A CLASS SPECIFIC DIMENSIONALITY REDUCTION
FRAMEWORK FOR CLASS IMBALANCE PROBLEM: CPC SMOTE
T. Maruthi Padmaja
1,2
, Bapi S. Raju
2
1
IDRBT Masab Tank, Hyderabad 500 057, (AP), India
2
DCIS, University of Hyderabad, Hyderabad 500 057, (AP), India
P. Radha Krishna
SET Labs, Infosys Technologies Ltd, Hyderabad 500 057, (AP), India
Keywords:
Class imbalance problem, Principle component analysis, SMOTE, Decision tree.
Abstract:
The performance of the conventional classification algorithms deteriorates due to the class imbalance prob-
lem, which occurs when one class of data severely outnumbers the other class. On the other hand the data
dimensionality also plays a crucial role in performance deterioration of classification algorithms. Principal
Component Analysis (PCA) is a widely used technique for dimensionality reduction. Due to unsupervised
nature of PCA, it is not adequate enough to hold class discriminative information for classification problems.
In case of unbalanced datasets the occurrence of minority class samples are rare or obtaining them are costly.
Moreover, the misclassification cost associated with minority class samples is higher than non-minority class
samples. Capturing and validating labeled samples, particularly minority class samples, in PCA subspace is
an important issue. We propose a class specific dimensionality reduction and oversampling framework named
CPC SMOTE to address this issue. The framework is based on combining class specific PCA subspaces to
hold informative features from minority as well as majority class and oversample the combined class specific
PCA subspace to compensate lack of data problem. We evaluated the proposed approach using 1 simulated
and 5 UCI repository datasets. The evaluation show that the framework is effective when compared to PCA
and SMOTE preprocessing methods.
1 INTRODUCTION
Classification usually utilizes a supervised learning
method and learns a model from already labeled his-
torical data and uses this model to predict the class
of unseen test data. Class imbalance problem betides
in those datasets, where one class (majority class) of
data severely outnumbers the other class of data (mi-
nority class). For two-class classification problem,
many real world applications such as fraud detection
(Phua and Lee, 2004), and rare disease prediction in
medical diagnosis (Yoon and Kwek, 2007) that re-
flect the nature of class imbalance problem. In the
literature, effect of class imbalance problem is investi-
gated over many classification algorithms such as de-
cision tree, neural networks, support vector machines,
k nearest neighbor classifier and proved that the clas-
sification algorithms bias towards predicting the ma-
jority class in class imbalance scenario (Japkowicz
and Stephen, 2002). This bias nature of the classifier
performance towards the majority class leaves huge
error rates from the minority class.
For high-dimensional data, classification process
may also include dimensionality reduction to increase
the class discrimination, better data representation
and for attaining good computational efficiency. The
misclassification rate for classification process dras-
tically increases due to spurious dimensions in the
original high dimensional data space, known as the
curse of dimensionality problem (Duda and Strok,
2001). Hence there is a need for dimensionality re-
duction. Principal Component Analysis (PCA) is a
quite popularly used technique for dimensionality re-
duction. PCA linearly transforms high dimensional
data into lower dimensional space by maximizing
the global variance of the data as well as minimiz-
ing least square error for that transformation. How-
ever, PCA is an unsupervised dimensionality reduc-
tion technique. Therefore, it is not adequate enough
to hold the discriminative information for classifica-
237
Maruthi Padmaja T., S. Raju B. and Radha Krishna P..
A CLASS SPECIFIC DIMENSIONALITY REDUCTION FRAMEWORK FOR CLASS IMBALANCE PROBLEM: CPC SMOTE.
DOI: 10.5220/0003092502370242
In Proceedings of the International Conference on Knowledge Discovery and Information Retrieval (KDIR-2010), pages 237-242
ISBN: 978-989-8425-28-7
Copyright
c
 2010 SCITEPRESS (Science and Technology Publications, Lda.)

tion problems when the maximum variance direction
of one class is different from another class i.e., un-
equal covariance matrices (Vaswani and Chellappa,
2006; Xudong, 2009). Finding principal axes direc-
tions i.e principle components (PCs) is one of the key
steps for PCA and it depends on spread of the data.
