Abstract
Background/Objectives: Cancer classification using microarray datasets
presents a significant challenge due to their extremely high
dimensionality. This complexity necessitates advanced optimization
methods for effective gene selection. Methods: This study introduces
and evaluates the Nuclear Reaction Optimization (NRO)—drawing
inspiration from nuclear fission and fusion—for identifying informative
gene subsets in six benchmark cancer microarray datasets. Employed as a
standalone approach without prior dimensionality reduction, NRO was
assessed using both Support Vector Machine (SVM) and k-Nearest
Neighbors (k-NN). Leave-One-Out Cross-Validation (LOOCV) was used to
rigorously evaluate classification accuracy and the relevance of the
selected genes. Results: Experimental results show that NRO achieved
high classification accuracy, particularly when used with SVM. In
select datasets, it outperformed several state-of-the-art optimization
algorithms. However, due to the absence of additional dimensionality
reduction techniques, the number of selected genes remains relatively
high. Comparative analysis with Harris Hawks Optimization (HHO),
Artificial Bee Colony (ABC), Particle Swarm Optimization (PSO), and
Firefly Algorithm (FFA) shows that while NRO delivers competitive
performance, it does not consistently outperform all methods across
datasets. Conclusions: The study concludes that NRO is a promising gene
selection approach, particularly effective in certain datasets, and
suggests that future work should explore hybrid models and feature
reduction techniques to further enhance its accuracy and efficiency.
Keywords: nuclear reaction optimization (NRO), gene selection, cancer
classification, microarray data, bioinformatics, optimization
algorithms
1. Introduction
Cancer remains a significant global health challenge, necessitating
advancements in diagnostic and treatment methodologies [[26]1]. The
advent of microarray technology has revolutionized molecular oncology,
enabling the simultaneous analysis of thousands of genes to identify
potential biomarkers for various cancers. This technological leap has
brought new opportunities for precision medicine [[27]2]. Still, these
high-dimensional datasets—often incorporating thousands of genes but
only a limited number of patient samples—can lead to complications,
including an elevated risk of overfitting, unwieldy computational
demands, and interpretive complexity.
A commonly used strategy to address these issues is feature (gene)
selection. The idea is to single out a smaller set of genes that
captures the essence of the data, thereby enhancing model accuracy,
reducing noise, and keeping the analysis computationally manageable
[[28]3]. Throughout this paper, the terms “feature” and “gene” refer to
the same concept, as each feature in the dataset represents the
expression level of a specific gene. Therefore, “feature selection” and
“gene selection” are used interchangeably. A variety of methods have
been developed for this task, including a growing list of bio-inspired
optimization algorithms. Particle Swarm Optimization (PSO) [[29]4] and
Harris Hawks Optimization (HHO) [[30]5] have demonstrated significant
capabilities in high-dimensional spaces. These methods have been
successfully applied to feature-selection tasks in cancer
classification, achieving high performance in identifying optimal gene
subsets.
This study is the first to explore the potential of the Nuclear
Reaction Optimization (NRO) algorithm [[31]6] for optimizing gene
selection in cancer microarray data. Introduced in 2019 as a general
optimization algorithm, NRO has been previously applied in gene
selection for cancer classification using RNA Sequencing (RNA-Seq) data
[[32]7]. However, its effectiveness with microarray data remains
unexamined. This research assessed the standalone capability of the NRO
algorithm to select gene subsets for cancer classification across six
benchmark microarray datasets.
Here, we focused solely on the inherent strengths of the NRO algorithm,
foregoing additional dimensionality-reduction techniques. By
dimensionality reduction, we refer to preprocessing methods—such as
filtering, statistical ranking, or feature extraction—that reduce the
number of genes before applying optimization algorithms. To provide a
clear baseline for NRO’s performance, we intentionally excluded prior
dimensionality reduction. This approach allowed us to evaluate NRO’s
capability in handling raw, high-dimensional microarray data without
external influence from filtering methods.
The selected gene subsets were subsequently evaluated using robust
machine learning classifiers, such as Support Vector Machines (SVMs)
and k-Nearest Neighbors (k-NNs), with a primary focus on classification
accuracy. The outcomes of this study were intended to position the NRO
algorithm as a formidable and reliable tool in bioinformatics feature
selection, thus facilitating the development of more effective and
interpretable diagnostic models.
The rest of this paper is organized as follows: [33]Section 2 provides
a detailed background on classification techniques and optimization
algorithms, with a focus on NRO. [34]Section 3 reviews related works in
the domain of gene selection for cancer classification. [35]Section 4
describes the materials and methods used in this study, including the
datasets, preprocessing techniques, and the implementation of the NRO
algorithm. [36]Section 5 presents the results and analysis, comparing
NRO’s performance with those of other optimization methods. Finally,
[37]Section 6 concludes the paper with key findings, limitations, and
directions for future research.
2. Background
2.1. Classification Techniques in Cancer Diagnosis
In the field of bioinformatics, particularly in cancer classification,
specific classifiers have proven to be especially effective due to
their adaptability and performance with complex, high-dimensional
datasets like those generated from microarray gene-expression profiles.
Here, we explore two widely used classifiers: Support Vector Machine
(SVM) and k-Nearest Neighbors (k-NNs), which have distinct
characteristics making them suitable for cancer detection and
classification.
