USING SUPPORT VECTOR MACHINES (SVMS) WITH REJECT
OPTION FOR HEARTBEAT CLASSIFICATION
Zahia Zidelmal, Ahmed Amirou
Mouloud Mammeri University (UMMTO), Tizi-Ouzou, Algeria
Adel Belouchrani
Electrical Engineering Department, Ecole Nationale Polytechnique, Algiers, Algeria
Keywords:
Premature Ventricular Contraction (PVC), Beat Classification, Support Vector Machines, Reject option.
Abstract:
In this paper, we introduce a new system for ECG beat classification using Support Vector Machines (SVMs)
classifier with a double hinge loss. This classifier has the option to reject samples that cannot be classified
with enough confidence. Specifically in medical diagnoses, the risk of a wrong classification is so high that
it is convenient to reject the sample. After ECG preprocessing, feature selection and extraction, our decision
rule uses dynamic reject thresholds following the cost of rejecting a sample and the cost of misclassifying a
sample. Significant performance enhancement is observed when the proposed approach was tested with the
MIT/BIH arrythmia database. The achieved results are represented by the error reject tradeoff and a sensitivity
higher than 99%, being competitive to other published studies.
1 INTRODUCTION
Premature Ventricular Contraction (PVC) is an ec-
topic contraction caused by ventricular cells erro-
neously acting as a pacemaker. PVCs are character-
ized by the premature occurrence of bizarre shaped
QRS complexes (see figure 1). Counting the occur-
rence of ectopic beats is of particular interest to sup-
port the detection of ventricular tachycardia and to
evaluate the regularity of the depolarization of the
ventricles.
As a large amounts of data are often analysed
and stored when examining cardiac signals, comput-
ers can be used to automate signal processing. Ac-
cordingly, several algorithms have been proposed for
the detection and classification of heartbeats together
with signal processing techniques.
Classical techniques extract heuristic ECG de-
scriptors, such as the QRS morphology (Chazal et al.,
2004) and interbeat R-R intervals (Chazal et al.,
2004), (Jecova et al., 2004). Other ECG descrip-
tors rely on QRS frequency components calculated
either by Fourier transform (Minami et al., 1999)
or by computationally efficient algorithms with filter
banks (Afonso and Tompkins, 1999). Some meth-
ods apply QRS template matching procedures, based
on different transforms, e.g., Karhunen-Loeve trans-
form (Gomez-Herrero et al., 2006), Hermite functions
(Lagerholm et al., 2000) to approximate the variety
of temporal and frequency characteristics of the QRS
complex waveforms. Other techniques for comput-
erized arrhythmia detection employ cross-correlation
with predefined ECG templates (Krasteva and Jecova,
2007)
Several discriminative techniques such as artifi-
cial neural networks have been developed and used
to exploit their natural ability in pattern-recognition
tasks for successful classification of ECG beat (Yeap,
1990). The latter include linear back-propagation net-
work discriminants, Self-organizing maps with learn-
ing vector quantization (Hu et al., 1997) where Hu
et al customized a heartbeat classifier to a specific
patient (local classifier) and then combined it with a
global classifier using a mixture of experts approach
(MOE). Among all these methods, Support Vector
Machines (SVMs) have enjoyed a strong success in
this application field (Osowski et al., 2004) where
Osowski et al. combined multiple classifiers by the
weighted voting principle. but leads to a computa-
tionally highly expensive approach.
Even though the performance of all these tech-
niques, misclassifications cannot be completely elim-
204
Zidemal Z., Amirou A. and Belouchrani A. (2009).
USING SUPPORT VECTOR MACHINES (SVMS) WITH REJECT OPTION FOR HEARTBEAT CLASSIFICATION.
