Graph-Based Feature Selection Approach for Molecular
Activity Prediction
In the construction of QSAR models for the prediction of molecular activity, feature selection is a common task aimed at improving the results and understanding of the problem. The selection of features allows elimination of irrelevant and redundant features, reduces the effect of dimensionality problems, and improves the generalization and interpretability of the models. In many feature selection applications, such as those based on ensembles of feature selectors, it is necessary to combine different selection processes. In this work, we evaluate the application of a new feature selection approach to the prediction of molecular activity, based on the construction of an undirected graph to combine base feature selectors. The experimental results demonstrate the efficiency of the graph-based method in terms of the classification performance, reduction, and redundancy compared to the standard voting method. The graph-based method can be extended to different feature selection algorithms and applied to other cheminformatics problems.
Introduction<!>Material and Methods<!>Data
Set Characteristics and Molecular Representation<!><!>Graph-Based Feature Selection
Method<!><!>Graph-Based Feature Selection
Method<!>Experimental Setup<!>Experimental Results<!><!>Experimental Results<!><!>Experimental Results<!><!>Experimental Results<!>Conclusions<!>Data and Software Availability<!>
<p>In the construction of quantitative structure–activity relationship
(QSAR) models based on classification or regression techniques, the
preprocessing step is a fundamental component to avoid the use of
data that yield an identical effect, no effect, or even a deceptive
effect.1 Feature selection is one of the
most common tasks used in this preprocessing step. The selection of
an optimal set of features from which a model can achieve maximum
performance is a nondeterministic polynomial problem (NP).</p><p>The
objective of feature selection is to eliminate, as much as
possible, the amount of irrelevant and/or redundant features to improve
the performance of the prediction algorithms, reducing the negative
effects related to high dimensionality, accelerating the learning
process, and improving the generalization and interpretability of
models. Feature selection methods can rank individual features according
to their importance (ranking methods) or evaluate complete sets of
features to select an optimal subset (feature subset selection methods).
This paper is only concerned with the latter.</p><p>From a taxonomic
point of view, feature selection methods are traditionally
divided into four categories: (i) filter methods, (ii) wrapper methods,
(iii) embedded methods, and (iv) hybrid methods.2 Filters methods select the features regardless of the algorithm
used in building the model. A large number of filter methods have
been described in the literature, and among the most used are the
following: information gain,3 gain ratio,4 minimum redundancy, maximum relevance,5 Chi-square,4 fast
correlation-based filter,6 correlation-based
feature selection,4 Fisher score,7 fast clustering-based feature selection (FAST),8 and Relief or ReliefF.9</p><p>Wrappers methods choose the optimal subset of features for
evaluating
the performance of the modeling algorithm as if it were a black box.
Wrappers require a higher computational cost compared to the filter
methods. Furthermore, the subsets of features are biased toward the
modeling algorithm used in the evaluation. For this reason, the use
of independent validation samples is necessary for the reliable estimation
of the error.</p><p>Embedded methods integrate the selection of features
within the
modeling algorithm, either as part of the predictive/descriptive method
or as an extended functionality, and thus, the selection of features
is accomplished during the execution of the modeling algorithm.</p><p>Hybrid methods combine the advantages of filter and wrapper methods.
Usually these methods initially apply a filter method to reduce the
number of features, obtaining in many cases several possible subsets.
A wrapper method is then used to obtain the best subset of features.</p><p>Previous literature on the construction of QSAR models shows that
the most used feature selection methods are the following: chi-square
(CS),10,11 gain ratio (GR),12,13 information gain (IG),14,15 unbalanced correlation
score (UCS),7 mutual information (MI),16,17 standard correlation score (Fisher score, FS),18,19 F-score (FS) base ranking,20−22 Shannon entropy (SE),23,24 recursive feature elimination (RFE),25−28 and the fast clustering-based
feature selection algorithm (FAST).29</p><p>Ensemble approaches, based on bagging and/or boosting, have been
proposed for feature selection.30 These
methods have two configurable components: the boosting scheme and
the base feature selection algorithm to be used. Ensembles of feature
selectors are constructed by repeatedly applying feature selection
algorithms and then combining their results. Ensembles of feature
selectors focused on overcoming class imbalance problems have also
been proposed.31</p><p>In the construction
of feature selection ensembles, the combination
of the results of the different base selectors is crucial.32 The set of methods for combining feature subset
selectors is usually limited to take into account the result of applying
each feature selector by storing in a vector the number of times that
each feature was selected; this vector is used to obtain the final
selection. The most straightforward methods are the intersection or
union of feature sets. However, both of them produce poor performance.
