Improved Binary Elk Herd Optimizer with Fitness Balance Distance for

Feature Selection Using Gene Expression Data

Mohamed Wajdi Ouertani

1 a

, Raja Oueslati

1 b

and Ghaith Manita

1,2 c

Laboratory MARS, LR17ES05, ISITCom, Sousse University, Sousse, Tunisia

ESEN, Manouba University, Manouba, Tunisia

Keywords:

Feature Selection, Optimization, Elk Herd Optimiser, Distance Balance Mechanism.

Abstract:

This research paper introduces an enhanced version of the Binary Elk Herd Optimizer (BEHO), integrated with

a Fitness Distance Balance (FDB) mechanism called FDB-BEHO, tailored for high-dimensional optimization

tasks. This study evaluates the performance of FDB-BEHO across multiple gene expression datasets, focusing

on feature selection in bioinformatics—a domain characterized by complex, high-dimensional data. The FDB

mechanism is designed to prevent premature convergence by maintaining an optimal balance between explo-

ration and exploitation, utilizing a diversity measure that adjusts dynamically based on the ﬁtness-distance

correlation among solutions. Comparative analyses demonstrate that FDB-BEHO surpasses traditional meta-

heuristic algorithms in ﬁtness values and classiﬁcation accuracy and reduces the number of selected fea-

tures, thereby enhancing model simplicity and interpretability. These results validate the effectiveness of

FDB-BEHO in navigating complex solution spaces efﬁciently and underscore its potential applicability in

other domains requiring robust feature selection capabilities. The study’s ﬁndings suggest that incorporating

diversity-enhancing mechanisms like FDB can signiﬁcantly improve the performance of binary optimization

algorithms, offering promising directions for future research in optimization technology.

1 INTRODUCTION

In medical research, DNA microarray technology has

revolutionized our ability to analyze gene expression

data, enabling the simultaneous observation of thou-

sands of genes in a single experiment. However, this

advancement also presents a signiﬁcant challenge:

the curse of dimensionality. With such vast amounts

of data, it becomes crucial to identify and select

the most relevant features—genes that signiﬁcantly

contribute to accurate disease prediction and classi-

ﬁcation. Feature selection (FS) methods are pivotal

in addressing this challenge, allowing researchers

to eliminate irrelevant or redundant genes, thereby

enhancing the performance of predictive models and

reducing computational complexity (Zebari et al.,

2020).

In recent years, the importance of FS has been un-

derscored in various studies focused on cancer pre-

https://orcid.org/0009-0000-6164-0069

https://orcid.org/0009-0002-5783-5722

https://orcid.org/0000-0003-0782-9658

diction (Haq et al., 2021), where identifying key ge-

netic markers is essential for early diagnosis and treat-

ment planning. Traditional FS methods, such as ﬁl-

ter, wrapper, and embedded approaches, have been

widely applied, each with its strengths and limitations

(Venkatesh and Anuradha, 2019). Filters are inde-

pendent of the learning algorithm but may overlook

interactions between features (Bommert et al., 2022);

wrappers are more accurate but computationally ex-

pensive (Maldonado and Weber, 2009); and embed-

ded methods integrate FS within the model training

process, offering a balanced approach (Wang et al.,

2015).

Given the complexity and non-linearity of gene

expression data, metaheuristic algorithms have

emerged as powerful tools for FS (Dokeroglu et al.,

2022). Inspired by natural, biological, or social pro-

cesses, these optimization approaches are designed to

solve complex problems efﬁciently (Ouertani et al.,

2022a; Oueslati et al., 2024). Metaheuristics excels

at exploring vast search spaces and avoiding subop-

timal solutions, unlike traditional optimization meth-

ods, which may be constrained by linearity or con-

tinuity requirements (Ouertani et al., 2022b). Com-

786

Ouertani, M. W., Oueslati, R. and Manita, G.

Improved Binary Elk Herd Optimizer with Fitness Balance Distance for Feature Selection Using Gene Expression Data.

DOI: 10.5220/0013367600003890

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 2, pages 786-797

ISBN: 978-989-758-737-5; ISSN: 2184-433X

mon metaheuristics include evolutionary algorithms

(Srinivas and Patnaik, 1994) such as Artiﬁcial Bee

Colony (ABC) (Karaboga and Basturk, 2007), Ge-

netic Algorithm (GA) (Holland, 1992), and Differ-

ential Evolution (DE) (Qin et al., 2008). Swarm-

based optimization methods such as Elk Herd Opti-

mizer (EHO) (Al-Betar et al., 2024), Particle Swarm

Optimization (PSO) (Kennedy and Eberhart, 1995),

and Social Spider Algorithm (SSA) (Mirjalili et al.,

2017) have proven effective in various applications.

