On the Improvement of Feature Selection Techniques: The Fitness Filter
Artur J. Ferreira
1,3 a
and M
ario A. T. Figueiredo
2,3 b
ISEL, Instituto Superior de Engenharia de Lisboa, Instituto Polit
ecnico de Lisboa, Portugal
IST, Instituto Superior T
ecnico, Universidade de Lisboa, Portugal
Instituto de Telecomunicac¸
oes, Lisboa, Portugal
Machine Learning, Feature Selection, Dimensionality Reduction, Relevance-Redundancy, Classification.
The need for feature selection (FS) techniques is central in many machine learning and pattern recognition
problems. FS is a vast research field and therefore we now have many FS techniques proposed in the liter-
ature, applied in the context of quite different problems. Some of these FS techniques follow the relevance-
redundancy (RR) framework to select the best subset of features. In this paper, we propose a supervised
filter FS technique, named as fitness filter, that follows the RR framework and uses data discretization. This
technique can be used directly on low or medium dimensional data or it can be applied as a post-processing
technique to other FS techniques. Specifically, when used as a post-processing technique, it further reduces
the dimensionality of the feature space found by common FS techniques and often improves the classification
The goal of feature selection (FS) can be stated as that
of finding the best subset of features for a given prob-
lem (Duda et al., 2001; Guyon et al., 2006; Guyon and
Elisseeff, 2003). The use of FS techniques mitigates
the effects of the “curse of dimensionality” (Bishop,
1995; Bishop, 2007) phenomenon and it improves the
accuracy of machine learning tasks.
In the past years, with the advent of big data and
as a consequence high-dimensional data being often
present for many problems, the use of FS techniques
is an important and active topic of research. In the
presence of high-dimensional data, the irrelevance of
many features and the redundancy among features is
often found. For many years, researchers in the ma-
chine learning and pattern recognition fields have car-
ried investigation efforts to develop adequate FS tech-
niques for many different problems. As a result from
these research efforts, we now have many FS tech-
niques available and suited for quite different prob-
lems, within research fields and applications such as
data mining, computer vision, and biomedical data.
Some of the existing FS techniques aim at finding
the relevant features, while discarding the irrelevant
and redundant ones. However, in some cases, the use
of one single FS technique on the input data yields
sub-optimal solutions, regarding the size of the re-
sulting subset and the remaining redundancy among
features. In many of these situations, this results
from non-optimal parametrization of the FS algorithm
used. It is also known from the literature that the use
of discretization of the data typically improves on the
performance of machine learning and data mining al-
gorithms. The use of feature discretization (FD) tech-
niques usually yields more compact feature subsets
with better performance, as compared to the use of
the original features (Garcia et al., 2013).
In this paper, we propose a supervised filter fea-
ture selection technique named as fitness filter (FF).
This technique can be used directly on low or medium
dimensional data or it can be applied as a post pro-
cessing technique after the use of another FS algo-
rithm. In both scenarios, FF attains adequate results
at reducing the dimensionality of the data.
The remainder of this paper is organized as fol-
lows. Section 2 overviews FS filter approaches with
the RR framework. The proposed FF technique is pre-
sented in Section 3 and evaluated in Section 4. Fi-
nally, Section 5 ends the paper with some concluding
remarks and directions for future work.
Ferreira, A. and Figueiredo, M.
On the Improvement of Feature Selection Techniques: The Fitness Filter.
DOI: 10.5220/0010396303650372
In Proceedings of the 10th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2021), pages 365-372
ISBN: 978-989-758-486-2
2021 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
The need for FS often arises in many machine learn-
ing problems. The use of a FS technique usually im-
proves the accuracy of a classifier learnt from data,
since it helps to mitigate the effects of the well-
known “curse of dimensionality”. By removing some
features from the original data, we have a speed-
up of the training time and an improvement of the
generalization ability of a classifier. Thus, the re-
search community has developed many methods to
perform FS. In Subsection 2.1, we review some as-
pects and taxonomy regarding FS techniques. The
contents of Subsection 2.2 refer to the relevance-
redundancy framework, which is the foundation of
our proposal in this paper. Subsection 2.3 briefly de-
scribes some successful supervised FS filters that re-
sort to the relevance-redundancy framework. Finally,
Subsection 2.4 briefly reviews the benefits of using
discretized data on FS methods.