The intersection often returns very small sets and thus results in
poor performance. The union generally achieves better performance,
but with the drawback of almost negligible reduction. A more efficient
solution is to use a vote threshold to obtain sets of features based
on the classification performance30 or
subject to a data complexity measure.33</p><p>The main drawback of these three approaches is that they disregard
relationships between features considered by the individual application
of the feature selection. To solve this problem, in this work, we
use an approach34 based on a graph where
the nodes represent the features and the links represent the features
co-occurrence in the same use of the algorithm, rather than storing
the repeated application of different selectors of features as a vote
vector. In this way, the method considers how many times a feature
was selected and also considers the sets of features that are selected
together each time the algorithm is used.</p><p>The rest of this work
has been organized as follows: Section 2 describes
the data set characteristics and molecular
representation, the graph-based feature selection method, and the
experimental setup. Section 3 describes the
experimental results, and finally, Section 4 provides a summary of the conclusions of this work.</p><!><p>In this section, we discuss the methodology
used in this work,
the data set, and the algorithms used.</p><!><p>In our study,
the data were collected from different sources to
yield a total of 24 data sets previously used for the construction
of binary prediction models for different molecular targets. Each
molecule in the data sets was represented using GSFrag,35,36 which considers 1138 molecular fragments (247 GSFrag + 891 GSFragl),
with the fragments consisting of one or more disconnected components.
Each component considers, among other factors, paths of length n, cycles on m vertices, or paths (cycles)
with a number of attached chains of unit length.</p><p>In the construction
of QSAR models, the diversities of the data sets play fundamental
roles for the generalization of the models. Thus, models built from
small or homogeneous compound sets offer poor generalization capacity.37−39 Recently González-Medina et al.40 proposed a new approach to study the diversity of molecular databases
from different perspectives, including fingerprint-based diversity
and the diversity of physicochemical properties. In our work, we have
studied the diversity of the data sets from four perspectives: (i)
fingerprint-based diversity, (ii) diversity of physicochemical properties,
(iii) minority class ratio diversity, and (iv) data set size diversity
(number of compounds in the data set). To evaluate the fingerprint-based
diversity (FpSim), we used the Tanimoto similarity index calculated
from the topological fingerprint (Morgan/Circular Fingerprints, radius
= 2)41 for all pairs of molecules in each
of the data sets. The diversity of physicochemical properties was
calculated using two properties: the octanol/water partition coefficient
(ALogPS) and molecular weight (MW). Fingerprint and physicochemical
properties were calculated with the RDKit library.42</p><p>Table 1 summarizes
the characteristics of the data sets. The information shown in the
table includes a unique identifier for each data set, the number of
total molecules, the number of active/inactive elements (positive/negative
class), the coefficient of variations for ALogPS, MW, and FpSim, and
a description of the molecular pathway end point. The coefficient
of variation (CV) was defined as follows:1with σ being the standard deviation,
μ the mean, and Me the parameter under study (ALogPS, MW, FpSim).</p><p>In the Supporting Information, Figure S1a and b shows the distribution of physicochemical properties ALogPS
and MW in each data set, while Figure S1c shows the cumulative distribution function using the pairwise similarity
values of the compounds in each data set, where both representations
exhibit the diversity of the data sets used. The similarity cumulative
distribution function using the pairwise Tanimoto similarity with
the Morgan fingerprint shows pariwise similarity values lower than
0.5 in 80% of the cases, highlighting the structural diversity of
the molecules. Moreover, CV (FpSim) (Table 1) shows values equal to or less than 0.3
for a large majority of the data sets, indicating that the mean of
the pairwise fingerprint (Avg (FpSim)) is representative of the data
set. Considering the 24 data sets, the values of Avg (FpSim) range
from 0.31 to 0.53, which confirms the structural diversity mentioned
above.</p><p>In terms of molecular properties, the CV (ALogPS) and
CV (MW) values
show greater diversity in terms of ALogPS compared to MW. Moreover,
the selected data sets present a wide range in terms of the number
of compounds, with the minimum size of 70 and a maximum size of 19,737.