In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing, pages 204-210
DOI: 10.5220/0001431602040210
Copyright
c
 SciTePress

inated and, thus, can produce severe penalties. This
motivates the introduction of a reject option in the
classifiers. Although of its practical interest, this op-
tion has not received enough attention since the pub-
lications of Chow on the error reject tradeoff (Chow,
1957)-(Chow, 1970). A notable proposal of a reject
rule explicitly designed for SVMs has been presented
in (Fumera and Roli, 2002), (Kwok, 1999), (Tor-
torella, 2004) and recently in (Herbei and Wegkamp,
2006) and (Bartlett and Wegkamp, 2007). To our
Knowledge, this option was not used in heartbeat
classifiers although of its particular interest in med-
ical field where the risk of a wrong classification is so
high that it is convenient to reject the sample.
In this paper, SVMs working in binary classifica-
tion mode and different preprocessing techniques of
ECG waveform are used for ectopic heartbeat recog-
nition. We introduce a cost-sensitive reject rule for
SVMs using a double hinge loss. This classifier is
based on the one provided by Bartlett and Wegkamp
(Bartlett and Wegkamp, 2007). This heuristic ap-
proach is coherent with the theoretical foundations of
SVMs, which are based on the structural risk mini-
mization (SRM) principle (Vapnik, 1995), (Cristian-
ini and Shawe-Taylor, 2000). Under this framework,
the rejection region must be determined during the
training phase of the classifier.
The paper is organized as follows. Section 2
presents the preprocessing techniques and how diag-
nostic features were selected and computed. Section
3 recalls Bayes’rule for binary classification with re-
jection. This section also presents the SVMs in this
framework and the learning criterion dedicated for the
problem at hand. This proposed method is tested em-
pirically in Section 4. Finally, Section 5 briefly con-
cludes the paper.
2 DATA PREPARATION
In this work, we concentrate on the classification of
normal beats and abnormal beats (PVC beats) see Fig-
ure 1.
Some records from the MIT-BIH arrhyth-
mia database (http://physionet.org/ phys-
iobank/database/mitdb) with 360 Hz sampling
frequency are used. Each record is accompanied by
an annotation file in which each ECG beat has been
identified by expert cardiologists. These labels, re-
ferred to as ’truth’ annotation and are used to develop
the SVMs classifier and to evaluate its performance.
Since this study is to evaluate the performance of
a binary classifier that can identify a premature
ventricular contraction, some records presenting a
amplitude (mV)
0 200 400 600
−1
0
1
2
Normal Beat
PVC Beats
R
R
R
time
Figure 1: Segment of ECG (record 208) showing the mor-
phology of PVC heartbeats (second and third ones).
high occurrence of PVC beats were selected.
2.1 Pre-processing
The objective of this paper is to classify the QRS beats
as normal or abnormal ones. Before performing this
task, several pre-processing steps were performed on
the raw data to study their effects upon the perfor-
mance of the classifier. In fact the electrocardiogram
(ECG) from body electrodes are corrupted by noise.
Usually, two principal sources of ECG noise can be
distinguished: first one caused by the physical pa-
rameters of the recording equipment and the second
one representing the bioelectrical activity also called
background activity or baseline wander. Several noise
removal techniques were recently developed. In this
work, we reduce the high frequency noise by thresh-
olding the ECG wavelet coefficients (Donoho, 1995).
The baseline wander is removed by setting to zero the
approximation coefficients vector of the sixth level of
wavelet decomposition.
2.2 Feature Extraction and Selection
The QRS complex in ECG signal varies with orig-
ination and conduction path of the activation pulse
in the heart. When the activation pulse originates in
the atrium and travels through the normal conduction
path, the normal QRS complex has a sharp and narrow
deflexion and the spectrum contains high frequency
components. When the activation pulse originates in
the ventricle and does not travel through the normal
path, the QRS becomes wide and the high frequency
components of the spectrum are attenuated.