2.1.1. Support Vector Machine (SVM)
The SVM is particularly effective for microarray data because of its
ability to work well in high-dimensional spaces and its robustness to
overfitting with small sample sizes. It identifies an optimal
hyperplane to separate data classes, maximizing classification margins,
which enhances generalization [[38]8]. The SVM has been extensively
used in cancer classification tasks to distinguish between cancer types
or between cancerous and non-cancerous samples based on gene-expression
profiles. Its capability to manage noisy, complex data makes it a
strong benchmark for evaluating gene-selection methods [[39]9]. The SVM
is highly effective in binary classification problems and can be
adapted for multiclass scenarios using strategies like one-vs.-all
(OVA) or one-vs.-one (OVO) [[40]10].
2.1.2. k-Nearest Neighbors (k-NNs)
The k-Nearest Neighbors (k-NNs) is a non-parametric classifier that
ranks among the most straightforward and effective methods for medical
diagnosis, including cancer classification. This method operates based
on a simple principle: it classifies each new case by analyzing the
most common class among its nearest neighbors, determined by a chosen
distance metric, typically Euclidean [[41]11]. In gene selection, the
k-NNs is valuable for assessing whether the selected genes preserve
meaningful data structures and class separability. Its sensitivity to
local data distributions allows it to highlight the effectiveness of
gene subsets in capturing disease-specific patterns in cancer datasets
[[42]12,[43]13].
2.2. Nuclear Reaction Optimization (NRO)
The Nuclear Reaction Optimization (NRO) algorithm is a physics-inspired
metaheuristic designed to solve optimization problems by mimicking the
natural processes of nuclear fission and nuclear fusion [[44]6]. These
processes simulate how nuclei split or merge to release energy. A
nuclide splitting into two or more smaller nuclides is known as nuclear
fission, as shown is [45]Figure 1. The opposite is nuclear fusion. A
bigger nuclide is created when two or more nuclides fuse together, as
shown in [46]Figure 2. Extremely large amounts of energy could be
released by nuclear fission or fusion. In this context, it translates
to exploring the solution space for optimal solutions. NRO employs a
dynamic balance between exploration (searching for new potential
solutions) and exploitation (refining existing solutions).
Figure 1.
[47]Figure 1
[48]Open in a new tab
Nuclear fission process, illustrating the inspiration for solution
diversification in the NRO algorithm.
Figure 2.
[49]Figure 2
[50]Open in a new tab
Nuclear fusion process, illustrating the inspiration for solution
refinement in the NRO algorithm.
2.2.1. Nuclear Fission Process
The nuclear fission process in NRO creates new candidate solutions by
splitting an existing solution into smaller parts. This helps the
algorithm explore different areas of the solution space and avoid
becoming stuck in one region. The process uses the following equation:
[MATH:
XiFiGau<
/mi>ssian(<
mrow>Xbest,σ1)+(ra
ndn·Xbest
−Pnes ·
Nei), if rand
≤ Pβ
,Gaussian(Xi,
σ2)+(rand<
/mi>n ·
Xbest<
/mi>−Pn
ee ·<
/mo> Nei), if
rand>Pβ,
:MATH]
(1)
Here,
*
[MATH:
Xi
Fi: :MATH]
the new solution generated during fission.
*
[MATH:
Xbe
st: :MATH]
the best solution found so far.
*
[MATH: Gaussian(X,σ
): :MATH]
a random value generated around
[MATH: X :MATH]
, with
[MATH: σ :MATH]
defining the spread.
*
[MATH:
σ1 , σ2 :MATH]
step sizes controlling the exploration range (see Equations (3) and
(4) below).
*
[MATH:
randn: :MATH]
random number introducing variability in the solution.
*
[MATH:
Pnes ,
Pnee: :MATH]
mutation factors determining the scale of adjustments for subaltern
and essential fission products, respectively.
*
[MATH:
Nei
: :MATH]
heated neutron, calculated as
[MATH:
Nei
=XiX
j, :MATH]
where
[MATH:
Xi and X<
mrow>j :MATH]
are two random solutions.
*
[MATH:
Pβ: :MATH]
the probability governing whether subaltern or essential fission
products are produced.
This approach ensures that some new solutions are close to the best one
[MATH:
Xbe
st :MATH]
(local search) while others are farther away (global search), providing
a good mix of exploration and refinement.
Step Size Adjustment in Fission
The effectiveness of the fission process depends on the step size,
which determines how far new solutions deviate from existing ones. The
step sizes were calculated as follows:
[MATH:
σ1
=loggg⋅Xi−Xbest,
:MATH]
(2)
[MATH:
σ2
=loggg⋅Xr−Xbest.
:MATH]
(3)
Here,
*
[MATH: g :MATH]
: current generation number; the term
[MATH: loggg :MATH]
ensures that step sizes decrease as iterations progress.
*
[MATH: Xi<
/mrow>−Xbest:<
/mo> :MATH]
distance between the current solution and the best-known solution.
*
[MATH: Xr<
/mrow>−Xbest:<
/mo> :MATH]
distance between a random solution and the best-known solution.
At first, large step sizes help explore the solution space. Over time,
the step sizes shrink, allowing the algorithm to refine the best
solutions and reach an optimal result.