The balance between the classes (active/inactive) also presents great
diversity, with the percentage of minority classes within the range
from 50% (perfectly balanced) to 0.8% (highly unbalanced).</p><p>Figure 1 shows a
more comprehensive multifactorial representation of data set diversity,
which simultaneously represents the diversity of chemical data sets
by fingerprint (x axis), physicochemical properties
(y axis), minority class ratio (color), and data
set size (mark size).</p><!><p>Multifactorial data set diversity representation: fingerprint
(x axis), physicochemical properties (y axis),
minority class ratio (color), and data set size (mark size).</p><!><p>The feature selection ensemble approach used in this
paper is based
on boosting.30 In every feature selection
boosting step, the selection is usually recorded by casting a vote
for each selected feature. These votes are weighted when the corresponding
boosting algorithm uses weighted classifiers as members of the ensemble.
Once the Tr rounds are finished, a vector of votes is obtained that
records how many times every feature has been selected, and a final
selection is determined using that vector.30 This approach does not take into account the relationship among
the features.</p><p>The algorithm utilized in this work uses a different
approach based on an undirected graph in the ensemble construction.34 The first step of the algorithm consists of
the construction of an undirected graph that allows storing the results
of each feature selection process. For this, the nodes of the graph
are used to store the features, and the edges represent the concurrent
selection of the two features in the same application of the algorithm.
In the second step, this graph is used to select a group of features
from the selection, which is performed for every step of the ensemble
construction process.</p><p>Figure 2 shows the
graph-based feature selection algorithm. The first step stores the
feature selection results using an undirected graph G(V, E), representing the vertices V = (1, 2, ..., M), the features Φ
= ϕ1, ϕ2, ...ϕM, and the links representing the selection of the
two features simultaneously. For simplicity, we consider that the
vertex i coincides with the feature ϕi and is assigned the value vi. Moreover, a member (i, j) of the set E corresponds to a link between i and j with an e(i, j) value.</p><!><p>Graph-based feature selection
algorithm.</p><!><p>Once a feature selection method
is applied in the ensemble construction
(FAST method8 in this work), the vertices
corresponding to the selected features and the edges linking these
vertices are increased by one. If the boosting method is used for
feature selection,30,52 the votes used for the tth member of the ensemble can be weighted by a value Î±t; in this way, the value is increased by
Î±t instead of increasing by one.