A set of algorithms from signal conditioning to mea-
surements of average wave amplitudes, durations,
morphology, and areas is usually adopted to perform
a quantitative description of a heartbeat and a parame-
ter extraction. In this study, we used some parameters
USING SUPPORT VECTOR MACHINES (SVMS) WITH REJECT OPTION FOR HEARTBEAT CLASSIFICATION
205

such as the instantaneous R-R interval, average R-R
interval, width, morphology and mobility of the QS
segment.
• The R peaks were detected using a robust method
based on wavelet coefficients (Lepage, 2003) witch
was compared to the well known Pan and Tompkins
algorithm (Pan and Tompkins, 1985). Once the R
peaks are detected, the instantaneous R-R interval is
calculated as the difference between the QRS peak of
the present beat and the previous one.
• The average R-R interval is calculated as the aver-
age R-R interval over the previous ten intervals. The
peaks Q and S are detected using simple peak detec-
tion method leading to the width of the QS segment.
• The morphology of the beat is captured by four Lin-
ear Predictive Coding (LPC) coefficients. The basic
idea of this technique is that future values of a dis-
crete signal are estimated as a linear function of pre-
vious samples. The most common representation is
b
y
n
=
p
∑
k=1
a
k
y
n−k
(1)
where a
k
is the k
th
linear prediction coefficient, p is
the order of the predictor and
b
y
n
the present predicted
sample.
• A frequency feature was extracted by comput-
ing mobility factor (MB) as defined in (Ramaswamy
et al., 2004)
MB(x) =
s
var(x
0
)
var(x)
(2)
where x is the original ECG signal from point Q to S,
var(x), the variance of x and x
0
the first derivative of x.
MB is basically a ratio of energy of higher frequency
signal over the energy of the signal. Since the ectopic
beats have longer QS segments, the higher frequency
energy will be lower. The information of each beat is
stored as a 7-element vector, with the first three ele-
ments representing the temporal parameters, the next
three elements representing the morphological infor-
mation and the last one is the mobility of the QS seg-
ment.
3 CLASSIFICATION WITH
REJECTION
Classification aims at predicting a class label y ∈ Y
from an observed pattern x ∈ X . For this purpose, we
construct a decision rule d that typically assigns a la-
bel to any x ∈ X . In binary problems, where the class
is tagged +1 or −1, the two types of possible errors
are: false positive (FP), where examples labeled −1
are categorized in the positive class, incurring a loss
c
−
; false negative (FN), where examples labeled +1
are categorized in the negative class, incurring a loss
c
+
. We consider here problems where some samples
may be not categorized. The classifier d based on the
one provided by (Bartlett and Wegkamp, 2007) has
the option to reject samples that cannot be classified
with enough confidence. This decision to abstain, will
be denoted r and incurs a loss r. The losses pertain-
ing to each possible decision are recapped in table 1
and illustrated on figure 2.
Table 1: Losses for each possible pair of label and decision.
d(x)
y
+1 −1
+1 0 c
−
0 r r
−1 c
+
0
losses
r
c+
0.5
0.5
c−
0
1
P−
P+
Posterior probability
Figure 2: Illustration of different risks vs. posterior proba-
bilities.
3.1 Bayes’ Rule with Reject Option
Bayes´s decision theory is the paramount framework
in statistical decision theory, where decisions are
taken to minimize the expected loss
L(d) = c
+
P(Y = 1, d(X) = −1) +
c
−
P(Y = −1, d(X) = 1) +
r P(d(X ) = r) . (3)
From figure 2, one gets that rejection is a viable
option if and only if
0 ≤ r ≤
c
−
c
+
c
−
+ c
+
. (4)
BIOSIGNALS 2009 - International Conference on Bio-inspired Systems and Signal Processing
206

If we assume that c
−
= c
+
= 1, the condition (4) be-
comes simply 0 ≤ r ≤
1
2
. In particular, this implies
that the loss of rejecting a pattern should be lower
than the loss of making an error. Bayes rule can then
be expressed simply, using two thresholds p
−
= r and
p
+
= 1 − r (see Figure 3).
f(x)=−1 R f(x)=+1
0 r 0.5 1−r 1
Figure 3: Bayes’ rule with reject option.
