ResearchArticle CEP Calculation Based on Weighted...

Research ArticleCEP Calculation Based on Weighted Bayesian Mixture Model

Shengdi Zhang1 Xiaojun Duan1 Chang Li1 Xiaojun Peng2 and Qiang Zhang2

1College of Science National University of Defense Technology Changsha Hunan 410073 China2Rocket Force Equipment Academy Beijing 100085 China

Correspondence should be addressed to Xiaojun Duan xj duan163com

Received 16 June 2016 Revised 29 November 2016 Accepted 26 December 2016 Published 24 January 2017

Academic Editor Alessandro Tasora

Copyright copy 2017 Shengdi Zhang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

CEP is an important accuracy index in the test evaluation for guidance weapon systems As for the samples origin from diversepopulations which is a general case in the practice traditional method with one population is inaccurate to give estimate directlyA weighted method considering the credibility of the prior information is proposed for Bayesian estimation algorithm and theweighted estimation of normal distribution parameters is provided And the statistical diversity between the weighted and classicalmethods is quantified and upper bound of the calculation errors caused by prior distortion is deduced for mixture of finite normaldistributions Taking the estimation of the dispersion of the guided weapon impact points as an example the conclusion is drawnthat our method is credible

1 Introduction

Circular Error Probable (CEP) is a common measure usedin the evaluation test of guidance weapon systems whichcan integrate precision with dispersion to assess the hitaccuracy of projectiles [1] In traditional CEP assessmentprocess the impact point deviation is assumed to followbivariate normal distribution and based on the formulasimplified by decorrelation coordinate transformation theMLE (maximum likelihood estimator) of normal distributionparameters are calculated by specific samples which areemployed in the forthcoming point estimation confidencebounds computation and hypothesis testing of CEP [2]

There are two restrictions for above CEP evaluationprocedure One is the concrete computational difficulty Itis hard to solve the complex CEP equation precisely sosome approximative expressions are adopted in practicalapplications with different assumptions [2] In [3] 12methodsof calculating CEP are summarized and classified into threecategories univariate parameterization binary computationand numerical integration

Another limitation is adaptability of the estimate processThe basis of bivariate normal distribution assumption is thecentral-limit theorem and engineering experience accumu-lated for ages [4] Compatibility and normality test are calledfor the hypothesis based on a certain confidence level But

the assumption is not suitable for complicated data whichdemands expanding the ranges of CEP application to obtainmore adaptable and reasonable results of hit accuracy [1 25 6] A bias-corrected estimator of CEP is provided basedon Cornish-Fisher expansion in [2] And in [5 6] CEPis extended to ACEP (Area Circular Error Probability) toovercome the disadvantage thatCEP is not combinedwith theself-characteristics of surface targets Reference [7] considersthe case that missile impact samples is not iid and providesa Bayesian procedure for estimating CEP

For INS (Inertial Navigation System)SAR (SyntheticAperture Radar) integrated system the system biases ofimpact point errors are quite distinct in diverse experi-ment conditions or scene features Under the circumstancesclassical CEP evaluation is not applicable because it istough to treat various samples as generated by the samenormal population In addition the stochastic quality ofscene matching also increases computational difficulty Inorder to use the samples following diverse populations toassess accuracy of projectiles this paper provides the CEPevaluation procedure under the circumstance that impactsamples are not iid data In Section 2 classical CEP isgeneralized to the CEP of multiconditional probability basedon nonparametric mixture model Then specific computa-tional method for mixed CEPmodel is designed in Section 3which includes simplification of general equation point and

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 8070786 7 pageshttpsdoiorg10115520178070786

2 Mathematical Problems in Engineering

interval estimation The algorithms of mixed and classicalCEP are compared and analyzed in Section 4 In Section 5some simulation experiments are implemented the resultsof which indicate that the mixed CEP algorithm is morereasonable than traditional one under the circumstances thatdiversity of accuracy samples is obvious

2 Modeling of Bayes Model of Scene Matchingof Multiconditional Probability

21 Classical CEPDefinition Thedefinition of CEP is distinctin kinds of literature and a common one is adopted in thispaper CEP is the radius of a circle where center is targetand the probability of an impact point inside is 50 [1]The corresponding mathematical description is as followsEstablish a rectangular coordinate system centered on targetwith (0 0) and assume the deviation of downrange and cross-range noted as 119883 and 119885 respectively both follows normaldistribution then the probability density function of 120578 =(119883 119885)119879 is

119891 (119909 119911) = 12120587120590119909120590119911radic1 minus 1205882

sdot expminus 12 (1 minus 1205882) [(119909 minus 120583119909)2

1205902119909minus 2120588 (119909 minus 120583119909) (119911 minus 120583119911)120590119909120590119911 + (119911 minus 120583119911)21205902119911 ]

(1)

where 120590119909 120590119911 are standard deviations of119883119885 120583119909 120583119911 aremeansof119883 119885 120588 is correlation coefficient of 119883 and 119885 such that 0 le|120588| lt 1 Then 119877 in the following formula is the CEP of 120578 =(119883 119885)119879

∬1199092+1199112⩽1198772

119891 (119909 119911) 119889119909 119889119911 = 05 (2)

where 119891(119909 119911) is the same as (1) And (2) is called the generalform of CEP equation

Suppose u119894 | u119894 = (119909119894 119911119894)119879119878119894=1 are the impact deviationsamples to be used In previous CEP algorithms they areutilized to calculate120590119909120590119911120583119909120583119911120588 in (1) which are employedto estimate 119877 by numerical integration or approximateapproaches in (2) But samples collected in different testenvironments and influenced by complicate factors such asscene features are distinct which is against the assumption ofsame population and make the traditional CEP evaluationinvalid Under this condition we propose the mixed CEPmodel and algorithm of multiconditional probability