Finally, the vertex values in the created graph reflect how many times
each feature was selected and the link values how many times two features
were selected together.</p><p>The method is based on the assumption
that the features that represent
the nonredundant information are those that are most frequently selected
together and not those that rarely occur. The second step of the algorithm
uses the created graph to select a set of features; for this purpose,
it is necessary to establish two thresholds: the first for the vertices
(τϕ) and the second for the edges (τϵ). Using these thresholds, it is possible to select
a set of features Φ′(τϕ, τϵ) by applying different strategies.</p><p>To select
a subset of Φ′(τϕ, τϵ) features, these two terms were evaluated
using reduction, r(Φ′(τϕ, τϵ)), and the performance in classification,
measured by Cohen's κ value, κ(Φ′(τϕ, τϵ)). Thus, considering m selected features from all M features
set, the reduction was measured as , and both metrics were combined using the
following equation2</p><p>The highest performance, J, is chosen by evaluating
all possible vertex and edge threshold pairs. As possible thresholds,
all the different values for vertices and edges or a fixed number
can be considered, and in this case, the values can be divided into
equal subintervals. Once the values for a pair (τϕ, τϵ) have been set, the feature selection
process proceeds in the following way: initially, for the vertex vi with a value greater than
or equal to τϕ, the features ϕi are selected. Then, for every pair of selected features
(ϕi, ϕj), if the corresponding edge, e(i, j), is below the edge threshold, e(i, j) < τϵ, feature ϕi is removed if vi < vj or feature ϕj is removed otherwise. With a vertex threshold of 0, the method
includes the particular case of a standard voting scheme in which
all voting combinations are considered.</p><p>Using this first strategy,
two variants of the algorithms are considered:
(i) the mT variant, where τϕ = 0 and τϵ ≠ 0, and (ii) the MmT variant, where τϕ ≠ 0 and τϵ ≠ 0.
These two methods are shown in the left-hand side of the algorithm
shown in Figure 2.</p><p>The second strategy of the algorithm (right-hand side in step 2
of the algorithm, Figure 2) is based on forming a chain of features using the following
procedure. First, the feature corresponding to the vertex with the
largest value is selected. Then, for these vertex edges, the largest
value above the given threshold τϵ is chosen,
and the corresponding feature is added to the set of selected features.
In this way, this feature becomes the new starting point, and the
process ends when all of the edges above the threshold from the current
last member of the chain are linked to already-selected vertices.
In this second strategy, two variants were also considered: mTC, for
τϕ = 0 and τϵ ≠
0, and MmTC, for τϕ ≠ 0 and τϵ ≠ 0.</p><!><p>The experiments
were performed following the protocol shown in Figure S2 of the Supporting Information. Each data set was
divided randomly into two disjoint sets: one to build the model and
the other to perform the external validation. Thus, following a repeated
double cross-validation procedure,53,54 five external
validation rounds (external loop in Figure S2) were completed. The feature selection process was conducted using
the subset of molecules that were used to build the model.</p><p>The
graph-based method was evaluated with three different well-known classifiers,
a decision tree (DT), a support vector machine (SVM), and a random
forest (RF). To set the hyperparameters of the classifiers in the
model's construction (inner loop in Figure S2), we used a 10-fold internal cross-validation process.</p><p>For each classifier, we identified the best hyperparameter values
from a set of possible values. For RF, we used a size of 100 trees
and the Gini impurity criterion to measure the quality of a split.
The nodes were expanded until all leaves were pure or until all leaves
contained less than two samples, and bootstrap samples were used for
building the trees. For SVMs, three parameters were set: the kernel
type, the C value, and for the Gaussian kernel, the
γ value. Thus, we tested a linear kernel with C ∈ {0.1, 1, 10} and a Gaussian kernel with C ∈ {0.1, 1, 10} and γ ∈ {0.0001, 0.001, 0.01,
0.1, 1, 10}. All 21 possible combinations were evaluated. For decision
trees, we used 1 and 10 trials with the option of softening the thresholds
and tested all four possible combinations.</p><p>The performance achieved
by a classifier was measured using the
geometric mean (G-Mean) of sensitivity and specificity,55 defined as3where TP, TN, FP, and FN
are the true positives,
true negatives, false positives, and false negatives, respectively.
