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Applied Soft Computing
j ourna l ho me p age: www.elsev ier .com/ l ocate /asoc
stimation of elastic constant of rocks using an ANFIS approach
ajesh Singh ∗, Ashutosh Kainthola, T.N. Singhepartment of Earth Sciences, Indian Institute of Technology Bombay, Mumbai, Maharastra, India
r t i c l e i n f o
rticle history:eceived 25 June 2010eceived in revised form 12 August 2011ccepted 18 September 2011vailable online 24 September 2011
eywords:NFISuzzy inference systemensityoint load
a b s t r a c t
The engineering properties of the rocks have the most vital role in planning of rock excavation and con-struction for optimum utilization of earth resources with greater safety and least damage to surroundings.The design and construction of structure is influenced by physico-mechanical properties of rock mass.Young’s modulus provides insight about the magnitude and characteristic of the rock mass deformationdue to change in stress field. The determination of the Young’s modulus in laboratory is very time con-suming and costly. Therefore, basic rock properties like point load, density and water absorption havebeen used to predict the Young’s modulus. Point load, density and water absorption can be easily deter-mined in field as well as laboratory and are pertinent properties to characterize a rock mass. The artificialneural network (ANN), fuzzy inference system (FIS) and neuro fuzzy are promising techniques whichhave proven to be very reliable in recent years. In, present study, neuro fuzzy system is applied to pre-
oung’s modulusAF
dict the rock Young’s modulus to overcome the limitation of ANN and fuzzy logic. Total 85 dataset wereused for training the network and 10 dataset for testing and validation of network rules. The networkperformance indices correlation coefficient, mean absolute percentage error (MAPE), root mean squareerror (RMSE), and variance account for (VAF) are found to be 0.6643, 7.583, 6.799, and 91.95 respectively,which endow with high performance of predictive neuro-fuzzy system to make use for prediction of
complex rock parameter.
. Introduction
The rock engineering properties have most importance role ineological, mining, petroleum engineering, geomechanical inves-igation. These properties are requisite for exploration, planning,esign of a project and for the optimal utilization of earth resources.he designing and construction of structure is influenced byesponse of physico-mechanical properties of rock mass underarious stress fields. The strength of the rock mass is adverselynfluenced by the anisotropy present within rock mass and theeconfiguring of active stresses due to loading, unloading andemoval of support from an area. An essential part of the site inves-igation is the determination of the deformational characteristicss well as the magnitude of the deformation in the rock mass. Thisan be achieved, by many methods like measuring the uniaxialompressive strength and the more complex methods of the mea-urement of the static moduli in laboratory doing tests on intactock specimens [1] and correlating them for the rock mass using
he field data.
Natural disasters like landslides, slope mass movements andarthquakes are unpredictable phenomena and magnitude of
destruction depends on elastic properties of the rock mass. Theelastic wave velocities in the rocks are primarily governed by theelastic constants. The most significance elastic constant is exten-sional stress–strain ratio, knows as Young’s modulus [2]. It iscomplicated, time consuming and tedious to determine Young’smodulus in the field as well as in the laboratory. The laboratoryexperiment for Young’s modulus is wearisome and need goodinstrumentation which is costly too [3]. A plenty of laboratory dataof mechanical properties is required to provide valuable insight forgeotechnical site characterization. Keeping in mind, the disconti-nuity and the anisotropic nature of rock, it is difficult to directlyobtain all the parameters of consideration. The rock engineers useempirical or analytical relationship among the various physicaland mechanical strength properties of rock mass of interest [4].Many researchers have proposed meaningful relationship usingadaptive neuro-fuzzy inference system ANFIS to characterize therock properties [5–7]. Gokceoglu et al. [6] determined modulus ofdeformation of jointed rock mass. Aali et al. [5] estimated satu-ration percentage (SP) of soil using multiple regression, ANN, andANFIS. They found that ANN and ANFIS are superior in estimatingSP. Therefore, fuzzy rule based ANN approach is a good and flexibleto define uncertainties in rock properties.