2.1 Notation and Concepts
We start by introducing some notation. Let X =
,... ,x
} be a dataset, with n patterns/examples.
Each pattern x
is a d-dimensional vector, thus d de-
notes the number of features, that is, the dimension-
ality of the data. The dataset X can be seen as n × d
matrix, in which the rows hold the patterns and the
columns are the features, designated as X
. Let c de-
note the number of distinct class labels represented
by c
and y = {c
,. .. ,c
} is the class label vector. FS
techniques can be placed into one of four categories:
wrapper, embedded, filter, and hybrid. For recent sur-
veys on FS techniques, please see (Chandrashekar and
Sahin, 2014) and (Miao and Niu, 2016).
2.2 Relevance-Redundancy
The relevance-redundancy (RR) framework (Peng
et al., 2005; Yu and Liu, 2004; Ooi et al., 2005; Zhang
et al., 2006) is followed by some FS methods. Let S
be some subset of selected features and p(c|S) be the
conditional probability of the class given S. The con-
cepts of relevance and redundancy can be formalized
as follows (John et al., 1994):
a feature X
/ S is said to be relevant iff p(c|S) 6=
); otherwise, feature X
is considered as
a feature X
/ S is said to be redundant iff
,S) = p(c|S), and p(c|X
) 6= p(c|S
), with
being a subset of S.
Thus, a feature X
is considered relevant, if its con-
catenation to the current feature set S, changes the
conditional probability of the class given the result-
ing feature set. A feature is redundant, in the pres-
ence of others, when there exists a smaller subset of
features holding the same the conditional probability
of the class given the feature set. However, this fea-
ture can be added to a smaller subset, without degrad-
ing this conditional probability. A feature can become
redundant due to the existence of other relevant fea-
tures, which provide similar prediction power. In this
case, it is of no worth to add a redundant feature to
the existing subset of features.
The RR framework for FS aims to remove the
redundant features, while keeping the most relevant
ones, thus expecting to improve the prediction accu-
racy. In (Yu and Liu, 2003; Yu and Liu, 2004), it is
shown that feature relevance alone is insufficient for
a good performance of FS methods, when the dimen-
sionality of the data increases. Therefore, redundancy
analysis is also necessary. Figure 1 depicts the RR
framework approach with two key steps:
(1) compute the relevance of each feature (or a subset
of features) and sort the resulting values into a list
by decreasing order, keeping the most relevant;
(2) remove the redundant features from the list.
Thus, we check for redundancy among the most rele-
vant features. In high-dimensional data, features can
be categorized into four subsets (Yu and Liu, 2004),
as depicted in Figure 2. A FS technique should find
the subset composed by both the weakly relevant and
the non-redundant features (part III) and the strongly
relevant features (part IV).
Figure 1: The relevance-redundancy framework key steps
for feature selection, as proposed by (Yu and Liu, 2004).
Figure 2: Feature categorization, as proposed by (Yu and
Liu, 2004). The optimal subset of features is given by part
III + part IV. Notice the size proportion between the subsets.
ICPRAM 2021 - 10th International Conference on Pattern Recognition Applications and Methods
2.3 Supervised Filter Methods
In this Subsection, we briefly describe three success-
ful supervised filter FS techniques that follow the RR
framework, as depicted in Figure 1, and have been
proven effective for many machine learning problems.
The maximal relevance minimum redundancy
(MRMR) method (Peng et al., 2005) computes both
the redundancy between features and the relevance of
each feature. The relevance is measured by the mutual
information (MI) between each feature and the class
labels. The redundancy between pairs of features is
also computed by MI. For two discrete random vari-
ables, U and V , MI is defined as
MI(U ;V ) =
(i, j) log
(i, j)
( j)
in which p
(i, j) is the joint probability of U and V .
