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A hybrid model of Maximum Margin Clustering method and support vector regression for solving the inverse ECG problem

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Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2012, Article ID 436281, 9 pagesdoi:10.1155/2012/436281
Research Article
AHybridModelofMaximumMarginClusteringMethodandSupportVectorRegressionforNoninvasiveElectrocardiographicImaging
MingfengJiang,
1
FengLiu,
2
YamingWang,
1
GuofaShou,
3
WenqingHuang,
1
andHuaxiongZhang
1
1
School of Information Science and Technology, Zhejiang Sci-Tech University, Hangzhou 310018, China
2
School of Information Technology and Electrical Engineering, University of Queensland, St. Lucia, Brisbane,QLD 4072, Australia
3
School of Electrical and Computer Engineering, University of Oklahoma, Norman, OK 73019, USA
Correspondence should be addressed to Mingfeng Jiang, jiang.mingfeng@hotmail.comReceived 3 August 2012; Accepted 8 October 2012Academic Editor: Dingchang ZhengCopyright © 2012 Mingfeng Jiang et al.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal work is properly cited.Noninvasive electrocardiographic imaging, such as the reconstruction of myocardial transmembrane potentials (TMPs)distribution, can provide more detailed and complicated electrophysiological information than the body surface potentials (BSPs).However, the noninvasive reconstruction of the TMPs from BSPs is a typical inverse problem. In this study, this inverse ECGproblem is treated as a regression problem with multi-inputs (BSPs) and multioutputs (TMPs), which will be solved by theMaximum Margin Clustering- (MMC-) Support Vector Regression (SVR) method. First, the MMC approach is adopted to clusterthe training samples (a series of time instant BSPs), and the individual SVR model for each cluster is then constructed. For eachtesting sample, we ﬁnd its matched cluster and then use the corresponding SVR model to reconstruct the TMPs. Using testingsamples, it is found that the reconstructed TMPs results with the MMC-SVR method are more accurate than those of the singleSVR method. In addition to the improved accuracy in solving the inverse ECG problem, the MMC-SVR method divides thetraining samples into clusters of small sample sizes, which can enhance the computation e
ﬃ
ciency of training the SVR model.
1.Introduction
Thetechniqueofnoninvasiveimagingoftheheart’selectricalactivity from the body surface potentials (BSPs) constitutesoneformoftheinverseproblemofECG[1,2].Approachesto
solving the inverse ECG problem have been usually based oneither an activation-based model or a potential-based model,which includes epicardial, endocardial, or transmembranepotentials. Activation-based models are used to investigatethe arrival time of the propagation wavefront within themyocardium [3, 4]. The potential-based models are used to
evaluate the potential values on the cardiac surface [5–7] or
within the myocardium [8] at certain time instants. In this
study, we explore a new solution for ECG inverse problemusing the potential-based approach.Due to its inherent ill-posed property, the inverse ECGproblem is usually solved by “regularization” techniques.In the last decades, numerous regularization methods havebeen proposed to solve this ill-posed problem, includingtruncated total least squares (TTLS) [9], GMRes [10],
and the LSQR [11, 12]. Most of them are essentially L2-
norm based regularization schemes, which inherently lead toconsiderable smoothness of the inverse solutions. L1-normregularization method can overcome this drawback of L2-norm regularization method, which has been applied forepicardial potential reconstruction [13–15]. Although the
above-mentioned regularization methods can more or lessdeal with the geometry and measurement noises for theECG inverse problems, which depends on the regularizationparameters, the robustness of the inverse solution is notalways guaranteed. In this paper, without seeking assistance
2 Computational and Mathematical Methods in Medicine
Input dataTraining dataTesting dataClustering by MCC algorithmTrainingTrainingTrainingTrainingdata 1data 1data 2data 2data 3data 3data 4data 4SVR SVR SVR SVR
model 1model 2model 3model 4model 1model 2model 3 model 4PredictingPredictingPredictingPredicting
Finial resultsTestingTestingTestingTesting
Figure
1: The framework of the proposed MCC-SVR method.
fromtheregularizationtechniques,weexploreanalternative,more robust approach to solve the inverse ECG problem.The method is called Support Vector Regression (SVR)[16]. To ﬁnd the solution for the inverse ECG problem, a
regression model will be set up with multi-inputs (BSPs)and multioutputs (transmembrane potentials, TMPs). Thisstatistic method based solution will be assessed with thequality of the inversely predicated TMPs from the measuredBSPs. Compared with conventional regularization methods(e.g., zero order Tikhonov and LSQR), the SVR method canproduce more accurate results in terms of reconstructionof the transmembrane potential distributions on epi- andendocardial surface. In addition, when the PCA and KPCAareadoptedtoextractusefulfeaturesfromthesrcinalinputsfor building the SVR model, the SVR method with featureextraction (PCA-SVR and KPCA-SVR) outperforms thatwithout the extract feature extraction (single SVR) in termsof the reconstruction of the TMPs [17].