Mutation Factors in Fission
Mutation factors, defined as follows:
[MATH: Pnes=roundrand+
1, :MATH]
(4)
[MATH: Pnee=roundrand+
2, :MATH]
(5)
control the magnitude of adjustments applied to fission products. These
factors ensure variability and are critical for balancing exploration
and exploitation:
*
[MATH:
rand:
:MATH]
a random number uniformly distributed between 0 and 1.
* The integer rounding ensures discrete adjustment levels for the
mutation process.
These factors are incorporated into the fission equation to refine or
diversify the search space, avoiding premature convergence and ensuring
effective optimization.
2.2.2. Nuclear Fusion Process
The nuclear fusion process refines solutions by combining promising
candidates. It consists of two sub-phases: ionization and fusion.
Ionization Step
In this step, solutions are adjusted based on the differences between
randomly selected ones. The formula is as follows:
[MATH:
Xi,<
mi>dIonX<
/mrow>r1,dF<
/mi>i+ran
d · (X<
/mi>r2,d<
mi>Fi−X
i,dFi
), if rand
≤ 0.5,
Xr1,dFi−rand · (Xr2,dFi−Xi,d<
/mi>Fi) if rand>0.5, :MATH]
(6)
Here,
*
[MATH:
Xr1,dFi, Xr2,dFi :MATH]
: components of two randomly selected fission solutions.
*
[MATH:
Xi,dFi
: :MATH]
current solution.
*
[MATH:
rand: :MATH]
random value for diversity.
This step prevents the algorithm from converging too early by adding
diversity. If the two selected solutions are too similar, Levy flight
[[51]14] is applied to make bigger changes and avoid stagnation:
[MATH:
Xi<
mo>,dIon=
Xi,dFi+α⊗Levyβd⋅Xi,dFi−Xbest,d<
mi
mathvariant="italic">Fi, :MATH]
(7)
Here,
*
[MATH: α :MATH]
: a scaling factor controlling the magnitude of jumps.
*
[MATH: Levyβ: :MATH]
heavy-tailed random step size, introducing both small and large
adjustments.
*
[MATH: ⊗: :MATH]
indicates element-wise multiplication.
*
[MATH: Xbest,d
Fi :MATH]
: best-known solution in the
[MATH:
dth
:MATH]
dimension.
Levy flight is specifically applied when the difference term
[MATH:
Xr2<
mo>,dFi−Xi,d
Fi
:MATH]
approaches zero, which could lead to stagnation in the search process.
By introducing varying step sizes, this adjustment ensures that the
algorithm explores new areas of the solution space, avoiding local
optima.
Fusion Step
The fusion step combines the strengths of promising solutions to refine
the search. It is expressed as follows:
[MATH:
XiFu=<
mi>XiIon+ra
nd⋅Xr1Ion−Xbest+
rand⋅Xr2Ion−Xbest,
:MATH]
(8)
Here,
*
[MATH:
Xi
Fu :MATH]
: refined solution after fusion.
*
[MATH: Xbest: :MATH]
best-known solution guiding the search.
*
[MATH:
Xr1Ion, <
msubsup>Xr2Ion: :MATH]
ionized solutions selected for comparison.
*
[MATH:
rand: :MATH]
random value for diversity.
When solutions become too similar
[MATH:
Xr1<
/mrow>Ion=
Xr2Ion :MATH]
, the algorithm may stagnate. To avoid this, Levy flight introduces
random jumps, helping the algorithm explore new areas and escape local
optima. This is governed by the following equation:
[MATH:
XiFu=<
mi>XiIon+α⊗
Levyβ⊗XiIon−
XbestIon, :MATH]
(9)
Here, Levy flight ensures that the fused solutions are not constrained
to a narrow region of the search space, especially in situations in
which the algorithm might otherwise fail to differentiate between
nearly identical candidates.
3. Related Works
Gene selection for cancer classification is critical for addressing the
challenges of high-dimensional microarray datasets. Numerous
bio-inspired optimization algorithms have been proposed to enhance
dimensionality reduction and improve classification accuracy, some of
which apply dimensionality reduction before optimization [[52]15] while
others rely solely on the optimization algorithm without any prior
filtering. This section reviews recent advancements in
optimization-based gene-selection approaches, focusing on their
methodologies and relevance to this study.
AlMazrua and AlShamlan [[53]16] proposed Harris Hawks Optimization
(HHO) for gene selection combined with SVM and k-NN classifiers to
tackle the dimensionality of microarray datasets. Their approach
integrated redundancy analysis and relevance scoring for preprocessing,
followed by HHO-based feature selection. Evaluated on six datasets, the
method achieved superior classification accuracy and reduced gene
subsets compared with traditional algorithms, showcasing the efficiency
of HHO in bioinformatics applications.
Alweshah et al. [[54]17] introduced the Monarch Butterfly Optimization
(MBO) algorithm as a wrapper-based feature-selection method, utilizing
the k-nearest neighbor (k-NN) classifier to enhance classification
accuracy and reduce computational complexity. Evaluated across 18
benchmark datasets, the MBO approach demonstrated superior performance
compared with other metaheuristic algorithms, achieving an average
classification accuracy of 93% while significantly reducing the number
of selected features. This study highlights MBO’s effectiveness in
balancing global and local search capabilities for feature-selection
tasks.