In case of the unbalanced datasets, the spread is dom-
inated by majority class as its prior probabilities are
much higher than minority class samples. Moreover,
in real world domains such as intrusion detection sys-
tems, the occurrence of intrusion transactions are rare
and generating them is a costly process. Miss pre-
dicting these rarely occurred intrusion transactions is
risky and could lead to financial loss for organiza-
tions. Therefore, Capturing and validating labeled
samples, particularly non-majority class samples in
PCA subspace for classification task is a challenging
issue.
In this paper, we propose a class specific di-
mensionality reduction and oversampling framework
named CPC SMOTE to address class imbalance is-
sue in Principle Component Analysis subspace where
there is directional difference in Principal Compo-
nents (PCs) of two classes. The proposed framework
based on capturing class specific features in order to
hold major variance directions from individual class
and oversampling is to compensate lack of data in
under-represented class. Proposed approach is eval-
uated over decision tree classifier using accuracy and
F-measure as evaluation metrics. Experimental ev-
idence show that proposed approach yields superior
performance on simulated and real world unbalance
datasets compared with classifier learned on reduced
dimensions of whole unbalanced datasets as well as
on oversampled datasets.
The rest of the paper is organized as follows.
Section 2 discusses work related to class imbalance
problem. Section 3 provides proposed CPC SMOTE
framework. Section 4 presents the experimental eval-
uation based on a comparative study, which is done
with applying PCA on whole unbalanced dataset as
well as applying SMOTE on unbalanced datasets. Fi-
nally, conclusions are given in section 5.
2 RELATED WORK
There are several ways to handle class imbalance
problem. Among them cost sensitive learning, one
class classification and resampling the class distri-
bution are frequently used. However most of the
research that addresses the class imbalance prob-
lem centered on balancing the class distributions.
Resampling the data either by random oversamp-
ing or by random undersampling in order to make
approximately balance class distributions are quite
popularly adopted solutions. But for discriminative
learners such as decision tree classifier oversampling
causes overfitting, where as the undersampling leads
to performance degradation due to loss of informa-
tive instances from majority class (Drummond and
Holte, 2003). (Weiss and Provost, 2003) concluded
that the natural distribution is not usually the best
distribution for learning. Study of ”whether over-
sampling is more effective than under-sampling” and
”which over-sampling or under-sampling rate should
be used” was done by (Estabrooks and Japkowicz,
2004), who concluded that combining different ex-
pressions of the resampling approach is an effective
solution. (Kubat and Matwin, 1997) did selective
under-sampling of majority class by keeping minor-
ity classes fixed. They categorized the minority sam-
ples into some noise overlapping, the positive class
decision region, borderline samples, redundant sam-
ples and safe samples. By using Tomek links con-
cept, which is a type of data cleaning procedure they
deleted the borderline majority samples.
(Chawla and Kegelmeyer, 2002), proposed Syn-
thetic Minority Over-sampling Technique (SMOTE).
It is an oversampling approach in which the minor-
ity sample is over-sampled by creating synthetic (or
artificial) samples rather than by oversampling with
replacement. The minority class is over-sampled by
taking each minority class sample and introducing
synthetic samples along the line segments joining
any/all of the k minority class’ nearest neighbors. De-
pending upon the amount of oversampling required,
neighbors from the k nearest neighbors are randomly
chosen. This approach effectively forces the decision
region of the minority class to become more general.
(Villalba and Cunningham, 2008) evaluate un-
supervised dimensionality reduction techniques over
one-class classification methods and concluded that
Principle Component Analysis (PCA) damages the
performance on most of the datasets. (Xudong, 2009)
analyzed the role of PCA over unbalanced training
sets and concluded that the PCA subspace is biased by
the majority class eigen vectors. Further the authors
proposed Asymmetric Principle component Analysis
(APCA), a weighted PCA to combat the bias issue
in PCA subspace. In this paper we propose a class
specific dimensionality reduction and oversampling
framework in the context of two class classification
to combat this problem. Proposed approach yielded
superior performance on those datasets where there is
directional difference between two classes’ principle
components.