MI is non-negative, being zero only when U and V are
statistically independent (Cover and Thomas, 2006).
The fast correlation-based filter (FCBF) was pro-
posed in (Yu and Liu, 2003; Yu and Liu, 2004). The
algorithm computes the feature-class and the feature-
feature correlations. It starts by selecting a set of fea-
tures that is highly correlated to the class with a cor-
relation value above some threshold set by the user.
The features with higher correlation with the class are
called predominant, in the first step. This correlation
is assessed by the symmetrical uncertainty (SU) (Yu
and Liu, 2003) measure, defined as
SU(U,V ) =
2MI(U ;V )
H(U) + H(V )
, (2)
where H(.) denotes the Shannon entropy of the ran-
dom variable (Cover and Thomas, 2006). The SU is
zero for independent random variables and one for de-
terministically dependent random variables.
In the second step, a redundancy detection proce-
dure finds redundant features among the predominant
ones. The set of redundant features is further split in
order to remove the redundant ones and keep those
that are the most relevant to the class. In order to re-
move the redundant features, three heuristic criteria
are applied.
The relevance-redundancy feature selection
(RRFS) method (Ferreira and Figueiredo, 2012)
follows the RR framework. RRFS, first finds the
most relevant subset of features and then searches for
redundancy among some pairs of the most relevant
features. In the end, it keeps only features with high
relevance and low redundancy among themselves
(below some threshold, named as maximum sim-
ilarity, M
). RRFS uses a generic (unsupervised
or supervised) relevance measure and the abso-
lute cosine between feature vectors to assess the
2.4 The Benefits of Discrete Data
Many datasets have features with continuous (real)
values. The use of feature discretization (FD) tech-
niques aims to yield representations of each feature
that contain enough information for the subsequent
machine learning task, discarding minor fluctuations
that may be irrelevant (Garcia et al., 2013; Garcia
et al., 2016). Thus, the use of FD techniques aims at
finding compact and more adequate representations of
the data for learning purposes. The use of FD usually
leads to a set of features yielding both better accu-
racy and lower training time, as compared to the use
of the original features. It has been found that the
use of FD techniques, with or without a coupled FS
technique, may improve the results of many learning
methods (Witten et al., 2016).
Moreover, some machine learning and classifi-
cation algorithms can only deal with discrete fea-
tures, thus at a certain point a discretization proce-
dure is necessary in these cases, as a pre-processing
stage (Hemada and Lakshmi, 2013).
In this section, we present in detail the proposed fit-
ness filter (FF) method. Subsection 3.1 describes the
key ideas behind the proposal of the method. Sub-
section 3.2 presents the FF method in an algorithmic
style and finally Subsection 3.3 shows some analysis
on the fitness and redundancy values for two datasets.
3.1 The Key Ideas
The FF method is tailored to be applied in two differ-
ent scenarios:
(i) direct use on the input data - acting as a supervised
FS filter, suited for low and medium dimensional
datasets, with, say, d < 100;
(ii) as a post-processing technique that further re-
duces the size of the subset of features found by
(any) common FS method.
The method relies on the RR framework as depicted
in Figure 1, in the sense that it computes the fitness
of each feature as the sum of its relevance to the class
label vector, subtracted by the average redundancy of
the feature, in the presence of all the other features.
On the Improvement of Feature Selection Techniques: The Fitness Filter
The goodness of a feature X
is directly proportional
to the value of its fitness, given by
f (X
) = rel(X
| {z }
d 1
j=1 j6=i
| {z }
, (3)
(i) rel(X
,y) denotes the relevance of feature X
, re-
garding the class label vector y; rel(.) is a generic
relevance function;
(ii) red(X
) denotes the redundancy among the
pair of features X
and X
; red(.) is a generic re-
dundancy (similarity) function.
These functions are generic and thus we can chose
different ways to compute the relevance and the re-
dundancy. However, care must be taken on the dy-
namic range of the values produced by these mea-
sures. The rel(.) and red(.) functions should return
non-negative values on the same range, e.g. 0 to r,
for instance with r = 1. When using functions that
do not produce values in the same range, then it is
necessary to apply some normalization in order to as-
sure that both functions meet this requirement. For
instance, the MI and the SU measures presented in
equations (1) and (2), respectively, are adequate for
this purpose. Moreover, other quantities from infor-
mation theory can also be applied.
