Research Article Quasi-Stochastic Integration Filter for...

Research ArticleQuasi-Stochastic Integration Filter for Nonlinear Estimation

Yong-Gang Zhang Yu-Long Huang Zhe-Min Wu and Ning Li

College of Automation Harbin Engineering University No 145 Nantong Street Nangang District Harbin 150001 China

Correspondence should be addressed to Yu-Long Huang heuedu163com

Received 21 October 2013 Revised 18 May 2014 Accepted 24 May 2014 Published 23 June 2014

Academic Editor Dan Simon

Copyright copy 2014 Yong-Gang Zhang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In practical applications numerical instability problem systematic error problem caused by nonlinear approximation and nonlocalsampling problem for high-dimensional applications exist in unscented Kalman filter (UKF) To solve these problems a quasi-stochastic integration filter (QSIF) for nonlinear estimation is proposed in this paper nonlocal sampling problem is solved based onthe unbiased property of stochastic spherical integration rule which can also reduce systematic error and improve filtering accuracyIn addition numerical instability problem is solved by using fixed radial integration rule Simulations of bearing-only trackingmodel and nonlinear filtering problem with different state dimensions show that the proposed QSIF has higher filtering accuracyand good numerical stability as compared with existing methods and it can also solve nonlocal sampling problem effectively

1 Introduction

Nonlinear filtering has been widely used in many appli-cations Generally filtering problem can be addressed byusing Bayesian estimation theory which provides an optimalsolution for dynamic state estimation problem by computingthe posterior probability density function (PDF) [1 2] How-ever multidimensional integrals involved in the Bayesianestimation are typically intractable and closed form solutionto the Bayesian estimation is available only for a few specialcases [3] for example for a linear Gaussian system whichleads to thewell-knownKalman filter (KF) [4] In other casesapproximate methods are necessary to obtain suboptimalnonlinear filters These methods can be divided into twogroups global and local methods [5 6]

Global methods do not make any explicit assumptionabout PDFs which are computed directly by using approxi-matemethods [6] such as the point-mass filter using adaptivegrids [7] the Gaussianmixture filter [8] the particle filter [9]and Quasi-Monte Carlo filter [10] Typically global methodssuffer from enormous computational demands In additionthe performances of the particle filter andQuasi-Monte Carlofilter depend highly on the selection of proposal distributions[10]

Local methods derive nonlinear filters by assuming PDFsto be Gaussian which leads to Gaussian filter (GF) [11]

The unscented transformation- (UT-) based unscentedKalman filter (UKF) is a typical Gaussian approximate filterand has been widely used due to its ease of implementationmodest computational cost and appropriate performance[12 13] However UKF suffers from three main problemsin its practical applications numerical instability problem[6 11] systematic error problem caused by nonlinearapproximation [14ndash16] and nonlocal sampling problem forhigh-dimensional applications [17]

In order to address the numerical instability problemArasaratnam and Haykin proposed the cubature Kalmanfilter (CKF) based on the third-degree spherical-radial cuba-ture rule [6] However CKF is virtually a special case ofUKF with parameter 120581 = 0 and can only capture the thirdorder information of Taylor series expansion for nonlinearapproximation thus large systematic errorsmay be producedIn addition CKF suffers the nonlocal sampling problem forhigh-dimensional applications

To reduce systematic errors caused by nonlinear approx-imation and improve filtering accuracy Jia et al proposedhigh-degree CKF which can capture the fifth order infor-mation of Taylor series expansion for nonlinear approx-imation by generalizing the third-degree spherical-radialcubature rule to arbitrary degree spherical-radial cubaturerule [18] However high order systematic errors still existin high-degree CKF and the nonlocal sampling problem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 967127 10 pageshttpdxdoiorg1011552014967127

2 Mathematical Problems in Engineering

is not addressed in high-dimensional applications Duniket al indicated that systematic errors exist in all Gaussianapproximate filters for nonlinear estimation [14] To eliminatesystematic errors caused by nonlinear approximation Strakaet al proposed the stochastic integration filter (SIF) [14ndash16]based on the stochastic integral rule (SIR) [19]The posteriorimean and covariance computed by SIR tend to true posteriorimean and covariance in the sense of statistics because it canprovide asymptotically unbiased integral computation thusfiltering accuracy is improved However a large number ofround-off errors are introduced into integration computationusing the third-degree SIR (3rd-SIR) because the negativeweight is utilized in the 3rd-SIR thus the filtering numericalstability cannot be ensured and the filtering accuracy willdegrade greatly even filtering divergence may appear in thethird-degree SIF (3rd-SIF)

To address nonlocal sampling problem for high-dimensional applications Chang et al proposed thetransformed UKF (TUKF) by using the transformed sigmapoints to compute the nonlinear posteriori mean andcovariance in the Gaussian filtering frame [17] As comparedwith CKF and UKF high order terms of nonlinear posteriorimean and covariance computed by TUKF are much smallerand the nonlocal sampling problem is addressed to someextent However similar to CKF algorithm TUKF canonly capture the third order information of Taylor seriesexpansion for nonlinear approximation thus large systematicerrors still exist in TUKF Besides the high order informationof transformed sigma points computingmean and covariancemay be in the opposite direction of the true terms in somecases which causes TUKF to producemuch larger systematicerrors

In conclusion the above-mentioned existing typical non-linear filtering methods including UKF CKF high-degreeCKF TUKF and 3rd-SIF cannot simultaneously addressnumerical instability problem systematic error problem andnonlocal sampling problem To simultaneously address theseproblems a quasi-stochastic integration rule (QSIR) witharbitrary degree accuracy is proposed in this paper basedon fixed radial integral rule (FRIR) and stochastic sphericalintegral rule (SSIR) Anovel quasi-stochastic integration filter(QSIF) with arbitrary degree accuracy is then obtained byapplying the proposed QSIR to compute multidimensionalintegrals involved in GF QSIF addresses the nonlocal sam-pling problem for high-dimensional applications and reducessystematic errors by SSIR The numerical instability problemis solved by FRIR thus filtering accuracy is improved

To illustrate the superiority of the proposed QSIF algo-rithm two numerical simulations are presented includingbearings-only tracking and nonlinear filtering problem withdifferent state dimensions As can be seen from simulationresults of bearings-only tracking the proposed QSIF hashigher filtering accuracy and better numerical stability thanexisting filtering algorithms Simulation results of nonlinearfiltering problem with different state dimensions show thatthe nonlocal sampling problem can be effectively addressedby the proposed QSIF Note that as a special case of SIF thefirst-degree SIF (1st-SIF) proposed in [16] can also addressnumerical instability problem systematic error problem and

nonlocal sampling problemHowever its estimation accuracycan be further improved and as shown in our simulationsthe proposed 5th-QSIF has higher estimation accuracy thanthe 1st-SIF and it provides a new way to solve the above-mentioned problems

The remainder of the paper is organized as follows TheGF for nonlinear estimation and its problems are introducedin Section 2 The QSIR based on FRIR and SSIR is proposedin Section 3 The specific third-degree QSIF (3rd-QSIF)and fifth-degree QSIF (5th-QSIF) are given in Section 4Simulations are provided in Section 5 Concluding remarksare drawn in Section 6

2 Gaussian Filter and Problem Statement

21 Gaussian Filter Consider a discrete nonlinear systemwritten in the form of dynamic state space model as follows

x119896 = f (x119896minus1) + w119896minus1

z119896 = h (x119896) + v119896(1)

where x119896 isin R119899 and z119896 isin R119898 represent the immeasurable stateof the system and measurement at time 119896 respectively f(sdot)and h(sdot) are any known functions w119896minus1 is Gaussian processnoise with zero mean and known covariance Q119896minus1 v119896 isGaussian measurement noise with zero mean and knowncovariance R119896 and it is uncorrelated with the process noiseThe nonlinear filter is to obtain the minimum varianceestimate of the true but unknown system state based on thenoisy observations at time 119895(119895 le 119896) that is E[x119896 | Z119896] withZ119896 = z119895 1 le 119895 le 119896 [11]

If the PDF is well approximated by the Gaussian distri-bution a Gaussian filter in the Kalman-like structure (usinglinear update rule) can be used to address the estimation taskTheGaussian distribution is uniquely characterized by its firsttwo-order moments (mean and covariance) and the generalGaussian filter is formulated as [6]

x119896|119896 = x119896|119896minus1 +W119896 (z119896 minus z119896|119896minus1) (2)

P119896|119896 = P119896|119896minus1 minusW119896P119911119911119896|119896minus1WT119896 (3)

W119896 = P119909119911119896|119896minus1Pminus1

119911119911119896|119896minus1 (4)

wherex119896|119896minus1 = E [f (x119896minus1) | Z119896minus1]

= int

R119899119909f (x119896minus1) 119901 (x119896minus1 | Z119896minus1) 119889x119896minus1

= int

R119899119909f (x119896minus1)119873 (x119896minus1 x119896minus1|119896minus1P119896minus|119896minus1) 119889x119896minus1

P119896|119896minus1 = E [(x119896 minus x119896|119896minus1) (x119896 minus x119896|119896minus1)T| Z119896minus1]

= int

R119899119909f (x119896minus1) f

T(x119896minus1)

times 119873 (x119896minus1 x119896minus1|119896minus1P119896minus1|119896minus1) 119889x119896minus1

minus x119896minus1|119896minus1xT119896minus1|119896minus1

+Q119896minus1

Mathematical Problems in Engineering 3

z119896|119896minus1 = E [h (x119896) | Z119896minus1]

= int

R119899119911h (x119896)119873 (x119896 x119896|119896minus1P119896|119896minus1) 119889x119896

P119911119911119896|119896minus1 = E [(z119896 minus z119896|119896minus1) (z119896 minus z119896|119896minus1)T| Z119896minus1]

= int

R119899119909h (x119896) h

T(x119896)119873 (x119896 x119896|119896minus1P119896|119896minus1) 119889x119896

minus z119896|119896minus1zT119896|119896minus1

+ R119896

P119909119911119896|119896minus1 = E [(x119896 minus x119896|119896minus1) (z119896 minus z119896|119896minus1)T| Z119896minus1]