The use of this metric is recommended for both balanced and unbalanced
data sets because it takes into account the uneven distribution of
class instances.</p><p>The reduction capacity for feature selection
methods can be defined
as follows4where m is the number of
selected features, and M denotes the total number
of features.</p><p>Redundancy was evaluated using two different metrics:
one based
on mutual information and the second based on the correlation.56 Mutual information can be defined as follows5where X and Y depict feature vector and class
vector, respectively, and P(·) represents probability.</p><p>Consider S to be a vector of a given set of features
and h a class variable. The redundancy based on mutual
information (MI) was measured as6where |S| is the number of
features in S.</p><p>Redundancy based on correlation
(AcRed) was evaluated by replacing
mutual information with a correlation coefficient.56 For this purpose, the mean (meanC) of the correlation coefficient
was calculated. Thus, AcRed was defined as follows7where fi, fj ∈ S∀i, j =
1, 2, ..., m, and Cor() is a correlation coefficient
function and abs() an absolute
value function.</p><p>To guarantee a rigorous comparison between the
graph-based feature
selection method and the standard method, it is necessary to use specially
designed statistical tests to evaluate multiple algorithms on multiple
data sets. To do this, it is first necessary to know whether there
is a significant difference among the methods. The Iman–Davenport
test is recommended to determine the existence of these statistical
differences. It is based on the χF2 Friedman test, which compares the average ranks (R) of k algorithms for N data sets, but it is more powerful.57</p><p>After applying the Iman–Davenport test, it is necessary
to apply some of the general procedures for controlling the family-wise
error in multiple testing. The Holm test57 is designed to compare in a stepwise manner the algorithm with the
best performance in terms of Friedman ranges with the rest of the
methods under study. The statistical test for comparing the ith and jth methods is defined as follows8</p><p>Using the normal
distribution table, the values of z were used to
find the corresponding probability. Step-down procedures
sequentially test the hypotheses in order of their significance; the
ordered p values were denoted by p1, p2..., such that p1 ≤ p2...
≤ pk–1.
The Holm method compares each pi with α/(k – 1). The step-down
procedure starts with the most significant p value.
If p1 is below α/(k – 1), the corresponding hypothesis is rejected, and we compare p2 with α/(k –
1). If the second hypothesis is rejected, the test proceeds to the
third and so on. When a null hypothesis cannot be rejected, all remaining
hypotheses are retained as well. For all tests, we used a significance
level α = 0.05.</p><p>To compare multiple algorithms, we used
the Nemenyi58 test. This test considers
the performances of
two algorithms to be significantly different if the corresponding
average Friedman's ranks differ by at least the critical difference.
The critical difference (CD) for N data sets and k algorithms was formulated as follows57,589</p><!><p>In order to assess the effectiveness of the proposed feature selection
approach in the construction of binary QSAR models for molecular activity
prediction, we performed different experiments with the three classifiers
mentioned above with respect to 24 data sets with different molecular
activities. As stated, the algorithm supports the use of different
feature selection methods. We used fast clustering-based feature selection
(FAST) in the tests.8 The standard voting
AdaBoost method (T) was used as a reference to make the comparisons.
The experimental results are included in Tables S1–S3 of the Supporting Information.</p><p>In this section,
we present a graphical representation (Figures 3–5) of the results for ease of presentation
and discussion. These figures include two types of representation:
the first (panel (a) in Figures 3–5) represents the average
values of four metrics G-Mean (y axis), AcRed (x
axis), MI (color), and R (mark size), while the second (panels (b)–(e)
in Figures 3–5) represents the performance of two metrics (G-Mean,
AcRed) for each data set.59 To construct
this 2D representation, the value of each axis represents the difference
of the graph-based methods (MmT, MmTC, mT, mTC) with respect to the
base method (T) for the same data set. In this way, the arrows pointing
downward-left represent the data sets for which the base T algorithm
outperformed our graph-based method for G-Mean and was worse in terms
of AcRed, the arrows pointing upward-left indicate that our graph-based
method improved the G-Mean and AcRed. The arrows pointing upward-right
show data sets for which our graph-based method improved the G-Mean
but had an inferior AcRed, and arrows pointing downward-right show
the data sets for which the base T algorithm outperformed our graph-based
method with respect to both G-Mean and AcRed. In the figures, the
values of the differences are represented as percentages.</p><!><p>Four metrics
results (a) and the moment diagrams (b–e) representing
the differences (G-Mean) of the proposals against the base method
(T) for the DT classifier.</p><p>Four metrics
results (a) and the moment diagrams (b–e) representing
the differences (G-Mean) of the proposals against the base method
(T)) for the RF classifier.</p><p>Four metrics
results (a) and the moment diagrams (b–e) representing
the differences (G-Mean) of the proposals against the base method
(T) for the SVM classifier.</p><!><p>Figure 3 shows the
results for the DT classifier. The results in terms of G-Mean (Figure 3a) showed a better
performance for the graph-based methods compared to the base method
T. This better behavior in terms of G-Mean was also shown by the number
of data sets for which the use of some of the graph-based methods
outperformed T. For example, with the application of MmT (Figure 3b), G-Mean was improved
in 17 of the 24 data sets evaluated. MmTC (Figure 3c) improved it in 19 of them. mT (Figure 3d) improved it in
20 of them, and mTC (Figure 3e) improved it in 21 of them.</p><p>Regarding redundancy,
the mean values obtained for MI and AcRed
also showed better results (lower values) for the graph-based method
compared to the base method (x axis and color scale
values in Figure 3a).