Almugren and Alshamlan [[55]18] proposed FF-SVM, a wrapper-based
gene-selection algorithm combining the Firefly Algorithm (FFA) with a
Support Vector Machine (SVM) classifier. The method aims to optimize
cancer classification by identifying the most informative genes from
high-dimensional microarray datasets. Using FFA for feature selection,
followed by SVM classification with Leave-One-Out Cross-Validation, the
algorithm achieved high classification accuracy with a minimal subset
of genes across five benchmark datasets. Comparative experiments
demonstrated the FF-SVM’s superior performance over several
state-of-the-art methods in terms of accuracy and dimensionality
reduction, highlighting its effectiveness in bioinformatics
applications.
Nssibi et al. [[56]19] introduced a hybrid optimization approach called
iBABC-CGO, which combines an island-based Artificial Bee Colony (iABC)
algorithm with Chaos Game Optimization (CGO) for gene selection. The
method addresses the challenges of high-dimensional microarray datasets
by using a binary representation to identify informative genes while
maintaining classification accuracy. The hybrid algorithm leverages CGO
principles to improve convergence and avoid local optima during the
migration process, and its binary version, iBABC-CGO, ensures efficient
exploration and exploitation. Experimental results on 15 biological
datasets demonstrated the approach’s superior performance compared with
state-of-the-art methods, highlighting its ability to achieve high
accuracy with minimal feature subsets.
AlMazrua and AlShamlan [[57]20] proposed a novel feature-selection
approach for cancer classification using the Gray Wolf Optimizer (GWO).
This bio-inspired optimization algorithm mimics the leadership
hierarchy and cooperative hunting behavior of gray wolves to
effectively explore and exploit high-dimensional datasets. Their
framework utilized the GWO to identify the most significant features,
achieving a balance between classification accuracy and dimensionality
reduction. Their experimental results demonstrated that the method
successfully reduced the number of selected features while maintaining
high classification performance, highlighting its potential for
bioinformatics and medical diagnostic applications.
Overall, the reviewed studies demonstrate the effectiveness of
bio-inspired optimization algorithms in gene selection for cancer
classification. Methods such as HHO, MBO, FFA, iBABC-CGO, and GWO have
achieved high classification accuracy while selecting minimal and
informative gene subsets. This study evaluates the Nuclear Reaction
Optimization (NRO) algorithm as a novel contribution to further enhance
gene selection and classification performance.
4. Materials and Methods
This study explores the application of the Nuclear Reaction
Optimization (NRO) algorithm for gene selection in cancer
classification. The methodology consists of dataset preprocessing,
optimization using NRO, evaluation through machine learning
classifiers, and fitness assessment. The implementation was carried out
using Python (version 3.x), with numpy and pandas for preprocessing
gene-expression data, scipy.io.arff for handling microarray datasets,
and sklearn for feature scaling and classification. The NRO algorithm
was implemented by coding its mathematical foundations, including
nuclear fission and fusion processes, Lévy flight-based step size
adjustments, and mutation mechanisms for optimal gene selection.
[58]Figure 3 shows the flowchart summarizing the entire methodology,
from data preprocessing and optimization to fitness evaluation and
termination.
Figure 3.
[59]Figure 3
[60]Open in a new tab
Flowchart of the methodology.
4.1. Dataset and Preprocessing
Using six well-known binary and multiclass microarray cancer datasets
that we obtained from [61]http://www.gems-system.org/ (accessed on 15
February 2025), we assessed the overall effectiveness of the
gene-selection techniques. In the discipline of bioinformatics, these
datasets are frequently used to compare the effectiveness of
gene-selection techniques. SRBCT [[62]21], Lymphoma [[63]22], and
Leukemia2 [[64]23] are multiclass microarray datasets, whereas the
binary-class microarray datasets are Lung [[65]24], Colon [[66]25], and
Leukemia1 [[67]25,[68]26]. We provide a thorough description of these
six benchmark microarray gene-expression datasets in [69]Table 1,
including information on the number of classes, samples, and genes.
Table 1.
Statistics of microarray cancer datasets.
Microarray Dataset Classes Samples Total Genes
Colon [[70]25] 2 62 2000
Leukemia1 [[71]26] 2 72 7129
Lung [[72]24] 2 96 7129
SRBCT [[73]21] 4 83 2308
Lymphoma [[74]22] 3 62 4026
Leukemia2 [[75]23] 3 72 7129
[76]Open in a new tab
To prepare the datasets for gene selection and classification, we
applied a series of preprocessing techniques commonly used in
bioinformatics to ensure the datasets were clean, consistent, and
suitable for machine learning algorithms. Firstly, any missing values
in the datasets were handled by replacing them with the mean value of
the respective gene’s expression levels using mean imputation. In our
analysis, missing values were found only in the Lymphoma dataset,
accounting for 4.91% of its total data, affecting 2796 genes. Given
this relatively low percentage, mean imputation was applied to maintain
data integrity without significantly impacting the classification
performance. To standardize the features and eliminate the effects of
varying scales across genes, we applied Z-score normalization,
transforming the expression levels of each gene to have a mean of 0 and
a standard deviation of 1. Additionally, for multiclass datasets, we
employed label encoding to convert categorical class labels into a
numerical format, making them compatible with classification
algorithms. These preprocessing steps ensured that the datasets were
optimally prepared for the subsequent feature-selection and
classification phases.