= int

R119899119909x119896h


minus x119896|119896minus1zT119896|119896minus1

(5)

It can be seen from the above update that the general GFconsists of one-step prediction of state and measurement in(5) and measurement update in the sense of linear minimummean square error (MSE) in (2)ndash(4) The heart of the GF isto compute Gaussian weighted integrals in (5) and differentGaussian filters will be obtained by using different numericalmethods For example the UKF can be obtained by using theUT to compute Gaussian weighted integrals in (5) Howeverthe numerical instability problem systematic error problemand nonlocal sampling problem exist in the UKF Next wewill show systematic errors and nonlocal sampling problemby introducing the computation of the mean and covariancein cubature transform (CT) and the numerical instabilityproblem by introducing the 3rd-SIR

22 Problems Statement

221 Systematic Errors Suppose that x is an 119899-dimensionalGaussian random vector with mean x and covariance P119909Another 119899119910-dimensional random vector y is related to xthrough the nonlinear function y = g(x) The CT can beused to calculate themean and covariance of yThemean andcovariance of y using CT that is yCT and PCT

119910 are formulated

as

yCT = 1

2119899

119899

sum

119895=1

[g (radic119899P119909119890119895 + x) + g (minusradic119899P119909119890119895 + x)]

PCT119910=

1

2119899

119899

sum

119895=1

[g (radic119899P119909119890119895 + x) gT (radic119899P119909119890119895 + x)

+ g (minusradic119899P119909119890119895 + x) gT (minusradic119899P119909119890119895 + x)]

minus yCT(yCT)T

(6)

where radicP119909 is the square root matrix of P119909 that isradicP119909radicP119909

T= P119909 119890119895 = [0 0 1

119895 0]

T denotes a unit

vector to the direction of coordinate axis 119895

Taylor series expansion is the most commonly used toolfor the error analysis of local methods Firstly computationalerror of mean using CT will be considered The Taylorexpansion of the true mean of y is as follows [12 13]

y = g (x) + 12

119899

sum

119894=1

119899

sum

119895=1

119875119909 (119894 119895) nabla2

119894119895g

+ E [D4Δ119909g (x)4

+

D6Δ119909g (x)6

+ sdot sdot sdot ]

(7)

where

nabla2

119894119895g = 120597

2

120597119909119894120597119909119895

g (x)1003816100381610038161003816100381610038161003816100381610038161003816x=x

1

119895

D119895Δ119909g (x) = 1

119895

(

119899

sum

119894=1

Δ119909119894

120597

120597119909119894

)

119895

g (x)10038161003816100381610038161003816100381610038161003816100381610038161003816x=x

Δx = x minus x

(8)

Δx119895 is the 119895th element of the vector ΔxThe Taylor expansion of computational mean of y can be

formulated as follows [14 15 20]

yCT = g (x) + 12

119899

sum

119894=1

119899

sum

119895=1

119875119909 (119894 119895) nabla2

119894119895g

+

1

2119899

2119899

sum

119901=1

(

D4120576119901

g (120594119901)4

+

D6120576119901

g (120594119901)6

+ sdot sdot sdot )

(9)

where

D119895120576119901

g (120594119901)119895

=

1

119895

(

119899

sum

119894=1

(120594119901 (119894) minus 119909119894)

120597

120597120594119901 (119894)

)

119895

g (120594119901)100381610038161003816100381610038161003816100381610038161003816100381610038161003816120594119901=x

120594119901 =radic119899P119909119890119901 + x 120576119901 =

radic119899P119909119890119901(10)

120594119901(119894) is the 119894th element of the vector 120594119901The systematic error 120576CT is defined as the difference

between the true mean value shown in (7) and the estimatedmean value calculated by CT in (9)

120576CT= E[

D4Δ119909g (x)4

+

D6Δ119909g (x)6

+ sdot sdot sdot ]

minus

1

2119899

2119899

sum

119901=1

(

D4120576119901

g (120594119901)4

+

D6120576119901

g (120594119901)6

+ sdot sdot sdot )

(11)

Generally the error 120576CT is nonzero [14] Similarly it canbe shown that the covariance matrix computations using theCT also contain systematic errors thus systematic errors exist


in CKF Finally it should be noted that systematic errors canbe found in all local filters To reduce systematic errors high-degree CKF captures the fifth order information of nonlinearTaylor series expansion in (7) by using more sigma pointsbut high order errors still exist Systematic errors in (11) inlocal filters are because of bias of the deterministic numericalintegration for computing multidimensional integrals [14ndash16] To address this problem Genz and Monahan proposedthe SIR by using stochastic radial integral rule (SRIR) andSSIR to fulfill the Gaussian weighted integrals computationBecause the SIR is unbiased that is E[120576SIR] = 0 hencethe SIR can eliminate the systematic errors in (11) andimprove filtering accuracy However the 3rd-SIF suffers fromnumerical instability problem which will be specified inSection 223

222 Nonlocal Sampling Problem The fourth and higher-order moments (hom) of the 119895th component for the compu-tational posterior mean of CT in (9) can also be written as[17]

hom =

1

2119899

2119899

sum

119901=1

[

[

infin

sum

119897=2

D2119897120576119901

g(120594119901)(2119897)

]

]119895

=

1

2119899

2119899

sum

119901=1

[

[

infin

sum

119897=2

[120576119901119895]

2119897

(2119897)

]

]

=

infin

sum

119897=2

119899119897minus1[

1

(2119897)

2119899

sum

119901=1

P119909 (119901 119895)]

(12)

As can be seen from (12) hom is proportional to theindex power of state dimension 119899119897minus1 and increases as statedimension increases More seriously the computational pos-terior mean and covariance using CT will have large highorder information for high-dimensional problem coupledwith strong nonlinear functions such as the exponent andtrigonometric which will degrade the filtering accuracy ofCKF [17] This problem is regarded as the nonlocal samplingproblem whose essence is that sigma points used in CKF arefar away from the mean as the state dimension increaseswhich degrades the ability of approximating PDFThe TUKFproposed in [17] mitigates the nonlocal sampling problem byconstructing transformed sigma points based on orthogonaltransformand improves the filtering accuracy ofCKForUKFHowever similar to CKF algorithm TUKF can only capturethe third order information of Taylor series expansion fornonlinear approximation thus large systematic errors stillexist Besides the high order information of transformedsigma points computing mean and covariance may be in theopposite direction of the true terms in some cases whichcauses TUKF to produce much larger systematic errors

223 Numerical Instability Problem Dunik et al proposedthe so-called SIF using the SIR to compute multidimensionalintegrals in (5) based on the GF frame [14ndash16] Next we willshow the numerical instability problem by introducing the3rd-SIR [14ndash16]

The mean of y computed by the 3rd-SIR is formulated as

ySIR3 = 1199080g (x)

+

119899

sum

119894=1

119908119894 [g (x + radicP119909120588Qe119894) + g (x minus radicP119909120588Qe119894)]

(13)

where1199080 = 1minus1198991205882 and119908119894 = 12120588

2 with 120588 sim 1205942(119899+2)Q is auniformly random orthogonal matrix 120588 is randomly chosenfrom 120594

2(119899 + 2) thus its value will be in the range [0 radic 119899]

with a certain probability Consequently 1199080 = 1 minus 1198991205882

will be negative with a certain probability In particular if120588 rarr 0 then 1199080 rarr minusinfin which leads to sum2119899119909

119894=0|119908119894| ≫

1 As a result large round-off errors are introduced forintegration computation [6 11 21 22] which may result indivergence of the 3rd-SIF method when the system noiseis large This problem is regarded as numerical instabilityproblem and it also exists in UKF [6] CKF method whichis based on deterministic spherical-radial cubature rule canavoid round-off errors of numerical computation and hasgood numerical stability However the accuracy of CKF islow and nonlocal sampling problem exists in CKF for high-dimensional applications

To mitigate numerical instability problem systematicerror problem and nonlocal sampling problem in nonlinearestimation a QSIR with arbitrary degree accuracy is pro-posed based on the FRIR and the SSIR in the paper Thenthe QSIF with arbitrary degree accuracy will be obtained byapplying the proposed QSIR to compute multidimensionalintegrals involved in GF Next we will introduce the QSIR

3 Quasi-Stochastic Integral Rules

Definition 1 We introduce the following integral

int

R119899g (x) 119908119891 (x) 119889x asymp sum

119894

119908119894g (120574119894) (14)

where x = [11990911199092 sdot sdot sdot 119909119899]Tisin R119899 and119908119891(x) is a given weighting

function Equation (14) is a 119889th-degree rule if it is exact for allmonomials 1199091205721

11199091205722

2sdot sdot sdot 119909120572119899

119899with the total degree up to 119889 that

is 119909120572111199091205722

2sdot sdot sdot 119909120572119899

119899| sum119899

119894=1120572119894 le 119889 and there is at least one

monomial of degree 119889 + 1 for which (14) is not exactIt is clear from the analysis in Section 21 that the key

problem of GF is how to compute multidimensional integralsformulated as

I (g) = intR119899g (x) exp (minusxTx) 119889x (15)

A crucial step before applying the spherical-radial cuba-ture rule is to transform the integration variable from Carte-sian coordinate system to spherical-radial system Define x =119903s with sTs = 1 then

I (g) = intinfin

0

int

119880119899

g (119903s) 119903119899minus1 exp (minus1199032) 119889120590 (s) 119889119903 (16)


where s = [1199041 1199042 119904119899]T119880119899 = s isin R119899 1199042

1+1199042

2+sdot sdot sdot+119904

2

119899= 1

and 120590(s) is the spherical surface measure or an area elementon 119880119899 Two types of integrals are contained in (16) that isradial integralintinfin

0g119903(119903)119903119899minus1 exp(minus1199032)119889119903 and spherical integral

int119880119899

g(119903s)119889120590(s)The integration rule proposed in this paper is QSIR in

which the spherical integral int119880119899

g(119903s)119889120590(s) is computed byusing SSIR and radial integral intinfin

0g119903(119903)119903119899minus1 exp(minus1199032)119889119903 is

computed by using FRIR Next we will introduce SSIR andFRIR respectively

31 Stochastic Spherical Integral Rule

Lemma 2 (see [18]) For the spherical integral I119880119899

(g119904) =

int119880119899

g119904(s)119889120590(s) the (2119898 + 1)th-degree full symmetric sphericalintegral rule is formulated as

