910
119894) (5)
where 120572119894(119894 = 1 119897 minus 1 119897) are the Lagrange multiplier The
conditions for optimality are given by
120597119871
120597120596= 0
120597119871
120597119887= 0
120597119871
120597120577119894
= 0120597119871
120597120572119894
= 0 (6)
After elimination of the variables 120596 and 120577 a set of linearequations can be obtained
(0 1198681015840
119879
1198681015840 Ω + 120574minus1119868)(
119887
120572) = (
0
119884) (7)
where 1198681015840 = (1 1 1)119879 isin 119877119897 119868 isin 119877119897times119897 denotes a unitmatrix 120572 = (120572
1 120572
119897minus1 120572119897)119879 119884 = (119910
1 119910
119897minus1 119910119897)119879 and
Ω119894119895= Φ(119883
119894)119879Φ(119883
119895) (119894 119895 = 1 119897 minus 1 119897)
Then LS-SVM regression model is expressed as
119891 (119883) =
119897
sum119894=1
120572119894Φ(119883119894)119879
Φ(119883119895) + 119887 (8)
The mapping function Φ(sdot) can be paraphrased by a kernelfunction 119870(sdot sdot) because of the application of Mercerrsquos theo-rem whichmeans that119870(sdot sdot) (119894 = 1 119897minus1 119897) are any kernelfunctions satisfying the Mercer condition and the Mercerscondition has been applied
119870(119883119894 119883119895) = Φ (119883
119894)119879
Φ(119883119895) (119894 119895 = 1 119897 minus 1 119897) (9)
This finally results in the following LS-SVM model forfunction regression
119891 (119883) =
119899
sum119894=1
120572119894119870(119883119883
119894) + 119887 (10)
As shown in Figure 1 the prediction model of phasespace reconstruction and LS-SVM regression mainly has twosteps First select the delay time (120591) embedding dimension(119898) and LS-SVM parameters (120574 and 120590) The phase spacereconstruction technique is used to determine the trainingsample pairs based on the parameters 120591 and 119898 which aredetermined Assuming the time series is 119909
1 1199092 119909
119873+1
the training sample set of attributes is as follows
119894in the future Select the attribute sample of the
previous time as input in the phase space and use the trainedLS-SVMmodel to obtain the predicted value of the moment
22 Membrane Computing Membrane computing (namely119901 systems) arises as a newmodel of computation inspired bythe way that cells are structured into vesicles and abstractingthe chemical reactions taking place inside them [44] It is abranch of molecular computing that aims to develop modelsandparadigms that are biologicallymotivatedThere has beena flurry of research activities in this area in recent years[45] Because of the built-in nature of maximal parallelisminherent on the models 119901 systems have a great potential forimplementing massively concurrent systems in an efficientway that would allow us to solve currently intractable prob-lems
A membrane system with degree 119889 (119889 gt 0) can beexpressed as
prod = (119881 119879 119862 1205831198821 119882
119889 (1198771 1205881) (119877
119889 120588119889)) (12)
where 119881 is an alphabet whose elements are called objects119879 denotes the output alphabet 119862 is a catalyst which doesnot exhibit any change in the course of evolution but somereaction must have its participation 120583 is the membranestructure which can be shown by []119882
119894denotes multiple sets
of objects in the membrane structure and (119877119894 120588119894) are the set
of rules in which 119877119894and 120588119894denote rule and the priority of the
rule respectivelyIn general 119901 system contains three core elements mem-
brane structure object multiple sets and evolution rules Amembrane system with givenmembrane structure evolutionrules and decided objects will be performed in the form ofnondeterministic and maximum parallel for the evolutionrules When all the objects are exhausted the rules are nolonger executed the system downtime A typical membranesystem consists of cell-likemembranes placed inside a uniqueldquoskinrdquo membrane Multisets of objectsmdashusually strings ofsymbolsmdashand a set of evolution rules are placed inside theregions delimited by the membranes Each object can betransformed into other objects can pass through a mem-brane or can dissolve or create membranes The evolution
4 The Scientific World Journal
1
234
Envi
ronm
ent
Skin
h
d
Regions
Elementary membrane
Envi
ronm
ent
Rules
ObjectsElementary membrane
Membrane
h rarr hh
d rarr dy
Figure 2 Simple membrane structure diagram
between system configurations is done nondeterministicallyby applying the rules in parallel for all objects able to evolve[46] As shown in Figure 2 a simple membrane structurediagram can be shown by [[[]
3]2[]4]1 The skin membrane
which is the outermost membrane of this structure separatesthe system from its environment Several membranes each ofwhich defines a region are placed inside the skin membraneElementary membranes do not contain any membrane Eachregion forms a different compartment of the membranestructure and contains a multiset of objects or membranesWhere ℎ and 119889 denote objects ℎ rarr ℎℎ and 119889 rarr 119889119910 arerules
3 Parameters Joint Optimization AlgorithmBased on Membrane Computing
The optimization algorithm based on cell-like membranecomputing is an important branch of membrane computingIt is an intelligent optimization algorithm inspired by themechanism and the function of biological cells and basedon the existing framework of membrane computing Thesteps generally are membrane structure establishment theobjects generation and evolution and so forth Shown inFigure 3 is the structure of P-LSSVM prediction modelwith the initial objects as initial parameters of predictionmodel these parameters are substituted into the phase spacereconstruction and LS-SVM model Then parameters jointoptimization algorithm based on membrane computing isused to decide the best combination of parameters Algo-rithm specific process is as shown in Figure 3
31The Establishment of the Cellular Membrane Structure andthe Generation of Objects As shown in Figure 4 this paperadopts two layers structure for membrane a skin contains 119861basic membrane generate initial objects in each membraneGenerally 119901 system uses character or character string toencode real number encoding are adopted in here which canreduce the trouble of decode For instance119874 = (119900
1 1199002 1199003 1199004)
where 119874 is an object and 1199001 1199002 1199003 and 119900
4denote 120591 119898
120574 and 120590 respectively We see each object as a solutionof the optimization problem Evolution of each membraneaccording to its own rules all the membrane are executed in
parallelThe final optimal results are output through the skinthat is the optimal solution
32 Construct the Fitness Function The goal of cell-likemembrane computing optimization algorithm is to find themost suitable combination of parameters (120591 119898 120574 and 120590)in order to establish the optimal forecasting model In thispaper we used the rootmean square prediction error (RMSE)to construct the fitness function That is 119891 = 1RMSERMSE = radic(1119885)sum119885
119894=1(119910119894minus 119910119894)2 where119885 denotes the number
of prediction points and 119910119894 119910119894represent the real values and
predicted values respectively
33 Operation Rules The basic rules of cellular membranecomputing optimization method are selection crossovermutation and communication [47] The specific form is asfollows
(1) Selection rule the rule of selection copies the objectsto the next generation according to the size of the string Thesize of the string is not the three-dimensional size of particlesin biological cells but the value of the fitness function Herewheel disk method is used to select objects to the nextgeneration
(2) Crossover rule for any two objects 119874119894
=
(1199001198941 1199001198942 1199001198943 1199001198944) and 119874
119895= (119900
1198951 1199001198952 1199001198953 1199001198954) use cross
rule to obtain new object 119874119896= (1199001198961 1199001198962 1199001198963 1199001198964)
1199001198961= 119903 times 119900
1198941+ (1 minus 119903) times 119900
1198951
1199001198962= 119903 times 119900
1198942+ (1 minus 119903) times 119900
1198952
1199001198963= 119903 times 119900
1198943+ (1 minus 119903) times 119900
1198953
1199001198964= 119903 times 119900
1198944+ (1 minus 119903) times 119900
1198954
(13)
where 119903 is a random number in (0 1)(3) Mutation rule in evolution according to a certain
mutation probability replace the worst 119905 objects with ran-domly generated 119905 objects Mutating rule is described asfollows
where []119894denotes membrane 119894 119902min 1 119902min 2 119902min 119905 are 119905
objects where fitness is the smallest in membrane 119894 and119902init1 119902init2 119902init119905 are randomly generated 119905 objects
(4)Communication rule each membrane 119894will transportthe best 119871 objects out of the membrane while the best 119871objects of foreign membrane are brought into the membrane119894 [48] This rule can be expressed as follows
119877119894communication = 119877119894communication1 cup 119877119894communication2
Parameters joint optimization based on membrane computing
Phase space reconstruction LS-SVM Chaotic time series
prediction and output
120591 m 120574 120590
Figure 3 The structure of P-LSSVM prediction model
0
1 2 B
Elementary membrane
Skin
O11 O
12 O
1G O2
1 O22 O
2G OB
1 OB2 O
BG
O01 O
02 O
0G
middot middot middot
Figure 4 Membrane structure
119877119894communication2 []119894 119902
1015840
max 1 1199021015840
max 2 1199021015840
max119871
997888rarr [1199021015840
max 1 1199021015840
max 2 1199021015840
max119871]119894
(15)
where []119894denotesmembrane 119894 119902max 1 119902max 2 119902max119871 are the
best 119871 objects in membrane 119894 and 1199021015840max 1 1199021015840
max 2 1199021015840
max119871 arethe best 119871 objects out of membrane 119894
34 Parameters Joint Optimization Algorithm Specific Stepsin P-LSSVM Model First of all generate the initial objectsas initial parameters of prediction model then apply evo-lutionary rules to evolve until the stop conditions are metall membranes are operating in parallel Finally output thefitness of the best object by the skin membrane that is theoptimal solution Specifically consider the following
Step 1 Initialize parameters and build cellular membranestructure
(1) Initialization the number of elementary membranesis 119861 the number of objects in eachmembrane is119866 the largestnumber of iterations is Max119879 crossover probability is 119875
119888
mutation probability is 119875119898 and the current iteration number
is 119896 and so forth(2) Create membrane structure as shown in Figure 4
generating randomly 119866 objects in each membrane eachobject represents a set of parametersrsquo combination expressedin decimal coding
Step 2 Optimize each membrane in turn(1) Every object in the membrane as a set of parameters
(120591 119898 120574 and 120590) of P-LSSVM model calculate the fitness ofeach object by training data and save the optimal object andits fitness
(2) Use the reproduction crossover and mutation rulesto evolve
Step 3 Make use of communication rules each membranewill transport the best 119871 objects out of the membrane at thesame time the best 119871 objects outside the membrane will beshipped into the membrane
Step 4 Determine whether the termination condition issatisfied that is whether it reaches the maximum numberof iterations when the number of iterations is less than themaximum number of iterations to continue iteration or stopiteration
Step 5 Theoptimal object is output from the skinmembrane
4 Electromagnetic Environment ParametersPredictions Based on P-LSSVM Model
Electromagnetic spectrum is a fundamental strategicresource to support the national economy and nationaldefense construction along with the rapid development ofinformation technology and it is widely used in various fieldssuch as economic development national defense constru-ction and social life [49] Strategic value and basic role
6 The Scientific World Journal