4.2. Apply Nuclear Reaction Optimization (NRO) Algorithm
The Nuclear Reaction Optimization (NRO) algorithm was adapted in this
study specifically for gene selection in cancer classification using
microarray data. Microarray datasets are characterized by their high
dimensionality, with thousands of genes but relatively few samples,
posing challenges for classification and optimization algorithms. NRO
addresses these challenges by iteratively refining subsets of genes,
balancing the need for compact feature sets with high classification
accuracy.
Each candidate solution in the NRO algorithm is a binary vector, where
1 indicates that a gene is selected, and 0 indicates exclusion. The
algorithm initialized a population of 500 such solutions, each
representing a potential subset of genes. We chose a population size of
500 to ensure a diverse set of candidate solutions, which helps in
effectively exploring the high-dimensional gene space while maintaining
computational efficiency. A smaller population might limit diversity
and lead to premature convergence, whereas a significantly larger one
would increase computational cost without substantial accuracy gains.
Over a maximum of 30 generations, the population evolved through
nuclear fission and fusion processes. This number of generations was
selected as a balance between exploration and convergence, allowing the
algorithm sufficient iterations to optimize gene selection while
avoiding excessive computational overhead. In our empirical tests,
increasing the generations beyond 30 provided no improvements in
classification accuracy.
In the context of microarray data, the nuclear fission phase introduced
diversity by splitting solutions into smaller fragments, which
correspond to alternative gene subsets. Equation (1) governs this
process, generating new solutions by mutating the existing ones. Step
sizes, dynamically adjusted using Equations (2) and (3), control the
degree of variability, enabling broad exploration in early generations
to identify different sets of candidate genes. As the algorithm
progresses, these step sizes decrease, focusing on the refinement of
the most promising gene subsets. Mutation factors from Equations (4)
and (5) further adjust the solutions, ensuring that the algorithm does
not converge prematurely and continues to explore the potential of
unselected genes.
The nuclear fusion phase refines gene subsets by combining promising
solutions. During the ionization step, solutions are adjusted using
differences between randomly selected subsets, following Equation (6).
This step ensures the algorithm can evaluate combinations of genes that
might not have been initially included in a single solution. For
microarray data, in which many genes have weak but complementary
contributions to classification, this step is crucial. When the
differences between the selected solutions are small, Lévy flight from
Equation (7) is applied to introduce large jumps, allowing the
algorithm to escape local optima and explore new subsets of genes.
In the fusion step, refined subsets are combined using Equation (8),
producing solutions that integrate information from the best-performing
gene subsets of previous generations. Lévy flight, applied when
necessary through Equation (9), ensures that the fusion phase continues
to explore new regions of the search space rather than stagnating on
similar subsets. This capability was critical for handling the
complexity of microarray data, in which interactions among genes can
lead to nonlinear relationships affecting the classification accuracy.
The algorithm evaluated each solution using the fitness function
described in [77]Section 4.4, which combines LOOCV-based classification
accuracy with a penalty for large subsets. The process continued until
the maximum number of generations was reached. The final output was an
optimized subset of genes that balanced high classification accuracy
with minimal dimensionality, ensuring that the selected genes were both
computationally efficient and biologically informative for cancer
classification. The pseudo code of the NRO algorithm is shown in
Algorithm 1.
Algorithm 1: Nuclear Reaction Optimization (NRO) Algorithm for Gene
Selection
>
Require: Population size N = 500, Maximum generations T = 30
Ensure: Optimized subset of gene
▷ Initialization
1: Initialize a population of 500 binary solutions
[MATH:
xi
:MATH]
, where
[MATH:
xi
:MATH]
[j] = 1 if gene
j is selected, otherwise
[MATH:
xi
:MATH]
[j] = 0
2: Set bounds [0, 1] for solutions
3: Initialize global best solution
[MATH: Xbest :MATH]
4: Compute initial fitness of
[MATH: Xbest :MATH]
using LOOCV
5: for g = 1 to T do
▷ Fission Phase: Introduce Diversity
6: for each solution
[MATH:
xi
:MATH]
in the population do
7: Generate new solutions as per Equation (1)
8: Adjust step sizes dynamically as per Equations (2) and (3)
9: Apply mutation factors as per Equations (4) and (5)
10: end for
▷ Fusion Phase: Refine Solutions
11: for each solution
[MATH:
xi
:MATH]
in the population do
12: Adjust solutions through ionization as per Equation (6)
13: if Ionized solutions are nearly identical then
14: Apply Lévy flight adjustment as per Equation (7)
15: end if
16: Combine solutions through fusion as per Equation (8)
17: if Fused solutions are nearly identical then
18: Apply Lévy flight adjustment as per Equation (9)
19: end if
20: end for
▷ Fitness Evaluation using LOOCV
21: for each solution
[MATH:
xi
:MATH]
in the population do
22: Compute classification accuracy using LOOCV
23: Compute fitness: Fitness(
[MATH:
xi
:MATH]
) = Accuracy − Penalty(size(
[MATH:
xi
:MATH]
))
24: if
[MATH:
Fitness(xi :MATH]
[MATH:
) > Fitness(Xbest :MATH]
) then
25:
[MATH:
Update Xbest :MATH]
←
[MATH:
xi
:MATH]
26: end if
27: end for
28: end for
29: Return
[MATH: Xbest :MATH]
Optimized subset of genes
[78]Open in a new tab
4.3. Apply Classifiers
To evaluate the performance of the gene subsets selected by the NRO
algorithm, two machine learning classifiers were employed: Support
Vector Machine (SVM) and k-Nearest Neighbors (k-NNs). These classifiers
were chosen for their established effectiveness in handling
high-dimensional data, making them particularly suitable for microarray
datasets with thousands of genes but relatively few samples.