I1198801198992119898+1 (g119904) asymp sum

|p|=119898119908pG up (17)

where I119880119899

denotes a spherical integral and I1198801198992119898+1 denotes

(2119898 + 1)th-degree full symmetric spherical integral rule Here119908p and Gup are defined as

119908p = I119880119899

(

119899

prod

119894=1

119901119894minus1

prod

119895=0

1199042

119894minus 1199062

119895

1199062119901119894

minus 1199062

119895

)

G up = 2minus119888(up)

sum

kg119904 (V1119906119901

1

V21199061199012

V119899119906119901119899

)

(18)

where 119901119894 is a nonnegative integer p = [1199011 1199012 119901119899] and|p| = 1199011+1199012+sdot sdot sdot+119901119899 119888(up) is the number of nonzero entries inup = (119906119901

1

1199061199012

119906119901119899

) where 119906119901119894

= radic119901119894119898 (119901119894 = 0 119898)The points of the spherical integral rule I119880

1198992119898+1 are given by

[V11199061199011

V21199061199012

V119899119906119901119899

] with weights 2minus119888(up)119908p where V119894 =plusmn1 Hence the arbitrary degree spherical integral rules can beobtained through Lemma 2

Lemma3 (see [19 23]) IfQ is a uniformly randomorthogonalmatrix I119880

1198992119898+1(g119904) asymp sum

119873119904

119895=1119908119895g119904(s119895) is (2119898 + 1)th-degree full

symmetric spherical integral rule of I119880119899

(g119904) = int119880119899

g119904(s)119889120590(s)then IQ119880

1198992119898+1(g119904) asymp sum

119873119904

119895=1119908119895g119904(Qs119895) is an unbiased (2119898 +

1)th-degree full symmetric SSIR

According to Lemma 2 the third-degree full symmetricspherical integral rule can be formulated as [18]

I1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (e119895) + g119904 (minuse119895)) (19)

Based on Lemma 3 the unbiased third-degree full sym-metric SSIR can be formulated as

IQ1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895)) (20)

Note that (20) is based on the third-degree full symmetricspherical integral rule and can capture the third orderinformation at least and it needs 2119899 spherical integral pointsthat is plusmnQe119895 (119895 = 1 2 119899) The existing first-degree SIR(1st-SIR) is based on the first-degree full symmetric sphericalintegral rule and can capture the first order information atleast and it needs 2 spherical integral points that is plusmnQ120589where 120589 is any point on 119880119899 Similarly the unbiased fifth-degree full symmetric SSIR can be formulated as

IQ1198801198995 (g119904)

asymp 1199081199041

119899(119899minus1)2

sum

119895=1

(g119904 (Qs+

119895)+g119904 (minusQs

+

119895)+g119904 (Qs

minus

119895) + g119904 (minusQs

minus

119895))

+ 1199081199042

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895))

(21)

where

1199081199041 =

119860119899

119899 (119899 + 2)

1199081199042 =

(4 minus 119899)119860119899

2119899 (119899 + 2)

s+119895= radic

1

2

(e119896 + e119897) 119896 lt 119897 119896 119897 = 1 2 119899

sminus119895= radic

1

2

(e119896 minus e119897) 119896 lt 119897 119896 119897 = 1 2 119899

(22)

In the calculation of the unbiased fifth-degree SSIR weneed 21198992 spherical integral points

32 Fixed Radial Integral Rule The generalized Gauss-Laguerre quadrature rule (GGLQR) can be used to computeradial integral If we define 119905 = 1199032 the radial integral can berewritten as

int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903 = 1

2

int

infin

0

g119903 (119905) t(1198992minus1) exp (minus119905) 119889119905

(23)

where g119903(119905) = g119903(radic119905) The right-hand side of (23) is thegeneralizedGauss-Laguerre integral with theweighting func-tion 119905(1198992minus1) exp(minus119905) and can be approximated byGGLQRThe(2119873119903 minus 1)th-degree GGLQR is formulated as

int

infin

0

g119903 (119905) 119905(1198992minus1) exp (minus119905) 119889119905 asymp

119873119903

sum

119894=1

119908119891119894g119903 (119903119891119894) (24)

where 119908119891119894 and 119903119891119894 can be obtained because (23) is exactfor g119903(119905) = 1 119905 119905

2 119905

2119873119903minus1 119908119903119894 and 119903119894 can be obtained

by using 119908119903119894 = (12)119908119891119894 119903119894 = radic119903119891119894 (119894 = 1 119873119903)Finally the GGLQR formulated by (24) is exact for g119903(119903) =1 1199032 1199034 119903

2(2119873119903minus1) Next two radial integral rules which are

the most commonly used in nonlinear filtering will be given


When 119873119903 = 1 the radial integral rule that is exact forg119903(119903) = 1 1199032 is formulated as [18]

int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903 asymp Γ (1198992)

2

g119903 (radic119899

2

) (25)

When 119873119903 = 2 the radial integral rule that is exact forg119903(119903) = 1 1199032 1199034 is formulated as [18]

int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903

asymp

1

119899 + 2

Γ (

1

2

119899) g119903 (0) +119899

2 (119899 + 2)

Γ (

1

2

119899) g119903 (radic1

2

119899 + 1)

(26)

As can be seen from (24) and (25) the above given FRIRsare not exact for odd-degree polynomials such as g119903(119903) =

119903 1199033 Fortunately when the FRIRs are combined with the

SSIRs to compute the integral (15) the combined spherical-radial rule vanishes for all odd-degree polynomials becauseSSIR vanishes by symmetry for any odd-degree Hence thearbitrary degree QSIR can be obtained by combining SSIRintroduced in Section 31 and FRIR introduced in Section 32Next we will formulate two QSIF methods based on theproposed QSIR

4 Quasi-Stochastic Integral Filtering Methods

The third-degree quasi-stochastic integration algorithm (3rd-QSIA) and fifth-degree QSIA (5th-QSIA) will be specifiedin this section We use them to compute Gaussian weightedmultidimensional integrals in (5) and obtain the correspond-ing 3rd-QSIF and 5th-QSIF

41 Third-Degree and Fifth-Degree Quasi-Stochastic Integra-tion Filters The third-degree QSIR (3rd-QSIR) is used in the3rd-QSIF Combining (16) (20) and (25) the 3rd-QSIR isgiven by

I = intR119899g (x)119873 (x120583Σ) 119889x

= int

R119899g (radicΣradic2x + 120583) exp (minusxTx) 119889x

=

1

1205871198992

int

infin

0

int

119880119899

g (radicΣradic2119903s + 120583) 119903119899minus1 exp (minus1199032) 119889120590 (s) 119889119903

asymp

1

1205871198992

119899

sum

119895=1

Γ (1198992)

2

119860119899

2119899

times [g(radicΣradic2radic1198992

Qe119895 + 120583) + g(minusradicΣradic2radic1198992

Qe119895 + 120583)]

asymp

1

2119899

119899

sum

119895=1

[g (radic119899ΣQe119895 + 120583) + g (minusradic119899ΣQe119895 + 120583)]

(27)

As can be seen from (27) in the calculation of the 3rd-QSIR we need 2119899 cubature points with the same weight(12119899) gt 0 Hence the 3rd-QSIR is numerically stablebecause its weights are all positive

Similarly combining (16) (21) and (26) the fifth-degreeQSIR (5th-QSIR) is given by

I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

2

119899 + 2

g (120583) + 1

(119899 + 2)2

times

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)

+g (minusradic(119899 + 2)ΣQsminus119895+ 120583))

+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583)) (28)

As can be seen from (28) in the calculation of the 5th-QSIR we need 21198992 + 1 cubature points and it has the sameweights as that of deterministic fifth-degree cubature ruleThe 3rd-QSIA and 5th-QSIA are summarized as follows

Step 1 Choose a maximum number of iterations119873max

Step 2 Set the number of iterations 119873 = 0 initial computa-tional integral values of the 3rd-QSIA and 5th-QSIA 1198683 = 0

and 1198685 = 0 and initial computational variance values of the3rd-QSIA and 5th-QSIA 1198813 = 0 and 1198815 = 0

Step 3 Repeat (until119873 = 119873max) the following loop

(a) Set119873 = 119873 + 1 1198781198773 = 0 and 1198781198775 = 0

(b) Generate a uniformly randomorthogonalmatrixQ ofdimension 119899 times 119899


(c) Compute the values 1198781198773 and 1198781198775 at current iteration

1198781198773 =

1

2119899

119899

sum

119895=1

[g (radic119899ΣQe119895 + 120583) + g (minusradic119899ΣQe119895 + 120583)] (29)

1198781198775 =

2

119899 + 2

g (120583)

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)

+ g (minusradic(119899 + 2)ΣQsminus119895+ 120583))

+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583))

(30)

and use them to update the approximate variance andintegral value as

119881119894 =

(119873 minus 2)119881119894

119873

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(119878119877119894 minus 119868119894)

119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

(119894 = 3 5)

119868119894 = 119868119894 +

(119878119877119894 minus 119868119894)

119873

(119894 = 3 5)

(31)

Step 4 Output the approximate integral value 119868119894 and standarddeviation 120590119894 = radic119881119894 of the 3rd-QSIA and 5th-QSIA

Note that the variable 119873max provides a tradeoff betweenefficiency and accuracy and its selection depends on applica-tion requirements For the calculation of (b) Stewart (1980)gave an algorithm for generating from the uniform distribu-tion (invariant Haar measure) over orthogonal matrices [23]Essentially the approach is to generate an 119899 times 119899 matrix Xof standard norm variables and form the QR factorizationX = QR Then Q has the right distribution The algorithmfor computing 119868119894 and 119881119894 is a modified version of a stable one-pass algorithm [19] The output standard deviations 1205903 and1205905 can be used to evaluate the exactness of the 3rd-QSIA and5th-QSIA respectively

Based on GF frame and using the above 3rd-QSIA and5th-QSIA to compute Gaussian weighted multidimensionalintegrals in (5) the 3rd-QSIF and 5th-QSIF can be developed