In this paper, an attempt was made to determine the Young’smodulus from basic properties like point load, density, and waterabsorption of rock using neuro-fuzzy system. The principle con-stituents of the modeling approach are fuzzy set, fuzzy logic and
R. Singh et al. / Applied Soft Computing 12 (2012) 40–45 41
zzy in
nmco
1
pIpsdtTogwd
1
agmadspvragF
1
tfFcT
Fig. 1. Simple fu
eural network. These are combined together and called hybridodeling framework (neuro-fuzzy). The aim of study is also to
onstruct the model and evaluate the reliability of predictabilityf model for determination of Young’s modulus.
.1. The dataset
In this study, an elastic property of rock (Young’s modulus) isredicted using physico-mechanical properties of rock using ANFIS.
t is difficult and expensive to determine all the geo-mechanicalarameters of the rocks. Hence, for investigation authors have con-idered easily determinable rock properties like point load test (PL),ensity and water absorption which were taken as input parame-ers and Young’s modulus as output parameter for ANFIS model.o construct the model, different physico-mechanical propertiesf different type of rocks were determined as per ISRM (1980) sug-ested method. Total 95 dataset have been used in which 85 datasetere taken as training data and 10 data were taken as the testingata.
.2. Neuro-fuzzy system
Zadeh [8] proposed a fuzzy set theory in which the set bound-ries were not precisely defined, but in fact boundaries wereradational. Such a set is characterized by continuum of grades ofembership (characteristic) function which assigns to each object
grade of membership ranging between zero and one [8]. Theseays, techniques in artificial neural networks, fuzzy set and fuzzyystem have been combined together which is termed as soft com-uting or intelligence technique. The fuzzy theory has been used inarious fields such as plant process control, automatization, patternecognition and decision making tool. The fuzzy inference systemsre also widely used in the areas of rock mechanics and engineeringeology [9–13]. The simple fuzzy inference system is illustrated inig. 1.
.3. Disadvantage of fuzzy logic
It was found that construction of a well performing fuzzy sys-em is not always easy as it seems. In the fuzzy logic, membership
unction and rules are determined by the trial and error process.or complex system, it required a significant time to find out theorrect membership function and rules to obtain a reliable solution.he generalization capability of the fuzzy logic is very poor because
ference system.
it uses the heuristic algorithms for defuzzification, rule evolutionand antecedent processing.
1.4. Disadvantage of neural networks
The neural network is most popular and widely used intelli-gence technique. But it also has disadvantages. The main drawbackof neural network is to determine proper size and optimal struc-ture of the neural net. The relationships of weight changes withinput output behavior during the training and use of trained systemto generate correct output using the weights is very complicatedto understand like a “Black box”. Manipulating learning parame-ters for learning and convergence is very difficult task because itworks on Bayesian regulation where over estimation and underestimation are not possible. Neural networks are costly to hardwareimplementation.
1.5. Advantage of neuro-fuzzy
Combining fuzzy logic and neural network is preeminent ideato overcome the disadvantage of both techniques. Neural networksare used to tune the membership functions of fuzzy systems evenfor complex systems. Communicating weight of the neural netusing fuzzy rules provides deep insight into the neural net, thuseasier to a design of better neural networks. The non-linear mem-bership function of neuro-fuzzy approach reduces the rule basedand saved memories, hence reduces implementation cost. Neuro-fuzzy hybrid systems integrate the advantages of fuzzy systems fordealing with explicit knowledge which can be explained and under-stood, and neural networks for dealing with implicit knowledgewhich can be acquired by learning.
Therefore, combination of fuzzy system and neural networkshandles limitations of both methods and offers an excellent data-mining opportunity to solve the critical and complex problemin geosciences [7,13,14]. The details of neuro-fuzzy system weredescribed by Takagi and Hayashi [15]. One of most common toolsis ANFIS which combined both FL and Neural network.