The SVM was configured with a linear kernel and a penalty parameter (C
= 1), which is effective for binary classification tasks in which the
data are high-dimensional and sparse. Its ability to maximize the
margin between classes ensures robustness in separating cancerous and
non-cancerous samples, as well as in distinguishing between different
cancer subtypes.
The k-NN classifier was configured with k = 5, using Euclidean distance
to measure similarity between samples. Its non-parametric nature allows
it to adapt well to the inherent clustering within microarray datasets,
particularly when the dimensionality has been reduced through gene
selection.
Both classifiers were integrated into the fitness evaluation process,
in which they were used to calculate the classification accuracy
through Leave-One-Out Cross-Validation (LOOCV). By applying SVM and
k-NNs to the gene subsets generated by NRO, the methodology ensures
that the selected subsets are rigorously evaluated for their ability to
distinguish between cancer classes.
4.4. Fitness Evaluation
The fitness evaluation process measures the quality of each candidate
gene subset by balancing classification performance with dimensionality
reduction. The fitness of a solution is computed as follows:
[MATH:
Fitness=LOOCV Accuracy−Penalty,
:MATH]
(10)
where Equation (10) outlines the basic computation. The classification
accuracy was calculated using LOOCV (Leave-One-Out Cross-Validation), a
robust cross-validation technique suited to the small sample sizes
typical of microarray datasets. In LOOCV, each sample serves as the
test set exactly once, while the remaining samples form the training
set. This ensures that the evaluation is both comprehensive and
unbiased.
The penalty term was designed to discourage the selection of
excessively large gene subsets, promoting dimensionality reduction. It
was defined as follows:
[MATH:
Penalty=0.0001×Number of Selected Genes :MATH]
(11)
where Equation (11) indicates how the penalty scales with both the size
of the gene subset and the progress of the optimization process,
ensuring that the algorithm focuses on selecting compact subsets in
later generations.
During the evaluation of alternative cross-validation frameworks, both
5-fold and 10-fold methods were tested across all datasets. These
approaches resulted in classification accuracies approximately 1–2%
lower than those obtained using Leave-One-Out Cross-Validation (LOOCV).
Although LOOCV required significantly more computational time—typically
2 to 10 times longer depending on the dataset—it consistently produced
higher accuracy. Therefore, LOOCV was selected as the cross-validation
method in this study. This reflects a deliberate decision to prioritize
accuracy over computational efficiency, especially in the context of
small sample sizes in which each data point is critical. Moreover,
numerous studies in bioinformatics have demonstrated that LOOCV
provides better model reliability and generalization for
high-dimensional microarray data [[79]27,[80]28,[81]29,[82]30], further
supporting its adoption.
5. Results and Analysis
This section presents the results of applying the Nuclear Reaction
Optimization (NRO) algorithm for gene selection in cancer
classification across six benchmark microarray datasets. The
performance of NRO was evaluated in terms of the number of selected
genes and classification accuracy, followed by an analysis of
precision, recall, and F1-score to further assess classification
robustness. Additionally, the results are compared with
state-of-the-art gene-selection algorithms to assess the relative
effectiveness of NRO.
The results presented in [83]Table 2 reveal that the NRO algorithm
effectively selects informative gene subsets for cancer classification,
yielding high classification accuracies across multiple datasets.
However, due to the absence of dimensionality-reduction or filtering
techniques, the number of selected genes remains relatively high. This
was expected, as no feature-elimination techniques were applied before
the optimization process.
Table 2.
Summary of accuracy results using NRO.
Dataset Total Genes Classifier Selected Genes Accuracy CI (95%)
Best Average Worst
Colon 2000 SVM 494 82.16% 69.53% 60.74% [0.6904, 0.6933]
k-NNs 615 76.11% 62.57% 54.23% [0.6197, 0.6216]
Leukemia1 7129 SVM 308 95.53% 55.90% 27.37% [0.5341, 0.5368]
k-NNs 171 78.84% 38.96% 10.65% [0.3731, 0.3792]
Leukemia2 7129 SVM 198 92.47% 52.28% 24.55% [0.5046, 0.5083]
k-NNs 411 79.22% 37.32% 9.35% [0.3564, 0.3625]
Lung 7129 SVM 463 95.36% 56.53% 29.11% [0.5482, 0.5515]
k-NNs 117 97.78% 55.33% 27.67% [0.5395, 0.5437]
Lymphoma 4026 SVM 119 98.81% 75.25% 59.74% [0.7437, 0.7469]
k-NNs 72 99.28% 75.79% 59.74% [0.7439, 0.7486]
SRBCT 2308 SVM 74 99.26% 85.89% 76.96% [0.8535, 0.8557]
k-NNs 259 80.54% 67.42% 57.83% [0.6708, 0.6732]
[84]Open in a new tab
Notably, the classification accuracy achieved with SVM consistently
outperformed that of k-NNs across all the datasets. This aligned with
the expectation that SVM, being more robust to high-dimensional data,
would be better suited for the gene-expression datasets. The best
accuracy for SVM was observed in the SRBCT dataset (99.26%), followed
by Lymphoma (98.81%) and Lung (95.36%). Meanwhile, k-NNs yielded its
highest accuracy in the Lung dataset (97.78%), but its overall
performance was less stable, particularly for high-dimensional datasets
like Leukemia1 and Leukemia2, where the classification accuracy dropped
significantly for lower-performing runs.