22 Bayes Nonparametric Mixture Model The Bayes mixturemodel combined with random density function is as follows

119891 (119910) = int119870 (119910 120579) 119889119875 (120579) (3)

where119870(119910 120579) is density function with different parameters 120579And119875(120579) is randomdensity function as the following discreteform usually

119875 (119889120579) = infinsum119897=1

120596119897120575120579119897 (119889120579) (4)

where (120596119897 120579119897)infin119897=1 is prior information and 120575120579119897 is characteristicfunction with various 12057923 Mixed CEP Model for Scene Matching of MulticonditionalProbability Define mixed CEP as the radius of a circlecentered on the target such that the probability of animpact point inside the circle is 50 while impact deviationdoes not follow the same bivariate normal population ForINSSAR integrated navigation system the impact biases areinfluenced by scene matching of which the error sourcessuch as scene features types and matching number are quitedistinct and follow diverse populations [8] In order to assesshit accuracy of the system by mixed CEP modeling forscene matching is essential beforehand Since the types andoccurring probability of scene in one test are unknown beforethe experiment implement they can be regarded as randomvariables while modeling based on Bayes nonparametricmodel frame

Suppose synthetical test conditions of scene matching areclassified into 119873 types denoted as ℎ1 ℎ2 ℎ119873 of whichoccurring probability is 119901(ℎ1) 119901(ℎ2) 119901(ℎ119873) 119901(ℎ119894)(119894 =1 2 119873) are random variables in [0 1] such that

119873sum119896=1

119901 (ℎ119896) = 1 (5)

Consider the conditional probability density function ofimpact errors on ℎ119896 is normal as follows

119891119896 (119909 119911 | ℎ119896) = 12120587120590119909119896120590119911119896radic1 minus 1205881198962

sdot expminus12 (1 minus 1205881198962) [[

(119909 minus 120583119909119896)21205902119909119896minus 2120588119896 (119909 minus 120583119909119896) (119911 minus 120583119911119896)120590119909119896120590119911119896 + (119911 minus 120583119911119896)21205902119911119896 ]]

(6)

where 120590119909119896 120590119911119896 are standard deviations of 119883 119885 on conditionℎ119896120583119909119896 120583119911119896 aremeans of119883119885 on conditionℎ119896120588119896 is correlationcoefficient of 119883 119885 on condition ℎ119896 such that 0 le |120588119896| lt 1Then the joint probability density function of impact errorsis

119891 (119909 119911) = int119891119896 (119909 119911 | ℎ119896) 119901 (ℎ119896) 119889ℎ119896 (7)

The discrete form is

119891 (119909 119911) = 119873sum119896=1

119901 (ℎ119896) 119891119896 (119909 119911 | ℎ119896) (8)

Mathematical Problems in Engineering 3

Substitute (8) for 119891(119909 119911) in (2) and 119877 in the followingequation is the mixed CEP of multiconditional probability

∬1199092+1199112⩽1198772

119873sum119896=1

119901 (ℎ119896) 119891119896 (119909 119911 | ℎ119896) 119889119909 119889119911 = 05 (9)

3 Mixed CEP Algorithm of MulticonditionalProbability

31 Simplification of General CEP Equation For condition ℎ119896take orthogonal transformation for (6) as follows

(119906V) = ( cos 120585119896 sin 120585119896minus sin 120585119896 cos 120585119896)(

119909119911) (10)

where

120585119896 = 12119905119892minus1 21205881198961205901199091198961205901199111198961205901199091198962 minus 1205901199111198962 (11)

Then (119883 119885) is transformed to (119880 119881) of which 119880 and 119881are independent of condition ℎ119896 And probability densityfunction of (119880 119881) on condition ℎ119896 is

119891119896 (119906 V | ℎ119896) = 12120587120590119906119896120590V119896sdot expminus

12 [[(119906 minus 120583119906119896)21205902119906119896 + (V minus 120583V119896)21205902V119896 ]]

(12)

where 120590119906119896 120590V119896 are standard deviations of (119880 119881) on conditionℎ119896 120583119906119896 120583V119896 are means of (119880 119881) on condition ℎ119896 Then (9) isturned into

∬1199062+V2⩽1198772

119873sum119896=1

119901 (ℎ119896) 119891119896 (119906 V | ℎ119896) 119889119906 119889V = 05 (13)

Take polar coordinate transformation as follows119906 = 119903 cos120593V = 119903 sin120593 (14)

and then (13) is simplified from the form of double integral tothe following quadratic integral

12120587 int119877

0int21205870119903 119873sum119896=1

119901 (ℎ119896) sdot 119888119896sdot exp[minus1198871198961199032 + 1198861198961199032 cos 2120593 + 119903(1205831199061198961205902119906119896 cos120593 +

120583V1198961205902V119896 sin120593)]119889120593119889119903= 05(15)

where

119886119896 = 14 ( 11205902V119896 minus11205902119906119896 )

119887119896 = 14 ( 11205902V119896 +11205902119906119896 )

119888119896 = 1120590119906119896120590V119896 expminus12 [(120583119906119896120590119906119896 )

2 + (120583V119896120590V119896 )2]

(16)

32 Mixed CEP Point Estimates of Multiconditional Probabil-ity Suppose there are119873 types of impact error samples u(119896)119894 |u(119896)119894 = (119909(119896)119894 119911(119896)119894 )119879119899119896119894=1 (119896 = 1 2 119873) such that sum119873119894=1 119899119896 =119878 Assume u(119896)119894 119899119896119894=1 follow normal populations 119873(120583119896 Σ119896)respectively then the estimates of normal parameters are asfollows

119906119896 = 119909119896 cos 119896 + 119911119896 sin 119896V119896 = 119911119896 cos 119896 minus 119909119896 sin 119896119906119896= radic2119909119896cos2 119896 + 2119911119896sin2 119896 + 2119896119909119896 119911119896 cos 119896 sin 119896V119896= radic2119911119896cos2 119896 + 2119909119896sin2 119896 minus 2119896119909119896 119911119896 cos 119896 sin 119896

