research article study on parameter optimization for...

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 158056 9 pageshttpdxdoiorg1011552013158056

Research ArticleStudy on Parameter Optimization for Support VectorRegression in Solving the Inverse ECG Problem

Mingfeng Jiang1 Shanshan Jiang1 Lingyan Zhu2 Yaming Wang1

Wenqing Huang1 and Heng Zhang1

1 School of Information Science and Technology Zhejiang Sci-Tech University Hangzhou 310018 China2The Dongfang College Zhejiang University of Finance and Economics Hangzhou 310018 China

Correspondence should be addressed to Mingfeng Jiang mjiangzstueducn

Received 9 May 2013 Revised 26 June 2013 Accepted 2 July 2013

Academic Editor Kayvan Najarian

Copyright copy 2013 Mingfeng Jiang 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

The typical inverse ECG problem is to noninvasively reconstruct the transmembrane potentials (TMPs) from body surfacepotentials (BSPs) In the study the inverse ECG problem can be treated as a regression problem with multi-inputs (body surfacepotentials) and multi-outputs (transmembrane potentials) which can be solved by the support vector regression (SVR) methodIn order to obtain an effective SVR model with optimal regression accuracy and generalization performance the hyperparametersof SVR must be set carefully Three different optimization methods that is genetic algorithm (GA) differential evolution (DE)algorithm and particle swarm optimization (PSO) are proposed to determine optimal hyperparameters of the SVR model In thispaper we attempt to investigate which one is the most effective way in reconstructing the cardiac TMPs from BSPs and a fullcomparison of their performances is also provided The experimental results show that these three optimization methods are wellperformed in finding the proper parameters of SVR and can yield good generalization performance in solving the inverse ECGproblem Moreover compared with DE and GA PSO algorithm is more efficient in parameters optimization and performs betterin solving the inverse ECG problem leading to a more accurate reconstruction of the TMPs

1 Introduction

The inverse ECG problem is to obtain myocardial trans-membrane potential (TMPs) distribution from body surfacepotentials (BSPs) thus noninvasively imaging the electro-physiological information on the cardiac surface [1 2]Generally approaches to solving this inverse ECG problemcan be relied on potential-based model including epicardialendocardial or transmembrane potentials which is used toevaluate the potential values on the cardiac surface [3] atcertain time instantsMoreover the cardiac electrophysiolog-ical information is closely associatedwith the transmembranepotentials (TMPs) of the myocardial cells Compared to bodysurface potentials (BSPs) recordings TMPs can providemoredetailed and complicated electrophysiological informationIn this study we focus on implementing the reconstructionof TMPs from BSPs

To study inverse ECGproblems various numericalmeth-ods have been proposed In the last decades regularizationmethods have been engaged for dealing with the inherentill-posed property The regularization techniques includetruncated total least squares (TTLS) [4] GMRes [5] andthe LSQR [6] which require an appropriate selection ofregularization parameters so as to relax the ill-posednessof the inverse ECG problem and to produce a well-posedproblem However the robustness and the quality of theinverse solution are not always guaranteed despite that theycan more or less deal with the geometry and measurementnoises for the inverse ECG problems In addition duringthe solution procedure the inverse ECG problem can betreated as a regression problem with multi-inputs (BSPs)andmulti-outputs (TMPs)Therefore an alternative methodsupport vector regression (SVR)method [7] was proposed tosolve the inverse ECGproblem Comparedwith conventional

2 Computational and Mathematical Methods in Medicine

regularization methods SVR method can produce moreaccurate results in terms of reconstruction of the trans-membrane potential distributions on epi- and endocardialsurfaces [1 8] SVR is an extension version of support vectormachine (SVM) to solve nonlinear regression estimationproblems which aims at minimizing an upper bound ofthe generalization error instead of the training error [9] byadhering to the principle of structural risk minimization(SRM)

Although SVR is a powerful technique to solve thenonlinear regression problem it has received less attention inrelation to the inverse ECGproblems due to the fact that SVRalgorithm is sensitive to usersrsquo defined free parameters Theinvolved hyperparameters of SVR model consist of penaltyparameter 119862 insensitive loss function parameter 120576 and theparameter 120590 for kernel function Inappropriate parameters inSVR can lead to overfitting or underfitting problems How toproperly set the hyperparameters is a major task which has asignificant impact on the optimal generalization performanceand the SVR regression accuracy [10] especially when itcomes to predicting the diagnosis of cardiac diseases becauseeven a slight improvement of prediction accuracy could havea significance impact on the patientrsquos diagnosis [11 12]

In early development stage of the algorithm a grid searchoptimizing method [13] and cross-validation method [8] areemployed to optimize the hyperparameters However thesemethods are computationally expensive and data intensiveRecently a number of new algorithms have been proposedfor the optimization of the SVR parameters [14 15] ForexampleWang et al [16] has proposed a hybrid load forecast-ing model combining differential evolution (DE) algorithmand support vector regression to deal with the problemof annual load forecasting In the analysis of predictingtourism demand the study [17] applies genetic algorithm(GA) to seek the SVRrsquos optimal parameters and then adoptsthe optimal parameters to construct the SVR models Theexperimental results demonstrate that SVR outperforms theother two neural network techniques Another study [18]tries a new technology particle swarmoptimization (PSO) toautomatically determine the parameters of SVR then appliesthe hybridmodel (PSO-SVR) to grid resource prediction andthe experimental results indicate high predictive accuracy

In this paper the above mentioned optimization algo-rithms (GA DE and PSO) are all adopted to dynamicallyoptimize the hyperparameters of SVR model in solving theinverse ECG problem For convenience the SVR model withGA parameter selection is referred to as GA-SVR methodand the other two are termed DE-SVR and PSO-SVRrespectively In this paper we attempt to investigate whichone is the most effective in reconstructing the cardiac TMPsfrom BSPs and a full comparison of the performance forsolving the inverse ECG problem will be evaluated

2 Theory and Methodology

21 Brief Overview of the SVR In this section the basicSVR concepts are concisely described for detailed descrip-tion please see [19 20] Suppose a given training data of

119873 elements (119909119894 119910119894) 119894 = 1 2 119873 where 119909

119894denotes

the 119894th element in 119899-dimensional space that is 119909119894

=

1199091119894 119909

119899119894 isin 119877

119899 and 119910119894

isin 119877 is the output valuecorresponding to119909

119894 According tomathematical notation the

SVR algorithm builds the linear regression function as thefollowing form

119891 (119909 120596) = (120596 sdot 120593 (119909) + 119887)

120593 119877119899997888rarr 119865 120596 isin 119865

(1)

where 120596 and 119887 are the slope and offset coefficients 120593(119909)

denotes the high-dimensional feature space which is non-linearly mapped from the input space 119909 And the previousregression problem is equivalent to minimizing the followingconvex optimization problem [see (2)]

min 1

21205962

subject to 119910119894minus 120596120593 (119909

119894) minus 119887 le 120576

120596120593 (119909119894) + 119887 minus 119910

119894le 120576

(2)

