quasi-uniform theta graph

Integrated Intelligent Research (IIR) International Journal of Computing Algorithm
Volume: 06 Issue: 02 December 2017 Page No.64-68
ISSN: 2278-2397
64
Stability of Indexed Microarray and Text Data
T.Velmurugan1
S.Deepa Lakshmi2
1
Associate Professor, PG and Research Department of Computer Science, D.G.Vaishnav College, Chennai, India.
2
Research Scholar, Bharathiar University, Coimbatore, India.
Email: velmurugan_dgvc@yahoo.co.in, deepa.dgvc@gmail.com
Abstract: The common challenge for machine learning and
data mining tasks is the curse of High Dimensionality.
Feature selection reduces the dimensionality by selecting the
relevant and optimal features from the huge dataset. In this
research work, a clustering and genetic algorithm based
feature selection (CLUST-GA-FS) is proposed that has three
stages namely irrelevant feature removal, redundant feature
removal, and optimal feature generation. The performance
of the feature selection algorithms are analyzed using the
parameters like classification accuracy, precision, recall and
error rate. Recently, an increasing attention is given to the
stability of feature selection algorithms which is an indicator
that requires that similar subsets of features are selected
every time the algorithm is executed on the same dataset.
This work analyzes the stability of the algorithm on four
publicly available dataset using stability measurements
Average normal hamming distance(ANHD), Dices
coefficient, Tanimoto distance, Jaccards index and
Kuncheva index.
Keywords:Stability measurements, Jaccard Index,
Kuncheva Index, Tanimoto Distance, Dice Coefficient.
I. INTRODUCTION
Data mining and machine learning tools were proposed to
automate pattern recognition and knowledge discovery
process. The growth of technology has led to exponential
growth of data with respect to dimensionality. Storing and
processing high dimensional data has become more
challenging[1]. Dimensionality reduction is a technique to
reduce the size of the dataset and can be categorized into
feature extraction and selection[2]. Feature extraction
projects the features into new space with low dimension.
Examples of feature extraction are principal component
analysis, linear discriminant analysis etc. Feature selection
aims to select a small subset of relevant features[3]. Feature
selection is classified into three methods namely filter
method, wrapper method and embedded method[4]. The
proposed algorithm CLUST-GA-FS is an embedded feature
selection method. Commonly used evaluation metrics
include classification accuracy, precision, recall and error
rate. The selection stability has drawn an increasing
attention recently.
The selection stability is desired characteristic for feature
selection algorithms. Selection stability is the sensitivity of a
feature selection algorithm to perturbation in the training
data[5]. Stability metrics are used to assess the stability of
multiple feature selection results. A feature selection
algorithm is considered as stable if the features selected are
consistent from multiple execution of the algorithm with the
same dataset[6]. Petr Somol et al. proposed a modified form
of Average Normalized Hamming Index and the family of
inter measures was introduced [7]. An extension to Rapid
Miner was proposed in which Least Angle Regression
algorithm was used. Kuncheva index was used to measure
the stability of MRMR and an ensemble version of MRMR
[8]. Suphakit et al. proposed a method for measuring the
similarity between words using Jaccard Coefficient which
was developed using Prolog language[9]. Yvan Saeys et al.
used multiple feature methods to produce more robust result
on Microarray datasets. Stability metrics Spearman
correlation and Jaccard index were used to determine the
robustness of the feature selectors[10]. A Graph based
feature selection method that assess the nodes using Eigen
Vector Centrality was proposed and the stability was
assessed using Kuncheva index[11]. Iman Kamkar et al.
proposed a new method that uses intrinsic graph structure of
the electronic medical records in which the graph structure
was exploited using Jaccard similarity[12]. David
Dernoncourt et al. introduced ATIPA stability measure
which is a modification of ATI that aims to avoid a bias on
the number of selected features [13]. Sarah et al. illustrated
the benefits of stability measures and showed that Pearson’s
correlation similarity measure is a generalization of
Kuncheva index[14]. Kehan Gao et al. used six filter feature
selection techniques , three sampling approaches and the
stability measure Consistency index on software
measurement data to study the impact of data sampling on
Stability of Feature Selection[15]. The work presented in
this paper analyses the different stability metrics and applies
to the proposed algorithm.
The paper is organized as follows: the various
stability measurements are explained in chapter 2, the
proposed algorithm CLUST-GA-FS is discussed in chapter
3. Chapter 4 discusses the stability of the proposed
algorithm.
II. STABILITY MEASUREMENTS
The Stability of the feature selection algorithm is the
sensitivity of the selection to the variation of the dataset.