increasingly highlight in the electromagnetic spectrum withfrequency contradictions increasingly prominent betweencountries departments and military and space businesses[50] It is an important basis for spectrum management tocontrol comprehensively the change trend of parametersin the electromagnetic environment of country or region[51] It is the basis to master the frequency informationfor the frequency planning frequency allocation andsharing service frequency recovery work The situationof electromagnetic environment can be reflected by theelectromagnetic environment indicator parameters theseparameters mainly include band occupancy rate channeloccupancy rate large-signal ratio frequency offset and thefield strength A large number of experiment shown that timeseries data with chaotic in the electromagnetic environmentHence we used the proposed predictionmodel to predict theindicator parameters of the electromagnetic environmentThe experimental results show that the prediction modelproposed in this paper is reasonable and effective
Here we chose the band occupancy rate to do the testBand occupancy rate is calculated as follows extracting allthe signal points in the spectrum data the signals pointare merged with distance less than bandwidth by the belowformula to calculate the band occupancy rate (OccupyFreband)
OccupyFreband =119878119899lowast 119865119908
119865end minus 119865begin (16)
where 119878119899denotes the total number of signals judged 119865
119908is
necessary bandwidth in this band for the type of specifiedbusiness 119865begin is the start frequency point and 119865end is thecutoff frequency point
41 Experimental Data Sources In this paper we adoptdigital receiver EM100whichwas provided byGermanRohdeamp Schwarz Company and fixed radio monitoring stationof Xihua University to collect data for the experiment Wecollected data including frequency modulation (FM) broad-casting band and interphone band As shown in Figures 5 and6 in which the vertical axis denotes band occupancy rate thehorizontal axis represents the collection time and left pictureshows the data of band occupancy rate in FM broadcastingband we collected for 680 hours that is obtaining 680pieces of data Right figure indicates acquisition data ofband occupancy rate in interphone band we continuouslycollected for 187 hours that is gaining 187 pieces of data Inorder to facilitate narration here we put the band data of FMbroadcasting band and interphone band denoted by ldquodataset 1rdquo and ldquodata set 2rdquo respectively Use the method of smallamount of data to calculate the maximum Lyapunov indexof two groups of data which are 120582
1= 0126 and 120582
2= 014
respectively which show the time series with chaos
42 Data Preprocessing This paper mainly uses the Grubbscriteria to deal with the abnormal data the method is asfollows let 119902(ℎ 119889) be the sequence of the collected datawith the time interval between two data collections 119905 = 1
hour where ℎ = 0 22 23 denote 24 hours of a day119889 = 1 119867 minus 1119867 represents date code in total days of data
025035045055065075
1 71 141 211 281 351 421 491 561 631
Occ
upan
cy ra
te
t (hour)
Figure 5 Band occupancy rate data of FM radio band
000500070009001100130015001700190021
1 21 41 61 81 101 121 141 161 181
Occ
upan
cy ra
te
t (hour)
Figure 6 Band occupancy rate data of interphone band
collection119867 and 119902 denotes the collected data Using data setdenoted by 119876 = 119902
1 1199022 119902
119905 for each time point ℎ we
can get the expectation and variance of data sequence 119902(ℎ 119889)the formula is as follows
119864 (ℎ) =1
119878
119878
sum119896=1
119902 (ℎ 119889)
119863 (ℎ) = 1205902
119894=1
119878
119878
sum119896=1
[119902 (ℎ 119889) minus 119864 (ℎ)]2
(17)
where 119878 denotes the length of a unitAccording to the above two formulas combined with
Grubbs criteria if the sample point meet to1003816100381610038161003816119902 (ℎ 119889) minus 119864 (ℎ)
1003816100381610038161003816 ge 119866 (119899 120576) 120590119894 (18)
The sample point should be removed where 119866(119899 120576) is thecritical value of Grubbs criteria it can be obtained by lookingat Grubbs table 120576 denotes the significance level usuallysignificance level 120576 = 005
The Grubbs criteria are used to deal with ldquodata set 1rdquo andldquodata set 2rdquo respectively For the ldquodata set 1rdquo after processingwith Grubbs criteria the remaining 653 pieces of data we usethe front 600 pieces of data as the training data determiningthe best parameters combination and the surplus 53 pieces ofdata as test data testing the prediction accuracy of themodelFor the ldquodata set 2rdquo after processing with Grubbs criteria theremaining 180 pieces of data we use the front 150 pieces ofdata as the training data determining the best parameterscombination and the surplus 30 pieces of data as test datatesting the prediction accuracy of the model
43 Reference Model and Evaluation Criteria In order toverify the validity of the model this paper will comparethe prediction model (P-LSSVM) proposed in this paperwith conventional similar prediction model The first ref-erence model is the parameters joint optimization based
The Scientific World Journal 7
on genetic algorithm for chaos time series prediction (GA-LSSVM) [34] The second reference model uses the mutualinformation method and Cao method to get the best delaytime 120591 and embedding dimension 119898 respectively And thenuse grid search method to obtain LS-SVM parameters (120574and 120590) (denoted as M-C-LSSVM) [19] The third referencemodel uses the mutual information method and false nearestneighbor method to calculate the optimal delay time 120591 andembedding dimension119898 respectively And then use geneticalgorithm to get the optimal combination parameters of LS-SVM (120574 and 120590) (denoted as M-F-LSSVM) [27] The fourthreference model uses C-Cmethod to seek simultaneously thebest delay time 120591 and embedding dimension 119898 Then theoptimal parameters of LS-SVM (120574 and 120590) by using geneticalgorithm (denoted as C-C-LSSVM) [27 52]
Meanwhile this paper uses three evaluation criterianormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)NMSE RMSE and MAPE are defined respectively as fol-lows
where 119885 is the number of prediction points 119910 is the averagevalue and 119910
119894and 119910
119894denote the real value and the predicted
value of 119894th point respectively
44 Experimental Results In this paper the scope of parame-ters 120591119898 120574 and 120590 is [1 8] [3 17] [1 1000] and [01 10000]respectively In the process of evolution the other parametersare set as follows the number of elementary membranes119861 = 20 the number of objects in each membrane 119866 =
100 evolution algebra Max119879 = 1000 crossover probability119875119888= 085 and mutation probability 119875
119898= 005 The optimal
parameters combinations of eachmodel are shown in Tables 1and 2
441 Single-Step Prediction Selecting the first point as inputto obtain first predicted value then the real value of the firstpoint is added to the historical data predicting the next pointAnd so obtain the predicted value of all points Predictionresults of five models are shown in Tables 3 4 5 6 7 and 8and Figures 7 8 9 and 10
442 Multistep Forecast Selecting a point as input to obtainpredicted value then the prediction value of the first point is
Table 1The optimal parameters combination of fivemodels for FMbroadcasting band
added to the historical data predicting next point And soobtain the predicted value of all points Predicted results offive models are shown in Tables 9 10 11 12 13 and 14 andFigures 11 12 13 and 14
45 Analysis of Experimental Results The optimal parame-ters combinations of five models for FM broadcasting bandand interphone band are shown inTables 1 and 2 respectivelyAs seen from experimental results we can find that theparameters 120591 119898 120574 and 120590 are very sensitive to predictionaccuracy the optimal parameters combination is P-LSSVMmodel FM broadcasting bands are 7 14 1631 and 66475Interphone bands are 3 8 1709 and 21621 It can be seenfrom predicted results diagram (Figures 7 to 14) that whethersingle-step prediction or multistep prediction five models getvery good results However the P-LSSVM model predictscurve best fit to real data and other curves relative deviationfrom far away For five prediction models respectively run10 times computing the maximum minimum mean andvariance of error As can be seen from predicted results inTables 3 to 14 three kinds of models evaluation standard areRMSE NMSE andMAPE the model proposed in this paperis the minimum This shows that not only is the P-LSSVMmodel reasonable and correct but prediction accuracy is alsoenhanced
Comparing single-step prediction with multistep predic-tion it can be found that the error of multistep predictionis larger than the single-step prediction indicating that theeffect of single-step prediction is better than multistep pre-diction The reason is that errors exist in every step andthe accumulation of error will lead to decline in the overallprediction accuracy
5 Conclusion
Modeling and prediction of chaotic time series has become ahot spot in the research field of the chaotic signal processing
8 The Scientific World Journal
Table 3 Five models predicted error based on RMSE for FM broadcasting band
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
where Φ(sdot) is a nonlinear mapping from the input space tothe feature space 120596 is a vector of weight coefficients and 119887 isa bias constant
The optimal hyperplane will be determined by the maxi-mum geometry interval Hence the LS-SVR problem can betransformed as follows [43]
min 119869 (120596 120577) =1
2120596119879120596 +
1
2120574
119897
sum119894=1
120577119894
st 119910119894= 120596119879Φ(119883119894) + 120577119894+ 119887 119894 = 1 119897 minus 1 119897
(4)
where 120577119894are the error variables and 120574 is hyperparameter
The process of finding the optimal decision function is todetermine the process parameters 120596 and 119887
Introducing Lagrange multipliers one can establishLagrange functions as follows
119871 =1
2120596119879120596 +
1
2120574
119897
sum119894=1
120577119894minus
119897
sum119894=1
120572119894(120596119879Φ(119883119894) + 120577119894+ 119887 minus 119910
119894) (5)
where 120572119894(119894 = 1 119897 minus 1 119897) are the Lagrange multiplier The
conditions for optimality are given by
120597119871
120597120596= 0
120597119871
120597119887= 0
120597119871
120597120577119894
= 0120597119871
120597120572119894
= 0 (6)
After elimination of the variables 120596 and 120577 a set of linearequations can be obtained
(0 1198681015840
119879
1198681015840 Ω + 120574minus1119868)(
119887
120572) = (
0
119884) (7)
where 1198681015840 = (1 1 1)119879 isin 119877119897 119868 isin 119877119897times119897 denotes a unitmatrix 120572 = (120572
1 120572
119897minus1 120572119897)119879 119884 = (119910
1 119910
119897minus1 119910119897)119879 and
Ω119894119895= Φ(119883
119894)119879Φ(119883
119895) (119894 119895 = 1 119897 minus 1 119897)
Then LS-SVM regression model is expressed as
119891 (119883) =
119897
sum119894=1
120572119894Φ(119883119894)119879
Φ(119883119895) + 119887 (8)
The mapping function Φ(sdot) can be paraphrased by a kernelfunction 119870(sdot sdot) because of the application of Mercerrsquos theo-rem whichmeans that119870(sdot sdot) (119894 = 1 119897minus1 119897) are any kernelfunctions satisfying the Mercer condition and the Mercerscondition has been applied
119870(119883119894 119883119895) = Φ (119883
119894)119879
Φ(119883119895) (119894 119895 = 1 119897 minus 1 119897) (9)
This finally results in the following LS-SVM model forfunction regression
119891 (119883) =
119899
sum119894=1
120572119894119870(119883119883
119894) + 119887 (10)
As shown in Figure 1 the prediction model of phasespace reconstruction and LS-SVM regression mainly has twosteps First select the delay time (120591) embedding dimension(119898) and LS-SVM parameters (120574 and 120590) The phase spacereconstruction technique is used to determine the trainingsample pairs based on the parameters 120591 and 119898 which aredetermined Assuming the time series is 119909
1 1199092 119909
119873+1
the training sample set of attributes is as follows
119894in the future Select the attribute sample of the
previous time as input in the phase space and use the trainedLS-SVMmodel to obtain the predicted value of the moment