As described in Subsection 2.4, there are many
benefits of using discrete data for learning tasks.
Thus, the proposed FF method uses a supervised FD
= disc(X
,y), that discretizes individu-
ally each feature X
, given the class label vector y.
This is the first step of the algorithm and it aims to
attain an adequate representation of the data, before
the fitness is computed for each feature. There are
many adequate supervised FD algorithms in the liter-
ature (Garcia et al., 2013).
3.2 The Algorithm
The generic proposed technique is presented as Al-
gorithm 1. After computing the fitness of each fea-
ture, the algorithm keeps those with fitness above the
fit parameter. By assuring that the rel(.) and red(.)
functions output values in the same range, the mid-
point 0 may be a meaningful crossing point to assess
the goodness of each feature. A zero-fitness feature
means that the amount of relevance of that feature
equals the average redundancy of that feature with
all the features. Regarding the choice of the disc(.),
rel(.), and red(.) functions, we recommend the fol-
Algorithm 1: Fitness Filter (supervised).
Input: X, n × d matrix, n patterns of a d-dimensional
training set.
y, 1 × n class label vector.
disc(.), a discretizer function.
rel(.), a relevance function.
red(.), a redundancy function.
f it, the minimum fitness threshold.
Output: idx: mdimensional vector with the indexes of
the selected features.
1: Discretize each feature
= disc(X
,y), (columns of
X), for i {1, ..., d}, obtaining a discretized version
of the training data.
2: Compute the fitness of each feature f (
) (columns of
X), for i {1, ..., d}, using equation (3), with appro-
priate rel(.) and red(.) functions.
3: idx Keep the indexes of the features, for which
f (
) f it.
disc(.) - a supervised discretizer using measures
from statistical or information theory to compute
the discretization bins; for instance, the well-
known method by (Fayyad and Irani, 1993);
rel(.) and red(.) - a statistical measure of resem-
blance, such as correlation coefficients or indexes;
a measure from information theory such as MI or
SU, as defined in equations (1) and (2), respec-
3.3 Fitness and Redundancy Analysis
In order to get some insight into how the fitness and
redundancy values change for different datasets, Fig-
ure 3 shows the fitness values, computed as in (3) and
the red values using MI, for the Dermatology and
Lung datasets (described in Table 1). On the left-
hand-side, we have the fitness values for the d = 34
features, while on right-hand-side, we have the re-
dundancy values for all pairs of features, displayed
as a d × d image (the main diagonal corresponds to
the similarity of a feature with itself). We consider
MI for both rel and red functions. We have distinc-
tive fitness values, with all of the features exhibiting
positive values. On the redundancy analysis, we can
identify small groups of features as being more sim-
ilar between themselves and thus may be redundant
for a given learning task. For the Lung dataset, we ob-
serve a much more distinctive variation on the fitness
values, since in this case only m = 27 features have
a positive fitness value. The redundancy pattern (the
right-hand-side image) is also quite different from the
corresponding image on the Dermatology dataset.
ICPRAM 2021 - 10th International Conference on Pattern Recognition Applications and Methods
Figure 3: Fitness and redundancy of all the features: top - Dermatology dataset (d = 34); bottom - Lung dataset (d = 56).
In this section, we report the experimental evaluation
of the proposed method. Subsection 4.1 describes
the public domain datasets considered on the experi-
ments. The learning task and its assessment measures
are described in Subsection 4.2. On Subsection 4.3
and Subsection 4.4, we report some experimental re-
sults on standard datasets, on the two scenarios de-
scribed in Section 3.1.
4.1 Public Domain Datasets
Table 1 briefly describes the public domain bench-
mark datasets from the University of California
at Irvine (UCI) (Dua and Graff, 2019) and the
knowledge extraction based on evolutionary learning
(KEEL) repositories (Alcal
a-Fdez et al., 2011), which
were considered in our experiments. We chose several
well-known datasets with different kinds of data.