Compared with using single SVR model, the hybridmodels by integrating di
ﬀ
erence methods show better per-formance. The self-organizing map (SOM) is an unsuper-vised and competitive learning algorithm, which can beviewed as clustering techniques [18]. Combining the SOM
with SVR or LS-SVM, the proposed hybrid method hasthe potential to ﬁnd better inverse solutions than usinga single SVR model [19, 20]. Xu et al. [21] proposed
the Maximum Margin Clustering (MMC) method, whichperforms clustering by simultaneously ﬁnding the largemargin separating hyperplane between clusters. The MMCmethod has been successfully applied to many clusteringproblems [22]. However, its e
ﬃ
ciency is an issue of concern.Recently,Zhang et al. proposed [23, 24] an e
ﬃ
cient approachfor solving the MMC via an alternative optimization proce-dure,which wasimplemented by using the SVRmethod withthe Laplacian loss in the inner optimization subproblem.The modiﬁed MMC algorithm is more accurate, much fasterand therefore more practical for solving engineering inverseproblems. In this paper, the hybrid model of modiﬁed MMCmethod and SVR is proposed to solve the inverse ECGproblem, which is referred to as an MCC-SVR method.The conference version of this submission has appeared inCINC 2011 [25]. This submission has undergone substantial
revisions and o
ﬀ
ers extended experiment results.The main purpose of this study is to use an MCC-SVR model to investigate the reconstruction capability of TMPs.In this study, based on our previously developed realisticheart-torso model, the equivalent double layer (EDL) sourcemodel method was applied to generate the data set fortraining and testing the SVR model. The proposed algorithmwas also compared with a single SVR model for noninvasiveECG imaging.
2.TheoryandMethodology
The framework of the proposed MCC-SVR method is shownin Figure 1. The MCC method is used to classify the inputdata; the SVR is then applied to construct the regressionmodel of each cluster.
Computational and Mathematical Methods in Medicine 3
(1) Initialize the labels
y
by using the
k
-means;(2) Fix
y
, and perform SVR with Laplacian loss;(3) Compute
ω
from the Karush-Kuhn-Tucker (KKT) conditions;(4) Compute the bias
b
as described above;(5) Assign the labels as
y
i
=
sign(
ω
T
ϕ
(
x
i
) +
b
);(6) Repeat steps 2–5 until convergence.
Algorithm
1: Iterative SVR procedure for MCC method.
2.1. Maximum Margin Clustering (MMC) Method [ 23 , 24].
The clustering principle is to ﬁnd a labeling to identify dominant structures in the data and to group similarinstancestogether,sothemarginobtainedwouldbemaximalover all possible labelings, that is, given a training set
{
(
x
i
,
y
i
)
}
ni
=
1
, where
x
i
∈
χ
is the input and
y
i
∈ {±
1
}
is theoutput.TheSVMﬁndsalargemarginhyperplanetoseparatepatterns of opposite classes by the classify function
f
(
x
) [26]:
f
(
x
)
=
ω
T
ϕ
(
x
) +
b
, (1)where
ϕ
(
x
) denotes the high-dimensional feature space,which is nonlinearly mapped from the input space
x
by thekernel function
k
,
ω
is the normal vector of the hyperplane,and
b
is the o
ﬀ
set of the hyperplane. Computationally, thisleads to the following optimization problem [24, 26]:
min
ω
,
b
,
ξ
ω
2
+ 2
Cξ
T
e
subject to
y
i
ωϕ
(
x
i
) +
b
≥
1
−
ξ
i
ξ
i
≥
0,
i
=
1,
...
,
n
,(2)where
ξ
=
[
ξ
1
,
...