The stability can be assessed by pair wise comparison
between the results. If the similarity is greater, the stability
is higher. There are three different representations of the
output of the feature selection algorithms namely indexing,
ranking and weighting.
A. Stability by Index
Let the dataset be X and the set of features be F. The
selected subset of features is represented as a vector of
indices that correspond to the features f ⊂ F. Let m be the
number of features in the dataset and k be the number of
features selected and k ≤ m. Stability measurement assesses
the amount of overlap between results.
Average Normal Hamming Distance(ANHD): ANHD
measures the amount of overlap between the features

ISSN: 2278-2397
65
selected on multiple execution of the algorithm using the
same dataset[6]. It represents the selected feature subset as
fij, where fik has a value 1 if the kth
feature was selected in
the ith
run and a value 0 if the kth
feature was not selected in
the ith
run. The stability between two set of features selected
by the algorithm fi and fj is given by
𝐻(𝑓𝑖, 𝑓𝑗) =
1
𝑚
∑ |𝑓𝑖𝑘 − 𝑓𝑗𝑘
𝑚
𝑘=1 | (1)
H value is in the interval [0,1], where 0 indicates more
stability and 1 means that it is not stable. As the value of m
is large for high dimensional data, the value of H is small
and it indicates a more stable algorithm. The property of
ANHD will mislead to higher stability when k<m where
majority of features are not selected. It cannot deal with
different size of selected features.
Dice’s Coefficient: Dice’s coefficient is a similarity
measure used to calculate the overlap between two set of
features[16]. It takes values between [0,1] where 0 means
no overlap and 1 means the two set of features are identical.
Dice’s coefficient is given by
𝐷𝑖𝑐𝑒(𝑓𝑖, 𝑓𝑗) =
2|𝑓𝑖∩𝑓𝑗|
|𝑓𝑖|+|𝑓𝑗|
(2)
Where fi and fj are the two feature subsets generated by the
feature selection algorithm.
Tanimoto Distance: Tanimoto measures the overlap
between two set of features and has value in the same range
as Dice[17]. It produces values in the range [0,1]. Tanimot
distance is given by
𝑇𝑎𝑛𝑖𝑚𝑜𝑡𝑜(𝑓𝑖, 𝑓𝑗) = 1 −
|𝑓𝑖|+|𝑓𝑗|−2|𝑓𝑖∩𝑓𝑗|
|𝑓𝑖|+|𝑓𝑗|−|𝑓𝑖∩𝑓𝑗|
(3)
Jaccard’s Index: Jaccard index known as intersection over
union is used for comparing the similarity of feature
subsets[18]. It produces values in the range [0,1]. The
advantage is that it can deal with sets of different
cardinalities and Dice, Tanimoto and Jaccard do not take the
dimensionality into account. Jaccard index is given by
𝐽𝑎𝑐𝑐𝑎𝑟𝑑(𝑓𝑖, 𝑓𝑗) =
|𝑓𝑖∩𝑓𝑗|
|𝑓𝑖∪𝑓𝑗|
(4)
Kuncheva Index KI: The drawback of the stability
measurements is that the overlap between the selected
feature subset is more due to chance as the cardinality of the
feature subset is larger[19]. Kuncheva index contains a
correction term to avoid intersection of the selected subset
of features by chance. It produces values in the range [-1,1]
where -1 means there is no intersection between the subsets
and 1 means the subsets are identical. Kuncheva index is
given by
𝐾𝐼(𝑓𝑖, 𝑓𝑗) =
|𝑓𝑖∩𝑓𝑗|.𝑚−𝑘2
𝑘(𝑚−𝑘)
(5)
B. Stability by Ranking
Stability by rank evaluates the correlation between the
ranking vectors. They can handle vectors with different
cardinality or vectors that have different set of features.
Some of the stability measurements are Spearman’s rank
correlation coefficient and Canberra distance[16]. Filter
methods can be assessed using rank based metrics and
Wrapper methods can be assessed using set based stability
metrics. In rank based metrics, there are two types
depending on whether the list obtained is full ranking or
partial ranking. For full ranking list, the stability is assessed
based on the rank of all features and for partial list, a
threshold is specified and the features greater than the
threshold are used for stability assessment. Spearman’s
correlation is the widely used full ranking metric and its
value ranges between -1 to +1 with 0 indicating no
correlation, +1 indicating positive correlation and -1
indicating negative correlation. Partial ranking metrics used
are Jaccard index and Kuncheva index.
C. Stability by Weighting
Stability by weight deals with the weight of the feature set.