Another key observation is that the number of selected genes varies
across datasets, with some requiring a larger subset for optimal
classification. For example, while only 74 genes were selected in the
SRBCT dataset for SVM, the Colon dataset required 494 genes for its
best-performing result. This indicates that NRO was highly adaptive in
its selection process, though the lack of dimensionality reduction
results in relatively large gene subsets, which could be further
refined in future research.
NRO’s performance varied across datasets due to differences in gene
count, sample size, and data complexity. It achieved the highest
accuracy on SRBCT and Lymphoma, which had moderate gene counts and
yielded smaller gene subsets. Lung also performed well despite high
dimensionality, likely due to its larger sample size. Leukemia1 and
Leukemia2 showed more variability, reflecting the challenge of
optimizing in high-dimensional, small-sample datasets. Colon, despite
having fewer genes, resulted in the lowest accuracy and the largest
subsets, suggesting that feature selection was more difficult, possibly
due to dataset-specific complexity.
To complement the evaluation of classification accuracy, a detailed
analysis of the 95% confidence intervals (CIs) offers valuable insights
into the stability and reliability of the results presented in
[85]Table 2. Confidence intervals provide a statistical range within
which the true average accuracy is expected to fall, offering a measure
of how consistently the NRO algorithm performs across multiple runs. In
this study, the CIs were remarkably narrow, with widths ranging from
0.0019 to 0.0061 and an average width of 0.0036, indicating minimal
variability and high repeatability in classification outcomes. For
instance, the narrowest CI was observed for the Colon dataset using
k-NNs ([0.6197, 0.6216]), while the widest CI appeared in Leukemia1
with k-NNs ([0.3731, 0.3792]); even in this case, the interval remained
tight and statistically sound. Additionally, all the CIs were logically
bounded between 0 and 1, validating the accuracy of the calculations.
These results underscore the consistency of NRO’s performance and
affirm that the reported accuracies are not outliers but represent
stable, repeatable outcomes. Including confidence intervals, therefore,
enriches the analysis by quantifying uncertainty and reinforcing the
robustness of the findings.
While accuracy provides an overall measure of classification
performance, it does not fully capture the model’s ability to minimize
false positives and false negatives. To further evaluate the
effectiveness of NRO-selected genes, [86]Table 3 presents the
precision, recall, and F1-score, which offer a deeper understanding of
classification reliability.
Table 3.
Precision, recall, and F1-score Using NRO.
Dataset Total Genes Classifier Precision Recall F1-Score
Colon 2000 SVM 81.25% 81.46% 81.28%
k-NNs 75.70% 74.39% 71.34%
Leukemia1 7129 SVM 98.60% 98.57% 98.57%
k-NNs 84.23% 82.22% 80.67%
Leukemia2 7129 SVM 95.81% 95.72% 95.70%
k-NNs 83.37% 80.70% 80.10%
Lung 7129 SVM 100% 100% 100%
k-NNs 99.11% 99.02% 99.04%
Lymphoma 4026 SVM 100% 100% 100%
k-NNs 100% 100% 100%
SRBCT 2308 SVM 100% 100% 100%
k-NNs 84.05% 81.27% 79.70%
[87]Open in a new tab
Precision measures the proportion of correctly predicted positive cases
out of all predicted positives. A higher precision means fewer false
positives, which is crucial in medical applications to avoid
unnecessary treatments. As shown in [88]Table 3, SVM achieved perfect
precision (100%) in the Lung, Lymphoma, and SRBCT datasets, indicating
that it classifies all positive cases correctly without false
positives. In other datasets, SVM maintained high precision, such as in
Leukemia1 (98.60%) and Leukemia2 (95.81%). Meanwhile, k-NNs generally
had lower precision, with its best performance in Lymphoma and Lung but
lower values in other datasets, such as Colon (75.70%).
Recall, or sensitivity, measures the ability of the classifier to
correctly identify all actual positive cases. A high recall reduces the
risk of missing cancer cases, which is critical in medical diagnostics.
SVM achieved perfect recall (100%) in the Lung, Lymphoma, and SRBCT
datasets, ensuring that no positive cases are overlooked. In other
datasets, recall remained high, such as in Leukemia1 (98.57%) and
Leukemia2 (95.72%). The k-NNs also achieved 100% recall in the Lymphoma
dataset, matching SVM, but in other cases, it performed worse than SVM,
such as in Leukemia2 (80.70%) and Colon (74.39%).