In this equation an implicit assumption is that a function119891 essentially approximates all pairs (119909

119894 119910119894) with 120576 precision

but sometimes this may not be the case Therefore by intro-ducing two additional positive slack variables 120585

119894and 120585lowast

119894 the

minimization is reformulated as the following constrainedoptimization problem [shown in (3)]

min 119877 (120596 120585 120585lowast) =

1

21205962+ 119862

119873

sum

119894=1

(120585119894+ 120585119894

lowast)

st 119910119894minus (120596 120593 (119909)) minus 119887 le 120576 + 120585

lowast

119894

(120596 120593 (119909)) + 119887 minus 119910119894le 120576 + 120585

119894

120585119894 120585119894

lowastge 0 119894 = 1 2 119873 120576 ge 0

(3)

where the parameter 119862 is the regulator which is determinedby the user and it influences a tradeoff between an approxi-mation error and the weights vector norm ||120596|| 120585


119894are

slack variables that represent the distance from actual valuesto the corresponding boundary values of 120576-tube

According to the strategy outlined by Scholkopf andSmola [10] by applying Lagrangian theory and the KKTcondition the constrained optimization problem can befurther restated as the following equation

119891 (119909 120596) =

119873

sum

119894=1

(120572119894minus 120572lowast

119894)119870 (119909

119894 119909) + 119887

st119873

sum

119894=1

(120572119894minus 120572119894

lowast) = 0

(4)

Here 120572119894and 120572

lowast

119894are the Lagrange multipliers The term

119870(119909119894 119909) is defined as the kernel function The nonlinear

separable cases could be easily transformed to linear casesby mapping the original variable into a new feature spaceof high dimension using 119870(119909

119894 119909) The radial basis function

Computational and Mathematical Methods in Medicine 3

(RBF) was applied in the study which has the ability touniversally approximate any distribution in the feature spaceWith an appropriate parameter RBF usually provides a betterprediction performance so it is adopted in this study asshown in (5)

119870(119909119894 119909119895) = exp(minus

10038171003817100381710038171003817119909119894minus 119909119895

10038171003817100381710038171003817

2

21205902) (5)

where 119909119894and 119909

119894are input vector spaces and 120590

2 is the band-width of the kernel function

In the above equations there exist three hyper-param-eters to be determined in advance that is the penaltyparameter 119862 insensitive parameter 120576 and the related kernelfunction parameters 120590

2 They heavily affect the regressionaccuracy and computation complexity of SVR The penaltyparameter 119862 controls the degree of punishing the sampleswhose errors go beyond the given value The insensitiveparameter 120576 controls the width of the 120576-insensitive zoneused to fit the training data The value of 120576 can enhancethe generalization capability with the increase of 120576 thenumber of support vectors will decrease and the algorithmiccomputation complexity will also reduceThe bandwidth 120590 ofthe kernel function has a great influence on the performanceof learning machine

In this study three optimization methods that is GADE and PSO are presented to determine the optimal hyper-parameters of the SVR model According to [15] the generalrange of119862 1205902 and 120576 has been given In the trial operation wenarrowed it to avoid blindness in the optimization process Inthis paper the set of hyperparameter (119862 120590

2 120576) is initialized

in the given range 119862 isin [0 10000] 1205902 isin [0 2] and 120576 isin

[0 00001] where optimization methods (GA DE and PSO)are to seek the global optimal solutions

In order to calculate the error uniformly we adopt thesame fitness function which plays a critical role in measuringthese algorithms performance The fitness function is deter-mined as follows

min119891 =sum119899119905

119894=1(1003816100381610038161003816119886119894 minus 119901

119894

1003816100381610038161003816 119886119894)

119899119905

lowast 100 (6)

where 119899119905is the number of training data samples 119886

119894is the

actual TMPs of train data and 119901119894is the predicted TMPs The

solution with a smaller fitness 119891 of the training dataset has abetter chance of surviving in the successive generations Themain tool LIBSVM was used for training and validating theSVR model [21]

22 Genetic Algorithm (GA) Optimization Method The GA[22] is a biologically motivated optimization techniqueguided by the concepts of biological evolution and Darwinrsquosprinciple of survival of the fittest It is a computer model ofan evolution of a population of artificial individuals In thisstudy for the specific optimizing problemof hyperparameters(119862 1205902 120576) the process is defined as follows

Step 1 (initialize population) The population size NP isequal to 20 the dimension of parameter vectors 119863 is 3

The termination criterion is set as follows the number ofiterations is set as 30 and the fitness tolerance value is setas 0001 The probabilities of selection (Stochastic universalsampling) crossover (multipoint crossover) and mutation(mutation operator of the Breeder Genetic Algorithm) thatwere used herein are 09 08 and 005 respectively

Step 2 (encode chromosomes) According to the possiblerange of parameters 119862 120590

2 and 120576 given before the GA utilizesbinary encoding method and each parameter is encodedby 20 bits of binary number Therefore the search space isdefined as the solution space in which each potential solutionis encoded as a distinct chromosome

Step 3 The parameters 119862 1205902 and 120576 of each individual

are used to build the SVR model With the BSPs of thetraining data the cardiac TMPs are reconstructed Then theperformances of individuals in the generation are evaluatedby the specific objective fitness function according to (6)Theindividual with the minimum fitness value will be selectedand then the chromosome of selected individual is preservedas the best result

Step 4 Thebasic genetic search operators comprise selectionmutation and crossover which are applied orderly to obtaina new generation where the new individual (119862 120590

2 120576) with

the best performance is retained

Step 5 The new best fitness value will be compared to that ofthe best result and then select the better one to update thebest result

Step 6 This process will not come to an end until thetermination criterion is met and the best chromosome ispresented as the final solution otherwise go back to Step 4

23 Differential Evolution (DE) Optimization Method Dif-ferential evolution [23] is a population-based and paralleldirect searchmethod which is used to approximate the globaloptimal solution As is mentioned above the optimization ofhyperparameter (119862 120590

2 120576) can be transformed into solving

the minimization of the fitness function Min 119891(119909) 119909 =

119909119894119866 119894119866

= 1 2 andNP is three-dimensional parametervectors including (119862 120590

2 and 120576) Subsequently each gen-eration evolves by employing the evolutionary operatorsinvolving mutation crossover and selection operations toproduce a trail vector for each individual vector [24] and thedetailed evolutionary strategies can be described as follows

Step 1 (initialize DE parameters) The population size NP isset as 20 [25] and the dimension of parameter vectors 119863 is3 The termination criterion is set as follows the number ofiterations is set as 30 and the fitness tolerance value is set as0001 The mutation factor 119865 is selected in [05 1] and thecrossover rate CR is selected in [0 1]

Step 2 (initialize population) Set 119866 = 0 Generate an NPlowastDgeneration which consists of individuals (119862 120590