22 Membrane Computing Membrane computing (namely119901 systems) arises as a newmodel of computation inspired bythe way that cells are structured into vesicles and abstractingthe chemical reactions taking place inside them [44] It is abranch of molecular computing that aims to develop modelsandparadigms that are biologicallymotivatedThere has beena flurry of research activities in this area in recent years[45] Because of the built-in nature of maximal parallelisminherent on the models 119901 systems have a great potential forimplementing massively concurrent systems in an efficientway that would allow us to solve currently intractable prob-lems
A membrane system with degree 119889 (119889 gt 0) can beexpressed as
prod = (119881 119879 119862 1205831198821 119882
119889 (1198771 1205881) (119877
119889 120588119889)) (12)
where 119881 is an alphabet whose elements are called objects119879 denotes the output alphabet 119862 is a catalyst which doesnot exhibit any change in the course of evolution but somereaction must have its participation 120583 is the membranestructure which can be shown by []119882
119894denotes multiple sets
of objects in the membrane structure and (119877119894 120588119894) are the set
of rules in which 119877119894and 120588119894denote rule and the priority of the
rule respectivelyIn general 119901 system contains three core elements mem-
brane structure object multiple sets and evolution rules Amembrane system with givenmembrane structure evolutionrules and decided objects will be performed in the form ofnondeterministic and maximum parallel for the evolutionrules When all the objects are exhausted the rules are nolonger executed the system downtime A typical membranesystem consists of cell-likemembranes placed inside a uniqueldquoskinrdquo membrane Multisets of objectsmdashusually strings ofsymbolsmdashand a set of evolution rules are placed inside theregions delimited by the membranes Each object can betransformed into other objects can pass through a mem-brane or can dissolve or create membranes The evolution
4 The Scientific World Journal
1
234
Envi
ronm
ent
Skin
h
d
Regions
Elementary membrane
Envi
ronm
ent
Rules
ObjectsElementary membrane
Membrane
h rarr hh
d rarr dy
Figure 2 Simple membrane structure diagram
between system configurations is done nondeterministicallyby applying the rules in parallel for all objects able to evolve[46] As shown in Figure 2 a simple membrane structurediagram can be shown by [[[]
3]2[]4]1 The skin membrane
which is the outermost membrane of this structure separatesthe system from its environment Several membranes each ofwhich defines a region are placed inside the skin membraneElementary membranes do not contain any membrane Eachregion forms a different compartment of the membranestructure and contains a multiset of objects or membranesWhere ℎ and 119889 denote objects ℎ rarr ℎℎ and 119889 rarr 119889119910 arerules
3 Parameters Joint Optimization AlgorithmBased on Membrane Computing
The optimization algorithm based on cell-like membranecomputing is an important branch of membrane computingIt is an intelligent optimization algorithm inspired by themechanism and the function of biological cells and basedon the existing framework of membrane computing Thesteps generally are membrane structure establishment theobjects generation and evolution and so forth Shown inFigure 3 is the structure of P-LSSVM prediction modelwith the initial objects as initial parameters of predictionmodel these parameters are substituted into the phase spacereconstruction and LS-SVM model Then parameters jointoptimization algorithm based on membrane computing isused to decide the best combination of parameters Algo-rithm specific process is as shown in Figure 3
31The Establishment of the Cellular Membrane Structure andthe Generation of Objects As shown in Figure 4 this paperadopts two layers structure for membrane a skin contains 119861basic membrane generate initial objects in each membraneGenerally 119901 system uses character or character string toencode real number encoding are adopted in here which canreduce the trouble of decode For instance119874 = (119900
1 1199002 1199003 1199004)
where 119874 is an object and 1199001 1199002 1199003 and 119900
4denote 120591 119898
120574 and 120590 respectively We see each object as a solutionof the optimization problem Evolution of each membraneaccording to its own rules all the membrane are executed in
parallelThe final optimal results are output through the skinthat is the optimal solution
32 Construct the Fitness Function The goal of cell-likemembrane computing optimization algorithm is to find themost suitable combination of parameters (120591 119898 120574 and 120590)in order to establish the optimal forecasting model In thispaper we used the rootmean square prediction error (RMSE)to construct the fitness function That is 119891 = 1RMSERMSE = radic(1119885)sum119885
119894=1(119910119894minus 119910119894)2 where119885 denotes the number
of prediction points and 119910119894 119910119894represent the real values and
predicted values respectively
33 Operation Rules The basic rules of cellular membranecomputing optimization method are selection crossovermutation and communication [47] The specific form is asfollows
(1) Selection rule the rule of selection copies the objectsto the next generation according to the size of the string Thesize of the string is not the three-dimensional size of particlesin biological cells but the value of the fitness function Herewheel disk method is used to select objects to the nextgeneration
(2) Crossover rule for any two objects 119874119894
=
(1199001198941 1199001198942 1199001198943 1199001198944) and 119874
119895= (119900
1198951 1199001198952 1199001198953 1199001198954) use cross
rule to obtain new object 119874119896= (1199001198961 1199001198962 1199001198963 1199001198964)
1199001198961= 119903 times 119900
1198941+ (1 minus 119903) times 119900
1198951
1199001198962= 119903 times 119900
1198942+ (1 minus 119903) times 119900
1198952
1199001198963= 119903 times 119900
1198943+ (1 minus 119903) times 119900
1198953
1199001198964= 119903 times 119900
1198944+ (1 minus 119903) times 119900
1198954
(13)
where 119903 is a random number in (0 1)(3) Mutation rule in evolution according to a certain
mutation probability replace the worst 119905 objects with ran-domly generated 119905 objects Mutating rule is described asfollows
where []119894denotes membrane 119894 119902min 1 119902min 2 119902min 119905 are 119905
objects where fitness is the smallest in membrane 119894 and119902init1 119902init2 119902init119905 are randomly generated 119905 objects
(4)Communication rule each membrane 119894will transportthe best 119871 objects out of the membrane while the best 119871objects of foreign membrane are brought into the membrane119894 [48] This rule can be expressed as follows
119877119894communication = 119877119894communication1 cup 119877119894communication2
Parameters joint optimization based on membrane computing
Phase space reconstruction LS-SVM Chaotic time series
prediction and output
120591 m 120574 120590
Figure 3 The structure of P-LSSVM prediction model
0
1 2 B
Elementary membrane
Skin
O11 O
12 O
1G O2
1 O22 O
2G OB
1 OB2 O
BG
O01 O
02 O
0G
middot middot middot
Figure 4 Membrane structure
119877119894communication2 []119894 119902
1015840
max 1 1199021015840
max 2 1199021015840
max119871
997888rarr [1199021015840
max 1 1199021015840
max 2 1199021015840
max119871]119894
(15)
where []119894denotesmembrane 119894 119902max 1 119902max 2 119902max119871 are the
best 119871 objects in membrane 119894 and 1199021015840max 1 1199021015840
max 2 1199021015840
max119871 arethe best 119871 objects out of membrane 119894
34 Parameters Joint Optimization Algorithm Specific Stepsin P-LSSVM Model First of all generate the initial objectsas initial parameters of prediction model then apply evo-lutionary rules to evolve until the stop conditions are metall membranes are operating in parallel Finally output thefitness of the best object by the skin membrane that is theoptimal solution Specifically consider the following
Step 1 Initialize parameters and build cellular membranestructure
(1) Initialization the number of elementary membranesis 119861 the number of objects in eachmembrane is119866 the largestnumber of iterations is Max119879 crossover probability is 119875
119888
mutation probability is 119875119898 and the current iteration number
is 119896 and so forth(2) Create membrane structure as shown in Figure 4
generating randomly 119866 objects in each membrane eachobject represents a set of parametersrsquo combination expressedin decimal coding
Step 2 Optimize each membrane in turn(1) Every object in the membrane as a set of parameters
(120591 119898 120574 and 120590) of P-LSSVM model calculate the fitness ofeach object by training data and save the optimal object andits fitness
(2) Use the reproduction crossover and mutation rulesto evolve
Step 3 Make use of communication rules each membranewill transport the best 119871 objects out of the membrane at thesame time the best 119871 objects outside the membrane will beshipped into the membrane
Step 4 Determine whether the termination condition issatisfied that is whether it reaches the maximum numberof iterations when the number of iterations is less than themaximum number of iterations to continue iteration or stopiteration
Step 5 Theoptimal object is output from the skinmembrane
4 Electromagnetic Environment ParametersPredictions Based on P-LSSVM Model
Electromagnetic spectrum is a fundamental strategicresource to support the national economy and nationaldefense construction along with the rapid development ofinformation technology and it is widely used in various fieldssuch as economic development national defense constru-ction and social life [49] Strategic value and basic role
6 The Scientific World Journal
increasingly highlight in the electromagnetic spectrum withfrequency contradictions increasingly prominent betweencountries departments and military and space businesses[50] It is an important basis for spectrum management tocontrol comprehensively the change trend of parametersin the electromagnetic environment of country or region[51] It is the basis to master the frequency informationfor the frequency planning frequency allocation andsharing service frequency recovery work The situationof electromagnetic environment can be reflected by theelectromagnetic environment indicator parameters theseparameters mainly include band occupancy rate channeloccupancy rate large-signal ratio frequency offset and thefield strength A large number of experiment shown that timeseries data with chaotic in the electromagnetic environmentHence we used the proposed predictionmodel to predict theindicator parameters of the electromagnetic environmentThe experimental results show that the prediction modelproposed in this paper is reasonable and effective
Here we chose the band occupancy rate to do the testBand occupancy rate is calculated as follows extracting allthe signal points in the spectrum data the signals pointare merged with distance less than bandwidth by the belowformula to calculate the band occupancy rate (OccupyFreband)
OccupyFreband =119878119899lowast 119865119908
119865end minus 119865begin (16)
where 119878119899denotes the total number of signals judged 119865
119908is
necessary bandwidth in this band for the type of specifiedbusiness 119865begin is the start frequency point and 119865end is thecutoff frequency point
41 Experimental Data Sources In this paper we adoptdigital receiver EM100whichwas provided byGermanRohdeamp Schwarz Company and fixed radio monitoring stationof Xihua University to collect data for the experiment Wecollected data including frequency modulation (FM) broad-casting band and interphone band As shown in Figures 5 and6 in which the vertical axis denotes band occupancy rate thehorizontal axis represents the collection time and left pictureshows the data of band occupancy rate in FM broadcastingband we collected for 680 hours that is obtaining 680pieces of data Right figure indicates acquisition data ofband occupancy rate in interphone band we continuouslycollected for 187 hours that is gaining 187 pieces of data Inorder to facilitate narration here we put the band data of FMbroadcasting band and interphone band denoted by ldquodataset 1rdquo and ldquodata set 2rdquo respectively Use the method of smallamount of data to calculate the maximum Lyapunov indexof two groups of data which are 120582