4.2 Learning Task and Assessment
As the learning task for the assessment of the pro-
posed FF method, we have considered the super-
vised classification scenario. We use the linear sup-
port vector machine (SVM) classifier, implemented
in the Waikato environment for knowledge analysis
(WEKA) tool (Frank et al., 2016). The classifica-
Table 1: UCI and KEEL datasets used in our experiments;
d, c, and n denote the number of features, classes, and pat-
terns, respectively.
Dataset Name d c n
Heart 13 2 270
Wine 13 3 178
Hepatitis 19 2 155
WBCD 30 2 569
Dermatology 34 6 358
Ionosphere 34 2 351
Lung 56 3 32
Sonar 60 2 208
Libras 90 15 360
On the Improvement of Feature Selection Techniques: The Fitness Filter
tion accuracy of the classifiers were evaluated using
a leave-one-out cross-validation (LOOCV) method-
ology. We have also considered the implementation
of the FS methods available at the Arizona State Uni-
versity (ASU) repository for FS (Zhao et al., 2010),
with their parameters set with the corresponding de-
fault values.
In our experiments, we have made the following
choice of functions for the FF method:
disc(.) - the supervised discretization method
by (Fayyad and Irani, 1993), named as informa-
tion entropy maximization (IEM), with its default
parameter values;
rel(.) and red(.) - the SU as defined in equa-
tion (2).
4.3 Direct Use on the Input Data
We start by analyzing the sensitivity of the FF method
with the changes on the fit parameter value in Fig-
ure 4. We display the test set error for the SVM clas-
sifier, using the LOOCV methodology, for different
values of the fit parameter. We compare the test er-
ror with the original baseline and the discretized (with
IEM) baseline dataset, for the SVM classifier.
For the Hepatitis dataset, we have that fit=-0.08 is
the optimal choice for the minimum test set error rate
and for fit values below 0.04, we attain lower test
error rate, as compared with the full set of features.
On the Dermatology dataset the optimal fit values are
0.04 and 0.05, to attain the lowest test set error rate.
On the Ionosphere dataset, it is not possible to achieve
lower test error rate, as compared with the full set of
features. However, using the chosen selected subsets
of features by FF for fit up to -0.01, the test error is
never worse than using the full set of features. For the
Sonar dataset, the optimal value for the fit parame-
ter is below -0.1. For all datasets, an excessive large
value of the fit parameter will lead to a reduced sub-
set size and as a consequence to a low test set error
We now consider the use of the FF method, as
compared to the MRMR, FCBF, and RRFS methods,
described in Section 2.3, to perform FS on low and
medium dimensional datasets. Thus, we perform a
comparison of these methods for the supervised FS
filter task. For each dataset, the FCBF method is used
first, with its default parameters, and it returns the
subset of features, with m features. Then, MRMR and
RRFS methods are applied using m as the size of the
subset of features to be selected, for a fair comparison.
Table 2 reports the experimental results for the SVM
classifier. The column entitled ‘Original’ means that
we consider all the features (the original dimensional-
ity of data), and ‘Original Q. means their discretized
(quantized) version with the IEM algorithm.
4.4 Post-processing
On the second set of experiments, we apply the FF
method after the use of the FCBF and RRFS methods
on the input data. We aim to check if the use of the
FF method further improves on the results of the first
FS method. Table 3 shows the results obtained by the
SVM classifier on the FCBF and RRFS methods with
and without improvement with FF. From these results,
we conclude that:
The discretization step improves the test error
rate, for most datasets.
The FF method after the FS filter never gets a
worst result than the first method. In some cases,
it also improves the results, like in the Heart
dataset (FCBF-FF provides the same test error
rate as FCBF, but using less features).
The development of feature selection techniques to
find adequate subsets of data adequate for machine
learning problems is an active research field, since
many problems in different domains require their use.