1.6. Adaptive neuro-fuzzy inference system (ANFIS)
The acronym ANFIS stands for adaptive neuro-fuzzy inferencesystem. Using a given input/output data set, the toolbox functionANFIS constructs a fuzzy inference system (FIS) whose mem-bership function parameters are tuned (adjusted) using either a
42 R. Singh et al. / Applied Soft Computing 12 (2012) 40–45
archit
bspcogn(MTsifatTimo(nc[
isb
f
f
O
ws
Fig. 2. (a) First order Sugeno fuzzy reasoning. (b) Equivalent ANFIS
ackpropagation algorithm alone or in combination with a leastquares type of method. Several fuzzy inference systems have beenroposed by various researchers. Agreeing to the structure of theonsequent parts and the inference method to compute the outputf the model, rule based model can be classified mainly in fourroups [9]: fuzzy relational models [16], linguistic models [17],eural network based models [18,19], and Takagi–Sugeno–KangTSK) fuzzy models [20,21]. The most commonly used systems are
amdani-type and Takagi–Sugeno-type which is also known asakagi–Sugeno–Kang-type [7]. A Mamdani-type fuzzy inferenceystem, both premise (if) and consequent (then) parts of a fuzzy,f–then rule are fuzzy propositions whereas a Takagi–Sugeno-typeuzzy inference system where the premise part of a fuzzy rule is
fuzzy proposition, the consequent part is a mathematical func-ion, usually a zero- or first-degree polynomial function [21]. TheSK model is simpler to identify because it needs less rules andts parameters can be estimation from numerical data using opti-
ization methods such as least-square algorithms [9]. Advantagesf the Sugeno method are as it works well with linear techniquese.g., PID control) as well as with optimization and adaptive tech-iques; it provides ensured continuity of the output surface; it isomputationally efficient and well suited to mathematical analysis22].
For simplicity, assume that FIS under consideration has twonputs (x, y) and one output (f) Fig. 2a and b. Then governing ruleet with two if–then rule of Takagi and Surgeon’s type as illustratedelow [23].
Rule I: if x is A1 and y is B1, then f1 = p1x + q1y + r1.Rule 2: if x is A2 and y is B2, then f2 = p2x + q2y + r2.
The node functions in the same layer are of the same functionamily as described:
Layer 1: every node i in this layer is a square node with a nodeunction:
1i = �Ai(x), (1)
here x is input to node i, O1i
is the membership grade of a fuzzyet Ai and it specifies the degree to which the given input x satisfies
ecture (after Jang [23]). The output of (a) is input into layer 4 of (b).
the quantifier A and �Ai is Gaussian membership function and it isgiven by
�Ai(x) = exp
[−(
x − ci
ai
)2]
(2)
where ai and ci is parameter set. Parameters in this layer arereferred to as premise parameters.
Layer 2: every node in this layer is a circle node labeled � whoseoutput is product of all incoming inputs:-
wi = �Ai(x) × �Bi(x), i = 1,2 (3)
Each node output represents the firing strength of a rule.Layer 3: every node labeled as encircled N. The ith node calcu-
lates the ratio of the ith rule’s firing strength to the sum of all rules’firing strengths:-
wi = wi
w1 + w2, i = 1, 2. (4)
Outputs of this layer will be called normalized firing strengths.Layer 4: including adaptive nodes:
O41 = wifi = wi(pix + qiy + ri) (5)
where wi is the output of layer 3, and (pi, qi, ri) is the parame-ter set. Parameters in this layer will be referred to as consequentparameters.
Layer 5: including a single labeled encircled � with function ofsummation.
Overall output = O51 =
∑i
wifi =
∑i
wifi
∑i
wi
(6)
The merit of the ANFIS is that it practices a hybrid learningprocess for the estimation of the premise and consequent param-
eters [23]. The hybrid algorithm splits learning process into twoindependent stages (1) the adaptation of learning weights and (2)adaptation of the nonlinear membership functions. This algorithmis able to decrease the complexity of algorithm and at the same
R. Singh et al. / Applied Soft Computing 12 (2012) 40–45 43
toc
2
hiG
Fa
Fig. 3. ANFIS model structure for Young’s modulus prediction.
ime increasing the learning efficiency [13]. A detailed discussionn ANFIS is described by Jang [24]. The illustrated Fig. 3 revealsonsidered ANFIS model structure for Young’s modulus estimation.
. Results and discussion
The Young’s modulus of the rock has been determined using
ybrid neuro-fuzzy model. The most significant step in model
s defining fuzzy membership function and corresponding value.aussian and bell membership functions are most popular methods
ig. 4. TSK membership function plot for input (a) “point load (PL)”, (b) “density”nd (c) “water absorption”.