The F1-score is the harmonic mean of the precision and recall,
providing a balanced measure of classification performance. A high
F1-score indicates that the classifier effectively minimizes both false
positives and false negatives. SVM achieved an F1-score of 100% in the
Lung, Lymphoma, and SRBCT datasets, reflecting its strong performance
in these cases. In Leukemia1 and Leukemia2, SVM also maintained high
F1-scores (98.57% and 95.70%, respectively). k-NNs reached 100% in the
Lymphoma dataset but struggled in others, particularly Colon (71.34%)
and SRBCT (79.70%), where its lower recall affected the overall
performance.
To further evaluate the performance of NRO, [89]Table 4 compared its
SVM-based classification accuracy with other gene-selection algorithms,
including Harris Hawks Optimization (HHO), Artificial Bee Colony (ABC),
Particle Swarm Optimization (PSO), and Firefly Algorithm (FF). This
comparison focuses solely on classification accuracy, as the majority
of studies on gene-selection algorithms evaluate performance based only
on accuracy. Other metrics such as precision, recall, and F1-score are
not included, as they are rarely reported in gene-selection research.
Accuracy remains the primary benchmark for assessing the effectiveness
of selected gene subsets in classification tasks.
Table 4.
Comparison of gene-selection algorithms’ accuracies with SVM classifier
for six microarray datasets.
Dataset NRO HHO ABC [[90]31] PSO [[91]32] PSO [[92]33] FF [[93]18]
Colon 82.16% 90.32% 95.61% 85.48% 87.01% 98.2%
Leukemia1 95.53% 97.22% 93.05% 94.44% 93.06% 100%
Leukemia2 92.47% 84.72% 97.22% - - 97.2%
Lung 95.36% 100% 97.91% - - 100%
Lymphoma 98.81% 100% 96.96% - - -
SRBCT 99.26% 92.77% 95.36% - - 98.8%
[94]Open in a new tab
The results show that NRO was competitive but did not consistently
outperform other methods. It achieved the highest accuracy only in the
SRBCT dataset (99.26%), surpassing FF (98.8%) and other algorithms.
However, in all other datasets, NRO was outperformed. For example, in
the Colon dataset, NRO’s accuracy (82.16%) was the lowest, while FF
achieved 98.2%. In Leukemia1, NRO reached 95.53%, which was lower than
HHO (97.22%) and FF (100%). In Lung, both HHO and FF achieved 100%
accuracy, surpassing NRO’s 95.36%. Similarly, in Leukemia2, NRO’s
92.47% was outperformed by ABC (97.22%) and FF (97.2%). In Lymphoma,
NRO achieved 98.81%, slightly below HHO (100%).
Despite not achieving the highest accuracy in all the datasets, NRO
consistently delivered a strong and reliable performance across diverse
cancer datasets. Its ability to outperform methods like HHO and ABC in
datasets such as Leukemia2 and SRBCT, and to achieve near-perfect
accuracy in Lymphoma (98.81%) and SRBCT (99.26%), demonstrates its
robustness in handling high-dimensional gene-expression data. These
results highlight NRO’s effectiveness and adaptability, validating its
optimization approach based on nuclear reaction mechanisms. Overall,
NRO proved to be a promising and competitive algorithm for gene
selection in cancer classification, with the potential for further
enhancement through hybrid or refined optimization strategies.
6. Conclusions
This study explored the application of the Nuclear Reaction
Optimization (NRO) algorithm for gene selection in cancer
classification using six benchmark microarray datasets. The results
demonstrated that NRO effectively identified relevant gene subsets,
leading to high classification accuracy, particularly when paired with
SVM. Compared with state-of-the-art optimization methods, including
HHO, ABC, PSO, and FF, NRO showed competitive performance,
outperforming some approaches in specific datasets such as SRBCT and
Leukemia2. NRO demonstrated strong adaptability in high-dimensional
feature spaces. Its ability to refine solutions through nuclear fission
and fusion highlights its potential as a powerful bioinformatics tool
for cancer classification. However, despite its promising results,
certain limitations hinder its full efficiency in gene-selection tasks.
A key limitation of this study is the large number of selected genes.
Since no dimensionality-reduction techniques were applied before
running NRO, this led to reduced interpretability and higher
computational costs. Additionally, the Leave-One-Out Cross-Validation
(LOOCV) method, while chosen for its superior accuracy, further
increased computational expense due to its iterative training process.
Future work should integrate filtering techniques before applying NRO
to remove irrelevant genes, improving both classification accuracy and
efficiency. Furthermore, hybrid optimization approaches, in which a
secondary metaheuristic is used before NRO to refine the initial
population, could help accelerate convergence, enhance
feature-selection quality, and reduce unnecessary computations. These
improvements would make the method more scalable, interpretable, and
effective for larger and more complex datasets.
Additionally, this study did not assess the biological relevance of the
selected genes. Future research will include pathway enrichment
analysis and comparison with known cancer biomarkers to enhance the
interpretability of the results. While NRO has previously been applied
to RNA-Seq data, this study demonstrates its effectiveness on
microarray data. Building on these findings, future studies will
evaluate NRO’s scalability and performance on larger and more complex
genomic datasets.
Acknowledgments