42 Properties of Quasi-Stochastic Integration Filters QSIFis obtained by using QSIR to compute Gaussian weightedmultidimensional integrals in (5) thus its properties dependcompletely on properties of QSIR

Firstly QSIF can reduce systematic errors and improvefiltering accuracy by using unbiased spherical integral com-putation The computational accuracy of Gaussian weighted

multidimensional integral mainly depends on the computa-tional accuracy of spherical integral [18 21 22] An unbiasedspherical integral computation is important in nonlinear GF

Secondly QSIF has good numerical stability similar tothat of CKF QSIR has the same weights as deterministicspherical-radial cubature rule because it uses FRIR to com-pute the radial integral The 3rd-QSIR is a completely stablenumerical integral rule because its weights are all positiveThe 5th-QSIR is also completely stable when state dimensionis less than four that is 119899 le 4 However it is not completelystable when 119899 ge 5 due to sum

21198992

119894=0|119908119894| gt 1 Fortunately

sum21198992

119894=0|119908119894| ≫ 1 will not happen in the 5th-QSIR because

as 119899 rarr +infin sum21198992

119894=0|119908119894| rarr 1 thus it does not suffer

from numerical instability problem Hence QSIF has goodnumerical stability

Thirdly high order information of computational meanand covariance of QSIF converges to true values and its highorder errors do not increase as state dimension increases soQSIF can mitigate the nonlocal sampling problem

Finally a comparison of computational complexitybetween the proposed filters and existing Gaussian approx-imate filters is shown in Table 1The computational complex-ity of the proposed QSIF and existing SIF are all dependenton the iteration numbers and the proposed 3rd-QSIF andexisting 3rd-SIF are almost consistent in computationalcomplexity Note that as a special case of SIF the 1st-SIF canbe deemed as MCKF with antithetic variates [16] Althoughits computational complexity in a single iteration is smallerthan the proposed QSIF algorithms it needs more iterationnumbers to achieve equivalent accuracy as compared with3rd-SIF 3rd-QSIF and 5th-QSIF As will be shown in latersimulations the proposed 5th-QSIF has higher estimationaccuracy than the 1st-SIF with equivalent computationalcomplexity by choosing different iteration numbers for bothalgorithms

In conclusion the proposed QSIF not only has highaccuracy and good numerical stability but also can effectivelymitigate the nonlocal sampling problem Next the advantagesof the proposedQSIF as comparedwith existingmethods willbe illustrated by two simulation examples

5 Simulations

The high accuracy and good numerical stability of theproposed QSIF are illustrated by a bearings-only trackingsimulation A nonlinear filtering problem with different statedimensions is used to illustrate that the proposed QSIF caneffectively mitigate the nonlocal sampling problem

51 Bearings-Only Tracking The considered nonlinearmodel describing the bearings-only tracking is of the form[24]

x119896 = [

09 0

0 1] x119896minus1 + n119896minus1

z119896 = tanminus1 (x2119896 minus sin (119896)x1119896 minus cos (119896)

) + v119896(32)


Table 1 Comparison of computational complexity

Filters 3rd-CKF and TUKF 5th-CKF 1st-SIF 3rd-SIF and 3rd-QSIF 5th-QSIFComputational complexity 119874(119899

3) 119874(119899

4) 119874(119873max119899

2) 119874(119873max119899

3) 119874(119873max119899

4)

Table 2 AMSEs and single step running times for bearings-only tracking

Filters TUKF MCKF 1st-SIF 3rd-SIF 3rd-CKF 5th-CKF 3rd-QSIF 5th-QSIFAMSEs (m) 879 6433 230 10787 879 207 686 179Time (msec) 038 364 365 185 035 072 181 362

where x119896 = [x1119896 x2119896]T= [119904 119905]

T denotes the positionsin the 119904-119905 plane (Cartesian coordinate system) The processnoise n119896 sim 119873(0Q119896) and Q119896 = [ 01 005005 01

] The measurementnoise v119896 sim 119873(0R119896) and R119896 = 0025 The true initialstate x0 = [20 5]

T and the associated covariance P0|0 =[01 0 0 01]

T Under the same initial conditions we givesimulation results of TUKF MCKF with sampling pointsof 50 1st-SIF with iteration numbers of 25 3rd-SIF withiteration numbers of 5 3rd-CKF 5th-CKF the proposed3rd-QSIF and 5th-QSIF with iteration numbers of 5 Notethat the 1st-SIF MCKF and the proposed 5th-QSIF havealmost consistent computational complexity in this simula-tion Besides we also give the conditional posterior CramerndashRao low bound (CPCRLB) [25] for better comparison Tocompare the performances of these filters simulation resultsare presented in the form of figures and tables In figures allfiltering methods are intuitively compared by using the MSEperformance index defined as follows

MSE (119896) = 1

119873

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(119894 = 1 2) (33)

In Table 2 all filtering methods are specifically compared byusing the average MSE (AMSE) performance index definedas follows

AMSE (119896) = 1

2119879119873

2

sum

119894=1

119879

sum

119896=1

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(34)

where x119899119894119896

is the 119894th component of the true state in the119899th simulation at time 119896 and x119899

119894119896119896is its filtering estimate

119873 denotes the number of Monte Carlo simulations and 119879denotes the simulation time We set 119879 as 100 seconds insimulation

For a fair comparison we do 1000 independent MonteCarlo runs MSEs and CPCRLB of positions are shown inFigures 1 and 2 and AMSEs and single step running times areshown in Table 2 (Note that TUKF is identical to 3rd-CKFwhen state dimension 119899 = 2)

As shown in Figures 1 and 2 the existing 3rd-SIF andMCKF diverge at time 30 s and 10 s respectively and theproposed 3rd-QSIF and 5th-QSIF outperform the existing3rd-CKF and 5th-CKF in terms of filtering accuracy respec-tively From Table 2 we can see that the proposed 3rd-QSIF has almost consistent computation demands with theexisting 3rd-SIF and the proposed 5th-QSIF provides higherestimation accuracy than the existing 1st-SIF with almostconsistent computation demands

0 20 40 60 80 1000

2

4

6

8

10

12

Time (s)

CPCRLBTUKFMCKF1st-SIF3rd-SIF

3rd-CKF5th-CKF3rd-QSIF5th-QSIF

Mea

n sq

uare

erro

r ofx

(1)

Figure 1 MSE of x(1)

52 Nonlinear Problem with Different State Dimensions Weconsider a nonlinear filtering problem with different statedimensions to show the nonlocal sampling problem fordifferent methods The nonlinear model is formulated as [17]

x119896 = 2 cos (x119896minus1) + q119896minus1

z119896 = radic1 + xT119896x119896 + k119896 119896 = 1 119879

(35)

where x119896 is assumed to be an 119899-dimensional Gaussianrandom variable q119896 sim 119873(119902 0119899times1 I119899) is the Gaussian processnoise 0119899times1 denotes an 119899-dimensional column vector with allits elements equal to 0 I119899 denotes an 119899-dimensional identitymatrix k119896 is Gaussian measurement noise with zero meanand unity varianceThe true initial state x0 = 01times1119899times1 where1119899times1 denotes an 119899-dimensional column vector with all itselements equal to 1 The initial conditions for all methods arex0 = 0119899times1 andP0|0 = I119899We aim to compare the performancesof 3rd-CKF 5th-CKF TUKF 1st-SIF 3rd-SIF 3rd-QSIF and5th-QSIF for different state dimensions 119899 (119899 = 1 30)


Table 3 Single step running times for 30-dimensional system model

Filters TUKF 1st-SIF 3rd-SIF 3rd-CKF 5th-CKF 3rd-QSIF 5th-QSIFTime (msec) 101 899 1168 097 1463 1033 13193

0 20 40 60 80 1000

5

10

15

20

25

30

Time (s)



Mea

n sq

uare

erro

r ofx

(2)


The performance is compared by using the MSE defined asfollows

MSE (119899) = 1

119872119879

119872

sum

119898=1

119879

sum

119896=1

(x1198981119896minus x1198981119896119896

)

2

(36)

where119872denotes the number ofMonteCarlo simulations and119879 denotes the simulation time We set 119879 as 100 seconds insimulation x119898

1119896is the first component of the true state in the

119898th simulation at time 119896 and x1198981119896119896

is its filtering estimateIn fact any element of x119896 can be chosen to illustrate theperformance due to their symmetric status in x119896 and x119896(1) ischosen in our simulation

We do 50 independent Monte Carlo runs under the sameinitial conditions MSEs for state dimensions 119899 = 1 30 areshown in Figure 3 and single step running times of all filtersfor 30-dimensional system model are shown in Table 3

As shown in Figure 3 TUKF outperforms the 1st-SIF 3rd-SIF 3rd-CKF and 5th-CKF especially for high state dimen-sions and the proposed 3rd-QSIF and 5th-QSIF considerablyoutperform TUKF for moderate state dimensions in termsof filtering accuracy and the proposed 3rd-QSIF 5th-QSIFand TUKF are almost consistent in filtering accuracy for highstate dimensions Besides from Figure 3 and Table 3 we cansee that the proposed 3rd-QSIF has higher filtering accuracythan the existing 1st-SIF with almost consistent computationdemands Simulation results illustrate that the proposedQSIFcan effectively mitigate the nonlocal sampling problem

0 5 10 15 20 25 30

26

28

3

32

34

36

38

4

42

Dimension nM

SE

3rd-CKF5th-CKFTUKF1st-SIF

3rd-SIF3rd-QSIF5th-QSIF

Figure 3 MSE with different state dimensions

6 Conclusion

A QSIF method is proposed based on unbiased SSIR andFRIR in the paper It can effectively solve the numericalinstability problem systematic error problem caused bynonlinear approximation and nonlocal sampling problemfor high-dimensional applications which exist in Gaussianfiltering Simulations of bearing-only tracking model andnonlinear filtering problem with different state dimensionsshow the advantages as compared with existing Gaussianfiltering algorithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grant nos 61001154 61201409and 61371173 China Postdoctoral Science Foundation no2013M530147 Heilongjiang Postdoctoral Fund LBH-Z13052and the Fundamental Research Funds for the Central Univer-sities of Harbin Engineering University no HEUCFX41307