,
ξ
n
]
T
is the vector of a slack variable forthe errors, and
C >
0 is the trade-o
ﬀ
parameter betweenthe smoothness
ω
2
and the ﬁtness (
ξ
T
e
) of the decisionfunction
f
(
x
).MMC attempts to extend large margin methods toallocate the input data points to di
ﬀ
erent classes, leadingto large separation between the di
ﬀ
erent classes. Here, thecase with two clusters is considered in this work. Since onecould simply assign all the data points to the same class andobtain an unbounded margin, a proper constraint on theclass balance needs to be imposed. Xu et al. [21] introduced
a class constraint that requires
y
to satisfy
−
ℓ
≤
e
T
y
≤
ℓ
, (3)where
ℓ
≥
0 is a user-deﬁned constant controlling the classimbalance. Then the margin is maximized with respect toboth unknown
y
and unknown SVM parameter (
ω
,
b
) asfollows:min
y
min
ω
,
b
,
ξ
ω
2
+ 2
Cξ
T
e
subject to
y
i
ωϕ
(
x
i
) +
b
≥
1
−
ξ
i
ξ
i
≥
0,
y
i
∈{±
1
}
,
i
=
1,
...
,
n
−
ℓ
≤
e
T
y
≤
ℓ.
(4)The srcin nonconvex MMC problem in (4) can be formu-lated as a sequence of QPs which can be solved by somee
ﬃ
cient QP solvers. However, it su
ﬀ
ers from a prematureconvergence and easily gets stuck in poor local optima.Zhang et al. [23, 24] proposed to replace the SVM by
SVR with Laplacian loss, which can lead to a signiﬁcantimprovement in the clustering performance compared tothat of iterative SVM procedure. The primal problem of SVR with Laplacian loss can be formulated asmin
ω
,
b
,
ξ
i
,
ξ
∗
i
ω
2
+ 2
C
n
i
=
1
ξ
i
+
ξ
∗
i
subject to
y
i
−
ω
T
ϕ
(
x
i
) +
b
≤
ξ
i
ω
T
ϕ
(
x
i
) +
b
−
y
i
≤
ξ
∗
i
for
i
=
1,
...
,
n
,
ξ
i
≥
0,
ξ
∗
i
≥
0,(5)where
ξ
i
and
ξ
∗
i
are slack variables. With the obtained labels,the MMC problem based on the iterative SVR with theLaplacian loss becomesmin
ω
,
b
,
ξ
i
,
ξ
∗
i
ω
2
+ 2
C
n
i
=
1
ξ
i
+
ξ
∗
i
subject to
y
i
−
ω
T
ϕ
(
x
i
) +
b
≤
ξ
i
ω
T
ϕ
(
x
i
) +
b
−
y
i
≤
ξ
∗
i
ξ
i
≥
0,
ξ
∗
i
≥
0 for
i
=
1,
...
,
n y
i
∈{±
1
}−
ℓ
≤
e
T
y
≤
ℓ.
(6)After
ω
is obtained from the optimization of SVR, theproblem in (6) is reduced to the formmin
y
,
bn
i
=
1
ω
T
ϕ
(
x
i
) +
b
−
y
i
subject to
y
i
∈{±
1
}
,
i
=
1,
...
,
n
−
ℓ
≤
e
T
y
≤
ℓ.
(7)According to Zhang’s proposition [24], for a ﬁxed
b
, theoptimal strategy to determine the
y
i
’s in (7) is to assign all
y
i
’s as
−
1 for those with
ω
T
ϕ
(
x
i
) +
b <
0 and assign
y
i
’sas 1 for those with
ω
T
ϕ
(
x
i
) +
b >
0. The bias
b
can bedetermined as follows. (i) we sort the
ω
T
ϕ
(
x
i
)’s and use theset of midpoints between any two consecutive sorted values
4 Computational and Mathematical Methods in Medicineas the candidates of
b
; (ii) from these sorted
b
’s, the ﬁrstand the last (
n
−
ℓ
)
/
2 of them can be dropped, and themiddle
ℓ
can be remained; (iii) foreach remaining candidate,we determine the
y
i
’s according to the above propositionand compute the corresponding objective value in (7); (iv)ﬁnally, we choose the
b
that has the smallest objective. Thecomplete iterative SVR procedure for MCC method is shownin Algorithm 1.