KDIR 2010 - International Conference on Knowledge Discovery and Information Retrieval
238

Figure 1: Flow diagram for CPC SMOTE framework.
3 THE CPC SMOTE
FRAMEWORK
The main goal behind this framework is to combat
class imbalance problem in PCA subspace, where
the principle component analysis predominantly rep-
resents majority class maximum variance directions
only. In order to accomplish this goal a class spe-
cific principle component analysis and oversampling
framework is proposed. Figure 1 depicts the flow di-
agram for proposed framework.
The class specific PCA is for extracting better in-
formative features from both classes, where there is
directional difference between two class PCs. This is
done by concatenating class specific principle compo-
nents that are extracted from each class by applying
PCA. Latter SMOTE is applied to this reduced fea-
ture space of minority class to alleviate the lack of
data problem.
The steps for proposed CPC SMOTE framework
are discussed below.
• Step 1: Extract minority class patterns (Mi)
p∗d
and majority class patterns (M j)
q∗d
from an un-
balanced data X
n∗d
. where n = p+ q,q > p and
d=number of features in unbalanced data X
n∗d
.
• Step 2: Apply classical PCA on each class, for
each class select r(< d) eigen vectors correspond-
ing to first r large eigen values. Let (E
c
)
d∗r
be
the corresponding eigen vector matrices selected
in this step. Where c = 1 and 2.
• Step 3: Concatenate eigen vectors (E
c
)
d∗r
ob-
tained in step 2 in order to facilitate class specific
informative directions.
(E
total
)
d∗2r
= [(E
1
)
d∗r
,(E
2
)
d∗r
].
• Step 4: Project unbalanced data X
n∗d
into
(E
total
)
d∗2r
to enable the informativefeature space
of majority and minority classes.
(NEW X)
n∗2r
= X
n∗d
∗ (E
total
)
d∗2r
.
• Step 5: Extract reduced minority class
(NEW X Mi)
p∗2r
and majority class
(NEW X M j)
q∗2r
patterns from the infor-
mative feature space (NEW X)
n∗2r
and do
SMOTE on (NEW X Mi)
p∗2r
.
(T NEW X Mi)
l∗2r
= SMOTE(NEW X Mi)
p∗2r
,where l = (q/p) ∗ p.
• Step 6: Combine minority class and majority class
patterns
(T NEW X)
t∗2r
= [(T NEW X Mi)
q∗2r
,
(NEW X M j)
q∗2r
], where t = 2q.
Let X
n∗d
be an unbalanced dataset with n num-
ber of records and d number of features. In order to
get better separated features we apply PCA indepen-
dently on each class distribution. Let us suppose that
(Mi)
p∗d
, (Mj)
q∗d
be the corresponding minority class
and majority class distributions, where q > p. From
each class distribution of (Mi)
p∗d
and (M j)
q∗d
equal
number of eigen vectors are extracted. Let r(< d)
are the selected number of eigen vectors and (E
c
)
d∗r
be the corresponding eigen vector matrix, and here
c = 1,2. The extracted eigen vectors are combined
horizontally to get class specific features directions
(E
total
)
d∗2r
. Now the selected number of features be-
comes 2r, wherer number of features from each class.
Then the whole unbalanced data X
n∗d
is projected into
these combined feature space (E
total
)
d∗2r
in order to
get combined reduced feature space (NEW X)
n∗2r
.