1= 0126 and 120582
2= 014
respectively which show the time series with chaos
42 Data Preprocessing This paper mainly uses the Grubbscriteria to deal with the abnormal data the method is asfollows let 119902(ℎ 119889) be the sequence of the collected datawith the time interval between two data collections 119905 = 1
hour where ℎ = 0 22 23 denote 24 hours of a day119889 = 1 119867 minus 1119867 represents date code in total days of data
025035045055065075
1 71 141 211 281 351 421 491 561 631
Occ
upan
cy ra
te
t (hour)
Figure 5 Band occupancy rate data of FM radio band
000500070009001100130015001700190021
1 21 41 61 81 101 121 141 161 181
Occ
upan
cy ra
te
t (hour)
Figure 6 Band occupancy rate data of interphone band
collection119867 and 119902 denotes the collected data Using data setdenoted by 119876 = 119902
1 1199022 119902
119905 for each time point ℎ we
can get the expectation and variance of data sequence 119902(ℎ 119889)the formula is as follows
119864 (ℎ) =1
119878
119878
sum119896=1
119902 (ℎ 119889)
119863 (ℎ) = 1205902
119894=1
119878
119878
sum119896=1
[119902 (ℎ 119889) minus 119864 (ℎ)]2
(17)
where 119878 denotes the length of a unitAccording to the above two formulas combined with
Grubbs criteria if the sample point meet to1003816100381610038161003816119902 (ℎ 119889) minus 119864 (ℎ)
1003816100381610038161003816 ge 119866 (119899 120576) 120590119894 (18)
The sample point should be removed where 119866(119899 120576) is thecritical value of Grubbs criteria it can be obtained by lookingat Grubbs table 120576 denotes the significance level usuallysignificance level 120576 = 005
The Grubbs criteria are used to deal with ldquodata set 1rdquo andldquodata set 2rdquo respectively For the ldquodata set 1rdquo after processingwith Grubbs criteria the remaining 653 pieces of data we usethe front 600 pieces of data as the training data determiningthe best parameters combination and the surplus 53 pieces ofdata as test data testing the prediction accuracy of themodelFor the ldquodata set 2rdquo after processing with Grubbs criteria theremaining 180 pieces of data we use the front 150 pieces ofdata as the training data determining the best parameterscombination and the surplus 30 pieces of data as test datatesting the prediction accuracy of the model
43 Reference Model and Evaluation Criteria In order toverify the validity of the model this paper will comparethe prediction model (P-LSSVM) proposed in this paperwith conventional similar prediction model The first ref-erence model is the parameters joint optimization based
The Scientific World Journal 7
on genetic algorithm for chaos time series prediction (GA-LSSVM) [34] The second reference model uses the mutualinformation method and Cao method to get the best delaytime 120591 and embedding dimension 119898 respectively And thenuse grid search method to obtain LS-SVM parameters (120574and 120590) (denoted as M-C-LSSVM) [19] The third referencemodel uses the mutual information method and false nearestneighbor method to calculate the optimal delay time 120591 andembedding dimension119898 respectively And then use geneticalgorithm to get the optimal combination parameters of LS-SVM (120574 and 120590) (denoted as M-F-LSSVM) [27] The fourthreference model uses C-Cmethod to seek simultaneously thebest delay time 120591 and embedding dimension 119898 Then theoptimal parameters of LS-SVM (120574 and 120590) by using geneticalgorithm (denoted as C-C-LSSVM) [27 52]
Meanwhile this paper uses three evaluation criterianormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)NMSE RMSE and MAPE are defined respectively as fol-lows
where 119885 is the number of prediction points 119910 is the averagevalue and 119910
119894and 119910
119894denote the real value and the predicted
value of 119894th point respectively
44 Experimental Results In this paper the scope of parame-ters 120591119898 120574 and 120590 is [1 8] [3 17] [1 1000] and [01 10000]respectively In the process of evolution the other parametersare set as follows the number of elementary membranes119861 = 20 the number of objects in each membrane 119866 =
100 evolution algebra Max119879 = 1000 crossover probability119875119888= 085 and mutation probability 119875
119898= 005 The optimal
parameters combinations of eachmodel are shown in Tables 1and 2
441 Single-Step Prediction Selecting the first point as inputto obtain first predicted value then the real value of the firstpoint is added to the historical data predicting the next pointAnd so obtain the predicted value of all points Predictionresults of five models are shown in Tables 3 4 5 6 7 and 8and Figures 7 8 9 and 10
442 Multistep Forecast Selecting a point as input to obtainpredicted value then the prediction value of the first point is
Table 1The optimal parameters combination of fivemodels for FMbroadcasting band
added to the historical data predicting next point And soobtain the predicted value of all points Predicted results offive models are shown in Tables 9 10 11 12 13 and 14 andFigures 11 12 13 and 14
45 Analysis of Experimental Results The optimal parame-ters combinations of five models for FM broadcasting bandand interphone band are shown inTables 1 and 2 respectivelyAs seen from experimental results we can find that theparameters 120591 119898 120574 and 120590 are very sensitive to predictionaccuracy the optimal parameters combination is P-LSSVMmodel FM broadcasting bands are 7 14 1631 and 66475Interphone bands are 3 8 1709 and 21621 It can be seenfrom predicted results diagram (Figures 7 to 14) that whethersingle-step prediction or multistep prediction five models getvery good results However the P-LSSVM model predictscurve best fit to real data and other curves relative deviationfrom far away For five prediction models respectively run10 times computing the maximum minimum mean andvariance of error As can be seen from predicted results inTables 3 to 14 three kinds of models evaluation standard areRMSE NMSE andMAPE the model proposed in this paperis the minimum This shows that not only is the P-LSSVMmodel reasonable and correct but prediction accuracy is alsoenhanced
Comparing single-step prediction with multistep predic-tion it can be found that the error of multistep predictionis larger than the single-step prediction indicating that theeffect of single-step prediction is better than multistep pre-diction The reason is that errors exist in every step andthe accumulation of error will lead to decline in the overallprediction accuracy
5 Conclusion
Modeling and prediction of chaotic time series has become ahot spot in the research field of the chaotic signal processing
8 The Scientific World Journal
Table 3 Five models predicted error based on RMSE for FM broadcasting band
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
between system configurations is done nondeterministicallyby applying the rules in parallel for all objects able to evolve[46] As shown in Figure 2 a simple membrane structurediagram can be shown by [[[]
3]2[]4]1 The skin membrane
which is the outermost membrane of this structure separatesthe system from its environment Several membranes each ofwhich defines a region are placed inside the skin membraneElementary membranes do not contain any membrane Eachregion forms a different compartment of the membranestructure and contains a multiset of objects or membranesWhere ℎ and 119889 denote objects ℎ rarr ℎℎ and 119889 rarr 119889119910 arerules
3 Parameters Joint Optimization AlgorithmBased on Membrane Computing
The optimization algorithm based on cell-like membranecomputing is an important branch of membrane computingIt is an intelligent optimization algorithm inspired by themechanism and the function of biological cells and basedon the existing framework of membrane computing Thesteps generally are membrane structure establishment theobjects generation and evolution and so forth Shown inFigure 3 is the structure of P-LSSVM prediction modelwith the initial objects as initial parameters of predictionmodel these parameters are substituted into the phase spacereconstruction and LS-SVM model Then parameters jointoptimization algorithm based on membrane computing isused to decide the best combination of parameters Algo-rithm specific process is as shown in Figure 3
31The Establishment of the Cellular Membrane Structure andthe Generation of Objects As shown in Figure 4 this paperadopts two layers structure for membrane a skin contains 119861basic membrane generate initial objects in each membraneGenerally 119901 system uses character or character string toencode real number encoding are adopted in here which canreduce the trouble of decode For instance119874 = (119900
1 1199002 1199003 1199004)
where 119874 is an object and 1199001 1199002 1199003 and 119900
4denote 120591 119898
120574 and 120590 respectively We see each object as a solutionof the optimization problem Evolution of each membraneaccording to its own rules all the membrane are executed in
parallelThe final optimal results are output through the skinthat is the optimal solution
32 Construct the Fitness Function The goal of cell-likemembrane computing optimization algorithm is to find themost suitable combination of parameters (120591 119898 120574 and 120590)in order to establish the optimal forecasting model In thispaper we used the rootmean square prediction error (RMSE)to construct the fitness function That is 119891 = 1RMSERMSE = radic(1119885)sum119885
119894=1(119910119894minus 119910119894)2 where119885 denotes the number
of prediction points and 119910119894 119910119894represent the real values and
predicted values respectively
33 Operation Rules The basic rules of cellular membranecomputing optimization method are selection crossovermutation and communication [47] The specific form is asfollows
(1) Selection rule the rule of selection copies the objectsto the next generation according to the size of the string Thesize of the string is not the three-dimensional size of particlesin biological cells but the value of the fitness function Herewheel disk method is used to select objects to the nextgeneration
(2) Crossover rule for any two objects 119874119894
=
(1199001198941 1199001198942 1199001198943 1199001198944) and 119874
119895= (119900
1198951 1199001198952 1199001198953 1199001198954) use cross
rule to obtain new object 119874119896= (1199001198961 1199001198962 1199001198963 1199001198964)
1199001198961= 119903 times 119900
1198941+ (1 minus 119903) times 119900
1198951
1199001198962= 119903 times 119900
1198942+ (1 minus 119903) times 119900
1198952
1199001198963= 119903 times 119900
1198943+ (1 minus 119903) times 119900
1198953
1199001198964= 119903 times 119900
1198944+ (1 minus 119903) times 119900
1198954
(13)
where 119903 is a random number in (0 1)(3) Mutation rule in evolution according to a certain
mutation probability replace the worst 119905 objects with ran-domly generated 119905 objects Mutating rule is described asfollows
where []119894denotes membrane 119894 119902min 1 119902min 2 119902min 119905 are 119905
objects where fitness is the smallest in membrane 119894 and119902init1 119902init2 119902init119905 are randomly generated 119905 objects
(4)Communication rule each membrane 119894will transportthe best 119871 objects out of the membrane while the best 119871objects of foreign membrane are brought into the membrane119894 [48] This rule can be expressed as follows
119877119894communication = 119877119894communication1 cup 119877119894communication2
Parameters joint optimization based on membrane computing
Phase space reconstruction LS-SVM Chaotic time series
prediction and output
120591 m 120574 120590
Figure 3 The structure of P-LSSVM prediction model
0
1 2 B
Elementary membrane
Skin
O11 O
12 O
1G O2
1 O22 O
2G OB
1 OB2 O
BG
O01 O
02 O
0G
middot middot middot
Figure 4 Membrane structure
119877119894communication2 []119894 119902
1015840
max 1 1199021015840
max 2 1199021015840
max119871
997888rarr [1199021015840
max 1 1199021015840
max 2 1199021015840
max119871]119894
(15)
where []119894denotesmembrane 119894 119902max 1 119902max 2 119902max119871 are the