2.2. Support Vector Regression (SVR) Model.
The SVR algo-rithm [26] is only brieﬂy described here; for details, see[16, 26]. As a linear regression model, the SVR algorithm
relies on an estimation of a linear regression function:
f
(
x
)
=
ω
,
x
+
b
, (
ω
,
x
∈ℜ
), (8)where
ω
and
b
aretheslopeando
ﬀ
setoftheregressionlinear,and
·
,
·
denotesthedotproductin
ℜ
.Theaboveregressionproblem can be written as a convex optimization problem:min 12
ω
2
subject to
y
i
−
ω
,
x
i
−
b
≤
ε
ω
,
x
i
+
b
−
y
i
≤
ε.
(9)In (9), an implicit assumption is that a function
f
essentially approximatesallpairs(
x
i
,
y
i
)with
ε
precision,butsometimesthis may not be the case. Therefore, one can introducetwo additional positive slack variables
ξ
i
,
ξ
∗
i
to reﬁne theestimation ofvariables
ω
and
b
.Now (9)canbe reformulated[16] asmin 12
ω
2
+
C
n
i
=
1
ξ
i
+
ξ
∗
i
subject to
y
i
−
ω
,
x
i
−
b
≤
ε
+
ξ
i
ω
,
x
i
+
b
−
y
i
≤
ε
+
ξ
∗
i
ξ
i
,
ξ
∗
i
≥
0,(10)where the constant
C
is a trade-o
ﬀ
parameter and
n
denotesthenumberofsamples;
ξ
i
representstheuppertrainingerror,and
ξ
∗
i
is the lower training error subject to
ε
intensivetube. According to the strategy outlined by Vapnik [26],using Lagrange multipliers, the constrained optimizationproblemshownin(3)canbefurtherrestatedasthefollowingequation:
f
x
,
α
i
,
α
∗
i
=
n
i
=
1
α
i
−
α
∗
i
K
(
x
i
,
x
) +
b
subject to
n
i
=
1
α
i
−
α
∗
i
=
0, 0
≤
α
i
,
α
∗
i
≤
C
,(11)where
α
i
and
α
∗
i
are the Lagrange multipliers. The term
K
(
x
i
,
x
j
) in (11) is deﬁned as the kernel function, whosevalues are the inner product of two vectors
x
i
and
x
j
in thefeature space
ϕ
(
x
i
) and
ϕ
(
x
j
). And bias
b
can be computed asfollows:
b
=
y
i
−
n
j
=
1
α
i
−
α
∗
i
K
x
j
,
x
i
−
ε
for
α
i
∈
(0,
C
)
y
i
−
n
j
=
1
α
i
−
α
∗
i
K
x
j
,
x
i
+
ε
for
α
∗
i
∈
(0,
C
)
.
(12)The kernel function handles any dimension feature spacewith no explicit calculation of
ϕ
(
x
). In this study, theGaussian kernel function is chosen as the SVR’s applicationmapping in this study:
K
x
i
,
x
j
=
exp
−
x
i
−
x
j
2
2
σ
2
, (13)where
x
i
and
x
j
are input vector spaces;
σ
2
is the bandwidthof the kernel function.In this study, an accurate and fast approach based onthe GA and the simplex search techniques is presented todetermine the optimal hyperparameters of the SVR model[17], as shown in Figure 2. The GA algorithm used here
is based on a GA toolbox developed by Chipperﬁeld et al.[27], and the simplex optimization method is implemented
using the MATLAB optimization toolbox. The developedSVR model was trained and validated with the softwareLIBSVM [28].