Since the combined class specific direction matrix
(E
total
)
d∗2r
covers the maximum variance directions
from all classes, the data can be better discriminate in
combined reduced feature space (NEW X)
n∗2r
. But
still there is lack of data problem in reduced fea-
ture space of (NEW X)
n∗2r
due to the unbalanced na-
ture of the original data. This lack of data problem
is alleviated by oversampling the minority class re-
duced feature space with synthetic samples using syn-
thetic minority oversampling technique (SMOTE). In
order to do so, extract minority class feature space
(NEW X Mi)
p∗2r
and majority class feature space
(NEW X M j)
q∗2r
from (NEW X)
n∗2r
. Generate l
number of synthetic samples where l = (q/p) ∗ p in
minority class feature space (NEW X Mi)
p∗2r
using
SMOTE. Finally combine both class reduced feature
spaces (T NEW X Mi)
q∗2r
, (NEW X M j)
q∗2r
to at-
tain balanced class reduced feature space distribu-
tion (T NEW X)
t∗2r
where t = 2q. The final ma-
trix (T NEW X)
t∗2r
obtained is the combined matrix
with combination of class specific features and over-
sampled reduced feature space. This matrix can be
directly used for classification tasks. Proposed ap-
proach is evaluated using decision tree classifier.
A CLASS SPECIFIC DIMENSIONALITY REDUCTION FRAMEWORK FOR CLASS IMBALANCE PROBLEM: CPC
SMOTE
239

4 EXPERIMENTAL EVALUATION
This section describes the datasets, presents evalua-
tion metric for estimating classifier performance and
elaborates the comparative study with other methods
for approximating the performance of the proposed
framework, in terms of experimental results and dis-
cussion.
4.1 Evaluation Metric
Generally the outcomes of any classification model
are:
• True positive(TP) Rate is the percentage of cor-
rectly classified positive samples.
• True negative(TN) rate is the percentage of cor-
rectly classified negative samples.
• False negative(FN)rate is the percentage of incor-
rectly classified positive samples.
• False positive(FP) rate is the percentage of nega-
tive examples predicted as positives.
The goal of any ideal classifier is to maximize TP and
TN rates. The following are normally applied mea-
sures for evaluating classification performance:
Accuracy =
(TP+ TN)
(TP+ TN + FP+ FN)
. (1)
TPrate = Recall =
TP
(TP+ FN)
. (2)
Precision =
TP
(TP+ FP)
. (3)
F − measure =
2∗ Precision∗ Recall
Precision+ Recall
. (4)
Proposed approach is evaluated using classifica-
tion accuracy (eq.1) and F-measure (eq 4). Classifi-
cation accuracy is designed to reduce the overall clas-
sification error rate. Moreover, for classification pro-
cess, n number of dimensions selected as reduced fea-
tures, based on improvement in overall accuracy over
original input space. Similarly n number of features
selected from PCA. But class imbalance point of view
accuracy is not an appropriate measure for evaluat-
ing the classifier performance. Consider that there are
only 6% of samples from minority class and 94% of
the samples are from majority class. If a classifier
miss predicts all minority class samples as majority
class samples then the accuracy becomes 94% with
the contribution of majority class samples only.
F-measure depicts the performance of the target
class in terms of tradeoff between precision and re-
call, where as recall is simply the TP rate of the tar-
get class and precision gives the trade-off between TP
and FP rates. If both Precision and Recall are high,
then F-measure is also high. For unbalanced datasets
the precision and recall goals are conflicting, increas-
ing Recall rates without disturbing the precision of the
minority class (target class) is a challenging issue.
4.2 Datasets
In order to show how the directional difference be-
tween principles axes of PCA affects the performance
of unbalanced datasets we have generated a synthetic
dataset. The synthetic dataset generated using multi-
variate normal distributions separately for each class:
Where p(x) is probability density function, x is a
M-dimensional vector of features, µ is feature vector
mean and Σ is a covariance matrix.