2 and 120576) with


uniform probability distribution random values within thecontrol variable bounds of parameter space

Step 3 With the parameters train the SVR model andforecast the training datasetsThen calculate the fitness targetof all the individuals in the generation and record theminimumvalues of the fitness function and set (119862 120590

2 and 120576)

correspondingly as the 119909best

Step 4 (mutation operation) For each target vector 119909119894119866 an

associated mutant vector is generated according to

V119894119866+1

= 119909best + 119865 [(1199091199031119866

minus 1199091199032119866) + (119909

1199033119866

minus 1199091199034119866)] (7)

The random indexes 1199031 1199032 1199033 and 119903

4have to be distin-

guished from each other and from the current trial index 119894

Step 5 (crossover operation) In order to increase the diver-sity of the perturbed parameter vectors crossover is intro-duced by the following formula

119906119895119894119866+1

= V119895119894119866+1

if (randb (119895) le CR) or 119895 = rnbr (119894)119909119895119894119866+1

if (randb (119895) gt CR) and 119895 = rnbr (119894)

(8)

where 119906119894119866+1

= (1199061119894119866+1

1199062119894119866+1

1199063119894119866+1

) randb(119895) isin [0 1] isthe 119895th evaluation of a uniform random number generatorand rnbr(119894) is a randomly chosen integer in the range in [1 3]to ensure that 119906

119894119866+1gets at least one element from V

119894119866+1

Step 6 (selection operation) After the fitness values of thenew generation being calculated then selecting the new bestindividual as 119906best119866+1 the selection operation is performedTo decide whether or not it could replace the 119909best the fitnessvalue of 119891(119906best119866+1) is compared to that of 119891(119909best) Theoperation is given as follows

119909best = 119906best119866+1 119891 (119906best119866+1) le 119891 (119909best)

119909best 119891 (119906best119866+1) gt 119891 (119909best) (9)

Step 7 Termination condition checking if max iterations119866 ismet or the fitness value of119891(119909best) is reachedwithin the fitnesstolerance value return the recorded global optimal parameter(119862 1205902 and 120576) otherwise go to Step 4

24 Particle Swarm Optimization (PSO) Optimization Meth-od The Particle swarm optimization (PSO) is a bioinspiredstochastic optimization technique which was basically devel-oped through simulating social behaviour of birds and insectsthat keep living by maintaining swarm actions A particleis considered as a bird in a swarm consisting of a numberof birds and all particles fly through the searching spaceby following the current optimum particle to find the finaloptimum solution of the optimization problem [26]

The detailed experimental procedure for selecting thebest hyperparameters (119862 120590

2 and 120576) is listed as follows

Step 1 Particle initialization andPSOparameters setting [27]set the PSO parameters including the scope of 119862 120590

2 and

120576 the number of particles is 20 particle dimension is 3The termination criterion is set as follows the number ofiterations is set as 30 and the fitness tolerance value as 0001

Step 2 Randomly generate primal population of randomparticles (119862 120590

2 and 120576) and velocities inside the searching

space

Step 3 Construct SVR model with the parameters in eachparticle and perform the prediction with the training dataThen evaluate the regression accuracy based on the definedfitness functionThen maintain the record of the best perfor-mance with the minimum error as the global best solution

Step 4 Particle manipulations suppose that 119883119894(119896) and 119881

119894(119896)

are the particle and velocity at step 119896 and they are updatedinto their new values 119883

119894(119896 + 1) and 119881

119894(119896 + 1) at step 119896 + 1 by

applying the two equations below

119881119894 (119896 + 1) = 120596119881

119894 (119896) + 11988811199031(119901119894 (119896) minus 119883

119894 (119896))

+ 11988821199032(119901119892 (119896) minus 119883

119894 (119896))

119883119894 (119896 + 1) = 119883

119894 (119896) + 119881119894 (119896 + 1) Δ119905

(10)

where 119881119894

isin [119881119894min 119881119894max] Δ119905 is the unit time step value

and 120596 is the inertia weight 119901119894(119896) and 119901

119892(119896) are the best

coordinates of particle number 119894 and the whole swarm untilstep 119896 respectively Positive constants 119888

1and 1198882are learning

rates while 1199031and 1199032are random numbers between 0 and 1

Step 5 Train the SVRmodel using the particles (119862 1205902 and 120576)

in the new generation and forecast the training datasets Afterevaluating the fitness function pick the new best particle asthe local best

Step 6 Compare the local best and global best and choosethe better one to update the global best

Step 7 Stop condition checking if a termination criterion(the number of iterations) is met or the fitness value of theglobal best is reached within the extent predefined returnthe recorded global best as the best parameter (119862 120590

2 and

120576) otherwise go to Step 4

25 Simulation Protocol and Data Set The SVR model istested with our previously developed realistic heart-torsomodel [11 12] In this simulation protocol a normal ven-tricular excitation is illustrated as an example to calculatethe data set for the SVR model The considered ventricularexcitation period from the first breakthrough to the end is357ms and the time step is 1ms and thus 358 BSPs 120593

119861and

TMPs 120593119898temporal data sets are calculated Sixty data sets

(120593119861and 120593

119898) are chosen at times of 3ms 9ms 15ms

357ms after the first ventricular breakthrough which will beused as testing samples to assess the SVR models predictioncapability and robustness The rest 298 in 358 data setsare employed as the training samples for the training andoptimal parameter selection procedures Moreover duringeach ventricular excitation period 412 potentials on the BSPs


and 478 potentials on the TMPs are captured That is to saythe matrix of BSPs is divided into the training data (298 times

412) and testing data (60 times 412) and the matrix of TMPs isalso divided into the training data (298 times 478) and testingdata (60 times 478) Each SVR regression model is built byusing the mapping relations between BSPs training data (298times 412) and one TMPs training point (298 times 1) Based onall the training data 478 different regression models werebuilt by usingGA-SVRDE-SVR and PSO-SVR respectivelyUsing the corresponding SVR model we can reconstruct thecorresponding TMPs (60 times 1) from the BSPs testing data (60times 412) For all testing samples we can reconstruct the TMPsfrom the all 478 regression SVR models

3 The Proposed System Framework

31 Overview of the SVRParametersOptimization FrameworkAs shown in Figure 1 in the first stage the original inputdata is preprocessed by scaling the training data and featureextraction In the second stage the hyperparameters (119862 1205902and 120576) of SVR method are set carefully by using the GADE and PSO optimization methods Moreover GA-SVRDE-SVR and PSO-SVR are applied to construct the hybridSVR model respectively Finally according to the effectivehybrid SVR models we can reconstruct the transmembranepotentials (TMPs) from body surface potentials (BSPs) byusing these testing samples

32 Preprocessing the Data Set

321 Scaling the Training Set During preprocessing stageeach input variable is scaled in order to avoid the potentialvalue from spanning a great numerical range and to preventnumeric difficulties Generally all input values including 358BSPs and TMPs temporal data sets are linearly scaled to therange (01) by the following formula [see (11)]