(17)

where

119909119896 = 1119899119896119899119896sum119894=1

119909(119896)119894119911119896 = 1119899119896

119899119896sum119894=1

119911(119896)119894119909119896 = radic 1119899119896 minus 1

119899119896sum119894=1

(119909(119896)119894 minus 119909119896)2

119911119896 = radic 1119899119896 minus 1119899119896sum119894=1

(119911(119896)119894 minus 119911119896)2119896 = sum119899119896119894=1 [(119909(119896)119894 minus 119909119896) (119911(119896)119894 minus 119911119896)]

radic[sum119899119896119894=1 (119909(119896)119894 minus 119909119896)2] [sum119899119896119894=1 (119911(119896)119894 minus 119911119896)2]

119896 = 12119905119892minus1 2119896119909119896 1199111198962119909119896 minus 2119911119896

(18)

Suppose prior information of text conditions is explicit thatis

119864 [119901 (ℎ119896)] = 119901119896 (19)

where 119901119896 are all constants such that

119873sum119896=1

119901119896 = 1 (20)

Substitute 119906119896 V119896 119906119896 V119896 119901119896 for 120583119906119896 120583V119896 120590119906119896 120590V119896 [119901(ℎ119896)] in(15) then mixed CEP can be calculated approximately withnumerical integration and bisection algorithm

33 Mixed CEP Interval Estimation of Multiconditional Prob-ability Bootstrap method [9 10] is called for calculating


mixed CEP confidence interval estimation of multicon-ditional probability here Suppose is the plug-in pointestimator of119877119862120572 denote120572 quantile of119877 that is119875(119877 le 119862120572) =120572 and corresponding estimator is 120572 Then the calculationformulas of confidence interval [119877119887119906 119877119887119897] and confidenceupper bound are given as follows

119877119887119906 = 1minus1205722119877119887119897 = 1205722119877119887 = 1minus120572

(21)

where 1minus1205722 1205722 120572 are approximately computed by thefollowing steps

(1) Calculate 119906119896 V119896 119906119896 V119896 by samples (119909(119896)119894 119911(119896)119894 )119899119896119894=1in (17) (119896 = 1 2 119873)

(2) Resample119872 times frombivariate normal distribution119873((119906119896 V119896)119879 diag(2119906119896 2V119896))with size 119899119896 respectively(119896 = 1 2 119873) Then compute119872 point estimatorsof 119877 by Section 32 and sort them from lowest as (1)(2) (119872) where 1000 ⩽ 119872 ⩽ 3000

(3) The INT[(1 minus 1205722)119872]th INT[(1205722)119872]th andINT[120572119872]th one of the sequences (1) (2) (119872)are 1minus1205722 1205722 120572 where INT[sdot] is roundingoperator

4 Performance Analysis of the Algorithm

41 Comparison of Statistical Properties The statistical prop-erties ofmixedCEP algorithmare comparedwith the classicalone for the bivariate mixed population of two normal dis-tributions as an example According to Section 31 the rela-tionship between the downrange and cross-range error canbe decorrelated by coordinate transformation Therefore thefollowing discussion focuses only on the independent case

Suppose there are119873 types of impact error samples u(119896)119894 |u(119896)119894 = (119909(119896)119894 119911(119896)119894 )119879119899119896119894=1 (119896 = 1 2 119873) such that sum119873119894=1 119899119896 =119878 Assume u(119896)119894 119899119896119894=1 follow the normal population 119873(120583119896 Σ119896)which are under the conditionsℎ119896 respectivelyThen the jointprobability density function of impact errors is as (8) where119891119896(119909 119911 | ℎ119896) is probability density function of119873(120583119896 Σ119896) (119896 =1 2 119873) And the estimates of normal parameters are asfollows

119896 = (119909119896119911119896) Σ119896 = (2119909119896 0

0 2119911119896) (22)

If the prior probabilities of test experiments are fixed as119864[119901(ℎ119896)] = 119901119896 (0 ⩽ 119901119896 ⩽ 1) and populations 119873(120583119896 Σ119896) areindependent then the mean and variance estimates of mixed

population 119891(119909 119911) will be [120578] = 119873sum

119896=1

119901119896119896Var [120578] = 119873sum

119896=1

1199012119896Σ119896(23)

If the impact error samples u(119896)119894 | u(119896)119894 = (119909(119896)119894 119911(119896)119894 )119879119899119896119894=1(119896 = 1 2 119873) are following the same normal population119873(120583119904 Σ119904) the estimates of normal parameters are

119904 = (119909119904119911119904) =1119878119873sum119896=1

119899119896119896Σ119904 = 1119878 minus 1

119873sum119896=1

119899119896sum119894=1

(119909(119896)119894 minus 119909119904 00 119911(119896)119894 minus 119911119904)

2 (24)

Thus the errors of single normal population approximated tothe bivariate mixed population of two normal distributionsare

Δ119864 = 119904 minus [120578] = 119873sum119896=1

(119899119896119878 minus 119901119896) 119896 (25)

Δ119863 = Σ119904 minus Var [120578] = 119873sum119896=1

(119899119896 minus 1119878 minus 1 minus 1199012119896) Σ119896 + 120576 (26)

where

120576 = 2119878 minus 1sdot 119873sum119896=1

119899119896sum119894=1

((119909(119896)119894 minus 119909119896) (119909119896 minus 119909119904) 00 (119911(119896)119894 minus 119911119896) (119911119896 minus 119911119904))= 0

(27)

By (25) it indicates that the means of two algorithms arethe same when 119896 = 0 or 119901119896 = 119899119896119878 forall119896 = 1 2 119873 Butas is shown in (26) variances of two algorithms are distinctunder these circumstances