According to the distribution of the differences for each data set
in terms of AcRed, the best result was obtained with the application
of the methods MmTC, mT, and mTC, where 16 of the 24 data sets obtained
better results compared to the base method T. Moreover, for reduction
(R), the results were very similar for all of the methods.</p><p>Figure 4 shows the
results for the RF classifier. The overall performance for RF in terms
of G-Mean was better than that using DTs. Higher average values were
obtained, and the advantage of using the graph-based methods with
respect to the base method T was retained. In terms of redundancy
(MI, AcRed), values very similar to those achieved by DT were obtained,
with better performance for graph-based methods regarding T. For RF,
the distribution by data set achieved the best results for the mT
and mTC methods, which have surpassed the T method by 20 data sets
in terms of G-Mean and by 18 data sets in terms of AcRed.</p><p>For
the SVM classifier (Figure 5), although the values of G-Mean were lower than RF,
they showed similar global behavior to those obtained by both DT and
RF, showing that the G-Mean had a better performance with the use
of MmT, MmTC, mT, and mTC compared to the application of T. The values
of R, MI, and AcRed also behaved in a similar way to the results presented
for the DT and RF classifiers, observing a decrease in redundancy
(MI, AcRed) when the MmT, MmTC, mT, and mTC methods were applied.</p><p>Although the results presented so far show the benefits of using
the graph-based method compared to the standard method, to obtain
conclusive results, the benefits must be validated by means of statistical
tests. First, we tested global significant differences using Iman–Davenport.
If this test rejected the null hypothesis, we used the Holm procedure
to compare the best method with the rest of the methods and then the
Nemenyi test to perform a global comparison of all the methods.</p><p>Tables 2 and 3 show the results of these statistical tests, and Figures 6 and 7 show a graphical representation of these results in order
to facilitate comparisons.57 Holm graphs
show the best of the algorithms on the y axis and
use a bar graph to represent the p values and a line
graph to represent the thresholds. The Nemenyi graphs connect with
a horizontal line the groups of algorithms that were not significantly
different and show the critical difference in the upper left corner
of the graph.</p><!><p>The best method
according to the
Holm test is indicated by the thumbs up hand symbol.</p><p>The best method
according to the
Holm test is indicated by the thumbs up hand symbol.</p><p>Performance statistical test results representation
in terms of
G-Mean.</p><p>Reduction and redundancy statistical test results
representation.</p><!><p>As shown in Table 2 and Figure 6 for
the performance of classifiers in terms of G-Mean, the result obtained
for the Iman–Davenport test was very close to zero, demonstrating
the existence of a significant difference between the evaluated methods.