1205931015840

119905=

120593119905minus 120593119905min

120593119905max minus 120593

119905min (11)

where 120593119905is the original potential value of each time 119905 1205931015840

119905is the

scaled potential value 120593119905max is the maximum potential value

of each time 119905 and 120593119905min is the minimum potential value of

each time 119905

33 Feature Extraction by Using KPCA The kernel principalcomponent analysis (KPCA) is a nonlinear feature extractionmethod which is one type of a nonlinear PCA developedby generalizing the kernel method into PCA Specificallythe KPCA initially maps the original inputs into a high-dimensional feature space 119865 using the kernel method andthen calculates the PCA in the high-dimensional featurespace 119865The detailed theory of KPCA can be consulted in thepaper [28]

Feature extraction is a significant task for preprocessingthe original input data in developing an effective SVRmodelAs is investigated in our previous study [12] the SVR withfeature extraction by using KPCA has superior performances

BSPs

Scaling data set

KPCA for feature extraction

Train data

Parameters selected by

GA

Testing data


DE


PSO

Build GA-SVR

model

Build DE-SVR

model

Build PSO-SVR

model

Reconstructing TMPs

Figure 1 The framework of proposed parameters optimizationmethod for SVR method in solving the inverse ECG problem

0 50 100 150 200 250 300 350 400

0

20

Time (ms)

minus100

minus80

minus60

minus40

minus20

Original TMPsGA-SVR method

DE-SVR methodPSO-SVR method

TMP

(mV

)

Figure 2 The temporal course of the TMPs for one representativesource point on epicardium (50th) over the 60 testing timesThe reconstruction TMPs with GA-SVR DE-SVR and PSO-SVRmethods are all compared with original TMPs

to that of using PCAor the single SVRmethod in reconstruct-ing the TMPs In this paper the kernel principal componentanalysis (KPCA) was proposed to implement the featureextraction And the dimension of BSPs dataset is reducedfrom 412 to 200 properly which can reduce the dimensionsof the inputs and improve the generalization performances ofthe SVR method

4 Results and Discussion

41 The Result of Parameters Selection In this study theGA DE and PSO algorithms were proposed to seek the


(a)

(mV)minus90 minus80 minus70 minus60 minus50 minus40 minus30 minus20 minus10 0 10

(b)

Figure 3 The TMP distribution on the ventricular surface attwo sequential testing time points (a) shows the comparison ofreconstructed TMP distributions when the time point is 27ms and(b) shows the comparison of reconstructed TMP distributions whenthe time point is 51ms Moreover the upper row shows the TMPsfrom an anterior view and the lower from a posterior view In eachrow the first figure is the original TMPs and the rest three are thereconstructed TMPs by using the GA-SVR DE-SVR and PSO-SVRmethods

corresponding optimal hyperparameter (119862 1205902 and 120576) of the

478 SVR models Table 1 displays the average value and thestandard deviations of three parameters in 478 SVR modelsby using GA DE and PSO optimization algorithms From itwe can find that the gaps among these optimal parameters areobvious by using three different optimization methods Thestandard deviations of 120576 are neglected because they are verysmall

42 Comparison of Reconstruction Accuracy This study com-pares the reconstruction performance based on the proposedthree intelligent optimization algorithms for selecting thehyperparameters of SVR By using GA-SVR DE-SVR andPSO-SVR hybrid models two types of experiments forreconstructing the TMPs have been conducted The firstexperiment reconstructs the TMPs for one representativesource point (the 50th point) on the epicardium over 60testing times as is depicted in Figure 2 Since the gaps among

Table 1 The average and the standard deviations of optimal hyper-parameters (119862 120590

2 and 120576) by using GA DE and PSO optimization

algorithms

Parameter MethodGA DE PSO

119862 144761 plusmn 59432 762414 plusmn 91367 531 plusmn 302

1205902

064 plusmn 055 159 plusmn 138 122 plusmn 102

120576 148119890 minus 005 180119890 minus 005 198119890 minus 005

Table 2 The mean values and standard deviations of these 60testing errors between the simulated TMPs and the reconstructedTMPs by using GA-SVR DE-SVR and PSO-SVR methods at onerepresentative source point (the 50th point on the epicardium)

Indexes MethodGA-SVR DE-SVR PSO-SVR

Mean value 296 269 155Standard deviation 192 167 103

these three methods are so slight that we have to add astatistical analysis to validate the results The mean valuesand standard deviations of these 60 testing errors betweenthe simulated TMPs and the reconstructed TMPs by usingGA-SVR DE-SVR and PSO-SVR methods are investigatedas shown in the Table 2 From Figure 2 and Table 2 thePSO-SVR outperforms the GA-SVR and DE-SVR methodsin reconstructing the TMPs for one representative sourcepoint over all the testing times Furthermore the DE-SVR issuperior to the GA-SVRmethod in reconstructing the TMPs

The second experiment reconstructs the TMPs on all theheart surface points at one time instant These inverse ECGsolutions are shown in Figure 3 in which two sequentialtesting time points (27 and 51ms after the first ventricularbreakthrough) are presented to illustrate the performances oftheGA-SVRDE-SVR and PSO-SVRmethods It can be seenthat among the three parameters optimization methods thehybrid PSO-SVR performs better thanGA-SVR andDE-SVRmethods because its reconstructed solutions are much closerto the simulated TMPs distributions

Based on the simulation information of TMPs we canevaluate the accuracy of the reconstructed TMPs of thesethree different intelligent optimization algorithms at thetesting time by mean square error (MSE) the relative error(RE) and the correlation coefficient (CC)

MSE =1

119899

119899

sum

119894=1

(120593119888minus 120593119890)2

RE =

1003817100381710038171003817120593119888 minus 120593119890

1003817100381710038171003817

120593119890

CC =sum119899

119894=1[(120593119888)119894minus 120593119888] [(120593119890)119894minus 120593119890]

1003817100381710038171003817120593119888 minus 120593119890

10038171003817100381710038171003817100381710038171003817120593119890 minus 120593

119890

1003817100381710038171003817

(12)

Here 120593119890is the simulated TMP distribution at time 119905 and 120593

119888

is the reconstructed TMPs The quantities 120593119890and 120593

119888are the


0 10 20 30 40 50 600

002

004

006

008

01

012

014

Time (ms)

Mea

n sq

uare

d er

ror (

MSE

)

MSE of GA-SVR methodMSE of DE-SVR methodMSE of PSO-SVR method

(a)

Rela

tive e

rror

s (RE

)

RE of GA-SVR methodRE of DE-SVR methodRE of PSO-SVR method

0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

14

(b)

Cor

relat

ion

coeffi

cien

t (CC

)

CC of GA-SVR methodCC of DE-SVR methodCC of PSO-SVR method

0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

minus02

(c)