Where, assuming that (4) holds, Bayes’ rule with
reject option can then be stated as
d(x) =



−1 if P(Y = 1|X = x) < p
−
,
+1 if P(Y = 1|X = x) > p
+
,
r otherwise .
(5)
Since the paper of Chow (Chow, 1970), this type of
rule is sometimes referred to as Chow’s rule.
3.1.1 Bayes Rule with Weighted Errors
Often, in practice, FN errors are more costly than FP
errors or vice-versa. It is the case in medical field
where misclassifying a sick patient as healthy is, in
general, far worse than the reverse. In order to ac-
commodate for that, we consider the risk function (3)
with c
+
> c
−
. Bayes rule with weighted errors can
then be expressed using new thresholds
p
−
= r
p
+
= 1 − θr (6)
where θ represents the ratio of the costs of the two
types of error.
One of the major inductive principle is the empirical
risk minimization, where one minimizes the empiri-
cal counterpart of the expected loss (3). In classifica-
tion, this principle is usually NP-hard to implement.
Hence, as Bayes decision rule is defined by condi-
tional probabilities, many classifiers first estimate the
conditional probability
b
P(Y = 1|X = x), and then plug
this estimate in (5) to build the decision rule.
d(x) =



−1 if
b
P(Y = 1|X = x) < p
−
,
+1 if
b
P(Y = 1|X = x) > p
+
,
r otherwise .
(7)
3.2 SVMs Classifier with a Double
Hinge Loss
In this section, we show how the standard SVM
optimization problem is modified when the hinge loss
is replaced by a double hinge loss. The optimization
problem is first written using a compact notation, and
the dual problem is then derived.
3.2.1 Double Hinge
The double hinge loss function φ
r
(y f (x)), displayed
in Figure 4 was proposed by (Bartlett and Wegkamp,
2007) in the context of binary classification with re-
jection. It is a positive, convex and piecewise linear
loss function.
φ
r
(y f (x)) =



1 −
1−r
r
y f (x) if y f (x) < 0 ,
1 − y f (x) if 0 ≤ y f (x) < 1 ,
0 otherwise ,
(8)
The function f is estimated by the minimization of
a regularized empirical risk on the training samples
T = {(x
i
, y
i
)}
n
i=1
n
∑
i=1
φ
r
(y
i
f (x
i
)) + λΩ( f ) , (9)
where φ
r
is the loss function and Ω(·) is a regular-
ization functional, such as the (squared) norm of f
in a Reproducing Kernel Hilbert Space (RKHS)H ,
Ω( f ) = k f k
2
H
.
φ
r
(y f (x))
0
1.3
2.7
f
+
f
−
0 1
y f (x)
Figure 4: Loss function φ
r
versus margin y f (x) for r = 0.1
f
+
and f
−
are the reject thresholds to be computed after the
training step.
USING SUPPORT VECTOR MACHINES (SVMS) WITH REJECT OPTION FOR HEARTBEAT CLASSIFICATION
207

3.2.2 Optimization Problem
As in standard SVMs, we consider the regularized
empirical risk on the training sample. Introducing the
double hinge loss (8) results in an optimization prob-
lem that is similar to the standard SVMs problem.
Let C a constant to be tuned by cross-validation,
we define D = C(
1−2r
r
); The optimization problem
reads
min
f ,b
1
2
k f k
2
H
+C
n
∑
i=1
|
1 − y
i
( f (x
i
+ b))
|
+
+
D
n
∑
i=1
|
−y
i
( f (x
i
+ b))
|
+
, (10)
where | · |
+
= max(·, 0). Minimizing ( 10) is a
quadratic problem. This is best seen with the intro-
duction of slack variables ξ and γ











min
f ,b,ξ,γ
1
2
k f k
2
H
+C
n
∑
i=1
ξ
i
+ D
n
∑
i=1
γ
i
,
s.t. ξ
i
≥ 1 − y
i
( f (x
i
) + b),
γ
i
≥ −y
i
( f (x
i
) + b),
ξ
i
≥ 0 , γ
i
≥ 0 i = 1, . . . , n .