Additionally, human-inspired metaheuristics such as

Teaching-Learning Based Optimization (TLBO) (Rao

et al., 2011), Soccer League Competition (SLC)

(Moosavian and Roodsari, 2014), and Brain Storm

Optimization (BSO) (Shi, 2011) leverage social be-

haviors and cognitive processes for optimization. Fi-

nally, physics-based metaheuristics such as Simulated

Annealing (SA) (Bertsimas and Tsitsiklis, 1993),

Atom Search Optimization (ASO) (Zhao et al., 2019),

and Equilibrium Optimizer (EO) (Faramarzi et al.,

2020) draw inspiration from physical phenomena like

thermodynamics and gravitational forces to guide the

search for optimal solutions. Their primary advan-

tages lie in their ﬂexibility and adaptability, making

them suitable for various problems, particularly com-

binatorial optimization challenges. However, they can

be computationally intensive and do not guarantee

globally optimal solutions, often requiring careful pa-

rameter tuning to balance solution quality and com-

putational cost (Nssibi et al., 2024).

In order to address the inherent challenges of

gene expression data analysis (Nssibi et al., 2023)

and as the ﬁeld continues to evolve, the application

of advanced metaheuristics in FS not only enhances

model accuracy but also offers valuable insights into

complex underlying processes, paving the way for

more targeted and effective decision-making strate-

gies across various domains in bioinformatics (Saeys

et al., 2007). This study introduces an enhanced

version of the Binary Elk Herd Optimizer (BEHO),

named FDB-BEHO, which incorporates the Fitness

Distance Balance (FDB) mechanism to overcome pre-

mature convergence and maintain an optimal balance

between exploration and exploitation. The proposed

algorithm is speciﬁcally designed to minimize the

number of selected features, making it well-suited for

addressing high-dimensional optimization challenges

effectively. To evaluate its effectiveness, FDB-BEHO

is tested on nine benchmark biological datasets for

feature selection. Its performance is compared against

state-of-the-art metaheuristic algorithms using met-

rics such as ﬁtness values, classiﬁcation accuracy, and

feature selection efﬁciency.

The main objectives and contributions of this

work are as follows:

• Introduction of the Binary EHO (BEHO): A novel

adaptation of the EHO algorithm tailored for fea-

ture selection problems. This binary variant en-

ables the direct application of EHO in solving dis-

crete optimization challenges associated with FS.

• Improvement of BEHO with the FDB Mecha-

nism: A further enhancement of BEHO, incorpo-

rating the FDB mechanism to address issues of

premature convergence. This improvement en-

sures a more effective balance between explo-

ration and exploitation. The enhanced algorithm

dynamically adjusts diversity to maintain robust

performance across varying optimization land-

scapes.

• Evaluation of FDB-BEHO on gene expression

data for FS: Assess the performance of the pro-

posed FDB-BEHO algorithm on nine benchmark

biological datasets for FS. The evaluation involves

a comparative analysis with other state-of-the-art

metaheuristics to validate its efﬁcacy and robust-

ness.

The remainder of this paper is structured as fol-

lows: Section 2 provides a detailed overview of meta-

heuristic optimization in feature selection, highlight-

ing the effectiveness of various algorithms in manag-

ing high-dimensional data. Section 3 introduces the

proposed Binary Elk Herd Optimizer (Binary EHO)

and its enhancement with the Fitness Distance Bal-

ance (FDB) mechanism, including the technical de-

tails of its implementation. Section 4 presents the ex-

perimental setup and the results of applying the pro-

posed method to gene expression datasets, comparing

its performance with other state-of-the-art algorithms.

Finally, Section 5 concludes the paper by discussing

the ﬁndings, their implications for feature selection

in bioinformatics, and potential avenues for future re-

search.

2 METAHEURISTIC

OPTIMIZATION IN FEATURE

SELECTION

In this section, we explore the role of metaheuristic

optimization in feature selection, offering a compre-

hensive overview of various algorithms and their suc-

cessful applications in addressing the challenges of

high-dimensional gene expression data.