Moreover, as shown in Figure 6, for all classifiers, the methods selected as the best by
Holm test showed significant differences with respect to the base
method (T), with no significant differences shown with respect to
the rest of the compared methods. The methods selected as the best
were the following: mTC for DT, mT for RF, and MmTC for SVM.</p><p>The Nemenyi test confirmed these results, with the base method
(T) performing worst for all classifiers. However a different behavior
was observed for DT as the differences were not significant between
MmT and T, indicating significant differences of mTC and MmTC with
T. RF showed significant differences for all of the methods with respect
to T, and SVM did not achieve significant differences among all the
compared methods.</p><p>As shown in Table 3 and Figure 7, in
terms of reduction (R), the results of the Iman–Davenport test
showed significant differences. For redundancy, a significant difference
was obtained for AcRed but not for MI.</p><p>The best method in terms
of Friedman's ranks for R and MI
was the mTC method, and the best in terms of AcRed was mT. In all
cases, the base method T produces the worst results, with a significant
difference (below the threshold) in terms of R and AcRed.</p><p>The
results of the Nemenyi test confirmed the worst results obtained
by T in terms of R, MI, and AcRed. Moreover, in terms of R, the results
did not show a significant difference between the T and MmT methods,
with the best performances observed for the mTC and MmTC methods with
significant differences compared to T. In terms of AcRed, the best
performance was obtained for the mT method, with significant differences
with respect to T, and in terms of MI, the results did not show a
significant difference.</p><p>Finally, the experiments were extended
evaluating their application
to the prediction of toxicity. For this purpose, the benchmark proposed
in the Tox21 project60−62 was used. Figures 8 and 9 show the results of the
Nemenyi test in terms of performance and of reduction and redundancy,
respectively. The experimental results are included in Tables S5–S9 of the Supporting Information.</p><!><p>Performance
Nemenyi test results for TOX-21 benchmark in terms
of G-Mean.</p><p>Reduction and redundancy Nemenyi test results
for TOX-21 benchmark.</p><!><p>In terms of G-Mean (Figure 8), the best results
for the DT and RF classifiers were observed
for the MmTC, MmT, and mT methods, obtaining significant differences
for these methods with respect to the base method T. For the case
of the SVM classifier, the best results were obtained for MmTC and
MmT methods, with significant differences with the base method T.</p><p>As Figure 9 shows,
in terms of reduction (R), the proposals outperform the base method.
In terms of redundancy, the proposals outperform the base method in
terms of MI and AcRed. The best results were obtained for the MmT
and mT methods.</p><!><p>In this
work, we evaluated the application to the prediction of
molecular activity of a new feature selection approach, based on the
construction of an undirected graph to combine the base application
selectors to different features sets. In contrast to the standard
voting approach, the method not only considers the frequency which
with a feature is selected but also the relationship with other features.
Compared with the use of the standard voting method (T), the experimental
results for different scenarios that include the DT, RF, and SVM classifiers
showed the advantages of the graph-based methods mTC, MmTC, mT, and
MmT in terms of classifier performance. Among the graph-based methods,
the mTC method showed a better overall performance. One of the main
advantages of the graph-based method is that any standard feature
selection algorithm can be applied, thus opening new lines of research.
Furthermore, the same idea could be adapted to the instance selection
problem or the joint selection of features and instances for the construction
of QSAR models.</p><!><p>All
the data on which the conclusions of the work are based have
been exhaustively presented in the manuscript. Data sets DS1–DS24
used in the paper are included as Supporting Information. The toxicity
data sets can be download from the following Tox21 project link: https://tripod.nih.gov/tox21/challenge/. The source code under GNU General Public License v3.0 can be downloaded
from the following link: http://cib.uco.es/wp-content/uploads/2021/11/source.zip</p><!><p>Experimental results in
terms of R, MI, AcRed, G-Mean
for each data set (Tables S1–S9); distribution of physicochemical
properties ALogPS and MW in each data set (Figure S1a, b); cumulative
distribution function using the pairwise similarity values (Figure
S1c); and experimental setup (Figure S2)(PDF)</p><p>SDF files with the molecular representation
and activity
(class) values for data sets DS1–DS24 (ZIP)</p><p>ci1c01578_si_001.pdf</p><p>ci1c01578_si_002.zip</p><p>The authors declare no
competing financial interest.</p>