best 119871 objects in membrane 119894 and 1199021015840max 1 1199021015840
max 2 1199021015840
max119871 arethe best 119871 objects out of membrane 119894
34 Parameters Joint Optimization Algorithm Specific Stepsin P-LSSVM Model First of all generate the initial objectsas initial parameters of prediction model then apply evo-lutionary rules to evolve until the stop conditions are metall membranes are operating in parallel Finally output thefitness of the best object by the skin membrane that is theoptimal solution Specifically consider the following
Step 1 Initialize parameters and build cellular membranestructure
(1) Initialization the number of elementary membranesis 119861 the number of objects in eachmembrane is119866 the largestnumber of iterations is Max119879 crossover probability is 119875
119888
mutation probability is 119875119898 and the current iteration number
is 119896 and so forth(2) Create membrane structure as shown in Figure 4
generating randomly 119866 objects in each membrane eachobject represents a set of parametersrsquo combination expressedin decimal coding
Step 2 Optimize each membrane in turn(1) Every object in the membrane as a set of parameters
(120591 119898 120574 and 120590) of P-LSSVM model calculate the fitness ofeach object by training data and save the optimal object andits fitness
(2) Use the reproduction crossover and mutation rulesto evolve
Step 3 Make use of communication rules each membranewill transport the best 119871 objects out of the membrane at thesame time the best 119871 objects outside the membrane will beshipped into the membrane
Step 4 Determine whether the termination condition issatisfied that is whether it reaches the maximum numberof iterations when the number of iterations is less than themaximum number of iterations to continue iteration or stopiteration
Step 5 Theoptimal object is output from the skinmembrane
4 Electromagnetic Environment ParametersPredictions Based on P-LSSVM Model
Electromagnetic spectrum is a fundamental strategicresource to support the national economy and nationaldefense construction along with the rapid development ofinformation technology and it is widely used in various fieldssuch as economic development national defense constru-ction and social life [49] Strategic value and basic role
6 The Scientific World Journal
increasingly highlight in the electromagnetic spectrum withfrequency contradictions increasingly prominent betweencountries departments and military and space businesses[50] It is an important basis for spectrum management tocontrol comprehensively the change trend of parametersin the electromagnetic environment of country or region[51] It is the basis to master the frequency informationfor the frequency planning frequency allocation andsharing service frequency recovery work The situationof electromagnetic environment can be reflected by theelectromagnetic environment indicator parameters theseparameters mainly include band occupancy rate channeloccupancy rate large-signal ratio frequency offset and thefield strength A large number of experiment shown that timeseries data with chaotic in the electromagnetic environmentHence we used the proposed predictionmodel to predict theindicator parameters of the electromagnetic environmentThe experimental results show that the prediction modelproposed in this paper is reasonable and effective
Here we chose the band occupancy rate to do the testBand occupancy rate is calculated as follows extracting allthe signal points in the spectrum data the signals pointare merged with distance less than bandwidth by the belowformula to calculate the band occupancy rate (OccupyFreband)
OccupyFreband =119878119899lowast 119865119908
119865end minus 119865begin (16)
where 119878119899denotes the total number of signals judged 119865
119908is
necessary bandwidth in this band for the type of specifiedbusiness 119865begin is the start frequency point and 119865end is thecutoff frequency point
41 Experimental Data Sources In this paper we adoptdigital receiver EM100whichwas provided byGermanRohdeamp Schwarz Company and fixed radio monitoring stationof Xihua University to collect data for the experiment Wecollected data including frequency modulation (FM) broad-casting band and interphone band As shown in Figures 5 and6 in which the vertical axis denotes band occupancy rate thehorizontal axis represents the collection time and left pictureshows the data of band occupancy rate in FM broadcastingband we collected for 680 hours that is obtaining 680pieces of data Right figure indicates acquisition data ofband occupancy rate in interphone band we continuouslycollected for 187 hours that is gaining 187 pieces of data Inorder to facilitate narration here we put the band data of FMbroadcasting band and interphone band denoted by ldquodataset 1rdquo and ldquodata set 2rdquo respectively Use the method of smallamount of data to calculate the maximum Lyapunov indexof two groups of data which are 120582
1= 0126 and 120582
2= 014
respectively which show the time series with chaos
42 Data Preprocessing This paper mainly uses the Grubbscriteria to deal with the abnormal data the method is asfollows let 119902(ℎ 119889) be the sequence of the collected datawith the time interval between two data collections 119905 = 1
hour where ℎ = 0 22 23 denote 24 hours of a day119889 = 1 119867 minus 1119867 represents date code in total days of data
025035045055065075
1 71 141 211 281 351 421 491 561 631
Occ
upan
cy ra
te
t (hour)
Figure 5 Band occupancy rate data of FM radio band
000500070009001100130015001700190021
1 21 41 61 81 101 121 141 161 181
Occ
upan
cy ra
te
t (hour)
Figure 6 Band occupancy rate data of interphone band
collection119867 and 119902 denotes the collected data Using data setdenoted by 119876 = 119902
1 1199022 119902
119905 for each time point ℎ we
can get the expectation and variance of data sequence 119902(ℎ 119889)the formula is as follows
119864 (ℎ) =1
119878
119878
sum119896=1
119902 (ℎ 119889)
119863 (ℎ) = 1205902
119894=1
119878
119878
sum119896=1
[119902 (ℎ 119889) minus 119864 (ℎ)]2
(17)
where 119878 denotes the length of a unitAccording to the above two formulas combined with
Grubbs criteria if the sample point meet to1003816100381610038161003816119902 (ℎ 119889) minus 119864 (ℎ)
1003816100381610038161003816 ge 119866 (119899 120576) 120590119894 (18)
The sample point should be removed where 119866(119899 120576) is thecritical value of Grubbs criteria it can be obtained by lookingat Grubbs table 120576 denotes the significance level usuallysignificance level 120576 = 005
The Grubbs criteria are used to deal with ldquodata set 1rdquo andldquodata set 2rdquo respectively For the ldquodata set 1rdquo after processingwith Grubbs criteria the remaining 653 pieces of data we usethe front 600 pieces of data as the training data determiningthe best parameters combination and the surplus 53 pieces ofdata as test data testing the prediction accuracy of themodelFor the ldquodata set 2rdquo after processing with Grubbs criteria theremaining 180 pieces of data we use the front 150 pieces ofdata as the training data determining the best parameterscombination and the surplus 30 pieces of data as test datatesting the prediction accuracy of the model
43 Reference Model and Evaluation Criteria In order toverify the validity of the model this paper will comparethe prediction model (P-LSSVM) proposed in this paperwith conventional similar prediction model The first ref-erence model is the parameters joint optimization based
The Scientific World Journal 7
on genetic algorithm for chaos time series prediction (GA-LSSVM) [34] The second reference model uses the mutualinformation method and Cao method to get the best delaytime 120591 and embedding dimension 119898 respectively And thenuse grid search method to obtain LS-SVM parameters (120574and 120590) (denoted as M-C-LSSVM) [19] The third referencemodel uses the mutual information method and false nearestneighbor method to calculate the optimal delay time 120591 andembedding dimension119898 respectively And then use geneticalgorithm to get the optimal combination parameters of LS-SVM (120574 and 120590) (denoted as M-F-LSSVM) [27] The fourthreference model uses C-Cmethod to seek simultaneously thebest delay time 120591 and embedding dimension 119898 Then theoptimal parameters of LS-SVM (120574 and 120590) by using geneticalgorithm (denoted as C-C-LSSVM) [27 52]
Meanwhile this paper uses three evaluation criterianormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)NMSE RMSE and MAPE are defined respectively as fol-lows
where 119885 is the number of prediction points 119910 is the averagevalue and 119910
119894and 119910
119894denote the real value and the predicted
value of 119894th point respectively
44 Experimental Results In this paper the scope of parame-ters 120591119898 120574 and 120590 is [1 8] [3 17] [1 1000] and [01 10000]respectively In the process of evolution the other parametersare set as follows the number of elementary membranes119861 = 20 the number of objects in each membrane 119866 =
100 evolution algebra Max119879 = 1000 crossover probability119875119888= 085 and mutation probability 119875
119898= 005 The optimal
parameters combinations of eachmodel are shown in Tables 1and 2
441 Single-Step Prediction Selecting the first point as inputto obtain first predicted value then the real value of the firstpoint is added to the historical data predicting the next pointAnd so obtain the predicted value of all points Predictionresults of five models are shown in Tables 3 4 5 6 7 and 8and Figures 7 8 9 and 10
442 Multistep Forecast Selecting a point as input to obtainpredicted value then the prediction value of the first point is
Table 1The optimal parameters combination of fivemodels for FMbroadcasting band
added to the historical data predicting next point And soobtain the predicted value of all points Predicted results offive models are shown in Tables 9 10 11 12 13 and 14 andFigures 11 12 13 and 14
45 Analysis of Experimental Results The optimal parame-ters combinations of five models for FM broadcasting bandand interphone band are shown inTables 1 and 2 respectivelyAs seen from experimental results we can find that theparameters 120591 119898 120574 and 120590 are very sensitive to predictionaccuracy the optimal parameters combination is P-LSSVMmodel FM broadcasting bands are 7 14 1631 and 66475Interphone bands are 3 8 1709 and 21621 It can be seenfrom predicted results diagram (Figures 7 to 14) that whethersingle-step prediction or multistep prediction five models getvery good results However the P-LSSVM model predictscurve best fit to real data and other curves relative deviationfrom far away For five prediction models respectively run10 times computing the maximum minimum mean andvariance of error As can be seen from predicted results inTables 3 to 14 three kinds of models evaluation standard areRMSE NMSE andMAPE the model proposed in this paperis the minimum This shows that not only is the P-LSSVMmodel reasonable and correct but prediction accuracy is alsoenhanced
Comparing single-step prediction with multistep predic-tion it can be found that the error of multistep predictionis larger than the single-step prediction indicating that theeffect of single-step prediction is better than multistep pre-diction The reason is that errors exist in every step andthe accumulation of error will lead to decline in the overallprediction accuracy
5 Conclusion
Modeling and prediction of chaotic time series has become ahot spot in the research field of the chaotic signal processing
8 The Scientific World Journal
Table 3 Five models predicted error based on RMSE for FM broadcasting band
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
Parameters joint optimization based on membrane computing
Phase space reconstruction LS-SVM Chaotic time series
prediction and output
120591 m 120574 120590
Figure 3 The structure of P-LSSVM prediction model
0
1 2 B
Elementary membrane
Skin
O11 O
12 O
1G O2
1 O22 O
2G OB
1 OB2 O
BG
O01 O
02 O
0G
middot middot middot
Figure 4 Membrane structure
119877119894communication2 []119894 119902
1015840
max 1 1199021015840
max 2 1199021015840
max119871
997888rarr [1199021015840
max 1 1199021015840
max 2 1199021015840
max119871]119894
(15)
where []119894denotesmembrane 119894 119902max 1 119902max 2 119902max119871 are the