p(x) =
1
(2π)
d/2
|Σ|
1/2
exp[−
1
2
(x− µ)
t
Σ
−1
(x− µ)]
(5)
The two class distribution are generated with equal
mean µ = 0, M =10 uncorrelated features and with
unequal covariance matrices. The two classes co-
variance matrices are generated in such a way that
one class variance dominates the other class variance
and with different maximum variance directions, so
(Σ
1
> Σ
2
). Moreover, generated distribution contains
ω
1
= 2000 samples from majority class and ω
2
= 100
samples from minority class with imbalance ratio of
20%. Figure 2 depicts the structure of gaussians gen-
erated for the evaluation purpose, where ω
1
is the ma-
jority class distribution, ω
2
is minority class distribu-
tion and probability density of p(ω
1
) > p(ω
2
)). The
diagonal elements for two classes’ covariance matri-
ces for which the half diagonal elements are all zeros
are depicted as
Σ
1
d
ii
= [7,5,1, 3,4,0.1, 0,0.5, 0.01,0.01],
Σ
2
d
ii
= [0.5, 1,2,0.9, 0.7,0, 0.5,0.1,0.01, 0.01] where
i = 1,2,...,10.
The rest of the datasets Waveform, Mfeat-pixel,
Satimage and Musk are taken from UCI reposi-
tory (http://archive.ics.uci.edu/ml/., ) where as Bron-
chiolitis dataset is taken from UCD repository
(http://mlg.ucd.ie/datasets, ). Out of these considered
data sets Musk and Bronchiolitis datasets are meant
for two-class classification problem. The rest of the
datasets have more than two classes, and so they are
converted into binary class datasets by consideringthe
class with fewer samples as minority (positive) class
and rest of the samples as majority (negative)class, as
suggested in (Villalba and Cunningham, 2008). Table
2 shows the description of the datasets used for the
experiments.
KDIR 2010 - International Conference on Knowledge Discovery and Information Retrieval
240

Table 1: Datasets description.
Dataset #Min #Maj Imb.ratio #Attri
Simulated 100 2000 20 10
Waveform-1 1647 3353 2 21
Bronchiolitis 37 81 2.18 22
Mfeat-pixel-8 200 1800 9 240
Satimage-4 626 5809 9.27 36
Musk 1017 5581 5.48 166
Figure 2: Data from classes ω
1
and ω
2
where the probabil-
ity of p(ω
1
) > p(ω
2
).
The vertical line indicate class separation.
4.3 Experimental Results and
Discussion
Science decision tree classifier prone to be sensi-
tive for class imbalance problem, in this work it
is used as a baseline for evaluating the proposed
CPC SMOTE and other methods considered for com-
parative study. Decision tree classifier learned on pro-
posed CPC SMOTE is compared with same classifier
learned on original data, PCA subspace and on the
oversampled data obtained using SMOTE.
Table 3 describes the obtained results on consid-
ered methods in terms of accuracy and F-measure.
Here θ enables the directional difference between
two class’s first principal components which contains
maximum variance, in terms of cosine angle. If θ is
near to one means both classes principle components
are in same direction else they are in different direc-
tions. The best performance in terms of F-measure
that is obtained for that dataset is represented in bold.
As mentioned in section 4.1 for evaluating PCA we
considered that n number of PC’s which can give bet-
ter accuracy than the decision tree learned on origi-
nal dataset. The results on simulated dataset reported
95% of accuracy for all methods, but minority class F-
measure varied accordingly.PCAon simulated dataset
performed badly compared with all methods. Even
though the overall accuracy is 95%, but the minor-
ity class prediction in terms of F-measure is left with
0.This might be because of two reasons. (1) Due to
maximum variance coverage from majority class by
ignoring minority class maximum variance direction
which contributes less variance to the whole distri-
bution. Here θ=0.137 which shows larger directional
difference between two class PC’s. (2) The lack of
data causes unnecessary overlap in the PCA subspace.
This result clearly substantiates the effect of unbal-
ance data on PCA. Further, classifier learned on orig-
inal data attained 49% of F-measure. As expected
SMOTE showed large improvement of 46% over
original class F-measure. Proposed CPC SMOTE
yielded superior performanceamong all methods with
95.2% F-measure.