Figure 4 The performances of the reconstructed TMPs over 60 sampling times by using GA-SVR DE-SVR and PSO-SVR respectively(a) The mean square error (MSE) of the reconstructed TMPs (b) The relative errors (RE) of the reconstructed TMPs (c) The correlationcoefficient (CC) of the reconstructed TMPs

mean value of 120593119890and 120593

119888over the whole ventricular surface

nodes at time 119905 119899 is the number of nodes on the ventricularsurface

The MSE RE and CC of the three sets of reconstructedTMPs over the 60 testing samples can be found in Figure 4In contrast to GA-SVR and DE-SVR the PSO-SVR methodcan yield rather better results with lower RE MSE and ahigher CC To validate the robustness of the three methodsin reconstructing the TMPs the quantities m-MSE m-REand m-CC that is the mean value of MSE RE and CC atdifferent testing time instants are presented Moreover std-MSE std-RE and std-CC that is the standard deviation ofMSE RE and CC over the whole testing samples are alsoinvestigated As shown in Figure 5 the mean value of theMSE RE and CC over the 60 testing samples obtained byGA-SVR DESVR and PSO-SVRmethods are presented andthe standard deviations of those 60 MSEs REs and CCs arealso provided

43 Comparison of Reconstruction Efficiency In solving theinverse ECG problem seeking the optimal parameters inthe settled range is time consuming Therefore how to suc-cessfully excogitate a relatively timesaving optimal methodmatters significantly in the practical application particularlyin the diagnosis of cardiac disease Here we figure out themean time of the three optimizationmethods in selecting theoptimal hyperparameters of SVR In order to forecast the 478TMPs over 60 testing samples on an epi- and endocardialsurface 478 sets of parameters (119862 120590

2 and 120576) have to beselected in each optimization method and with the testingdata 478models are built accordingly In this study themeantime for selecting the optimal parameters by adopting GADE and PSO is 20827 s 34253 s and 13016 s respectively Itcan be seen that PSO can lead to a convergence more quicklyand lessen more time than GA and DE Therefore accordingto the comparison of their efficiencies PSO-SVR methodsignificantly outperforms the GA-SVR and DE-SVR in the


GA-SVR DE-SVR PSO-SVR0

001

002

003

004

005

006m

-MSE

and

std-M

SE

(a)

GA-SVR DE-SVR PSO-SVR

05

04

03

02

01

0

m-R

E an

d std

-RE

(b)


11

1

09

08

07

06

05

04

m-C

C an

d std

-CC

(c)

Figure 5The m-MSE m-RE m-CC std-MSE std-RE and std-CC of 60 testing sampling times by using GA-SVR DE-SVR and PSO-SVRrespectively in terms of mean plusmn standard deviation (a) m-MSE and std-MSE (b) m-RE and std-RE and (c) m-CC and std-CC

reconstruction of the TMP which is the most efficient oneamong these three optimization methods

5 Conclusion

In the study of the inverse ECG problem SVR methodis a powerful technique to solve the nonlinear regressionproblem and can serve as a promising tool for performingthe inverse reconstruction of the TMPsThis study introducesthree optimization methods (GA DE and PSO) to deter-mine the hyperparameters of the SVR model and utilizesthese models to reconstruct the TMPs from the remoteBSPs Feature extraction using the KPCA is also adopted topreprocess the original input data The experimental resultsdemonstrate that PSO-SVR is a relatively effective methodin terms of accuracy and efficiency By comparison PSO-SVR method is superior to GA-SVR and DE-SVR methodswhose reconstructed TMPs are close to the simulated TMPsBesides PSO-SVR is much more efficient in determining theoptimal parameters and in building the predicting modelAccording to these results the PSO-SVR method can serveas a promising tool to solve the nonlinear regression problemof the inverse ECG problem

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (30900322 61070063 and 61272311)This work is also supported by the Research Foundation

of Education Bureau of Zhejiang Province under Grant noY201122145

References

[1] M Jiang J Lv C Wang L Xia G Shou and W HuangldquoA hybrid model of maximum margin clustering method andsupport vector regression for solving the inverse ECGproblemrdquoComputing in Cardiology vol 38 pp 457ndash460 2011

[2] C Ramanathan R N Ghanem P Jia K Ryu and Y RudyldquoNoninvasive electrocardiographic imaging for cardiac electro-physiology and arrhythmiardquo Nature Medicine vol 10 no 4 pp422ndash428 2004

[3] B F Nielsen X Cai and M Lysaker ldquoOn the possibility forcomputing the transmembrane potential in the heart with a oneshot method an inverse problemrdquo Mathematical Biosciencesvol 210 no 2 pp 523ndash553 2007

[4] G Shou L Xia M Jiang Q Wei F Liu and S CrozierldquoTruncated total least squares a new regularization method forthe solution of ECG inverse problemsrdquo IEEE Transactions onBiomedical Engineering vol 55 no 4 pp 1327ndash1335 2008

[5] C Ramanathan P Jia R Ghanem D Calvetti and Y RudyldquoNoninvasive electrocardiographic imaging (ECGI) applica-tion of the generalized minimal residual (GMRes) methodrdquoAnnals of Biomedical Engineering vol 31 no 8 pp 981ndash9942003

[6] M Jiang L Xia G Shou and M Tang ldquoCombination ofthe LSQR method and a genetic algorithm for solving theelectrocardiography inverse problemrdquo Physics in Medicine andBiology vol 52 no 5 pp 1277ndash1294 2007


[7] M Brown S R Gunn and H G Lewis ldquoSupport vectormachines for optimal classification and spectral unmixingrdquoEcological Modelling vol 120 no 2-3 pp 167ndash179 1999

[8] K Ito and R Nakano ldquoOptimizing Support Vector regressionhyper-parameters based on cross-validationrdquo in Proceedings ofthe International Joint Conference onNeural Networks vol 3 pp871ndash876 2005

[9] V N Vapnik ldquoAn overview of statistical learning theoryrdquo IEEETransactions on Neural Networks vol 10 no 5 pp 988ndash9991999

[10] B Scholkopf and A J Smola Learning with kernels [PhDthesis] GMD Birlinghoven Germany 1998

[11] M Jiang L Xia G ShouQWei F Liu and S Crozier ldquoEffect ofcardiac motion on solution of the electrocardiography inverseproblemrdquo IEEE Transactions on Biomedical Engineering vol 56no 4 pp 923ndash931 2009

[12] M Jiang L Zhu Y Wang et al ldquoApplication of kernel prin-cipal component analysis and support vector regression forreconstruction of cardiac transmembrane potentialsrdquo Physics inMedicine and Biology vol 56 no 6 pp 1727ndash1742 2011

[13] S S Keerthi ldquoEfficient tuning of SVM hyperparameters usingradiusmargin bound and iterative algorithmsrdquo IEEE Transac-tions on Neural Networks vol 13 no 5 pp 1225ndash1229 2002