42 Influence of Prior Distortion to Mixed CEP Errors Theselection of prior information is very influential to Bayesianposterior estimate If the prior is not accurate the error ofassessment results may be very large Here the influence ofprior distortion to mixed CEP calculation is discussed forfinite mixture model as sensitivity analysis

Let

∬1199092+1199112⩽1198772

119891119896 (119909 119911 | ℎ119896) 119889119909 119889119911 = 05 (28)

and note the solutions of (9) and (28) are 119877 and 119877119896 (119896 =1 2 119873) Suppose 119877119894 ⩽ 119877119895 (119894 lt 119895 119894 119895 = 1 2 119873) with-out loss of generality then we have the following equation

1198771 ⩽ 119877 ⩽ 119877119873 (29)


minus4minus2

02

4

X (m)Z (m)

minus4minus2

02

40

005

01

015

02

pdf

(a) 3 dimensions

minus3

minus2

minus1

0

1

2

3

4

minus3 minus2 minus1 0 1 2 3 4

Z (m

)

X (m)

002

004

006

008

01

012

014

(b) 2 dimensions

Figure 1 Probability density function of two normal mixture populations

In the equation above 1198771 and 119877119873 are the lower and upperbound of two normal mixture population of which thecorresponding values of the prior weights 119875119908 = (119901(ℎ1)119901(ℎ2) 119901(ℎ119873))119879 are (1 0 0)119879 and (0 0 1)119879 respec-tively It indicates that the CEP response reaches extremumsat the interval boundaries of prior weights Thus we have theproposition as follows

Proposition 1 Suppose random vector follows the mixturedistribution of which probability density function is the same as(8) Prior weights 119875119908 = (119901(ℎ1) 119901(ℎ2) 119901(ℎ119873))119879 is assumedas a random vector in [0 1]119873 such that 119864[119901(ℎ119896)] = 119901119896 (119896 =1 2 119873) and sum1198731 119901(ℎ119896) = sum1198731 119901119896 = 1 Note 119908 is theestimate of119875119908 andΔ119877 is the error of mixed CEPwith 119908Then

sup119908isin[01]

119873119908=1

Δ119877 (119908) ⩽ 10038161003816100381610038161003816100381610038161003816 max119896=12119873

119877119896 minus 119877010038161003816100381610038161003816100381610038161003816 (30)

where 1198770 is the solution of (9) with 119901(ℎ119896) = 119901119896 (119896 =1 2 119873) and 119877119896 denotes the solution of (9) with 119901(ℎ119896) = 1and 119901(ℎ119895) = 0 (119896 = 1 2 119873 119895 = 119896)5 Numerical Experiments

51 Example 1 Suppose synthetical test conditions of scenematching are ℎ1 and ℎ2 with 119864[119901(ℎ1)] = 04 and 119864[119901(ℎ2)] =06 And assume the impact errors (119883 119885) follow normaldistributions119873(( 00 ) ( 1 001001 1 )) and119873(( 11 ) ( 1 0505 1 )) underthe conditions ℎ1 and ℎ2 respectively Then the probabilitydensity function of two normal mixture population is shownin Figure 1

As displayed in Figure 1 the mixed population is stillunimodal of which the theoretical classical and mixed CEPare 1414m 1215m and 1420m Take sample from normalpopulations 1 and 2with sizes 20 and 30 respectively ofwhichthe dispersion is shown in Figure 2 In Figure 2 only 8 samples

Theoritical CEPMixture CEPClassical CEP

Sample by population 1Sample by population 2

1050 15 2minus1minus15minus2 minus05minus25X (m)

minus25

minus2

minus15

minus1

minus05

0

05

1

Z (m

)

Figure 2 Sample dispersion and comparison of CEP by differentalgorithms

(16) impact in the circle of classical CEP which does notaccord with the definition of CEP Meanwhile there are 28samples (56) in the circle of mixed CEP which illustratesthat the mixed CEP calculation algorithm in this paper ismore adaptable and accurate contrasted with theoretical andclassical CEP under the circumstance

In order to reduce the influence of random factors toCEP calculation result resample 10 and 100 times underthe circumstance that weights sample size and parametersof two normal populations are fixed and the averages ofcorresponding computing results are shown in Table 1

In Table 1 it indicates that the average precision of mixedCEP algorithm is larger than classical one on the condition


Table 1 Contrast average CEP mixed with classical algorithms by resampling

Sampling times CEP algorithms Samples in CEP circle Percentage of more accurate mixed CEPMixed Classical Mixed Classical

1 1420 1215 56 16 10010 1357 1176 47 24 100100 1355 1159 44 21 77

Table 2 CEP and corresponding errors with different prior information

1 Δ1 Δ0 04 1633 021901 03 1573 016002 02 1517 010403 01 1464 005004 0 1414 005 01 1367 minus004706 02 1323 minus009007 03 1283 minus013108 04 1245 minus016909 05 1210 minus02041 06 1177 minus0236

Table 3 Value of impact errors

Test condition Direction Test data (m)1 2 3 4 5 6 7 8 9

1 119883 0538 1834 minus2259 0862 0319 minus1308 minus0434 0343 3578119885 2769 minus1350 3035 0725 minus0063 0715 minus0205 minus0124 1490

2 119883 0641 0642 0567 0379 0572 0663 0549 0604 0573119885 0470 0529 0421 0589 0385 0393 0419 0206 0644

3 119883 1033 0925 1137 0829 0990 0976 1032 1031 0914119885 0997 0984 1063 1109 1111 0914 1008 0879 0889

Table 4 Actual and prior mean of probability of test conditions

Testcondition

Mean of realprobability

Prior mean ofprobability

1 13 3102 13 3103 13 25

of resampling and the probability when mixed CEP is moreprecise than typical one is large (about 77 in resampling100 times) under the circumstance that samples are fixedIn addition viewed from the perspective of containingsamples the proportion of samples in the circle of mixedCEP much more comply with the CEP definition of whichthe average is about 44 in resampling times Meanwhile thecorresponding value of classical CEP circle is far from theideal 50 of which the average is about 21