References

[1] H Jazwinski Stochastic Processing and Filtering Theory Aca-demic Press New York NY USA 1970

[2] Y C Ho and R C K Lee ldquoA Bayesian approach to problemsin Stochastic estimation and controlrdquo IEEE Transactions onAutomatic Control vol 9 no 4 pp 211ndash221 1975

[3] B D O Anderson and S B Moore Optimal Filtering PrenticeHall Englewood Cliffs NJ USA 1979

[4] D Simon Optimal State Estimation Kalman H Infinity andNonlinear Approaches John Wiley and Sons New Jersey NJUSA 2006

[5] H W Sorenson ldquoOn the development of practical nonlinearfiltersrdquo Information Sciences vol 7 pp 230ndash270 1974

[6] I Arasaratnam and S Haykin ldquoCubature Kalman filtersrdquo IEEETransactions on Automatic Control vol 54 no 6 pp 1254ndash12692009

[7] M Simandl J Kralovec and T Soderstrom ldquoAdvanced point-mass method for nonlinear state estimationrdquo Automatica vol42 no 7 pp 1133ndash1145 2006

[8] D L Alspach andHW Sorenson ldquoNonlinear Bayesian estima-tion usingGaussian sumapproximationsrdquo IEEETransactions onAutomatic Control vol AC-17 no 4 pp 439ndash448 1972

[9] N J Gordon D J Salmond and A F M Smith ldquoNovelapproach to nonlinearnon-gaussian Bayesian state estimationrdquoIEE Proceedings F Radar and Signal Processing vol 140 no 2pp 107ndash113 1993

[10] D Guo and XWang ldquoQuasi-Monte Carlo filtering in nonlineardynamic systemsrdquo IEEE Transactions on Signal Processing vol54 no 6 pp 2087ndash2098 2006

[11] Y Wu D Hu M Wu and X Hu ldquoA numerical-integrationperspective on gaussian filtersrdquo IEEE Transactions on SignalProcessing vol 54 no 8 pp 2910ndash2921 2006

[12] S Julier J Uhlmann and H F Durrant-Whyte ldquoA newmethodfor the nonlinear transformation of means and covariances infilters and estimatorsrdquo IEEE Transactions on Automatic Controlvol 45 no 3 pp 477ndash482 2000

[13] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[14] J Dunik O Straka and M Simandl ldquoThe development of arandomised unscented Kalman filterrdquo in Proceedings of the 18thIFACWorld Congress vol 18 pp 8ndash13 Milano Italy 2011

[15] O Straka J Dunık and M Simandl ldquoRandomized unscentedKalman filter in target trackingrdquo in Proceedings of the 15thInternational Conference on Information Fusion (FUSION rsquo12)pp 503ndash510 Singapore September 2012

[16] J Dunık O Straka and M Simandl ldquoStochastic integrationfilterrdquo IEEE Transactions on Automatic Control vol 58 no 6pp 1561ndash1566 2013

[17] L Chang B Hu A Li and F Qin ldquoTransformed unscentedKalman filterrdquo IEEE Transactions on Automatic Control vol 58no 1 pp 252ndash257 2013

[18] B Jia M Xin and Y Cheng ldquoHigh-degree cubature Kalmanfilterrdquo Automatica vol 49 no 2 pp 510ndash518 2013

[19] A Genz and J Monahan ldquoStochastic integration rules forinfinite regionsrdquo SIAM Journal on Scientific Computing vol 19no 2 pp 426ndash439 1998

[20] M Simandl and J Dunık ldquoDerivative-free estimation methodsnew results and performance analysisrdquo Automatica vol 45 no7 pp 1749ndash1757 2009

[21] P J Davis and P RabinowitzMethods of Numerical IntegrationAcademic Press New York NY USA 1975

[22] A H Stroud Approximate Calculation of Multiple IntegralsPrentice-Hall Englewood Cliffs NJ USA 1971

[23] G W Stewart ldquoThe efficient generation of random orthogonalmatrices with an application to condition estimatorsrdquo SIAMJournal on Numerical Analysis vol 17 no 3 pp 403ndash409 1980

[24] J Dunık M Simandl and O Straka ldquoUnscented Kalmanfilter aspects and adaptive setting of scaling parameterrdquo IEEETransactions on Automatic Control vol 57 no 9 pp 2411ndash24162012

[25] Y Zheng O Ozdemir R Niu and P K Varshney ldquoNewconditional posterior Cramer-Rao lower bounds for nonlinearsequential Bayesian estimationrdquo IEEE Transactions on SignalProcessing vol 60 no 10 pp 5549ndash5556 2012

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Algebra

Discrete Dynamics in Nature and Society



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Discrete MathematicsJournal of


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Stochastic AnalysisInternational Journal of


is not addressed in high-dimensional applications Duniket al indicated that systematic errors exist in all Gaussianapproximate filters for nonlinear estimation [14] To eliminatesystematic errors caused by nonlinear approximation Strakaet al proposed the stochastic integration filter (SIF) [14ndash16]based on the stochastic integral rule (SIR) [19]The posteriorimean and covariance computed by SIR tend to true posteriorimean and covariance in the sense of statistics because it canprovide asymptotically unbiased integral computation thusfiltering accuracy is improved However a large number ofround-off errors are introduced into integration computationusing the third-degree SIR (3rd-SIR) because the negativeweight is utilized in the 3rd-SIR thus the filtering numericalstability cannot be ensured and the filtering accuracy willdegrade greatly even filtering divergence may appear in thethird-degree SIF (3rd-SIF)

To address nonlocal sampling problem for high-dimensional applications Chang et al proposed thetransformed UKF (TUKF) by using the transformed sigmapoints to compute the nonlinear posteriori mean andcovariance in the Gaussian filtering frame [17] As comparedwith CKF and UKF high order terms of nonlinear posteriorimean and covariance computed by TUKF are much smallerand the nonlocal sampling problem is addressed to someextent However similar to CKF algorithm TUKF canonly capture the third order information of Taylor seriesexpansion for nonlinear approximation thus large systematicerrors still exist in TUKF Besides the high order informationof transformed sigma points computingmean and covariancemay be in the opposite direction of the true terms in somecases which causes TUKF to producemuch larger systematicerrors

In conclusion the above-mentioned existing typical non-linear filtering methods including UKF CKF high-degreeCKF TUKF and 3rd-SIF cannot simultaneously addressnumerical instability problem systematic error problem andnonlocal sampling problem To simultaneously address theseproblems a quasi-stochastic integration rule (QSIR) witharbitrary degree accuracy is proposed in this paper basedon fixed radial integral rule (FRIR) and stochastic sphericalintegral rule (SSIR) Anovel quasi-stochastic integration filter(QSIF) with arbitrary degree accuracy is then obtained byapplying the proposed QSIR to compute multidimensionalintegrals involved in GF QSIF addresses the nonlocal sam-pling problem for high-dimensional applications and reducessystematic errors by SSIR The numerical instability problemis solved by FRIR thus filtering accuracy is improved

To illustrate the superiority of the proposed QSIF algo-rithm two numerical simulations are presented includingbearings-only tracking and nonlinear filtering problem withdifferent state dimensions As can be seen from simulationresults of bearings-only tracking the proposed QSIF hashigher filtering accuracy and better numerical stability thanexisting filtering algorithms Simulation results of nonlinearfiltering problem with different state dimensions show thatthe nonlocal sampling problem can be effectively addressedby the proposed QSIF Note that as a special case of SIF thefirst-degree SIF (1st-SIF) proposed in [16] can also addressnumerical instability problem systematic error problem and

nonlocal sampling problemHowever its estimation accuracycan be further improved and as shown in our simulationsthe proposed 5th-QSIF has higher estimation accuracy thanthe 1st-SIF and it provides a new way to solve the above-mentioned problems

The remainder of the paper is organized as follows TheGF for nonlinear estimation and its problems are introducedin Section 2 The QSIR based on FRIR and SSIR is proposedin Section 3 The specific third-degree QSIF (3rd-QSIF)and fifth-degree QSIF (5th-QSIF) are given in Section 4Simulations are provided in Section 5 Concluding remarksare drawn in Section 6

2 Gaussian Filter and Problem Statement

21 Gaussian Filter Consider a discrete nonlinear systemwritten in the form of dynamic state space model as follows

x119896 = f (x119896minus1) + w119896minus1

z119896 = h (x119896) + v119896(1)

where x119896 isin R119899 and z119896 isin R119898 represent the immeasurable stateof the system and measurement at time 119896 respectively f(sdot)and h(sdot) are any known functions w119896minus1 is Gaussian processnoise with zero mean and known covariance Q119896minus1 v119896 isGaussian measurement noise with zero mean and knowncovariance R119896 and it is uncorrelated with the process noiseThe nonlinear filter is to obtain the minimum varianceestimate of the true but unknown system state based on thenoisy observations at time 119895(119895 le 119896) that is E[x119896 | Z119896] withZ119896 = z119895 1 le 119895 le 119896 [11]

If the PDF is well approximated by the Gaussian distri-bution a Gaussian filter in the Kalman-like structure (usinglinear update rule) can be used to address the estimation taskTheGaussian distribution is uniquely characterized by its firsttwo-order moments (mean and covariance) and the generalGaussian filter is formulated as [6]

x119896|119896 = x119896|119896minus1 +W119896 (z119896 minus z119896|119896minus1) (2)

P119896|119896 = P119896|119896minus1 minusW119896P119911119911119896|119896minus1WT119896 (3)

W119896 = P119909119911119896|119896minus1Pminus1

119911119911119896|119896minus1 (4)

wherex119896|119896minus1 = E [f (x119896minus1) | Z119896minus1]

= int

R119899119909f (x119896minus1) 119901 (x119896minus1 | Z119896minus1) 119889x119896minus1

= int

R119899119909f (x119896minus1)119873 (x119896minus1 x119896minus1|119896minus1P119896minus|119896minus1) 119889x119896minus1

P119896|119896minus1 = E [(x119896 minus x119896|119896minus1) (x119896 minus x119896|119896minus1)T| Z119896minus1]