The relevance-redundancy framework is suited to ad-
dress the taxonomy of the feature subspaces and to
be used as a foundation to develop successful feature
selection methods and approaches. Along the years,
some feature selection methods based on this frame-
work have been proposed, aiming to keep the relevant
features, while discarding the irrelevant and redun-
dant ones. When using these methods, after the fea-
ture selection stage, there are still some redundancies
on the remaining subset of features, in some cases.
In this paper, we have proposed a new supervised
filter method based on the relevance-redundancy
framework, that can further reduce the dimensionality
of the data, after the use of a common feature selec-
tion method. The proposed method is also adequate
to be directly applied on the data, for low and medium
dimensional datasets, since the redundancy check part
of the algorithm has nearly quadratic complexity with
the dimensionality of the data.
The experimental results using public domain
datasets and standard methodologies have shown that
the proposed approach attains adequate results, being
competitive with state-of-the art methods. The pro-
posed approach is also able to successfully reduce the
data dimensionality.
ICPRAM 2021 - 10th International Conference on Pattern Recognition Applications and Methods
Figure 4: Test set error of the SVM classifier, using the LOOCV methodology, with the original, original IEM discretized
data, and FF on the Hepatitis, Dermatology, Ionosphere, and Sonar datasets.
Table 2: Average number of features per fold (m) and test set error rate (Err) for LOOCV using the SVM classifier, with the
MRMR, FCBF, RRFS, and FF methods for feature selection. We use M
= 0.8 for RRFS. ‘Original’ means the baseline data
(with d features) and ‘Original Q. is the baseline data quantized with IEM, with default parameters. The best test set error
rate is in bold face. In case of a tie, the best result is the one with less features.
MRMR FCBF RRFS FF (fit=0) FF (fit=-0.1)
Dataset, d Original Original Q. m Err m Err m Err m Err m Err
Heart, 13 17.04 16.30 6 18.15 6 14.81 5 14.44 7 15.56 12 15.56
Wine, 13 1.12 1.12 9 3.37 9 1.69 7 5.06 13 1.12 13 1.12
Hepatitis, 19 24.52 20.00 6 16.77 6 20.65 5 16.13 5 16.77 18 20.00
WBCD, 30 2.28 3.16 6 5.98 6 5.45 2 11.25 23 3.34 27 3.51
Dermatology, 34 2.51 2.51 13 25.98 13 7.26 5 18.44 34 2.51 34 2.51
Ionosphere, 34 11.97 11.11 4 17.38 4 18.80 1 19.94 26 12.54 33 11.11
Sonar, 60 22.60 18.75 9 28.37 9 29.33 8 29.81 22 20.19 46 21.15
Libras, 90 26.67 24.44 9 60.00 9 46.11 1 95.56 90 24.44 90 24.44
Table 3: Average number of features per fold (m and k) and test set error rate (Err) for LOOCV using the SVM classifier, with
the FCBF, RRFS, FCBF-FF, and RRFS-FF methods for feature selection. We use M
= 0.9 for RRFS and fit= -0.1 for FF.
The best test set error rate is in bold face. In case of a tie, the best result is the one with less features.
Dataset, d Original Original Q. m Err k Err m Err k Err
Heart, 13 17.04 16.30 6 14.81 4 14.81 12 14.81 12 14.81
Wine, 13 1.12 1.12 9 1.69 9 1.69 11 1.69 11 1.69
Hepatitis, 19 24.52 20.00 6 20.65 5 20.65 13 18.71 11 17.42
WBCD, 30 2.28 3.16 6 5.45 5 5.45 22 3.69 22 3.69
Dermatology, 34 2.51 2.51 13 7.26 13 7.26 26 3.63 26 3.63
Ionosphere, 34 11.97 11.11 4 18.80 3 19.09 28 13.11 28 13.11
Sonar, 60 22.60 18.75 9 29.33 9 29.33 53 18.27 41 20.67
Libras, 90 26.67 24.44 9 46.11 9 46.11 48 29.72 48 29.72
On the Improvement of Feature Selection Techniques: The Fitness Filter
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unsupervised problems as well as fine tuning of the
threshold used by the algorithm. We also intend to
devise a strategy to lower the time taken to compute
the redundancy between features, which is the most
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