52 Example 2 The influence of prior distortion to mixedCEP errors for two normalmixture population is discussed inthis subsection of which the related parameters are the sameas Section 51 In this case the input of prior parameter is only1 isin [0 1] while 2 can be replaced by 2 = 1 minus 1 So therelationship between 1 and mixed CEP can be expressed asthe following equation

int1199092+1199112le1198772

[1198912 + 1 (1198911 minus 1198912)] 119889119909 119889119911 = 05 (31)

where 1198911 and 1198912 denote density functions of normal dis-tribution 1 and 2 Suppose 1 = 04 is the real priorinformation and the exact mixed CEP is 1414m Then CEPand corresponding errors are calculated on the conditions of1 = 01119896 (119896 = 0 1 10) which are shown in Table 2

As is shown in Table 2 the CEP error Δ increases asthe prior error Δ1 increases And we can see the maximumof CEP under prior distortion is achieved by the boundary


Table 5 CEP under different assumptions of the population

Assumptions of the population CEP point estimate CEP confidence upper bound CEP confidence intervalLevel 09 Level 08 Level 09 Level 08

Single 1416 1595 1527 [1220 1665] [1261 1595]Mixture with prior distortion 1356 1404 1383 [1299 1421] [1314 1404]Mixture without prior distortion 1340 1390 1369 [1281 1410] [1293 1390]

point 1 = 04 which is in accordance with Proposition 1In addition we can conclude that prior distortion candeviate the mixed CEP from theoretic value indeed but theupper bound of errors is estimable which is quantified inProposition 1

53 Example 3 The assessment of CEP in the case of smallsamples is discussed in this subsection Assume 9 impacterror samples are obtained after a group of flight tests anddivided into 3 classes under corresponding conditions whichis shown in Table 3 In addition real and prior mean ofprobability of test conditions are given in Table 4 Thenthe CEP point estimators and confidence intervals can becalculated after resampling 1000 times under the differentassumptions which is shown in Table 5

The CEP confidence upper bounds under differentassumptions are compared and analyzed here As is shownin Table 5 the assessment results by mixed CEP methodare closer to the mixed CEP of accurate prior informationcompared with classical single algorithm even with thedistorted prior It illustrates that the adaptive ability of mixedCEP algorithm is promising in the case of small samples

6 Conclusions

We have investigated the problem that classical CEP methodmay be unreasonable and may give inaccurate results forthe impact samples following different populations Forpurpose of addressing it the mixed CEP is modeled in thecontext that the errors caused by scene matching followsdiverse populations and corresponding algorithms of pointestimator and confidence interval estimation are designedTaking mixed population of two normal distributions as anexample the statistical properties of traditional and mixedCEP algorithm are compared and it concludes that priordistortion can deviate the mixed CEP from theoretic valueindeed but the upper bound of errors is estimable

In numerical experiments CEP calculated by mixedalgorithm is close to theoretical value in the case of the pop-ulation mixing two normal distributions while traditionalCEPmethod is unreasonable and unprecise which illustratesthat the mixed CEP algorithm designed in this paper isadaptable under the circumstance In addition the responserelationship of prior parameters and CEP is analyzed in thecase which verifies Proposition 1 from the perspective ofstatistical simulation And on the condition of small sampleswith mixture population of three normal distributions theconfidence upper bounds of mixed CEP with real and biased

prior parameters are calculated which indicates that themixed CEP algorithm is credible for small samples

In the case of impact samples following the mixedpopulation of finite normal distributions the mixed CEPalgorithm proposed in this paper is credible However theeffect of the method to other types of mixture models is notexplicit which is what we aim to investigate furthermore

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is partially supported by the program forNew Century Excellent Talents in University State EducationMinistry in China (no NCET10-0893) and the NaturalScience Foundation of China (no 6153367)

References

[1] C E Williams ldquoA Comparison of Circular Error ProbableEstimators for Small Samplesrdquo DTIC Document 1997

[2] J Zhang andWAn ldquoAssessing circular error probable when theerrors are elliptical normalrdquo Journal of Statistical Computationamp Simulation vol 82 no 4 pp 565ndash586 2012

[3] Y Wang G Yang D Yan Y Wang and X Song ldquoComprehen-sive assessment algorithm for calculating CEP of positioningaccuracyrdquoMeasurement vol 47 no 1 pp 255ndash263 2014

[4] A B Chatfield Fundamentals of High Accuracy Inertial Naviga-tion AIAA Reston Va USA 1997

[5] G Wang X Duan and Z Wang ldquoOptimal selection methodof aim points based on hitting vital index of area targetrdquo ActaAeronautica et Astronautica Sinica vol 29 no 5 pp 1258ndash12632008

[6] W Gang X Duan and Z Wang ldquoEstimation method offiring accuracy for area targets based on Monte Carlo integralmethodrdquo Systems Engineering and Electronics vol 7 article 362009 (Chinese)

[7] J C Spall and J L Maryak ldquoA feasible Bayesian estimator ofquantiles for projectile accuracy from non-iid datardquo Journal ofthe American Statistical Association vol 87 no 419 pp 676ndash6811992

[8] T Shi-Xue ldquoIntroduction to the scene matching missile guid-ance technologiesrdquo in Proceedings of the Conference for 30thAnniversary of CSAA Establishment 1994

[9] B Efron ldquoBootstrap methods another look at the jackkniferdquoThe Annals of Statistics vol 7 no 1 pp 1ndash26 1979

[10] J MacKinnon ldquoBootstrap hypothesis testingrdquo Working Papersvol 74 no 4 pp 183ndash213 2007