= int

R119899119909f (x119896minus1) f

T(x119896minus1)

times 119873 (x119896minus1 x119896minus1|119896minus1P119896minus1|119896minus1) 119889x119896minus1

minus x119896minus1|119896minus1xT119896minus1|119896minus1

+Q119896minus1



= int



= int

R119899119909h (x119896) h



+ R119896


= int

R119899119909x119896h



(5)





as

yCT = 1

2119899

119899

sum

119895=1


PCT119910=

1

2119899

119899

sum

119895=1



minus yCT(yCT)T

(6)


T= P119909 119890119895 = [0 0 1

119895 0]

T denotes a unit



y = g (x) + 12

119899

sum

119894=1

119899

sum

119895=1

119875119909 (119894 119895) nabla2

119894119895g

+ E [D4Δ119909g (x)4

+

D6Δ119909g (x)6

+ sdot sdot sdot ]

(7)

where

nabla2

119894119895g = 120597

2

120597119909119894120597119909119895

g (x)1003816100381610038161003816100381610038161003816100381610038161003816x=x

1

119895

D119895Δ119909g (x) = 1

119895

(

119899

sum

119894=1

Δ119909119894

120597

120597119909119894

)

119895

g (x)10038161003816100381610038161003816100381610038161003816100381610038161003816x=x

Δx = x minus x

(8)



yCT = g (x) + 12

119899

sum

119894=1

119899

sum

119895=1

119875119909 (119894 119895) nabla2

119894119895g

+

1

2119899

2119899

sum

119901=1

(

D4120576119901

g (120594119901)4

+

D6120576119901

g (120594119901)6

+ sdot sdot sdot )

(9)

where

D119895120576119901

g (120594119901)119895

=

1

119895

(

119899

sum

119894=1

(120594119901 (119894) minus 119909119894)

120597

120597120594119901 (119894)

)

119895

g (120594119901)100381610038161003816100381610038161003816100381610038161003816100381610038161003816120594119901=x

120594119901 =radic119899P119909119890119901 + x 120576119901 =

radic119899P119909119890119901(10)



120576CT= E[

D4Δ119909g (x)4

+

D6Δ119909g (x)6

+ sdot sdot sdot ]

minus

1

2119899

2119899

sum

119901=1

(

D4120576119901

g (120594119901)4

+

D6120576119901

g (120594119901)6

+ sdot sdot sdot )

(11)





hom =

1

2119899

2119899

sum

119901=1

[

[

infin

sum

119897=2

D2119897120576119901

g(120594119901)(2119897)

]

]119895

=

1

2119899

2119899

sum

119901=1

[

[

infin

sum

119897=2

[120576119901119895]

2119897

(2119897)

]

]

=

infin

sum

119897=2

119899119897minus1[

1

(2119897)

2119899

sum

119901=1

P119909 (119901 119895)]

(12)




ySIR3 = 1199080g (x)

+

119899

sum

119894=1


(13)

where1199080 = 1minus1198991205882 and119908119894 = 12120588





119894=0|119908119894| ≫





int

R119899g (x) 119908119891 (x) 119889x asymp sum

119894

119908119894g (120574119894) (14)



11199091205722

2sdot sdot sdot 119909120572119899


is 119909120572111199091205722

2sdot sdot sdot 119909120572119899

119899| sum119899






I (g) = intinfin

0

int

119880119899



where s = [1199041 1199042 119904119899]T119880119899 = s isin R119899 1199042

1+1199042


2

119899= 1



int119880119899








(g119904) =

int119880119899


I1198801198992119898+1 (g119904) asymp sum

|p|=119898119908pG up (17)

where I119880119899



119908p = I119880119899

(

119899

prod

119894=1

119901119894minus1

prod

119895=0

1199042

119894minus 1199062

119895

1199062119901119894

minus 1199062

119895

)


sum

kg119904 (V1119906119901

1

V21199061199012

V119899119906119901119899

)

(18)


1

1199061199012

119906119901119899

) where 119906119901119894


1198992119898+1 are given by

[V11199061199011

V21199061199012

V119899119906119901119899



1198992119898+1(g119904) asymp sum

119873119904



(g119904) = int119880119899

g119904(s)119889120590(s)then IQ119880

1198992119898+1(g119904) asymp sum

119873119904




I1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (e119895) + g119904 (minuse119895)) (19)


IQ1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895)) (20)


IQ1198801198995 (g119904)

asymp 1199081199041

119899(119899minus1)2

sum

119895=1

(g119904 (Qs+

119895)+g119904 (minusQs

+

119895)+g119904 (Qs

minus

119895) + g119904 (minusQs

minus

119895))

+ 1199081199042

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895))

(21)

where

1199081199041 =

119860119899

119899 (119899 + 2)

1199081199042 =

(4 minus 119899)119860119899

2119899 (119899 + 2)

s+119895= radic

1

2

(e119896 + e119897) 119896 lt 119897 119896 119897 = 1 2 119899

sminus119895= radic

1

2

(e119896 minus e119897) 119896 lt 119897 119896 119897 = 1 2 119899

(22)



int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903 = 1

2

int

infin

0


(23)


int

infin

0


119873119903

sum

119894=1

119908119891119894g119903 (119903119891119894) (24)


2 119905







int

infin

0


2

g119903 (radic119899

2

) (25)


int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903

asymp

1

119899 + 2

Γ (

1

2

119899) g119903 (0) +119899

2 (119899 + 2)

Γ (

1

2

119899) g119903 (radic1

2

119899 + 1)

(26)







I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

1

1205871198992

119899

sum

119895=1

Γ (1198992)

2

119860119899

2119899



Qe119895 + 120583)]

asymp

1

2119899

119899

sum

119895=1


(27)



I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

2

119899 + 2

g (120583) + 1

(119899 + 2)2

times

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583)) (28)






(a) Set119873 = 119873 + 1 1198781198773 = 0 and 1198781198775 = 0




1198781198773 =

1

2119899

119899

sum

119895=1


1198781198775 =

2

119899 + 2

g (120583)

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583))

(30)


119881119894 =

(119873 minus 2)119881119894

119873

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(119878119877119894 minus 119868119894)

119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

(119894 = 3 5)

119868119894 = 119868119894 +

(119878119877119894 minus 119868119894)

119873

(119894 = 3 5)

(31)








21198992

119894=0|119908119894| gt 1 Fortunately

sum21198992








5 Simulations



x119896 = [

09 0

0 1] x119896minus1 + n119896minus1


) + v119896(32)




3) 119874(119899

4) 119874(119873max119899

2) 119874(119873max119899

3) 119874(119873max119899

4)



where x119896 = [x1119896 x2119896]T= [119904 119905]





MSE (119896) = 1

119873

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(119894 = 1 2) (33)


AMSE (119896) = 1

2119879119873

2

sum

119894=1

119879

sum

119896=1

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(34)

where x119899119894119896






0 20 40 60 80 1000

2

4

6

8

10

12

Time (s)



Mea

n sq

uare

erro

r ofx

(1)




z119896 = radic1 + xT119896x119896 + k119896 119896 = 1 119879

(35)





0 20 40 60 80 1000

5

10

15

20

25

30

Time (s)



Mea

n sq

uare

erro

r ofx

(2)



MSE (119899) = 1

119872119879

119872

sum

119898=1

119879

sum

119896=1

(x1198981119896minus x1198981119896119896

)

2

(36)







0 5 10 15 20 25 30

26

28

3

32

34

36

38

4

42

Dimension nM

SE




6 Conclusion




Acknowledgments



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











= int



= int

R119899119909h (x119896) h



+ R119896


= int

R119899119909x119896h



(5)





as

yCT = 1

2119899

119899

sum

119895=1


PCT119910=

1

2119899

119899

sum

119895=1



minus yCT(yCT)T

(6)


T= P119909 119890119895 = [0 0 1

119895 0]

T denotes a unit



y = g (x) + 12

119899

sum

119894=1

119899

sum

119895=1

119875119909 (119894 119895) nabla2

119894119895g

+ E [D4Δ119909g (x)4

+

D6Δ119909g (x)6

+ sdot sdot sdot ]

(7)

where

nabla2

119894119895g = 120597

2

120597119909119894120597119909119895

g (x)1003816100381610038161003816100381610038161003816100381610038161003816x=x

1

119895

D119895Δ119909g (x) = 1

119895

(

119899

sum

119894=1

Δ119909119894

120597

120597119909119894

)

119895

g (x)10038161003816100381610038161003816100381610038161003816100381610038161003816x=x

Δx = x minus x

(8)



yCT = g (x) + 12

119899

sum

119894=1

119899

sum

119895=1

119875119909 (119894 119895) nabla2

119894119895g

+

1

2119899

2119899

sum

119901=1

(

D4120576119901

g (120594119901)4

+

D6120576119901

g (120594119901)6

+ sdot sdot sdot )

(9)

where

D119895120576119901

g (120594119901)119895

=

1

119895

(

119899

sum

119894=1

(120594119901 (119894) minus 119909119894)

120597

120597120594119901 (119894)

)

119895

g (120594119901)100381610038161003816100381610038161003816100381610038161003816100381610038161003816120594119901=x

120594119901 =radic119899P119909119890119901 + x 120576119901 =

radic119899P119909119890119901(10)



120576CT= E[

D4Δ119909g (x)4

+

D6Δ119909g (x)6

+ sdot sdot sdot ]

minus

1

2119899

2119899

sum

119901=1

(

D4120576119901

g (120594119901)4

+

D6120576119901

g (120594119901)6

+ sdot sdot sdot )

(11)





hom =

1

2119899

2119899

sum

119901=1

[

[

infin

sum

119897=2

D2119897120576119901

g(120594119901)(2119897)

]

]119895

=

1

2119899

2119899

sum

119901=1

[

[

infin

sum

119897=2

[120576119901119895]

2119897

(2119897)

]

]

=

infin

sum

119897=2

119899119897minus1[

1

(2119897)

2119899

sum

119901=1

P119909 (119901 119895)]

(12)




ySIR3 = 1199080g (x)