Comparing the results of real world datasets,
for Waveform CPC SMOTE yielded superior perfor-
mance of 92% in terms of both accuracy and F-
mesure than rest of the methods. The value θ = 0.781
clearly indicates the directional difference between
two clases’s PC directions. For this dataset PCA
showed an improvement of 9.6% F-measure on origi-
nal data. But compared to the accuracy, minority class
F-measure on PCA is less, showing the bias towards
majority class. In this dataset SMOTE did not per-
form well in improving the performance of original
dataset both in terms of F-measure and accuracy than
rest of the two methods. For Bronchiolitis and Mfeat-
pixel datasets PCA subspace did not improvethe clas-
sification accuracy of original data set. Moreover the
minority class prediction in terms of F-measure con-
siderably less than the original datasets as like sim-
ulated dataset. Corresponding directional differences
θ = 0.297,0.671 shows larger variation in principle
axis directions which substantiate the evidence for
lose performance in minority class data due to bias of
PCA subspace towards majority class.However, pro-
posed solution CPC SMOTE yields superior perfor-
mance on Bronchiolitis dataset with 80.2% minority
class prediction in terms of F-measure. But on Mfeat-
pixel data set SMOTE yielded superior performance
than CPC SMOTE. For this dataset compared with
PCA subspace on original data CPC SMOTE reduces
the bias caused by selecting the majority class maxi-
mum variance directions. Even though the two class’s
PC are in same direction with θ = 0.923 for Satimage
dataset CPC SMOTE is superior to rest of the two
methods with 94.7% minority class F-measure and
with 94.6% of overall accuracy. For this dataset PCA
on decision tree classifier showed consistency in im-
proving the performance on original data in terms of
F-measure with 5.7% as well as in total accuracy with
1.4% respectively. For this dataset, SMOTE exhib-
ited 38.3% improvement in minority class prediction
in terms of F-measure.
For Musk dataset where the directional differ-
A CLASS SPECIFIC DIMENSIONALITY REDUCTION FRAMEWORK FOR CLASS IMBALANCE PROBLEM: CPC
SMOTE
241

Table 2: Comparison of CPC SMOTE(C S) performance with Original Data(OD), PCA, SMOTE(S) in terms of accuracy and
minority class F-measure over decision tree classifier.
Dataset θ Accuracy% F-measure% Selected Features
OD PCA S C S OD PCA S C S PCA C S
Simulated 0.137 95 95 95 95 49.1 0 95 95.2 4 8
Waveform-1 0.781 85 91 88 92 77.4 87 88 92.2 5 10
Bronchiolitis 0.297 72 65 79.3 78.2 54.3 42.3 79.2 80.2 7 14
Mfeat-pixel-8 0.671 93.6 88.3 96.7 93.5 68.8 32.8 96.9 94 40 80
Satimage-4 0.923 91.8 93.2 94.4 94.6 56.1 61.8 94 94.4 7 14
Musk 0.901 96.8 97 97.5 96.5 89.8 90.9 97.5 96.2 35 70
ence θ = 0.901 is SMOTE achieved superior perfor-
mance than rest of the methods with 97.5% minor-
ity class prediction and with over all accuracy 97.5.
CPC SMOTE stood in second position with 96.2%
F-measure and with 96.5% overall accuracy. PCA
showed 1% and 2% consistent improvement on mi-
nority class F-meaure and on overall accuracy. From
our experiments we observed that the overall accuracy
of the classifier learned on original unbalanced data
as well as on PCA subspace is biased towards major-
ity class when there is directional difference between
principle components of the two classes. Though
CPC SMOTE is computationally costlier than PCA,
it is effective in alleviating the bias caused by major-
ity class in PCA subspace and enables better minor-
ity class prediction. From the considered 6 datasets
CPC SMOTE outperformed on 4 datasets for which
the directional difference is high.
5 CONCLUSIONS
This paper proposed a class specific dimensional-
ity reduction and oversampling framework to com-
bat the class imbalance issue occurred in Principle
Component Analysis while selecting maximum vari-
ance directions. Proposed approach is compared
with classical PCA for dimensionality reduction and
SMOTE for oversampling in terms of accuracy and
F-measure. Experimental evidence showed that pro-
posed approach yields superior performance in terms
of dimensionality reduction and classification of un-
balanced data where the maximum variance predom-
inantly represents majority class.
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