best 119871 objects in membrane 119894 and 1199021015840max 1 1199021015840
max 2 1199021015840
max119871 arethe best 119871 objects out of membrane 119894
34 Parameters Joint Optimization Algorithm Specific Stepsin P-LSSVM Model First of all generate the initial objectsas initial parameters of prediction model then apply evo-lutionary rules to evolve until the stop conditions are metall membranes are operating in parallel Finally output thefitness of the best object by the skin membrane that is theoptimal solution Specifically consider the following
Step 1 Initialize parameters and build cellular membranestructure
(1) Initialization the number of elementary membranesis 119861 the number of objects in eachmembrane is119866 the largestnumber of iterations is Max119879 crossover probability is 119875
119888
mutation probability is 119875119898 and the current iteration number
is 119896 and so forth(2) Create membrane structure as shown in Figure 4
generating randomly 119866 objects in each membrane eachobject represents a set of parametersrsquo combination expressedin decimal coding
Step 2 Optimize each membrane in turn(1) Every object in the membrane as a set of parameters
(120591 119898 120574 and 120590) of P-LSSVM model calculate the fitness ofeach object by training data and save the optimal object andits fitness
(2) Use the reproduction crossover and mutation rulesto evolve
Step 3 Make use of communication rules each membranewill transport the best 119871 objects out of the membrane at thesame time the best 119871 objects outside the membrane will beshipped into the membrane
Step 4 Determine whether the termination condition issatisfied that is whether it reaches the maximum numberof iterations when the number of iterations is less than themaximum number of iterations to continue iteration or stopiteration
Step 5 Theoptimal object is output from the skinmembrane
4 Electromagnetic Environment ParametersPredictions Based on P-LSSVM Model
Electromagnetic spectrum is a fundamental strategicresource to support the national economy and nationaldefense construction along with the rapid development ofinformation technology and it is widely used in various fieldssuch as economic development national defense constru-ction and social life [49] Strategic value and basic role
6 The Scientific World Journal
increasingly highlight in the electromagnetic spectrum withfrequency contradictions increasingly prominent betweencountries departments and military and space businesses[50] It is an important basis for spectrum management tocontrol comprehensively the change trend of parametersin the electromagnetic environment of country or region[51] It is the basis to master the frequency informationfor the frequency planning frequency allocation andsharing service frequency recovery work The situationof electromagnetic environment can be reflected by theelectromagnetic environment indicator parameters theseparameters mainly include band occupancy rate channeloccupancy rate large-signal ratio frequency offset and thefield strength A large number of experiment shown that timeseries data with chaotic in the electromagnetic environmentHence we used the proposed predictionmodel to predict theindicator parameters of the electromagnetic environmentThe experimental results show that the prediction modelproposed in this paper is reasonable and effective
Here we chose the band occupancy rate to do the testBand occupancy rate is calculated as follows extracting allthe signal points in the spectrum data the signals pointare merged with distance less than bandwidth by the belowformula to calculate the band occupancy rate (OccupyFreband)
OccupyFreband =119878119899lowast 119865119908
119865end minus 119865begin (16)
where 119878119899denotes the total number of signals judged 119865
119908is
necessary bandwidth in this band for the type of specifiedbusiness 119865begin is the start frequency point and 119865end is thecutoff frequency point
41 Experimental Data Sources In this paper we adoptdigital receiver EM100whichwas provided byGermanRohdeamp Schwarz Company and fixed radio monitoring stationof Xihua University to collect data for the experiment Wecollected data including frequency modulation (FM) broad-casting band and interphone band As shown in Figures 5 and6 in which the vertical axis denotes band occupancy rate thehorizontal axis represents the collection time and left pictureshows the data of band occupancy rate in FM broadcastingband we collected for 680 hours that is obtaining 680pieces of data Right figure indicates acquisition data ofband occupancy rate in interphone band we continuouslycollected for 187 hours that is gaining 187 pieces of data Inorder to facilitate narration here we put the band data of FMbroadcasting band and interphone band denoted by ldquodataset 1rdquo and ldquodata set 2rdquo respectively Use the method of smallamount of data to calculate the maximum Lyapunov indexof two groups of data which are 120582
1= 0126 and 120582
2= 014
respectively which show the time series with chaos
42 Data Preprocessing This paper mainly uses the Grubbscriteria to deal with the abnormal data the method is asfollows let 119902(ℎ 119889) be the sequence of the collected datawith the time interval between two data collections 119905 = 1
hour where ℎ = 0 22 23 denote 24 hours of a day119889 = 1 119867 minus 1119867 represents date code in total days of data
025035045055065075
1 71 141 211 281 351 421 491 561 631
Occ
upan
cy ra
te
t (hour)
Figure 5 Band occupancy rate data of FM radio band
000500070009001100130015001700190021
1 21 41 61 81 101 121 141 161 181
Occ
upan
cy ra
te
t (hour)
Figure 6 Band occupancy rate data of interphone band
collection119867 and 119902 denotes the collected data Using data setdenoted by 119876 = 119902
1 1199022 119902
119905 for each time point ℎ we
can get the expectation and variance of data sequence 119902(ℎ 119889)the formula is as follows
119864 (ℎ) =1
119878
119878
sum119896=1
119902 (ℎ 119889)
119863 (ℎ) = 1205902
119894=1
119878
119878
sum119896=1
[119902 (ℎ 119889) minus 119864 (ℎ)]2
(17)
where 119878 denotes the length of a unitAccording to the above two formulas combined with
Grubbs criteria if the sample point meet to1003816100381610038161003816119902 (ℎ 119889) minus 119864 (ℎ)
1003816100381610038161003816 ge 119866 (119899 120576) 120590119894 (18)
The sample point should be removed where 119866(119899 120576) is thecritical value of Grubbs criteria it can be obtained by lookingat Grubbs table 120576 denotes the significance level usuallysignificance level 120576 = 005
The Grubbs criteria are used to deal with ldquodata set 1rdquo andldquodata set 2rdquo respectively For the ldquodata set 1rdquo after processingwith Grubbs criteria the remaining 653 pieces of data we usethe front 600 pieces of data as the training data determiningthe best parameters combination and the surplus 53 pieces ofdata as test data testing the prediction accuracy of themodelFor the ldquodata set 2rdquo after processing with Grubbs criteria theremaining 180 pieces of data we use the front 150 pieces ofdata as the training data determining the best parameterscombination and the surplus 30 pieces of data as test datatesting the prediction accuracy of the model
43 Reference Model and Evaluation Criteria In order toverify the validity of the model this paper will comparethe prediction model (P-LSSVM) proposed in this paperwith conventional similar prediction model The first ref-erence model is the parameters joint optimization based
The Scientific World Journal 7
on genetic algorithm for chaos time series prediction (GA-LSSVM) [34] The second reference model uses the mutualinformation method and Cao method to get the best delaytime 120591 and embedding dimension 119898 respectively And thenuse grid search method to obtain LS-SVM parameters (120574and 120590) (denoted as M-C-LSSVM) [19] The third referencemodel uses the mutual information method and false nearestneighbor method to calculate the optimal delay time 120591 andembedding dimension119898 respectively And then use geneticalgorithm to get the optimal combination parameters of LS-SVM (120574 and 120590) (denoted as M-F-LSSVM) [27] The fourthreference model uses C-Cmethod to seek simultaneously thebest delay time 120591 and embedding dimension 119898 Then theoptimal parameters of LS-SVM (120574 and 120590) by using geneticalgorithm (denoted as C-C-LSSVM) [27 52]
Meanwhile this paper uses three evaluation criterianormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)NMSE RMSE and MAPE are defined respectively as fol-lows
where 119885 is the number of prediction points 119910 is the averagevalue and 119910
119894and 119910
119894denote the real value and the predicted
value of 119894th point respectively
44 Experimental Results In this paper the scope of parame-ters 120591119898 120574 and 120590 is [1 8] [3 17] [1 1000] and [01 10000]respectively In the process of evolution the other parametersare set as follows the number of elementary membranes119861 = 20 the number of objects in each membrane 119866 =
100 evolution algebra Max119879 = 1000 crossover probability119875119888= 085 and mutation probability 119875
119898= 005 The optimal
parameters combinations of eachmodel are shown in Tables 1and 2
441 Single-Step Prediction Selecting the first point as inputto obtain first predicted value then the real value of the firstpoint is added to the historical data predicting the next pointAnd so obtain the predicted value of all points Predictionresults of five models are shown in Tables 3 4 5 6 7 and 8and Figures 7 8 9 and 10
442 Multistep Forecast Selecting a point as input to obtainpredicted value then the prediction value of the first point is
Table 1The optimal parameters combination of fivemodels for FMbroadcasting band
added to the historical data predicting next point And soobtain the predicted value of all points Predicted results offive models are shown in Tables 9 10 11 12 13 and 14 andFigures 11 12 13 and 14
45 Analysis of Experimental Results The optimal parame-ters combinations of five models for FM broadcasting bandand interphone band are shown inTables 1 and 2 respectivelyAs seen from experimental results we can find that theparameters 120591 119898 120574 and 120590 are very sensitive to predictionaccuracy the optimal parameters combination is P-LSSVMmodel FM broadcasting bands are 7 14 1631 and 66475Interphone bands are 3 8 1709 and 21621 It can be seenfrom predicted results diagram (Figures 7 to 14) that whethersingle-step prediction or multistep prediction five models getvery good results However the P-LSSVM model predictscurve best fit to real data and other curves relative deviationfrom far away For five prediction models respectively run10 times computing the maximum minimum mean andvariance of error As can be seen from predicted results inTables 3 to 14 three kinds of models evaluation standard areRMSE NMSE andMAPE the model proposed in this paperis the minimum This shows that not only is the P-LSSVMmodel reasonable and correct but prediction accuracy is alsoenhanced
Comparing single-step prediction with multistep predic-tion it can be found that the error of multistep predictionis larger than the single-step prediction indicating that theeffect of single-step prediction is better than multistep pre-diction The reason is that errors exist in every step andthe accumulation of error will lead to decline in the overallprediction accuracy
5 Conclusion
Modeling and prediction of chaotic time series has become ahot spot in the research field of the chaotic signal processing
8 The Scientific World Journal
Table 3 Five models predicted error based on RMSE for FM broadcasting band
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
increasingly highlight in the electromagnetic spectrum withfrequency contradictions increasingly prominent betweencountries departments and military and space businesses[50] It is an important basis for spectrum management tocontrol comprehensively the change trend of parametersin the electromagnetic environment of country or region[51] It is the basis to master the frequency informationfor the frequency planning frequency allocation andsharing service frequency recovery work The situationof electromagnetic environment can be reflected by theelectromagnetic environment indicator parameters theseparameters mainly include band occupancy rate channeloccupancy rate large-signal ratio frequency offset and thefield strength A large number of experiment shown that timeseries data with chaotic in the electromagnetic environmentHence we used the proposed predictionmodel to predict theindicator parameters of the electromagnetic environmentThe experimental results show that the prediction modelproposed in this paper is reasonable and effective