+

119899

sum

119894=1


(13)

where1199080 = 1minus1198991205882 and119908119894 = 12120588





119894=0|119908119894| ≫





int

R119899g (x) 119908119891 (x) 119889x asymp sum

119894

119908119894g (120574119894) (14)



11199091205722

2sdot sdot sdot 119909120572119899


is 119909120572111199091205722

2sdot sdot sdot 119909120572119899

119899| sum119899






I (g) = intinfin

0

int

119880119899



where s = [1199041 1199042 119904119899]T119880119899 = s isin R119899 1199042

1+1199042


2

119899= 1



int119880119899








(g119904) =

int119880119899


I1198801198992119898+1 (g119904) asymp sum

|p|=119898119908pG up (17)

where I119880119899



119908p = I119880119899

(

119899

prod

119894=1

119901119894minus1

prod

119895=0

1199042

119894minus 1199062

119895

1199062119901119894

minus 1199062

119895

)


sum

kg119904 (V1119906119901

1

V21199061199012

V119899119906119901119899

)

(18)


1

1199061199012

119906119901119899

) where 119906119901119894


1198992119898+1 are given by

[V11199061199011

V21199061199012

V119899119906119901119899



1198992119898+1(g119904) asymp sum

119873119904



(g119904) = int119880119899

g119904(s)119889120590(s)then IQ119880

1198992119898+1(g119904) asymp sum

119873119904




I1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (e119895) + g119904 (minuse119895)) (19)


IQ1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895)) (20)


IQ1198801198995 (g119904)

asymp 1199081199041

119899(119899minus1)2

sum

119895=1

(g119904 (Qs+

119895)+g119904 (minusQs

+

119895)+g119904 (Qs

minus

119895) + g119904 (minusQs

minus

119895))

+ 1199081199042

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895))

(21)

where

1199081199041 =

119860119899

119899 (119899 + 2)

1199081199042 =

(4 minus 119899)119860119899

2119899 (119899 + 2)

s+119895= radic

1

2

(e119896 + e119897) 119896 lt 119897 119896 119897 = 1 2 119899

sminus119895= radic

1

2

(e119896 minus e119897) 119896 lt 119897 119896 119897 = 1 2 119899

(22)



int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903 = 1

2

int

infin

0


(23)


int

infin

0


119873119903

sum

119894=1

119908119891119894g119903 (119903119891119894) (24)


2 119905







int

infin

0


2

g119903 (radic119899

2

) (25)


int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903

asymp

1

119899 + 2

Γ (

1

2

119899) g119903 (0) +119899

2 (119899 + 2)

Γ (

1

2

119899) g119903 (radic1

2

119899 + 1)

(26)







I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

1

1205871198992

119899

sum

119895=1

Γ (1198992)

2

119860119899

2119899



Qe119895 + 120583)]

asymp

1

2119899

119899

sum

119895=1


(27)



I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

2

119899 + 2

g (120583) + 1

(119899 + 2)2

times

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583)) (28)






(a) Set119873 = 119873 + 1 1198781198773 = 0 and 1198781198775 = 0




1198781198773 =

1

2119899

119899

sum

119895=1


1198781198775 =

2

119899 + 2

g (120583)

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583))

(30)


119881119894 =

(119873 minus 2)119881119894

119873

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(119878119877119894 minus 119868119894)

119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

(119894 = 3 5)

119868119894 = 119868119894 +

(119878119877119894 minus 119868119894)

119873

(119894 = 3 5)

(31)








21198992

119894=0|119908119894| gt 1 Fortunately

sum21198992








5 Simulations



x119896 = [

09 0

0 1] x119896minus1 + n119896minus1


) + v119896(32)




3) 119874(119899

4) 119874(119873max119899

2) 119874(119873max119899

3) 119874(119873max119899

4)



where x119896 = [x1119896 x2119896]T= [119904 119905]





MSE (119896) = 1

119873

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(119894 = 1 2) (33)


AMSE (119896) = 1

2119879119873

2

sum

119894=1

119879

sum

119896=1

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(34)

where x119899119894119896






0 20 40 60 80 1000

2

4

6

8

10

12

Time (s)



Mea

n sq

uare

erro

r ofx

(1)




z119896 = radic1 + xT119896x119896 + k119896 119896 = 1 119879

(35)





0 20 40 60 80 1000

5

10

15

20

25

30

Time (s)



Mea

n sq

uare

erro

r ofx

(2)



MSE (119899) = 1

119872119879

119872

sum

119898=1

119879

sum

119896=1

(x1198981119896minus x1198981119896119896

)

2

(36)







0 5 10 15 20 25 30

26

28

3

32

34

36

38

4

42

Dimension nM

SE




6 Conclusion




Acknowledgments



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












hom =

1

2119899

2119899

sum

119901=1

[

[

infin

sum

119897=2

D2119897120576119901

g(120594119901)(2119897)

]

]119895

=

1

2119899

2119899

sum

119901=1

[

[

infin

sum

119897=2

[120576119901119895]

2119897

(2119897)

]

]

=

infin

sum

119897=2

119899119897minus1[

1

(2119897)

2119899

sum

119901=1

P119909 (119901 119895)]

(12)




ySIR3 = 1199080g (x)

+

119899

sum

119894=1


(13)

where1199080 = 1minus1198991205882 and119908119894 = 12120588





119894=0|119908119894| ≫





int

R119899g (x) 119908119891 (x) 119889x asymp sum

119894

119908119894g (120574119894) (14)



11199091205722

2sdot sdot sdot 119909120572119899


is 119909120572111199091205722

2sdot sdot sdot 119909120572119899

119899| sum119899






I (g) = intinfin

0

int

119880119899



where s = [1199041 1199042 119904119899]T119880119899 = s isin R119899 1199042

1+1199042


2

119899= 1



int119880119899








(g119904) =

int119880119899


I1198801198992119898+1 (g119904) asymp sum

|p|=119898119908pG up (17)

where I119880119899



119908p = I119880119899

(

119899

prod

119894=1

119901119894minus1

prod

119895=0

1199042

119894minus 1199062

119895

1199062119901119894

minus 1199062

119895

)


sum

kg119904 (V1119906119901

1

V21199061199012

V119899119906119901119899

)

(18)


1

1199061199012

119906119901119899

) where 119906119901119894


1198992119898+1 are given by

[V11199061199011

V21199061199012

V119899119906119901119899



1198992119898+1(g119904) asymp sum

119873119904



(g119904) = int119880119899

g119904(s)119889120590(s)then IQ119880

1198992119898+1(g119904) asymp sum

119873119904




I1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (e119895) + g119904 (minuse119895)) (19)


IQ1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895)) (20)


IQ1198801198995 (g119904)

asymp 1199081199041

119899(119899minus1)2

sum

119895=1

(g119904 (Qs+

119895)+g119904 (minusQs

+

119895)+g119904 (Qs

minus

119895) + g119904 (minusQs

minus

119895))

+ 1199081199042

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895))

(21)

where

1199081199041 =

119860119899

119899 (119899 + 2)

1199081199042 =

(4 minus 119899)119860119899

2119899 (119899 + 2)

s+119895= radic

1

2

(e119896 + e119897) 119896 lt 119897 119896 119897 = 1 2 119899

sminus119895= radic

1

2

(e119896 minus e119897) 119896 lt 119897 119896 119897 = 1 2 119899

(22)



int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903 = 1

2

int

infin

0


(23)


int

infin

0


119873119903

sum

119894=1

119908119891119894g119903 (119903119891119894) (24)


2 119905







int

infin

0


2

g119903 (radic119899

2

) (25)


int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903

asymp

1

119899 + 2

Γ (

1

2

119899) g119903 (0) +119899

2 (119899 + 2)

Γ (

1

2

119899) g119903 (radic1

2

119899 + 1)

(26)







I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

1

1205871198992

119899

sum

119895=1

Γ (1198992)

2

119860119899

2119899



Qe119895 + 120583)]

asymp

1

2119899

119899

sum

119895=1


(27)



I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

2

119899 + 2

g (120583) + 1

(119899 + 2)2

times

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583)) (28)






(a) Set119873 = 119873 + 1 1198781198773 = 0 and 1198781198775 = 0




1198781198773 =

1

2119899

119899

sum

119895=1


1198781198775 =

2

119899 + 2

g (120583)

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583))

(30)


119881119894 =

(119873 minus 2)119881119894

119873

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(119878119877119894 minus 119868119894)

119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

(119894 = 3 5)

119868119894 = 119868119894 +

(119878119877119894 minus 119868119894)

119873

(119894 = 3 5)

(31)








21198992

119894=0|119908119894| gt 1 Fortunately

sum21198992








5 Simulations



x119896 = [

09 0

0 1] x119896minus1 + n119896minus1


) + v119896(32)




3) 119874(119899

4) 119874(119873max119899

2) 119874(119873max119899

3) 119874(119873max119899

4)



where x119896 = [x1119896 x2119896]T= [119904 119905]





MSE (119896) = 1

119873

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(119894 = 1 2) (33)


AMSE (119896) = 1

2119879119873

2

sum

119894=1

119879

sum

119896=1

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(34)

where x119899119894119896






0 20 40 60 80 1000

2

4

6

8

10

12

Time (s)



Mea

n sq

uare

erro

r ofx

(1)




z119896 = radic1 + xT119896x119896 + k119896 119896 = 1 119879

(35)





0 20 40 60 80 1000

5

10

15

20

25

30

Time (s)



Mea

n sq

uare

erro

r ofx

(2)



MSE (119899) = 1

119872119879

119872

sum

119898=1

119879

sum

119896=1

(x1198981119896minus x1198981119896119896

)

2

(36)







0 5 10 15 20 25 30

26

28

3

32

34

36

38

4

42

Dimension nM

SE




6 Conclusion




Acknowledgments



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where s = [1199041 1199042 119904119899]T119880119899 = s isin R119899 1199042

1+1199042


2

119899= 1



int119880119899








(g119904) =

int119880119899


I1198801198992119898+1 (g119904) asymp sum

|p|=119898119908pG up (17)

where I119880119899



119908p = I119880119899

(

119899

prod

119894=1

119901119894minus1

prod

119895=0

1199042

119894minus 1199062

119895

1199062119901119894

minus 1199062

119895

)


sum

kg119904 (V1119906119901

1

V21199061199012

V119899119906119901119899

)