[14] J S Sartakhti M H Zangooei and K Mozafari ldquoHepatitisdisease diagnosis using a novel hybridmethod based on supportvectormachine and simulated annealing (SVM-SA)rdquoComputerMethods and Programs in Biomedicine vol 108 no 2 pp 570ndash579 2011

[15] B Ustun W J Melssen M Oudenhuijzen and L M CBuydens ldquoDetermination of optimal support vector regressionparameters by genetic algorithms and simplex optimizationrdquoAnalytica Chimica Acta vol 544 no 1-2 pp 292ndash305 2005

[16] J Wang L Li D Niu and Z Tan ldquoAn annual load forecastingmodel based on support vector regression with differentialevolution algorithmrdquo Applied Energy vol 94 pp 65ndash70 2012

[17] K-Y Chen and C-H Wang ldquoSupport vector regression withgenetic algorithms in forecasting tourism demandrdquo TourismManagement vol 28 no 1 pp 215ndash226 2007

[18] G Hu L Hu H Li K Li and W Liu ldquoGrid resourcesprediction with support vector regression and particle swarmoptimizationrdquo in Proceedings of the 3rd International JointConference on Computational Sciences and Optimization (CSOrsquo10) vol 1 pp 417ndash422 May 2010

[19] N Cristianini and J Taylor An Introduction to Support VectorMachines Cambridge University Press Cambridge UK 2000

[20] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[21] C C Chang andC J Lin ldquoLIBSVM a library for support vectormachinesrdquo 2001 httpwwwcsientuedutwsimcjlinlibsvm

[22] G Oliveri and A Massa ldquoGenetic algorithm (GA)-enhancedalmost difference set (ADS)-based approach for array thinningrdquoIET Microwaves Antennas and Propagation vol 5 no 3 pp305ndash315 2011

[23] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[24] M Varadarajan and K S Swarup ldquoNetwork loss minimizationwith voltage security using differential evolutionrdquoElectric PowerSystems Research vol 78 no 5 pp 815ndash823 2008

[25] K Zielinski P Weitkemper R Laur and K D KammeyerldquoParameter study for differential evolution using a power alloca-tion problem including interference cancellationrdquo EvolutionaryComputation vol 4 pp 16ndash21 2006

[26] R Poli K James and B Tim ldquoParticle swarm optimizationrdquoSwarm Intelligence vol 1 pp 35ndash57 2007

[27] L-P Zhang H-J Yu and S-X Hu ldquoOptimal choice of param-eters for particle swarm optimizationrdquo Journal of ZhejiangUniversity vol 6 no 6 pp 528ndash534 2005

[28] R Rosipal M Girolami L J Trejo and A Cichocki ldquoKernelPCA for feature extraction and de-noising in nonlinear regres-sionrdquoNeural Computing and Applications vol 10 no 3 pp 231ndash243 2001

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


regularization methods SVR method can produce moreaccurate results in terms of reconstruction of the trans-membrane potential distributions on epi- and endocardialsurfaces [1 8] SVR is an extension version of support vectormachine (SVM) to solve nonlinear regression estimationproblems which aims at minimizing an upper bound ofthe generalization error instead of the training error [9] byadhering to the principle of structural risk minimization(SRM)

Although SVR is a powerful technique to solve thenonlinear regression problem it has received less attention inrelation to the inverse ECGproblems due to the fact that SVRalgorithm is sensitive to usersrsquo defined free parameters Theinvolved hyperparameters of SVR model consist of penaltyparameter 119862 insensitive loss function parameter 120576 and theparameter 120590 for kernel function Inappropriate parameters inSVR can lead to overfitting or underfitting problems How toproperly set the hyperparameters is a major task which has asignificant impact on the optimal generalization performanceand the SVR regression accuracy [10] especially when itcomes to predicting the diagnosis of cardiac diseases becauseeven a slight improvement of prediction accuracy could havea significance impact on the patientrsquos diagnosis [11 12]

In early development stage of the algorithm a grid searchoptimizing method [13] and cross-validation method [8] areemployed to optimize the hyperparameters However thesemethods are computationally expensive and data intensiveRecently a number of new algorithms have been proposedfor the optimization of the SVR parameters [14 15] ForexampleWang et al [16] has proposed a hybrid load forecast-ing model combining differential evolution (DE) algorithmand support vector regression to deal with the problemof annual load forecasting In the analysis of predictingtourism demand the study [17] applies genetic algorithm(GA) to seek the SVRrsquos optimal parameters and then adoptsthe optimal parameters to construct the SVR models Theexperimental results demonstrate that SVR outperforms theother two neural network techniques Another study [18]tries a new technology particle swarmoptimization (PSO) toautomatically determine the parameters of SVR then appliesthe hybridmodel (PSO-SVR) to grid resource prediction andthe experimental results indicate high predictive accuracy

In this paper the above mentioned optimization algo-rithms (GA DE and PSO) are all adopted to dynamicallyoptimize the hyperparameters of SVR model in solving theinverse ECG problem For convenience the SVR model withGA parameter selection is referred to as GA-SVR methodand the other two are termed DE-SVR and PSO-SVRrespectively In this paper we attempt to investigate whichone is the most effective in reconstructing the cardiac TMPsfrom BSPs and a full comparison of the performance forsolving the inverse ECG problem will be evaluated

2 Theory and Methodology

21 Brief Overview of the SVR In this section the basicSVR concepts are concisely described for detailed descrip-tion please see [19 20] Suppose a given training data of

119873 elements (119909119894 119910119894) 119894 = 1 2 119873 where 119909

119894denotes

the 119894th element in 119899-dimensional space that is 119909119894

=

1199091119894 119909

119899119894 isin 119877

119899 and 119910119894

isin 119877 is the output valuecorresponding to119909

119894 According tomathematical notation the

SVR algorithm builds the linear regression function as thefollowing form

119891 (119909 120596) = (120596 sdot 120593 (119909) + 119887)

120593 119877119899997888rarr 119865 120596 isin 119865

(1)

where 120596 and 119887 are the slope and offset coefficients 120593(119909)

denotes the high-dimensional feature space which is non-linearly mapped from the input space 119909 And the previousregression problem is equivalent to minimizing the followingconvex optimization problem [see (2)]

min 1

21205962

subject to 119910119894minus 120596120593 (119909

119894) minus 119887 le 120576

120596120593 (119909119894) + 119887 minus 119910

119894le 120576

(2)

In this equation an implicit assumption is that a function119891 essentially approximates all pairs (119909

119894 119910119894) with 120576 precision

but sometimes this may not be the case Therefore by intro-ducing two additional positive slack variables 120585


119894 the

minimization is reformulated as the following constrainedoptimization problem [shown in (3)]

min 119877 (120596 120585 120585lowast) =

1

21205962+ 119862

119873

sum

119894=1

(120585119894+ 120585119894

lowast)

st 119910119894minus (120596 120593 (119909)) minus 119887 le 120576 + 120585

lowast

119894

(120596 120593 (119909)) + 119887 minus 119910119894le 120576 + 120585

119894

120585119894 120585119894

lowastge 0 119894 = 1 2 119873 120576 ge 0

(3)

where the parameter 119862 is the regulator which is determinedby the user and it influences a tradeoff between an approxi-mation error and the weights vector norm ||120596|| 120585