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


interval estimation The algorithms of mixed and classicalCEP are compared and analyzed in Section 4 In Section 5some simulation experiments are implemented the resultsof which indicate that the mixed CEP algorithm is morereasonable than traditional one under the circumstances thatdiversity of accuracy samples is obvious

2 Modeling of Bayes Model of Scene Matchingof Multiconditional Probability

21 Classical CEPDefinition Thedefinition of CEP is distinctin kinds of literature and a common one is adopted in thispaper CEP is the radius of a circle where center is targetand the probability of an impact point inside is 50 [1]The corresponding mathematical description is as followsEstablish a rectangular coordinate system centered on targetwith (0 0) and assume the deviation of downrange and cross-range noted as 119883 and 119885 respectively both follows normaldistribution then the probability density function of 120578 =(119883 119885)119879 is

119891 (119909 119911) = 12120587120590119909120590119911radic1 minus 1205882

sdot expminus 12 (1 minus 1205882) [(119909 minus 120583119909)2

1205902119909minus 2120588 (119909 minus 120583119909) (119911 minus 120583119911)120590119909120590119911 + (119911 minus 120583119911)21205902119911 ]

(1)

where 120590119909 120590119911 are standard deviations of119883119885 120583119909 120583119911 aremeansof119883 119885 120588 is correlation coefficient of 119883 and 119885 such that 0 le|120588| lt 1 Then 119877 in the following formula is the CEP of 120578 =(119883 119885)119879

∬1199092+1199112⩽1198772

119891 (119909 119911) 119889119909 119889119911 = 05 (2)

where 119891(119909 119911) is the same as (1) And (2) is called the generalform of CEP equation

Suppose u119894 | u119894 = (119909119894 119911119894)119879119878119894=1 are the impact deviationsamples to be used In previous CEP algorithms they areutilized to calculate120590119909120590119911120583119909120583119911120588 in (1) which are employedto estimate 119877 by numerical integration or approximateapproaches in (2) But samples collected in different testenvironments and influenced by complicate factors such asscene features are distinct which is against the assumption ofsame population and make the traditional CEP evaluationinvalid Under this condition we propose the mixed CEPmodel and algorithm of multiconditional probability

22 Bayes Nonparametric Mixture Model The Bayes mixturemodel combined with random density function is as follows

119891 (119910) = int119870 (119910 120579) 119889119875 (120579) (3)

where119870(119910 120579) is density function with different parameters 120579And119875(120579) is randomdensity function as the following discreteform usually

119875 (119889120579) = infinsum119897=1

120596119897120575120579119897 (119889120579) (4)

where (120596119897 120579119897)infin119897=1 is prior information and 120575120579119897 is characteristicfunction with various 12057923 Mixed CEP Model for Scene Matching of MulticonditionalProbability Define mixed CEP as the radius of a circlecentered on the target such that the probability of animpact point inside the circle is 50 while impact deviationdoes not follow the same bivariate normal population ForINSSAR integrated navigation system the impact biases areinfluenced by scene matching of which the error sourcessuch as scene features types and matching number are quitedistinct and follow diverse populations [8] In order to assesshit accuracy of the system by mixed CEP modeling forscene matching is essential beforehand Since the types andoccurring probability of scene in one test are unknown beforethe experiment implement they can be regarded as randomvariables while modeling based on Bayes nonparametricmodel frame

Suppose synthetical test conditions of scene matching areclassified into 119873 types denoted as ℎ1 ℎ2 ℎ119873 of whichoccurring probability is 119901(ℎ1) 119901(ℎ2) 119901(ℎ119873) 119901(ℎ119894)(119894 =1 2 119873) are random variables in [0 1] such that

119873sum119896=1

119901 (ℎ119896) = 1 (5)

Consider the conditional probability density function ofimpact errors on ℎ119896 is normal as follows

119891119896 (119909 119911 | ℎ119896) = 12120587120590119909119896120590119911119896radic1 minus 1205881198962

sdot expminus12 (1 minus 1205881198962) [[

(119909 minus 120583119909119896)21205902119909119896minus 2120588119896 (119909 minus 120583119909119896) (119911 minus 120583119911119896)120590119909119896120590119911119896 + (119911 minus 120583119911119896)21205902119911119896 ]]

(6)

where 120590119909119896 120590119911119896 are standard deviations of 119883 119885 on conditionℎ119896120583119909119896 120583119911119896 aremeans of119883119885 on conditionℎ119896120588119896 is correlationcoefficient of 119883 119885 on condition ℎ119896 such that 0 le |120588119896| lt 1Then the joint probability density function of impact errorsis

119891 (119909 119911) = int119891119896 (119909 119911 | ℎ119896) 119901 (ℎ119896) 119889ℎ119896 (7)

The discrete form is

119891 (119909 119911) = 119873sum119896=1

119901 (ℎ119896) 119891119896 (119909 119911 | ℎ119896) (8)



∬1199092+1199112⩽1198772

119873sum119896=1

119901 (ℎ119896) 119891119896 (119909 119911 | ℎ119896) 119889119909 119889119911 = 05 (9)




119909119911) (10)

where

120585119896 = 12119905119892minus1 21205881198961205901199091198961205901199111198961205901199091198962 minus 1205901199111198962 (11)



12 [[(119906 minus 120583119906119896)21205902119906119896 + (V minus 120583V119896)21205902V119896 ]]

(12)


∬1199062+V2⩽1198772

119873sum119896=1

119901 (ℎ119896) 119891119896 (119906 V | ℎ119896) 119889119906 119889V = 05 (13)



12120587 int119877

0int21205870119903 119873sum119896=1


120583V1198961205902V119896 sin120593)]119889120593119889119903= 05(15)

where

119886119896 = 14 ( 11205902V119896 minus11205902119906119896 )

119887119896 = 14 ( 11205902V119896 +11205902119906119896 )