(11)
As for standard SVMs, the dual formulation of
( 11) leads to efficient optimization algorithms. To
compute the solution, we use an active set algorithm
following a strategy that proved to be efficient for
standard SVMs. The SimpleSVM algorithm (Vish-
wanathan et al., 2003; Loosli et al., 2005) solves the
SVM training problem by a greedy approach in which
one solves a series of small problems. First, the train-
ing examples are assumed to be either support vectors
or not, and the training criterion is optimized consid-
ering that the partition of examples is fixed. This op-
timization results in a new partition of examples in
support and non-support vectors. These two steps are
iterated until some level of accuracy is reached. Note
that this algorithm compute the bias b in the same step
as computing the lagrange multipliers α
i
After training, we can represent f as the finite sum
f (x) =
n
∑
i=1
α
i
k(x
i
, x) + b (12)
where α
1
, ..., α
n
, b is the solution of the dual of prob-
lem (11)
3.3 Estimation of the Posterior
Probability
The SVM does not provide a probability measure.
Given a raw score value, the estimation of the prob-
ability is a post processing step. One method of pro-
ducing probabilistic outputs was proposed by (Platt,
2000). This method approximates the posterior prob-
ability by a two-parameter logistic function of the
form
P
A,B
(x) =
1
1 + exp(A f (x) + B)
, (13)
Where P
A,B
(x) ≈
b
P(Y = y|X = x). The best parame-
ters (A, B) are then estimated by minimizing the nega-
tive log likelihood of a validation set of labelled sam-
ples {(x
i
, y
i
)}
l
i=1
, which is a cross-entropy error func-
tion:
min
A,B
−
l
∑
i=1
t
i
log(p
i
) + (1 −t
i
)log(1 − p
i
) , (14)
where p
i
= P
A,B
(x
i
) and t
i
=
y
i
+1
2
. A pseudo code for
resolving (14) can be found in (Platt, 2000) and (Lin
et al., 2003).
After mapping the SVM outputs to posterior prob-
abilities, the decision rule (7) can be applied.
4 EXPERIMENT
From the clinical observations, we obtained 7 fea-
tures. Six temporal features and one spectral feature.
A support Vector Machine (SVMs) with reject option
was used to classify these features. The classifier was
constructed separately for each record selected in the
MIT/BIH database. The data set has been uniformly
divided into a training set, a validation set and a test
set.
To learn the classifier, we considered a Gaussian
kernel ( characterized by a width σ ) of the following
form K
σ
(x, y) = exp(−
kx−yk
2
σ
2
). The kernel parame-
ter σ and the penalization parameter C are firstly op-
timized by 5 fold cross-validation using SVMs with
double hinge loss. The training set is partitioned in
5 subsets where the proportion of positives exam-
ples is identical. Each subset is iteratively used as
a training set while the remaining ones are used as
test sets. Note that the features are normalized before
each training session. After learning, the best param-
eters (A, B) of the logistic function are estimated by
fitting on the validation set.
We can see on figure 5 the reject region produced
by the SVM classifier for symmetric misclassification
losses with r = 0.45 and for asymmetric misclassifi-
cation losses with r = 0.30.
As advocated in (Chow, 1970), a complete de-
scription of the performance of a recognition system
with reject option is given by the error-reject tradeoff.
Since the error rate E and the reject rate R are mono-
tonic functions of r, we can compute the tradeoff E
versus R from E(r) and R(r).