119894are

slack variables that represent the distance from actual valuesto the corresponding boundary values of 120576-tube

According to the strategy outlined by Scholkopf andSmola [10] by applying Lagrangian theory and the KKTcondition the constrained optimization problem can befurther restated as the following equation

119891 (119909 120596) =

119873

sum

119894=1

(120572119894minus 120572lowast

119894)119870 (119909

119894 119909) + 119887

st119873

sum

119894=1

(120572119894minus 120572119894

lowast) = 0

(4)

Here 120572119894and 120572

lowast

119894are the Lagrange multipliers The term

119870(119909119894 119909) is defined as the kernel function The nonlinear

separable cases could be easily transformed to linear casesby mapping the original variable into a new feature spaceof high dimension using 119870(119909

119894 119909) The radial basis function



119870(119909119894 119909119895) = exp(minus

10038171003817100381710038171003817119909119894minus 119909119895

10038171003817100381710038171003817

2

21205902) (5)

where 119909119894and 119909










min119891 =sum119899119905

119894=1(1003816100381610038161003816119886119894 minus 119901

119894

1003816100381610038161003816 119886119894)

119899119905

lowast 100 (6)


119894is the











2 120576) with







119909119894119866 119894119866





2 and 120576) with




2 and 120576)




V119894119866+1

= 119909best + 119865 [(1199091199031119866

minus 1199091199032119866) + (119909

1199033119866

minus 1199091199034119866)] (7)


4have to be distin-



119906119895119894119866+1

= V119895119894119866+1



(8)

where 119906119894119866+1

= (1199061119894119866+1

1199062119894119866+1

1199063119894119866+1



119894119866+1



119909best 119891 (119906best119866+1) gt 119891 (119909best) (9)






2 and




space



119894(119896)


119894(119896 + 1) and 119881

119894(119896 + 1) at step 119896 + 1 by


119881119894 (119896 + 1) = 120596119881

119894 (119896) + 11988811199031(119901119894 (119896) minus 119883

119894 (119896))

+ 11988821199032(119901119892 (119896) minus 119883

119894 (119896))

119883119894 (119896 + 1) = 119883

119894 (119896) + 119881119894 (119896 + 1) Δ119905

(10)

where 119881119894



119892(119896) are the best








2 and



119861and


(120593119861and 120593










1205931015840

119905=

120593119905minus 120593119905min

120593119905max minus 120593

119905min (11)


119905is the



each time 119905



BSPs

Scaling data set


Train data


GA

Testing data


DE


PSO

Build GA-SVR

model

Build DE-SVR

model

Build PSO-SVR

model

Reconstructing TMPs


0 50 100 150 200 250 300 350 400

0

20

Time (ms)

minus100

minus80

minus60

minus40

minus20



TMP

(mV

)






(a)


(b)







algorithms



1205902









MSE =1

119899

119899

sum

119894=1

(120593119888minus 120593119890)2

RE =

1003817100381710038171003817120593119888 minus 120593119890

1003817100381710038171003817

120593119890

CC =sum119899

119894=1[(120593119888)119894minus 120593119888] [(120593119890)119894minus 120593119890]

1003817100381710038171003817120593119888 minus 120593119890

10038171003817100381710038171003817100381710038171003817120593119890 minus 120593

119890

1003817100381710038171003817

(12)


119888


119888are the


0 10 20 30 40 50 600

002

004

006

008

01

012

014

Time (ms)

Mea

n sq

uare

d er

ror (

MSE

)


(a)

Rela

tive e

rror

s (RE

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

14

(b)

Cor

relat

ion

coeffi

cien

t (CC

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

minus02

(c)


mean value of 120593119890and 120593








001

002

003

004

005

006m

-MSE

and

std-M

SE

(a)


05

04

03

02

01

0

m-R

E an

d std

-RE

(b)


11

1

09

08

07

06

05

04

m-C

C an

d std

-CC

(c)



5 Conclusion


Acknowledgments



References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119870(119909119894 119909119895) = exp(minus

10038171003817100381710038171003817119909119894minus 119909119895

10038171003817100381710038171003817

2

21205902) (5)

where 119909119894and 119909










min119891 =sum119899119905

119894=1(1003816100381610038161003816119886119894 minus 119901

119894

1003816100381610038161003816 119886119894)

119899119905

lowast 100 (6)


119894is the











2 120576) with







119909119894119866 119894119866





2 and 120576) with




2 and 120576)




V119894119866+1

= 119909best + 119865 [(1199091199031119866

minus 1199091199032119866) + (119909

1199033119866

minus 1199091199034119866)] (7)


4have to be distin-



119906119895119894119866+1

= V119895119894119866+1



(8)

where 119906119894119866+1

= (1199061119894119866+1

1199062119894119866+1

1199063119894119866+1



119894119866+1



119909best 119891 (119906best119866+1) gt 119891 (119909best) (9)






2 and




space



119894(119896)


119894(119896 + 1) and 119881

119894(119896 + 1) at step 119896 + 1 by


119881119894 (119896 + 1) = 120596119881

119894 (119896) + 11988811199031(119901119894 (119896) minus 119883

119894 (119896))

+ 11988821199032(119901119892 (119896) minus 119883

119894 (119896))

119883119894 (119896 + 1) = 119883

119894 (119896) + 119881119894 (119896 + 1) Δ119905

(10)

where 119881119894



119892(119896) are the best








2 and



119861and


(120593119861and 120593










1205931015840

119905=

120593119905minus 120593119905min

120593119905max minus 120593

119905min (11)


119905is the



each time 119905



BSPs

Scaling data set


Train data


GA

Testing data


DE


PSO

Build GA-SVR

model

Build DE-SVR

model

Build PSO-SVR

model

Reconstructing TMPs


0 50 100 150 200 250 300 350 400

0

20

Time (ms)

minus100

minus80

minus60

minus40

minus20



TMP

(mV

)






(a)


(b)







algorithms



1205902









MSE =1

119899

119899

sum

119894=1

(120593119888minus 120593119890)2

RE =

1003817100381710038171003817120593119888 minus 120593119890

1003817100381710038171003817

120593119890

CC =sum119899

119894=1[(120593119888)119894minus 120593119888] [(120593119890)119894minus 120593119890]

1003817100381710038171003817120593119888 minus 120593119890

10038171003817100381710038171003817100381710038171003817120593119890 minus 120593

119890

1003817100381710038171003817

(12)


119888


119888are the


0 10 20 30 40 50 600

002

004

006

008

01

012

014

Time (ms)

Mea

n sq

uare

d er

ror (

MSE

)


(a)