119888119896 = 1120590119906119896120590V119896 expminus12 [(120583119906119896120590119906119896 )

2 + (120583V119896120590V119896 )2]

(16)



(17)

where

119909119896 = 1119899119896119899119896sum119894=1

119909(119896)119894119911119896 = 1119899119896

119899119896sum119894=1

119911(119896)119894119909119896 = radic 1119899119896 minus 1

119899119896sum119894=1

(119909(119896)119894 minus 119909119896)2

119911119896 = radic 1119899119896 minus 1119899119896sum119894=1



119896 = 12119905119892minus1 2119896119909119896 1199111198962119909119896 minus 2119911119896

(18)


119864 [119901 (ℎ119896)] = 119901119896 (19)


119873sum119896=1

119901119896 = 1 (20)





119877119887119906 = 1minus1205722119877119887119897 = 1205722119877119887 = 1minus120572

(21)








119896 = (119909119896119911119896) Σ119896 = (2119909119896 0

0 2119911119896) (22)



119896=1

119901119896119896Var [120578] = 119873sum

119896=1

1199012119896Σ119896(23)


119904 = (119909119904119911119904) =1119878119873sum119896=1

119899119896119896Σ119904 = 1119878 minus 1

119873sum119896=1

119899119896sum119894=1

(119909(119896)119894 minus 119909119904 00 119911(119896)119894 minus 119911119904)

2 (24)


Δ119864 = 119904 minus [120578] = 119873sum119896=1

(119899119896119878 minus 119901119896) 119896 (25)

Δ119863 = Σ119904 minus Var [120578] = 119873sum119896=1


where

120576 = 2119878 minus 1sdot 119873sum119896=1

119899119896sum119894=1


(27)



Let

∬1199092+1199112⩽1198772

119891119896 (119909 119911 | ℎ119896) 119889119909 119889119911 = 05 (28)


1198771 ⩽ 119877 ⩽ 119877119873 (29)


minus4minus2

02

4

X (m)Z (m)

minus4minus2

02

40

005

01

015

02

pdf

(a) 3 dimensions

minus3

minus2

minus1

0

1

2

3

4


Z (m

)

X (m)

002

004

006

008

01

012

014

(b) 2 dimensions




sup119908isin[01]

119873119908=1

Δ119877 (119908) ⩽ 10038161003816100381610038161003816100381610038161003816 max119896=12119873

119877119896 minus 119877010038161003816100381610038161003816100381610038161003816 (30)







minus25

minus2

minus15

minus1

minus05

0

05

1

Z (m

)








1 1420 1215 56 16 10010 1357 1176 47 24 100100 1355 1159 44 21 77






2 119883 0641 0642 0567 0379 0572 0663 0549 0604 0573119885 0470 0529 0421 0589 0385 0393 0419 0206 0644

3 119883 1033 0925 1137 0829 0990 0976 1032 1031 0914119885 0997 0984 1063 1109 1111 0914 1008 0879 0889


Testcondition



1 13 3102 13 3103 13 25



int1199092+1199112le1198772

[1198912 + 1 (1198911 minus 1198912)] 119889119909 119889119911 = 05 (31)










6 Conclusions





Competing Interests


Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











∬1199092+1199112⩽1198772

119873sum119896=1

119901 (ℎ119896) 119891119896 (119909 119911 | ℎ119896) 119889119909 119889119911 = 05 (9)




119909119911) (10)

where

120585119896 = 12119905119892minus1 21205881198961205901199091198961205901199111198961205901199091198962 minus 1205901199111198962 (11)



12 [[(119906 minus 120583119906119896)21205902119906119896 + (V minus 120583V119896)21205902V119896 ]]

(12)


∬1199062+V2⩽1198772

119873sum119896=1

119901 (ℎ119896) 119891119896 (119906 V | ℎ119896) 119889119906 119889V = 05 (13)



12120587 int119877

0int21205870119903 119873sum119896=1


120583V1198961205902V119896 sin120593)]119889120593119889119903= 05(15)

where

119886119896 = 14 ( 11205902V119896 minus11205902119906119896 )

119887119896 = 14 ( 11205902V119896 +11205902119906119896 )

119888119896 = 1120590119906119896120590V119896 expminus12 [(120583119906119896120590119906119896 )

2 + (120583V119896120590V119896 )2]

(16)



(17)

where

119909119896 = 1119899119896119899119896sum119894=1

119909(119896)119894119911119896 = 1119899119896

119899119896sum119894=1

119911(119896)119894119909119896 = radic 1119899119896 minus 1

119899119896sum119894=1

(119909(119896)119894 minus 119909119896)2

119911119896 = radic 1119899119896 minus 1119899119896sum119894=1



119896 = 12119905119892minus1 2119896119909119896 1199111198962119909119896 minus 2119911119896

(18)


119864 [119901 (ℎ119896)] = 119901119896 (19)


119873sum119896=1

119901119896 = 1 (20)





119877119887119906 = 1minus1205722119877119887119897 = 1205722119877119887 = 1minus120572

(21)








119896 = (119909119896119911119896) Σ119896 = (2119909119896 0

0 2119911119896) (22)



119896=1

119901119896119896Var [120578] = 119873sum

119896=1

1199012119896Σ119896(23)


119904 = (119909119904119911119904) =1119878119873sum119896=1

119899119896119896Σ119904 = 1119878 minus 1

119873sum119896=1

119899119896sum119894=1

(119909(119896)119894 minus 119909119904 00 119911(119896)119894 minus 119911119904)

2 (24)


Δ119864 = 119904 minus [120578] = 119873sum119896=1

(119899119896119878 minus 119901119896) 119896 (25)

Δ119863 = Σ119904 minus Var [120578] = 119873sum119896=1


where

120576 = 2119878 minus 1sdot 119873sum119896=1

119899119896sum119894=1


(27)