Here we chose the band occupancy rate to do the testBand occupancy rate is calculated as follows extracting allthe signal points in the spectrum data the signals pointare merged with distance less than bandwidth by the belowformula to calculate the band occupancy rate (OccupyFreband)
OccupyFreband =119878119899lowast 119865119908
119865end minus 119865begin (16)
where 119878119899denotes the total number of signals judged 119865
119908is
necessary bandwidth in this band for the type of specifiedbusiness 119865begin is the start frequency point and 119865end is thecutoff frequency point
41 Experimental Data Sources In this paper we adoptdigital receiver EM100whichwas provided byGermanRohdeamp Schwarz Company and fixed radio monitoring stationof Xihua University to collect data for the experiment Wecollected data including frequency modulation (FM) broad-casting band and interphone band As shown in Figures 5 and6 in which the vertical axis denotes band occupancy rate thehorizontal axis represents the collection time and left pictureshows the data of band occupancy rate in FM broadcastingband we collected for 680 hours that is obtaining 680pieces of data Right figure indicates acquisition data ofband occupancy rate in interphone band we continuouslycollected for 187 hours that is gaining 187 pieces of data Inorder to facilitate narration here we put the band data of FMbroadcasting band and interphone band denoted by ldquodataset 1rdquo and ldquodata set 2rdquo respectively Use the method of smallamount of data to calculate the maximum Lyapunov indexof two groups of data which are 120582
1= 0126 and 120582
2= 014
respectively which show the time series with chaos
42 Data Preprocessing This paper mainly uses the Grubbscriteria to deal with the abnormal data the method is asfollows let 119902(ℎ 119889) be the sequence of the collected datawith the time interval between two data collections 119905 = 1
hour where ℎ = 0 22 23 denote 24 hours of a day119889 = 1 119867 minus 1119867 represents date code in total days of data
025035045055065075
1 71 141 211 281 351 421 491 561 631
Occ
upan
cy ra
te
t (hour)
Figure 5 Band occupancy rate data of FM radio band
000500070009001100130015001700190021
1 21 41 61 81 101 121 141 161 181
Occ
upan
cy ra
te
t (hour)
Figure 6 Band occupancy rate data of interphone band
collection119867 and 119902 denotes the collected data Using data setdenoted by 119876 = 119902
1 1199022 119902
119905 for each time point ℎ we
can get the expectation and variance of data sequence 119902(ℎ 119889)the formula is as follows
119864 (ℎ) =1
119878
119878
sum119896=1
119902 (ℎ 119889)
119863 (ℎ) = 1205902
119894=1
119878
119878
sum119896=1
[119902 (ℎ 119889) minus 119864 (ℎ)]2
(17)
where 119878 denotes the length of a unitAccording to the above two formulas combined with
Grubbs criteria if the sample point meet to1003816100381610038161003816119902 (ℎ 119889) minus 119864 (ℎ)
1003816100381610038161003816 ge 119866 (119899 120576) 120590119894 (18)
The sample point should be removed where 119866(119899 120576) is thecritical value of Grubbs criteria it can be obtained by lookingat Grubbs table 120576 denotes the significance level usuallysignificance level 120576 = 005
The Grubbs criteria are used to deal with ldquodata set 1rdquo andldquodata set 2rdquo respectively For the ldquodata set 1rdquo after processingwith Grubbs criteria the remaining 653 pieces of data we usethe front 600 pieces of data as the training data determiningthe best parameters combination and the surplus 53 pieces ofdata as test data testing the prediction accuracy of themodelFor the ldquodata set 2rdquo after processing with Grubbs criteria theremaining 180 pieces of data we use the front 150 pieces ofdata as the training data determining the best parameterscombination and the surplus 30 pieces of data as test datatesting the prediction accuracy of the model
43 Reference Model and Evaluation Criteria In order toverify the validity of the model this paper will comparethe prediction model (P-LSSVM) proposed in this paperwith conventional similar prediction model The first ref-erence model is the parameters joint optimization based
The Scientific World Journal 7
on genetic algorithm for chaos time series prediction (GA-LSSVM) [34] The second reference model uses the mutualinformation method and Cao method to get the best delaytime 120591 and embedding dimension 119898 respectively And thenuse grid search method to obtain LS-SVM parameters (120574and 120590) (denoted as M-C-LSSVM) [19] The third referencemodel uses the mutual information method and false nearestneighbor method to calculate the optimal delay time 120591 andembedding dimension119898 respectively And then use geneticalgorithm to get the optimal combination parameters of LS-SVM (120574 and 120590) (denoted as M-F-LSSVM) [27] The fourthreference model uses C-Cmethod to seek simultaneously thebest delay time 120591 and embedding dimension 119898 Then theoptimal parameters of LS-SVM (120574 and 120590) by using geneticalgorithm (denoted as C-C-LSSVM) [27 52]
Meanwhile this paper uses three evaluation criterianormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)NMSE RMSE and MAPE are defined respectively as fol-lows
where 119885 is the number of prediction points 119910 is the averagevalue and 119910
119894and 119910
119894denote the real value and the predicted
value of 119894th point respectively
44 Experimental Results In this paper the scope of parame-ters 120591119898 120574 and 120590 is [1 8] [3 17] [1 1000] and [01 10000]respectively In the process of evolution the other parametersare set as follows the number of elementary membranes119861 = 20 the number of objects in each membrane 119866 =
100 evolution algebra Max119879 = 1000 crossover probability119875119888= 085 and mutation probability 119875
119898= 005 The optimal
parameters combinations of eachmodel are shown in Tables 1and 2
441 Single-Step Prediction Selecting the first point as inputto obtain first predicted value then the real value of the firstpoint is added to the historical data predicting the next pointAnd so obtain the predicted value of all points Predictionresults of five models are shown in Tables 3 4 5 6 7 and 8and Figures 7 8 9 and 10
442 Multistep Forecast Selecting a point as input to obtainpredicted value then the prediction value of the first point is
Table 1The optimal parameters combination of fivemodels for FMbroadcasting band
added to the historical data predicting next point And soobtain the predicted value of all points Predicted results offive models are shown in Tables 9 10 11 12 13 and 14 andFigures 11 12 13 and 14
45 Analysis of Experimental Results The optimal parame-ters combinations of five models for FM broadcasting bandand interphone band are shown inTables 1 and 2 respectivelyAs seen from experimental results we can find that theparameters 120591 119898 120574 and 120590 are very sensitive to predictionaccuracy the optimal parameters combination is P-LSSVMmodel FM broadcasting bands are 7 14 1631 and 66475Interphone bands are 3 8 1709 and 21621 It can be seenfrom predicted results diagram (Figures 7 to 14) that whethersingle-step prediction or multistep prediction five models getvery good results However the P-LSSVM model predictscurve best fit to real data and other curves relative deviationfrom far away For five prediction models respectively run10 times computing the maximum minimum mean andvariance of error As can be seen from predicted results inTables 3 to 14 three kinds of models evaluation standard areRMSE NMSE andMAPE the model proposed in this paperis the minimum This shows that not only is the P-LSSVMmodel reasonable and correct but prediction accuracy is alsoenhanced
Comparing single-step prediction with multistep predic-tion it can be found that the error of multistep predictionis larger than the single-step prediction indicating that theeffect of single-step prediction is better than multistep pre-diction The reason is that errors exist in every step andthe accumulation of error will lead to decline in the overallprediction accuracy
5 Conclusion
Modeling and prediction of chaotic time series has become ahot spot in the research field of the chaotic signal processing
8 The Scientific World Journal
Table 3 Five models predicted error based on RMSE for FM broadcasting band
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
on genetic algorithm for chaos time series prediction (GA-LSSVM) [34] The second reference model uses the mutualinformation method and Cao method to get the best delaytime 120591 and embedding dimension 119898 respectively And thenuse grid search method to obtain LS-SVM parameters (120574and 120590) (denoted as M-C-LSSVM) [19] The third referencemodel uses the mutual information method and false nearestneighbor method to calculate the optimal delay time 120591 andembedding dimension119898 respectively And then use geneticalgorithm to get the optimal combination parameters of LS-SVM (120574 and 120590) (denoted as M-F-LSSVM) [27] The fourthreference model uses C-Cmethod to seek simultaneously thebest delay time 120591 and embedding dimension 119898 Then theoptimal parameters of LS-SVM (120574 and 120590) by using geneticalgorithm (denoted as C-C-LSSVM) [27 52]
Meanwhile this paper uses three evaluation criterianormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)NMSE RMSE and MAPE are defined respectively as fol-lows
where 119885 is the number of prediction points 119910 is the averagevalue and 119910
119894and 119910
119894denote the real value and the predicted
value of 119894th point respectively
44 Experimental Results In this paper the scope of parame-ters 120591119898 120574 and 120590 is [1 8] [3 17] [1 1000] and [01 10000]respectively In the process of evolution the other parametersare set as follows the number of elementary membranes119861 = 20 the number of objects in each membrane 119866 =
100 evolution algebra Max119879 = 1000 crossover probability119875119888= 085 and mutation probability 119875
119898= 005 The optimal
parameters combinations of eachmodel are shown in Tables 1and 2
441 Single-Step Prediction Selecting the first point as inputto obtain first predicted value then the real value of the firstpoint is added to the historical data predicting the next pointAnd so obtain the predicted value of all points Predictionresults of five models are shown in Tables 3 4 5 6 7 and 8and Figures 7 8 9 and 10
442 Multistep Forecast Selecting a point as input to obtainpredicted value then the prediction value of the first point is
Table 1The optimal parameters combination of fivemodels for FMbroadcasting band
added to the historical data predicting next point And soobtain the predicted value of all points Predicted results offive models are shown in Tables 9 10 11 12 13 and 14 andFigures 11 12 13 and 14
45 Analysis of Experimental Results The optimal parame-ters combinations of five models for FM broadcasting bandand interphone band are shown inTables 1 and 2 respectivelyAs seen from experimental results we can find that theparameters 120591 119898 120574 and 120590 are very sensitive to predictionaccuracy the optimal parameters combination is P-LSSVMmodel FM broadcasting bands are 7 14 1631 and 66475Interphone bands are 3 8 1709 and 21621 It can be seenfrom predicted results diagram (Figures 7 to 14) that whethersingle-step prediction or multistep prediction five models getvery good results However the P-LSSVM model predictscurve best fit to real data and other curves relative deviationfrom far away For five prediction models respectively run10 times computing the maximum minimum mean andvariance of error As can be seen from predicted results inTables 3 to 14 three kinds of models evaluation standard areRMSE NMSE andMAPE the model proposed in this paperis the minimum This shows that not only is the P-LSSVMmodel reasonable and correct but prediction accuracy is alsoenhanced