BIOSIGNALS 2009 - International Conference on Bio-inspired Systems and Signal Processing
208

−0.11212
−0.11212
−0.11212
−0.11212
−0.11212
0
0
0
0
0
0.11212
0.11212
0.11212
0.11212
0.11212
−4 −3 −2 −1 0 1
−3
−2
−1
0
1
2
3
4
5
r = 0.45 θ = 1
−0.68117
−0.68117
−0.68117
0
0
0
0
0
0.3643
0.3643
0.3643
0.3643
−4 −3 −2 −1 0 1
−2
−1
0
1
2
3
4
5
r = 0.30 θ = 1.5
Figure 5: The reject region induced by the reject thresh-
olds in correspondence to the cost of rejecting samples.
r = 0.45 and θ = 1 (Top); r = 0.3 and θ = 1.5 (bottom).
The circles(◦) indicate the negative class N, wile the asts
(∗) indicate the positive class P.
Varying r between 0.5 and 0.1 with θ = 1, the
mean results obtained using the selected records are
reported on Figure 6.
The error rate E = E(R), as a function of the reject
rate Figure 6 (Top) decreases at a nearly constant rate
of roughly 5%.
For example, if we set our rejection thresholds to
exclude 5% of the cases, this means that the decision
rule can be specified to classify 95% of the cases with
a very low misclassification rate and identify the re-
maining 5% as hard cases that need special consider-
ations.
Another interesting performance criterion is the
sensitivity representing the fraction of real events that
are correctly detected. SE =
T P
T P+FN
where True Pos-
itive (TP) are the samples labelled +1 categorized in
the positive class , and False Negative (FN), are the
samples labelled +1 categorized in the negative class.
We show on Figure 6 (bottom) that we obtained more
than 98% of sensitivity with no rejection and more
than 99% of sensitivity after rejecting less than 5%
of instances. Note that the reject threshold follows
Sensitivity Error
0 0.05 0.3 0.6
0
0.014
0.028
0 0.05 0.3 0.6
0.98
0.99
1
Reject Rate
Figure 6: Error Reject curve obtained using the proposed
method (Top); Sensitivity vs. reject rate (bottom). While
varying r between 0.5 and 0.1 with θ = 1.
the real cost of rejecting a sample and the real cost of
misclassifying a sample to optimize the classification
cost. This, is the goal of this cost sensitive reject rule.
These results are very competitive to other pub-
lished studies e.g., (Krasteva and Jecova, 2007) get-
ting 98, 4% of sensitivity, (Osowski et al., 2004) ob-
taining 95, 9% of accuracy using a computationally
highly expensive approach and (Chazal et al., 2004)
obtaining 77, 7% of sensitivity for distinguishing Ven-
tricular Ectopic Beats (VEB) from non-VEBs.
5 CONCLUSIONS
This paper presents a new heartbeat classifier using
Support Vector Machines with an embedded reject
option. The proposed system accomplishes prepro-
cessing, feature extraction /selection and recognition
tasks for recognition of Premature Ventricular Con-
traction (PVC) beats.
For this purpose a cost-sensitive reject rule for
SVMs is used together with a double hinge loss for
asymmetric classification. For each class, the loss of
rejecting a pattern is assumed to be lower than the
loss of making an error. A training criterion based
on a convex and piecewise linear loss function is pro-
posed. Under this framework, the rejection region is
determined during the training phase of the classifier.
Our decision rule uses dynamic reject thresholds fol-
lowing the cost of rejecting a sample and the cost of
misclassifying a sample to optimize the classification
cost.
Our results shown above illustrate a good error re-
ject tradeoff and indicate that if we set our rejection
thresholds to exclude less than 5% of the cases, the
sensitivity of the classifier becomes higher than 99%,
being competitive to other published studies.
USING SUPPORT VECTOR MACHINES (SVMS) WITH REJECT OPTION FOR HEARTBEAT CLASSIFICATION
209

This paper has focused on binary classification
problems since only ECG records containing normal
beats and PVC beats were selected. Extension to
multi-category classification is also possible.
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