Let

∬1199092+1199112⩽1198772

119891119896 (119909 119911 | ℎ119896) 119889119909 119889119911 = 05 (28)


1198771 ⩽ 119877 ⩽ 119877119873 (29)


minus4minus2

02

4

X (m)Z (m)

minus4minus2

02

40

005

01

015

02

pdf

(a) 3 dimensions

minus3

minus2

minus1

0

1

2

3

4


Z (m

)

X (m)

002

004

006

008

01

012

014

(b) 2 dimensions




sup119908isin[01]

119873119908=1

Δ119877 (119908) ⩽ 10038161003816100381610038161003816100381610038161003816 max119896=12119873

119877119896 minus 119877010038161003816100381610038161003816100381610038161003816 (30)







minus25

minus2

minus15

minus1

minus05

0

05

1

Z (m

)








1 1420 1215 56 16 10010 1357 1176 47 24 100100 1355 1159 44 21 77






2 119883 0641 0642 0567 0379 0572 0663 0549 0604 0573119885 0470 0529 0421 0589 0385 0393 0419 0206 0644

3 119883 1033 0925 1137 0829 0990 0976 1032 1031 0914119885 0997 0984 1063 1109 1111 0914 1008 0879 0889


Testcondition



1 13 3102 13 3103 13 25



int1199092+1199112le1198772

[1198912 + 1 (1198911 minus 1198912)] 119889119909 119889119911 = 05 (31)










6 Conclusions





Competing Interests


Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877119887119906 = 1minus1205722119877119887119897 = 1205722119877119887 = 1minus120572

(21)








119896 = (119909119896119911119896) Σ119896 = (2119909119896 0

0 2119911119896) (22)



119896=1

119901119896119896Var [120578] = 119873sum

119896=1

1199012119896Σ119896(23)


119904 = (119909119904119911119904) =1119878119873sum119896=1

119899119896119896Σ119904 = 1119878 minus 1

119873sum119896=1

119899119896sum119894=1

(119909(119896)119894 minus 119909119904 00 119911(119896)119894 minus 119911119904)

2 (24)


Δ119864 = 119904 minus [120578] = 119873sum119896=1

(119899119896119878 minus 119901119896) 119896 (25)

Δ119863 = Σ119904 minus Var [120578] = 119873sum119896=1


where

120576 = 2119878 minus 1sdot 119873sum119896=1

119899119896sum119894=1


(27)



Let

∬1199092+1199112⩽1198772

119891119896 (119909 119911 | ℎ119896) 119889119909 119889119911 = 05 (28)


1198771 ⩽ 119877 ⩽ 119877119873 (29)


minus4minus2

02

4

X (m)Z (m)

minus4minus2

02

40

005

01

015

02

pdf

(a) 3 dimensions

minus3

minus2

minus1

0

1

2

3

4


Z (m

)

X (m)

002

004

006

008

01

012

014

(b) 2 dimensions




sup119908isin[01]

119873119908=1

Δ119877 (119908) ⩽ 10038161003816100381610038161003816100381610038161003816 max119896=12119873

119877119896 minus 119877010038161003816100381610038161003816100381610038161003816 (30)







minus25

minus2

minus15

minus1

minus05

0

05

1

Z (m

)








1 1420 1215 56 16 10010 1357 1176 47 24 100100 1355 1159 44 21 77






2 119883 0641 0642 0567 0379 0572 0663 0549 0604 0573119885 0470 0529 0421 0589 0385 0393 0419 0206 0644

3 119883 1033 0925 1137 0829 0990 0976 1032 1031 0914119885 0997 0984 1063 1109 1111 0914 1008 0879 0889


Testcondition



1 13 3102 13 3103 13 25



int1199092+1199112le1198772

[1198912 + 1 (1198911 minus 1198912)] 119889119909 119889119911 = 05 (31)










6 Conclusions





Competing Interests


Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus4minus2

02

4

X (m)Z (m)

minus4minus2

02

40

005

01

015

02

pdf

(a) 3 dimensions

minus3

minus2

minus1

0

1

2

3

4


Z (m

)

X (m)

002

004

006

008

01

012

014

(b) 2 dimensions




sup119908isin[01]

119873119908=1

Δ119877 (119908) ⩽ 10038161003816100381610038161003816100381610038161003816 max119896=12119873

119877119896 minus 119877010038161003816100381610038161003816100381610038161003816 (30)







minus25

minus2

minus15

minus1

minus05

0

05

1

Z (m

)








1 1420 1215 56 16 10010 1357 1176 47 24 100100 1355 1159 44 21 77






2 119883 0641 0642 0567 0379 0572 0663 0549 0604 0573119885 0470 0529 0421 0589 0385 0393 0419 0206 0644

3 119883 1033 0925 1137 0829 0990 0976 1032 1031 0914119885 0997 0984 1063 1109 1111 0914 1008 0879 0889


Testcondition



1 13 3102 13 3103 13 25



int1199092+1199112le1198772

[1198912 + 1 (1198911 minus 1198912)] 119889119909 119889119911 = 05 (31)










6 Conclusions





Competing Interests


Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 1420 1215 56 16 10010 1357 1176 47 24 100100 1355 1159 44 21 77






2 119883 0641 0642 0567 0379 0572 0663 0549 0604 0573119885 0470 0529 0421 0589 0385 0393 0419 0206 0644

3 119883 1033 0925 1137 0829 0990 0976 1032 1031 0914119885 0997 0984 1063 1109 1111 0914 1008 0879 0889


Testcondition



1 13 3102 13 3103 13 25



int1199092+1199112le1198772

[1198912 + 1 (1198911 minus 1198912)] 119889119909 119889119911 = 05 (31)










6 Conclusions





Competing Interests


Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















6 Conclusions





Competing Interests


Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ResearchArticle CEP Calculation Based on Weighted...

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