Comparing single-step prediction with multistep predic-tion it can be found that the error of multistep predictionis larger than the single-step prediction indicating that theeffect of single-step prediction is better than multistep pre-diction The reason is that errors exist in every step andthe accumulation of error will lead to decline in the overallprediction accuracy
5 Conclusion
Modeling and prediction of chaotic time series has become ahot spot in the research field of the chaotic signal processing
8 The Scientific World Journal
Table 3 Five models predicted error based on RMSE for FM broadcasting band
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
Figure 12 Five models predicted error diagram for FM broadcast-ing band
00005
0010015
002
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Occ
upan
cy ra
te
n
Prediction
RealP-LSSVMGA-LSSVM
M-C-LSSVMM-F-LSSVMC-C-LSSVM
Figure 13 Five models predicted diagram for interphone band
0
0005
001
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29n
Error
minus001
minus0005
P-LSSVMGA-LSSVMM-C-LSSVM
M-F-LSSVMC-C-LSSVM
Figure 14 Five models predicted error diagram for interphoneband
The Scientific World Journal 13
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
In this paper two defects were taken into considerationin the prediction model of LS-SVM for chaos time seriesprediction on the one hand ignoring the overall correlationof the parameters in predictionmodel and on the other handconsidering the contact between the parameters but theoptimization methods have some limitations For exampleuse genetic algorithm to solve the optimal parameter ofprediction model which itself has some limitations suchas falling into local optimum and iterative process compli-cation This paper puts forward a prediction model basedon membrane computing optimization algorithm for chaostime series prediction the model optimizes the parametersof phase space reconstruction and LS-SVM by using mem-brane computing optimization algorithm Then we used themodel to forecast band occupancy rate of FM broadcastingband and interphone band To show the applicability andsuperiority of the proposed model this paper will comparethe forecast model proposed in it with the traditional similarforecast model The experimental results show that whethersingle-step prediction or multistep prediction the proposedmodel performs best based on three error measures namelynormalized mean square error (NMSE) root mean squareerror (RMSE) and mean absolute percentage error (MAPE)For deficiency in multistep prediction in the next stage wewill further improve the prediction model or study otherprediction models to improve the multistep prediction
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is partially supported by the National NatureScience Foundation of China (61372187) Sichuan Key Tech-nology Research and Development Program (2012GZ00192013GXZ0155) and Graduate Innovation Foundation ofXihua University (ycjj2014038)
References
[1] R Ren X J Wang and S-H Zhu ldquoPrediction of chaotictime sequence using least squares support vector domainrdquo ActaPhysica Sinica vol 55 no 2 pp 555ndash563 2006
[2] M Ardalani-Farsa and S Zolfaghari ldquoChaotic time seriesprediction with residual analysis method using hybrid Elman-NARX neural networksrdquoNeurocomputing vol 73 no 13ndash15 pp2540ndash2553 2010
[3] H-Y Xing and T-L Jin ldquoWeak signal estimation in chaoticclutter using wavelet analysis and symmetric LS-SVM regres-sionrdquo Acta Physica Sinica vol 59 no 1 pp 140ndash146 2010
[4] AKazem E Sharifi F KHussainM Saberi andOKHussainldquoSupport vector regression with chaos-based firefly algorithmfor stock market price forecastingrdquo Applied Soft ComputingJournal vol 13 no 2 pp 947ndash958 2013
[5] W-Z Cui C-C Zhu W-X Bao and J-H Liu ldquoPrediction ofthe chaotic time series using support vector machines for fuzzy
[6] P Melin J Soto O Castillo and J Soria ldquoA new approachfor time series prediction using ensembles of ANFIS modelsrdquoExpert Systems with Applications vol 39 no 3 pp 3494ndash35062012
[7] P Samui and D P Kothari ldquoUtilization of a least square supportvector machine (LSSVM) for slope stability analysisrdquo ScientiaIranica vol 18 no 1 pp 53ndash58 2011
[8] Y B Sun V Babovic and E S Chan ldquoMulti-step-ahead modelerror prediction using time-delay neural networks combinedwith chaos theoryrdquo Journal of Hydrology vol 395 no 1-2 pp109ndash116 2010
[9] V Babovic S A Sannasiraj and E S Chan ldquoError correctionof a predictive ocean wavemodel using local model approxima-tionrdquo Journal of Marine Systems vol 53 no 1ndash4 pp 1ndash17 2005
[10] F-J Chang Y-M Chiang and L-C Chang ldquoMulti-step-aheadneural networks for flood forecastingrdquo Hydrological SciencesJournal vol 52 no 1 pp 114ndash130 2007
[11] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995
[12] V N Vapnik Statistical Leaning Theory John Wiley amp SonsNew York NY USA 1998
[13] L J Cao and F E H Tay ldquoSupport vector machine withadaptive parameters in financial time series forecastingrdquo IEEETransactions on Neural Networks vol 14 no 6 pp 1506ndash15182003
[14] K Chen and L Liu ldquoPrivacy preserving data classificationwith rotation perturbationrdquo in Proceedings of the 5th IEEEInternational Conference on Data Mining (ICDM rsquo05) pp 589ndash592 November 2005
[15] J W Cai S S Hu and H F Tao ldquoPrediction of chaotic timeseries based on selective support vector machine ensemblerdquoActa Physica Sinica vol 56 no 12 pp 6820ndash6827 2007
[16] J A K Suykens ldquoNonlinear modelling and support vectormachinesrdquo in Proceedings of the IEEE Instrumentation andMeasurement Technology Conference vol 1 pp 287ndash294 May2001
[17] B Jiang H Q Wang Y Fu X Li and G Guo ldquoBased on theLS-SVM chaotic prediction of sea clutterrdquo Progress in NaturalScience vol 17 no 3 pp 415ndash421 2007
[18] M ShenW-N Chen J ZhangH S-H Chung andO KaynakldquoOptimal selection of parameters for nonuniform embeddingof chaotic time series using ant colony optimizationrdquo IEEETransactions on Cybernetics vol 43 no 2 pp 790ndash802 2013
[19] Y Q Luo J B Xia and H B Wang ldquoApplication of chaos-support vector machine regression in traffic predictionrdquo Com-puter Science vol 36 no 7 pp 244ndash246 2009
[20] Y Benkler andH Nissenbaum ldquoCommons-based peer produc-tion and virtuerdquo The Journal of Political Philosophy vol 14 no4 pp 394ndash419 2006
[21] X Y Wang and M Han ldquoMultivariate chaotic time seriesprediction based onhierarchic reservoirsrdquo in IEEE InternationalConference on SystemsMan andCybernetics pp 14ndash17 October2012
[22] J A K Suykens J Vandewalle and B de Moor ldquoOptimalcontrol by least squares support vector machinesrdquo Neural Net-works vol 14 no 1 pp 23ndash35 2001
[23] H S He J Lu L F Wu and X J Qiu ldquoTime delay estimationvia non-mutual information among multiple microphonesrdquoApplied Acoustics vol 74 no 8 pp 1033ndash1036 2013
14 The Scientific World Journal
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
[24] H Ma X Li G Wang C Han J Xu and X Zhu ldquoSelectionof embedding dimension and delay time in phase space recon-structionrdquo Journal of Xirsquoan Jiaotong University vol 38 no 4 pp335ndash338 2004
[25] R Nath ldquoModified generalized autocorrelation based estimatorfor time delays in multipath environment-A tradeoff in esti-mator performance and number of multipathrdquo Computers andElectrical Engineering vol 37 no 3 pp 241ndash252 2011
[26] Z Q Li H Zheng and C M Pei ldquoA modified cao methodwith delay embeddedrdquo in Proceedings of the 2nd InternationalConference on Signal Processing Systems (ICSPS rsquo10) vol 3 ppV3458ndashV3460 July 2010
[27] S H Sun H Li and Z F Zhang ldquoOil price predicting based onunified solving by phase space reconstruction and parametersrdquoComputer Engineering and Applications vol 49 no 23 pp 247ndash251 2013
[28] I M Carrion E A Arias Antunez M M A Castillo andJ J M Canals ldquoParallel implementations of the false nearestneighbors method to study the behavior of dynamical modelsrdquoMathematical and Computer Modelling vol 52 no 7-8 pp1237ndash1242 2010
[29] C L Wu and K W Chau ldquoData-driven models for monthlystreamflow time series predictionrdquo Engineering Applications ofArtificial Intelligence vol 23 no 8 pp 1350ndash1367 2010
[30] C Wei M Chen C S Hui and Y S Chang ldquoStudy of basicmethod about WSD based on chaosrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo09) vol 3 pp 883ndash886 April2009
[31] C P Liu M Y Fan G W Wang and S L Ma ldquoOptimizingparameters of support vector machine based on gradientalgorithmrdquo Control and Decision vol 23 no 11 pp 1291ndash12962008
[32] WWei T Hui and X-P Ma ldquoRock burst chaotic prediction onmultivariate time series and LSSVRrdquo in Proceedings of the 25thChinese Control and Decision Conference (CCDC rsquo13) pp 1376ndash1381 May 2013
[33] Z Bo and A Shi ldquoLSSVM and hybrid particle swarm opti-mization for ship motion predictionrdquo in Proceedings of theInternational Conference on Intelligent Control and InformationProcessing (ICICIP rsquo10) pp 183ndash186 August 2010
[34] C Xiang Z Zhou X Yu and L-F Zhang ldquoStudy on chaotictime series prediction based on genetic algorithmrdquo ApplicationResearch of Computers vol 28 no 8 2011
[35] G Paun G Rozenberg and A Salomaa Handbook of Mem-brane Computing Oxford University Press Oxford UK 2009
[36] G Paun Membrane Computing Main Ideas Basic ResultsApplication Idea Group Publishing London UK 2004
[37] R C Muniyandi A M Zin and J W Sanders ldquoConvertingdifferential-equationmodels of biological systems tomembranecomputingrdquo BioSystems vol 114 no 3 pp 219ndash226 2013
[38] J Zhao and N Wang ldquoA bio-inspired algorithm based onmembrane computing and its application to gasoline blendingschedulingrdquo Computers amp Chemical Engineering vol 35 no 2pp 272ndash283 2011
[39] G-X Zhang C-X Liu and H-N Rong ldquoAnalyzing radaremitter signals with membrane algorithmsrdquo Mathematical andComputer Modelling vol 52 no 11-12 pp 1997ndash2010 2010
[40] D P Daniel P C Francisco and M A Gutierrez-Naranjo ldquoAparallel algorithm for skeletonizing images by using spikingneural P systemsrdquo Neurocomputing vol 115 pp 81ndash91 2013
[41] N H Packard J P Crutchfield J D Farmer and R S ShawldquoGeometry from a time seriesrdquo Physical Review Letters vol 45no 9 pp 712ndash716 1980
[42] F Takens ldquoDetecting strange attractors in turbulencerdquo LectureNotes in Mathematics vol 898 pp 361ndash381 1981
[43] J A K Suykens J de Brabanter L Lukas and J VandewalleldquoWeighted least squares support vector machines robustnessand sparce approximationrdquo Neurocomputing vol 48 no 3 pp85ndash105 2002
[44] A Riscos-Nunez ldquoA Framework for Complexity Classes inMembrane Computingrdquo Electronic Notes in Theoretical Com-puter Science vol 225 pp 319ndash328 2009
[45] OH Ibarra ldquoOnmembrane hierarchy in P systemsrdquoTheoreticalComputer Science vol 334 no 1ndash3 pp 115ndash129 2005
[46] C Teuscher ldquoFrom membranes to systems self-configurationand self-replication in membrane systemsrdquo BioSystems vol 87no 2-3 pp 101ndash110 2007
[47] LHuang IH Suh andAAbraham ldquoDynamicmulti-objectiveoptimization based on membrane computing for control oftime-varying unstable plantsrdquo Information Sciences vol 181 no11 pp 2370ndash2391 2011
[48] T M Taher R B Bacchus K J Zdunek and D A Rober-son ldquoLong-term spectral occupancy findings in Chicagordquo inProceedings of the IEEE International Symposium on DynamicSpectrum Access Networks (DySPAN rsquo11) pp 100ndash107 May 2011
[49] H Zhou ldquoAnalysis and resolved strategy of complicated electro-magnetic environment in battlefieldrdquo Journal of the Academy ofEquipment Command amp Technology vol 18 no 16 2007
[50] T ShaoHU Yihua S H Liang et al ldquoMethods for quantitativeevaluation of battlefield electromagnetic environment complex-ityrdquo Electronics Optics amp Control vol 17 no 1 2010
[51] ZWang and S Salous Spectrum occupancy analysis for cognitiveradio [PhD dissertation] Durham University Durham UK2009
[52] H Yin and H Wu ldquoStudy on time series forecasting based onleast squares support vector machinerdquo Computer Simulationvol 2 p 28 2011