Pearson’s correlation is the measurement that takes two set
of features and returns stability which is the correlation
between them. It cannot deal with different subset size. It
produces values in the range [-1,1] where 1 means the
feature subset are correlated, -1 means they are anti-
correlated and 0 means they are not correlated[16].
Table 1: Stability measurement Categories
Measures Stability by Different Size Range
ANHD Index Yes [1,0]
Dice Index Yes [0,1]
Tanimoto Index Yes [0,1]
Jaccard Index Yes [0,1]
Kuncheva Index Yes [-1,1]
Spearman Rank No [-1,1]
Canberra Rank No [0, ∞ ]
Pearson Weight No [-1,1]
III. CLUST-GA-FS ALGORITHM
The feature selection algorithm CLUST-GA-FS has three
components: irrelevant feature removal, redundant feature
removal, and optimal feature generation. The first
component removes the irrelevant features by using mutual
information, a filter method. The second component
removes the redundant features by choosing the
representatives from each cluster[20]. The genetic
algorithm is used as the third component to find the optimal
set of features. The irrelevant feature removal obtains
features relevant to the class by eliminating the features
which are irrelevant to the target class using mutual
information. Redundant feature removal removes features
that are redundant in 3 steps: Constructing a minimum
spanning tree from the relevant features, grouping the
features in the forest into clusters and selecting the
representative feature from each cluster. The set of a
representative feature from each cluster, the class variable
and the number of features desired is provided as input to a
genetic algorithm. Genetic algorithm (GA) is an adaptive
heuristic search that uses optimization techniques to find
true or approximate solutions based on the evolutionary
ideas of natural selection and genetics. GA begins with a

ISSN: 2278-2397
66
set of chromosomes called the population. New populations
are evolved by mutating solutions according to their fitness
value. The fitness function used in this proposed method is
based on the principle of min-redundancy and max-
relevance.
Proposed Algorithm: CLUST-GA-FS
Inputs: Data set S{f1,f2,….,fm C}, C-class, ϴ - threshold
value
Output: representative feature subset
//=====Part 1 : Irrelevant feature removal=======
Step 1: for each feature fi in the data set
compute MI(fi,C)
if MI(fi,C)<ϴ
remove the feature fi
end
end
relevant feature set F'={f'1,f'2,….,f'l}(l≤m).
//===== Part 2 : Removal of Redundant feature=====
Step 2: for each feature fi in F'
construct graph G=(V,E) with feature fi as
vertices
and MI(fi,fj) as edges
end
generate the minimum spanning tree of G
forest=minimum spanning tree of G
for each edge in forest
if MI(fi,fj )<MI(fi,C ) and MI(fi,fj
)<MI(fj,C)
remove edge from the forest
end
end
for each tree in the forest
find the maximum MI(fi,C)
select fi as the representative feature of the
cluster
end
representative feature set is F''={f'1,f'2,….,f'k}(k≤l).
//======= Part 3 : Generating Optimum set of features
using Genetic Algorithm======
Step 3: Input: Representative feature set F''={f'1,f'2,….,f'k},
Class C, desired no of features
Output: Optimal set of features
Max_gen=no.of generations desired
find the entropy of F''- Hf and C- HC and mutual
information between the features - MIff
and mutual information between the features and
class C- MIfC
generate the population consisting of the feature set
while generation is less than max_gen
find the fitness
function=𝑚𝑎𝑥𝑆 [
1
|𝐹′′|
∑ 𝑀𝐼(𝑓𝑖,𝑐)
𝑓𝑖∈𝐹′′
−
1
|𝑆|2
∑ 𝑀𝐼(𝑓𝑖,𝑓𝑗)
𝑓𝑖𝑓𝑗∈𝐹′′ ]
rearrange the population according to their
fitness values
create a new generation
if the chromosomes generated are identical
sel=population rearranged
end
end
sel=optimal set of features
Steps1 and 2 generate the relevant set of features and
remove the redundant feature. The representative set of
features is given as input to a genetic algorithm that
determines the optimal set of features.
IV. EXPERIMENTAL RESULTS
The proposed algorithm CLUST-GA-FS is executed on
microarray and text data. Sample sets are generated by
random sampling or l-fold cross validation from a given
dataset. To analyze the stability of the algorithm, the
algorithm is applied to each sample set and selects optimal
set of features. Similarity measures are applied to evaluate
the similarity between the features lists obtained. The
dataset used and the stability assessment are discussed
below.
A. Dataset Description
The implementation of CLUST-GA-FS algorithm was done
on four publicly available dataset. Three microarray
datasets and one text dataset are used in this work. The
number of features of the dataset varies from 1800 to 10000,
the instance also varies from 38 to 5000 and the number of
classes is 2.