Rela

tive e

rror

s (RE

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

14

(b)

Cor

relat

ion

coeffi

cien

t (CC

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

minus02

(c)


mean value of 120593119890and 120593








001

002

003

004

005

006m

-MSE

and

std-M

SE

(a)


05

04

03

02

01

0

m-R

E an

d std

-RE

(b)


11

1

09

08

07

06

05

04

m-C

C an

d std

-CC

(c)



5 Conclusion


Acknowledgments



References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















2 and 120576)




V119894119866+1

= 119909best + 119865 [(1199091199031119866

minus 1199091199032119866) + (119909

1199033119866

minus 1199091199034119866)] (7)


4have to be distin-



119906119895119894119866+1

= V119895119894119866+1



(8)

where 119906119894119866+1

= (1199061119894119866+1

1199062119894119866+1

1199063119894119866+1



119894119866+1



119909best 119891 (119906best119866+1) gt 119891 (119909best) (9)






2 and




space



119894(119896)


119894(119896 + 1) and 119881

119894(119896 + 1) at step 119896 + 1 by


119881119894 (119896 + 1) = 120596119881

119894 (119896) + 11988811199031(119901119894 (119896) minus 119883

119894 (119896))

+ 11988821199032(119901119892 (119896) minus 119883

119894 (119896))

119883119894 (119896 + 1) = 119883

119894 (119896) + 119881119894 (119896 + 1) Δ119905

(10)

where 119881119894



119892(119896) are the best








2 and



119861and


(120593119861and 120593










1205931015840

119905=

120593119905minus 120593119905min

120593119905max minus 120593

119905min (11)


119905is the



each time 119905



BSPs

Scaling data set


Train data


GA

Testing data


DE


PSO

Build GA-SVR

model

Build DE-SVR

model

Build PSO-SVR

model

Reconstructing TMPs


0 50 100 150 200 250 300 350 400

0

20

Time (ms)

minus100

minus80

minus60

minus40

minus20



TMP

(mV

)






(a)


(b)







algorithms



1205902









MSE =1

119899

119899

sum

119894=1

(120593119888minus 120593119890)2

RE =

1003817100381710038171003817120593119888 minus 120593119890

1003817100381710038171003817

120593119890

CC =sum119899

119894=1[(120593119888)119894minus 120593119888] [(120593119890)119894minus 120593119890]

1003817100381710038171003817120593119888 minus 120593119890

10038171003817100381710038171003817100381710038171003817120593119890 minus 120593

119890

1003817100381710038171003817

(12)


119888


119888are the


0 10 20 30 40 50 600

002

004

006

008

01

012

014

Time (ms)

Mea

n sq

uare

d er

ror (

MSE

)


(a)

Rela

tive e

rror

s (RE

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

14

(b)

Cor

relat

ion

coeffi

cien

t (CC

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

minus02

(c)


mean value of 120593119890and 120593








001

002

003

004

005

006m

-MSE

and

std-M

SE

(a)


05

04

03

02

01

0

m-R

E an

d std

-RE

(b)


11

1

09

08

07

06

05

04

m-C

C an

d std

-CC

(c)



5 Conclusion


Acknowledgments



References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of























1205931015840

119905=

120593119905minus 120593119905min

120593119905max minus 120593

119905min (11)


119905is the



each time 119905



BSPs

Scaling data set


Train data


GA

Testing data


DE


PSO

Build GA-SVR

model

Build DE-SVR

model

Build PSO-SVR

model

Reconstructing TMPs


0 50 100 150 200 250 300 350 400

0

20

Time (ms)

minus100

minus80

minus60

minus40

minus20



TMP

(mV

)






(a)


(b)







algorithms



1205902









MSE =1

119899

119899

sum

119894=1

(120593119888minus 120593119890)2

RE =

1003817100381710038171003817120593119888 minus 120593119890

1003817100381710038171003817

120593119890

CC =sum119899

119894=1[(120593119888)119894minus 120593119888] [(120593119890)119894minus 120593119890]

1003817100381710038171003817120593119888 minus 120593119890

10038171003817100381710038171003817100381710038171003817120593119890 minus 120593

119890

1003817100381710038171003817

(12)


119888


119888are the


0 10 20 30 40 50 600

002

004

006

008

01

012

014

Time (ms)

Mea

n sq

uare

d er

ror (

MSE

)


(a)

Rela

tive e

rror

s (RE

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

14

(b)

Cor

relat

ion

coeffi

cien

t (CC

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

minus02

(c)


mean value of 120593119890and 120593








001

002

003

004

005

006m

-MSE

and

std-M

SE

(a)


05

04

03

02

01

0

m-R

E an

d std

-RE

(b)


11

1

09

08

07

06

05

04

m-C

C an

d std

-CC

(c)



5 Conclusion


Acknowledgments



References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a)


(b)







algorithms



1205902









MSE =1

119899

119899

sum

119894=1

(120593119888minus 120593119890)2

RE =

1003817100381710038171003817120593119888 minus 120593119890

1003817100381710038171003817

120593119890

CC =sum119899

119894=1[(120593119888)119894minus 120593119888] [(120593119890)119894minus 120593119890]

1003817100381710038171003817120593119888 minus 120593119890

10038171003817100381710038171003817100381710038171003817120593119890 minus 120593

119890

1003817100381710038171003817

(12)


119888


119888are the


0 10 20 30 40 50 600

002

004

006

008

01

012

014

Time (ms)

Mea

n sq

uare

d er

ror (

MSE

)


(a)

Rela

tive e

rror

s (RE

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

14

(b)

Cor

relat

ion

coeffi

cien

t (CC

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

minus02

(c)


mean value of 120593119890and 120593








001

002

003

004

005

006m

-MSE

and

std-M

SE

(a)


05

04

03

02

01

0

m-R

E an

d std

-RE

(b)


11

1

09

08

07

06

05

04

m-C

C an

d std

-CC

(c)



5 Conclusion


Acknowledgments



References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















0 10 20 30 40 50 600

002

004

006

008

01

012

014

Time (ms)

Mea

n sq

uare

d er

ror (

MSE

)


(a)

Rela

tive e

rror

s (RE

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

14

(b)

Cor

relat

ion

coeffi

cien

t (CC

)


0 10 20 30 40 50 60Time (ms)

0

02

04

06

08

1

12

minus02

(c)


mean value of 120593119890and 120593








001

002

003

004

005

006m

-MSE

and

std-M

SE

(a)


05

04

03

02

01

0

m-R

E an

d std

-RE

(b)


11

1

09

08

07

06

05

04

m-C

C an

d std

-CC

(c)



5 Conclusion


Acknowledgments



References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















001

002

003

004

005

006m

-MSE

and

std-M

SE

(a)


05

04

03

02

01

0

m-R

E an

d std

-RE

(b)


11

1

09

08

07

06

05

04

m-C

C an

d std

-CC

(c)



5 Conclusion


Acknowledgments



References



































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of












































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















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