2.3. Simulation Protocol and Data Set.
The SVR model istested with our previously developed realistic heart-torsomodel[6,17].Inthisstudy,anequivalentdoublelayer(EDL)
source model is adopted to simulate the cardiac equivalentsource, which represents the cardiac electrical activity by meansofdoublelayersourceontheclosedsurface(includingthe endo- and epicardial surface of ventricle). For the ECGinverse problem studies, the ventricular surface TMPs andbody surface potentials (BSPs) are evaluated based on theEDLsourcemodel.Thetransfermatrix
A
betweenTMPsandBSPs is evaluated by the boundary element method (BEM),and it has the dimension of 412
×
478 and its conditionnumber (the ratio of largest and smallest singular values) is5.6
×
10
12
. As shown in Figure 3, the EDL source methodis used to obtain the BSPs
ϕ
B
and the TMPs
ϕm
. For theconstruction of the training and testing data set Di
ﬀ
erentAction Potentials (APs) for various myocardial cells andthe normal Ventricular Excitation Sequence (VES) are usedto calculate the TMPs (
ϕm
) at di
ﬀ
erent times; from thecalculated TMPs, the corresponding BSPs are deduced withthe transfer matrix
A
.In this study, a normal ventricular excitation data set isprepared for the setup of the SVR model. The consideredventricular excitation period from the ﬁrst breakthroughto the end is 357ms and the time step is 1ms, and,thus, 358 BSPs
ϕ
B
and TMPs
ϕm
temporal data sets arenumerically recorded; in addition, the 30dB simulatedGaussian white noise is added into the BSPs
ϕ
B
representingthe measurement noises. 60 datasets at times of 3ms, 9ms,
Computational and Mathematical Methods in Medicine 5
BSPs and TMPSClustering by MCCDi
ﬀ
erent clusterdata setGA optimize value of
γ
,
σ
2
,
ε
Simplex optimization Train SVR modelCalculate ﬁtness valueCreate new parameter populations by Optimal hyperparameters(1) Selection(2) Crossover(3) MutationNoYesInitial value of
γ
,
σ
2
,
ε
Satisfy stoppingcriteriaRandomize initialparameters populationCoding
γ
,
σ
2
,
ε
Figure
2: GA-Simplex optimization procedure for the parameter selection in the MCC-SVR model.
15ms,
...
,and357msaftertheﬁrstventricularbreakthroughare used as testing samples to evaluate the generalizationcapacity of the proposed SVR model. The rest 298 in 358data sets are employed as the training samples for buildingthe SVR model. With the consideration of a wide numericalrange of the
ϕ
B
values, for each time, the
ϕ
B
values can bescaled to the range (0, 1):
ϕ
BtN
=
ϕ
Bt
−
ϕ
Bt
min
ϕ
Bt
max
−
ϕ
Bt
min
, (14)where
ϕ
Bt
are the body surface potentials at time instant
t
,
ϕ
Bt
max
isthemaximumvalueofBSPsatthetime
t
,and
ϕ
Bt
min
is the minimum value of BSPs at the time
t
.As the TMPs are known in advance in the simulationstudy, the accuracy of reconstructed TMPs at the testing time
t
can be evaluated by either relative errors (REs):RE
=
ϕ
ct
−
ϕ
et
ϕ
et
, (15)or the correlation coe
ﬃ
cient (CC), given by CC
=
ni
=
1
ϕ
ct
i
−
ϕ
ct
ϕ
et
i
−
ϕ
et
ϕ
ct
−
ϕ
ct
ϕ
et
−
ϕ
et
, (16)where
n
is the number of nodes on the ventricular surface.
ϕ
et
denotes the simulated TMPs distribution at time
t
, and
ϕ
ct
are inversely computed. The quantities
ϕ
ct
and
ϕ
et
are themean value of
ϕ
ct
and
ϕ
et
over the whole ventricular surfacenodes at time
t
.
3.Results
According to the MCC method, the above 298 trainingsamples are classiﬁed four clusters as shown in Figure 4(a),and the numbers of the four clusters is 80, 74, 70, and 74,respectively. Then the individual SVR model is trained foreach cluster, and the hyperparameters are determined usingthe GA-Simplex method. For 60 testing samples, the MCCmethodisusedtoﬁndtheircorrespondingclusters,asshownin Figure 4(b).To illustrate the performances of the reconstructedTMPs, four sequential testing time points (3, 15, 27, and39ms after ventricle excitation) are presented. The inverseECG solutions are shown in Figure 5; in contrast to theconventional regularization methods, such as zero orderTikhonov regularization method and LSQR regularizationmethod,thesingleSVRmethodcanyieldratherbetterresultswith lower RE and higher CC. Moreover, it can be seenthat the MCC-SVR method o
ﬀ
ers superior performancesthan the single SVR method, as its solution is more close tothe simulated TMPs distributions. The time courses of thesimulated TMPs and reconstructions for one representativesource point on the heart surface are depicted in Figure 6.It can be found that, in reconstructing the TMPs for onerepresentative source point over all the testing times, theMCC-SVR method o
ﬀ
ers better solution compared withsingle SVR method.

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