Fig. 5. Number of epochs versus sum of square error for training and checking dataset.
for specifying the fuzzy set because of their smoothness and con-cise notation. Both membership functions have advantages of beingsmooth and non-zero at each point. The bell membership functionhas one more parameter than Gaussian membership function, soit can approach to non-fuzzy set if free parameter is tuned [22].Therefore the Gaussian membership function has been considered(Fig. 4a–c).
The hybrid algorithm has been applied to membership func-tion of each input. The advantage of hybrid method is that it usesback propagation for parameter associated with input member-ship function and least square estimation for parameters associatedwith output membership. Each input was normalized into range of[0,1] by the using formula-
Xnorm = X − Xmin
Xmax − Xmin(7)
where X is data which should be normalized, Xmax and Xmin arethe maximum and minimum value of original data respectively.Xnorm is normalized value of X. The three fuzzy “If–then rules” wereconsidered. The 30 epochs have been used to train the model. Fig. 5demonstrates the relation between sum of squared error and epoch.The plot shows that checking error decreases up to 3 epochs oftraining and then it increases. This increase suggests the pointsof model over fitting. Therefore, 3 epochs has been considered asappropriate for training of model.
The 85 dataset used to train model and 10 data were used tovalidate the predictability of the Young’s modulus and also seethe performance of the model. The observed and predicted val-ues of Young’s modulus along with percentage error are given in
Fig. 6. Cross-correlation between predicted and observed Young’s modulus values.
44 R. Singh et al. / Applied Soft Computing 12 (2012) 40–45
Table 1Observed and predicted values from hybrid neuro-fuzzy model along with percentage error.
S. no. Point load (MPa) Density (g/cc) Water absoprtion (%) Observed Young’s modulus (GPa) Predicted Young’s modulus GPa Error (%)
Fig. 7. Surface graph showing relationship of Young’s modulus with (a) point lo
able 1. To assess performance of the model, variance account forVAF), root mean square error (RMSE) and mean absolute percent-ge error (MAPE) were used as suggested by various researchers6,9,10,12–14,25]. Cross correlation of predicted and observedalue were determined and shown in Fig. 6. The correlation coef-cient between predicted and observed value is 0.6643. VAF andMSE were calculated using formula:
AF =(
1 − var(y − y′)var(y′)
)× 100 (8)
ar(y) = variance in set(y) = 1n
N∑i=1
(yi − y)
2
(9)
MSE =
√√√√ 1N
N∑i=1
(yi − y′i)2 (10)
d density (b) water absorption and point load (c) density and water absorption.
where y and y′ are the observed and predicted Young’s modulusvalues respectively, subscript i indicates ith data in the set, y is theaverage of the set y and N is the number of data.
The performance indices VAF, RMSE and MAPE, were calculatedas 91.95, 6.799 and 7.583 respectively. Theoretically, a predictionmodel is accepted as excellent when RMSE and MAPE are equalto zero and VAF is 100%. Performance indices VAF, RMSE andMAPE indicate that assessed result were highly correlated and pre-cise whereas correlation coefficient was also within the acceptablelimit.
The variations of Young’s modulus with any two inputs in theform of surface graph are shown in Fig. 7a–c. It can clearly be seen,variation of output (Young’s modulus) with input is found to be inan agreement with literature. This indicates the excellent identifi-cation capability of the neuro-fuzzy model.
3. Conclusion
The hybrid neuro-fuzzy system has been used to predict theYoung’s modulus using three geo-mechanical parameters which
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re easy and economical to determine in laboratory or field. Theotivation to prefer neuro-fuzzy modeling instead of other tech-
iques is primarily its capability of using vague and imprecisenformation of the rock. The calculated performance indices wereignifying that estimated results are very accurate and encourag-ng. The benefit of using ANFIS is that it combines advantages ofrtificial neural network and fuzzy logic with hybrid algorithm toeliver excellent modeling competency for complex, nonlinear andultivariable problems. This model can be used for prediction of
oung’s modulus in other field also.
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