(18)


1

1199061199012

119906119901119899

) where 119906119901119894


1198992119898+1 are given by

[V11199061199011

V21199061199012

V119899119906119901119899



1198992119898+1(g119904) asymp sum

119873119904



(g119904) = int119880119899

g119904(s)119889120590(s)then IQ119880

1198992119898+1(g119904) asymp sum

119873119904




I1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (e119895) + g119904 (minuse119895)) (19)


IQ1198801198993 (g119904) asymp

119860119899

2119899

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895)) (20)


IQ1198801198995 (g119904)

asymp 1199081199041

119899(119899minus1)2

sum

119895=1

(g119904 (Qs+

119895)+g119904 (minusQs

+

119895)+g119904 (Qs

minus

119895) + g119904 (minusQs

minus

119895))

+ 1199081199042

119899

sum

119895=1

(g119904 (Qe119895) + g119904 (minusQe119895))

(21)

where

1199081199041 =

119860119899

119899 (119899 + 2)

1199081199042 =

(4 minus 119899)119860119899

2119899 (119899 + 2)

s+119895= radic

1

2

(e119896 + e119897) 119896 lt 119897 119896 119897 = 1 2 119899

sminus119895= radic

1

2

(e119896 minus e119897) 119896 lt 119897 119896 119897 = 1 2 119899

(22)



int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903 = 1

2

int

infin

0


(23)


int

infin

0


119873119903

sum

119894=1

119908119891119894g119903 (119903119891119894) (24)


2 119905







int

infin

0


2

g119903 (radic119899

2

) (25)


int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903

asymp

1

119899 + 2

Γ (

1

2

119899) g119903 (0) +119899

2 (119899 + 2)

Γ (

1

2

119899) g119903 (radic1

2

119899 + 1)

(26)







I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

1

1205871198992

119899

sum

119895=1

Γ (1198992)

2

119860119899

2119899



Qe119895 + 120583)]

asymp

1

2119899

119899

sum

119895=1


(27)



I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

2

119899 + 2

g (120583) + 1

(119899 + 2)2

times

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583)) (28)






(a) Set119873 = 119873 + 1 1198781198773 = 0 and 1198781198775 = 0




1198781198773 =

1

2119899

119899

sum

119895=1


1198781198775 =

2

119899 + 2

g (120583)

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583))

(30)


119881119894 =

(119873 minus 2)119881119894

119873

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(119878119877119894 minus 119868119894)

119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

(119894 = 3 5)

119868119894 = 119868119894 +

(119878119877119894 minus 119868119894)

119873

(119894 = 3 5)

(31)








21198992

119894=0|119908119894| gt 1 Fortunately

sum21198992








5 Simulations



x119896 = [

09 0

0 1] x119896minus1 + n119896minus1


) + v119896(32)




3) 119874(119899

4) 119874(119873max119899

2) 119874(119873max119899

3) 119874(119873max119899

4)



where x119896 = [x1119896 x2119896]T= [119904 119905]





MSE (119896) = 1

119873

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(119894 = 1 2) (33)


AMSE (119896) = 1

2119879119873

2

sum

119894=1

119879

sum

119896=1

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(34)

where x119899119894119896






0 20 40 60 80 1000

2

4

6

8

10

12

Time (s)



Mea

n sq

uare

erro

r ofx

(1)




z119896 = radic1 + xT119896x119896 + k119896 119896 = 1 119879

(35)





0 20 40 60 80 1000

5

10

15

20

25

30

Time (s)



Mea

n sq

uare

erro

r ofx

(2)



MSE (119899) = 1

119872119879

119872

sum

119898=1

119879

sum

119896=1

(x1198981119896minus x1198981119896119896

)

2

(36)







0 5 10 15 20 25 30

26

28

3

32

34

36

38

4

42

Dimension nM

SE




6 Conclusion




Acknowledgments



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int

infin

0


2

g119903 (radic119899

2

) (25)


int

infin

0

g119903 (119903) 119903119899minus1 exp (minus1199032) 119889119903

asymp

1

119899 + 2

Γ (

1

2

119899) g119903 (0) +119899

2 (119899 + 2)

Γ (

1

2

119899) g119903 (radic1

2

119899 + 1)

(26)







I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

1

1205871198992

119899

sum

119895=1

Γ (1198992)

2

119860119899

2119899



Qe119895 + 120583)]

asymp

1

2119899

119899

sum

119895=1


(27)



I = intR119899g (x)119873 (x120583Σ) 119889x

= int


=

1

1205871198992

int

infin

0

int

119880119899


asymp

2

119899 + 2

g (120583) + 1

(119899 + 2)2

times

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583)) (28)






(a) Set119873 = 119873 + 1 1198781198773 = 0 and 1198781198775 = 0




1198781198773 =

1

2119899

119899

sum

119895=1


1198781198775 =

2

119899 + 2

g (120583)

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583))

(30)


119881119894 =

(119873 minus 2)119881119894

119873

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(119878119877119894 minus 119868119894)

119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

(119894 = 3 5)

119868119894 = 119868119894 +

(119878119877119894 minus 119868119894)

119873

(119894 = 3 5)

(31)








21198992

119894=0|119908119894| gt 1 Fortunately

sum21198992








5 Simulations



x119896 = [

09 0

0 1] x119896minus1 + n119896minus1


) + v119896(32)




3) 119874(119899

4) 119874(119873max119899

2) 119874(119873max119899

3) 119874(119873max119899

4)



where x119896 = [x1119896 x2119896]T= [119904 119905]





MSE (119896) = 1

119873

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(119894 = 1 2) (33)


AMSE (119896) = 1

2119879119873

2

sum

119894=1

119879

sum

119896=1

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(34)

where x119899119894119896






0 20 40 60 80 1000

2

4

6

8

10

12

Time (s)



Mea

n sq

uare

erro

r ofx

(1)




z119896 = radic1 + xT119896x119896 + k119896 119896 = 1 119879

(35)





0 20 40 60 80 1000

5

10

15

20

25

30

Time (s)



Mea

n sq

uare

erro

r ofx

(2)



MSE (119899) = 1

119872119879

119872

sum

119898=1

119879

sum

119896=1

(x1198981119896minus x1198981119896119896

)

2

(36)







0 5 10 15 20 25 30

26

28

3

32

34

36

38

4

42

Dimension nM

SE




6 Conclusion




Acknowledgments



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198781198773 =

1

2119899

119899

sum

119895=1


1198781198775 =

2

119899 + 2

g (120583)

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQs+119895+ 120583)

+ g (minusradic(119899 + 2)ΣQs+119895+ 120583))

+

1

(119899 + 2)2

119899(119899minus1)2

sum

119895=1

(g (radic(119899 + 2)ΣQsminus119895+ 120583)


+

4 minus 119899

2(119899 + 2)2

119899

sum

119895=1

(g (radic(119899 + 2)ΣQe119895 + 120583)

+g (minusradic(119899 + 2)ΣQe119895 + 120583))

(30)


119881119894 =

(119873 minus 2)119881119894

119873

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(119878119877119894 minus 119868119894)

119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

2

(119894 = 3 5)

119868119894 = 119868119894 +

(119878119877119894 minus 119868119894)

119873

(119894 = 3 5)

(31)








21198992

119894=0|119908119894| gt 1 Fortunately

sum21198992








5 Simulations



x119896 = [

09 0

0 1] x119896minus1 + n119896minus1


) + v119896(32)




3) 119874(119899

4) 119874(119873max119899

2) 119874(119873max119899

3) 119874(119873max119899

4)



where x119896 = [x1119896 x2119896]T= [119904 119905]





MSE (119896) = 1

119873

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(119894 = 1 2) (33)


AMSE (119896) = 1

2119879119873

2

sum

119894=1

119879

sum

119896=1

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(34)

where x119899119894119896






0 20 40 60 80 1000

2

4

6

8

10

12

Time (s)



Mea

n sq

uare

erro

r ofx

(1)




z119896 = radic1 + xT119896x119896 + k119896 119896 = 1 119879

(35)





0 20 40 60 80 1000

5

10

15

20

25

30

Time (s)



Mea

n sq

uare

erro

r ofx

(2)



MSE (119899) = 1

119872119879

119872

sum

119898=1

119879

sum

119896=1

(x1198981119896minus x1198981119896119896

)

2

(36)







0 5 10 15 20 25 30

26

28

3

32

34

36

38

4

42

Dimension nM

SE




6 Conclusion




Acknowledgments



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












3) 119874(119899

4) 119874(119873max119899

2) 119874(119873max119899

3) 119874(119873max119899

4)



where x119896 = [x1119896 x2119896]T= [119904 119905]





MSE (119896) = 1

119873

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(119894 = 1 2) (33)


AMSE (119896) = 1

2119879119873

2

sum

119894=1

119879

sum

119896=1

119873

sum

119899=1

(x119899119894119896minus x119899119894119896119896

)

2

(34)

where x119899119894119896






0 20 40 60 80 1000

2

4

6

8

10

12

Time (s)



Mea

n sq

uare

erro

r ofx

(1)




z119896 = radic1 + xT119896x119896 + k119896 119896 = 1 119879

(35)





0 20 40 60 80 1000

5

10

15

20

25

30

Time (s)



Mea

n sq

uare

erro

r ofx

(2)



MSE (119899) = 1

119872119879

119872

sum

119898=1

119879

sum

119896=1

(x1198981119896minus x1198981119896119896

)

2

(36)







0 5 10 15 20 25 30

26

28

3

32

34

36

38

4

42

Dimension nM

SE




6 Conclusion




Acknowledgments



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 20 40 60 80 1000

5

10

15

20

25

30

Time (s)



Mea

n sq

uare

erro

r ofx

(2)



MSE (119899) = 1

119872119879

119872

sum

119898=1

119879

sum

119896=1

(x1198981119896minus x1198981119896119896

)

2

(36)







0 5 10 15 20 25 30

26

28

3

32

34

36

38

4

42

Dimension nM

SE




6 Conclusion




Acknowledgments



References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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