Table 2: Data Set Description
S.
No.
Data set No. of
features
No. of
instances
No. of
Classes
Domain
1 Colon 2000 62 2 Microarray
2 Leukemia 7129 38 2 Microarray
3 Arcene 10001 200 2 Microarray
4 SMS
spam
1833 5574 2 Text
B. Stability Assessment
Sample dataset are obtained from the original dataset X by
using random sampling or l-fold cross validation and the
sample datasets are labeled X1 and X2. The algorithm is
executed on all the sample dataset and the optimal set of
features is generated. Let the feature list generated by the
proposed algorithm on the sample X1 be F1 and the feature
list generated by the proposed algorithm on the sample X2
be F2. Let S() be the measurement to assess the stability of
the algorithm. S(F1,F2) is the stability measure of the
algorithm. Since the output representation of the proposed
algorithm is an indexed set of optimal features, stability by
index measurements are used to assess the stability of the
algorithm. The Stability by index measurement used is
ANHD, Dice, Tanimoto, Jaccard and Kuncheva Index.
The assessment of stability of the proposed feature selection
algorithm CLUST-GA-FS is performed using the stability
metrics ANHD, Dice, Tanimoto, Jaccard and Kuncheva.
Different samples(X1 and X2) of the dataset(X) are obtained

ISSN: 2278-2397
67
using l-fold cross validation. The proposed algorithm is
executed on each sample of the dataset. The output of the
CLUST-GA-FS algorithm is an indexed list of optimal
features F1 and F2 respectively. The stability metric is
applied on the optimal features list obtained using two
samples of the dataset S(F1,F2).
Table 3 : Stability measurements
Dataset ANHD Dice Tanimot
o
Jaccar
d
Kunchev
a Index
Leukemi
a
0.0014 0.827 0.705 0.705 0.826
Colon 0.006 0.8 0.666 0.666 0.796
Arcene 0.0013
9
0.857
1
0.75 0.75 0.856
SMSSpa
m
0.0043
6
0.84 0.724 0.724 0.8377
Figure 1: ANHD and Dice Stability measurements
The Stability metric ANHD(F1,F2) produces values in the
range [0,1] where the value 0 indicates high stability and 1
indicates no stability. From the figure 1 a, it is found that
ANHD has produced stable results for all the datasets and
Arcene dataset in particular. The stability measurement of
all the datasets are close to the value 0 indicating high
stability. Dice(F1,F2) produces values in the range [0,1]
where 0 indicates no stability and 1 indicates high stability.
The stability measurement Dice produces high stability
values for all the datasets Leukemia,Colon, Arcene and
SMSSpam. A value of 0.8 indicates high stability and all
the datasets have a Dice stability close to 0.8 as shown in the
figure 1 b.
Figure 2: Tanimoto and Jaccard Stability measurements
Jaccard Stability measure deals with binary attributes and
Tanimoto is an extended version of Jaccard. The presence of
a feature in the optimal feature list has a value 1 and the
absence of a feature has a value 0. From figure 2, it is
evident that Tanimoto and Jaccard produce the same
similarity measure values for all the datasets. Tanimoto and
Jaccard produce values in the range [0,1] where 1 indicates
high stability. Tanimoto and Jaccard have produces high
stability for Leukemia, Arcene and SMSSpam datasets. The
stability measure produced for Colon dataset is
comaparatively less.
Figure 3: Kuncheva Index Stability measurements
Kuncheva index Stability measurement produces values in
the range [-1,1]. The higher the value, more stable is the
algorithm. The stability metric produced for all the datasets
varies between 0.79 and 0.85 which is closer to the value 1
indicating that the proposed algorithm has produced stable
optimal feature set.

ISSN: 2278-2397
68
V. CONCLUSION
The performance of an algorithm is analyzed using different
metrics classification accuracy, precision, error rate and
recall. In this research work, the quality of the feature set
obtained by the proposed algorithm is analyzed using
Stability measurements. The algorithm removes the
irrelevant features using mutual information, removes
redundant features by constructing a minimum spanning
tree, splitting the minimum spanning tree into clusters and
selecting a representative feature from each cluster. The
representative features are given to a genetic algorithm to
obtain an optimal set of features. Sample datasets are
obtained from the datasets and the Stability metrics ANHD,
Dice, Tanimoto, Jaccard and Kuncheva Index are applied to
the feature sets obtained using different sample datasets.
The Stability metrics have produced stable results for the
microarray and text datasets Leukemia, Colon, Arcene and
SMSSpam.
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