In (S

onmez et al., 2021), the study explores the use of

hybrid methods combining Genetic Algorithms (GA)

with Support Vector Machines (SVM) and k-Nearest

Improved Binary Elk Herd Optimizer with Fitness Balance Distance for Feature Selection Using Gene Expression Data

787

Neighbors (KNN) for feature selection and classiﬁ-

cation of gene expression datasets. The authors pro-

pose enhancing the GA through the integration of

ﬁlter methods such as Pearson’s correlation coefﬁ-

cient, Relief-F, and mutual information. The study

evaluates these methods across eight gene expres-

sion datasets, primarily related to cancer classiﬁca-

tion. The proposed GA-SVM and GA-KNN meth-

ods demonstrate superior accuracy compared to tra-

ditional approaches, highlighting the effectiveness of

hybrid techniques in reducing the dimensionality of

high-throughput data while maintaining or improv-

ing classiﬁcation accuracy. The research empha-

sizes the importance of hybrid methods in managing

the computational complexity of large gene expres-

sion datasets and improving predictive performance

in medical applications, particularly in cancer diag-

nosis and subtype classiﬁcation.

Qin et al. proposed a two-stage feature selection

framework tailored to classify high-dimensional gene

expression data, utilizing an improved Salp Swarm

Algorithm (SSA) (Qin et al., 2022). The ﬁrst stage

combines Weighted Gene Co-expression Network

Analysis (WGCNA), Random Forest (RF), and Max-

Relevance and Min-Redundancy (mRMR) to initially

reduce the feature space by selecting the most rele-

vant and non-redundant genes. In the second stage,

the improved binary SSA is employed to reﬁne the

feature set further, ensuring a balance between classi-

ﬁcation accuracy and the number of selected features.

The framework was tested across ten gene expres-

sion datasets and outperformed other intelligent opti-

mization algorithms like PSO, GWO, and WOA. The

results demonstrated that this method could achieve

high classiﬁcation accuracy with fewer selected fea-

tures, making it a robust solution for gene expression

data classiﬁcation in cancer diagnosis.

In (Alzaqebah et al., 2021), Alzaqebah et al. proposed

a Memory-Based Cuckoo Search (MBCS) algorithm

for feature selection in gene expression datasets, par-

ticularly focusing on cancer prediction. The study

addresses the challenges of high-dimensionality and

feature redundancy in microarray data, which can

hinder accurate classiﬁcation. The MBCS algorithm

enhances the traditional Cuckoo Search Algorithm

(Gandomi et al., 2013) by incorporating a memory

mechanism that records the best solutions found dur-

ing the search process. This memory helps the al-

gorithm avoid re-exploring suboptimal areas and fo-

cuses on promising regions of the search space. The

study tested the algorithm on twelve different mi-

croarray datasets and found that MBCS outperformed

other algorithms, such as Genetic Algorithms (GA),

Particle Swarm Optimization (PSO), and Gravita-

tional Search Algorithm (GSA), in terms of classiﬁ-

cation accuracy and the number of selected features.

The results indicate that MBCS is particularly effec-

tive in reducing the dimensionality of the data while

maintaining or improving the accuracy of cancer pre-

dictions.

Qu et al. introduced an improved Harris Hawks Op-

timization algorithm, tailored explicitly for feature

selection in gene expression data, called the Vari-

able Neighborhood Learning Harris Hawks Optimizer

(VNLHHO) (Qu et al., 2021). This approach was

designed to enhance the global exploration and lo-

cal exploitation capabilities of the standard Harris

Hawks Optimizer (HHO). The VNLHHO incorpo-

rates a dynamic neighborhood learning strategy and

mutation operations to increase population diversity

and prevent the algorithm from falling into local op-

tima. The effectiveness of VNLHHO was validated

across eight cancer gene expression datasets, demon-

strating superior classiﬁcation accuracy and conver-

gence speed compared to traditional algorithms like

PSO, GA, and the original HHO. The results showed

that VNLHHO not only improved classiﬁcation per-

formance but also effectively reduced the number of

selected features, making it a powerful tool for high-

dimensional biomedical data analysis.

3 PROPOSED APPROACH

This section presents the proposed approach, which

utilizes the Elk Herd Optimizer (EHO) (Al-Betar

et al., 2024) as the core algorithm for feature selec-

tion in high-dimensional gene expression data. The

approach is systematically organized into three main

components: (i) Overview of EHO: We begin by in-

troducing the EHO algorithm and highlighting its dis-

tinctive characteristics that make it suitable for com-

plex optimization tasks, (ii) Transformation to Binary

EHO: Next, we describe the adaptation of EHO into

a binary format (Binary EHO), speciﬁcally tailored

to meet the unique demands of feature selection prob-

lems, and (iii) Integration of Fitness Distance Balance

(FDB): Finally, we incorporate the Fitness Distance

Balance (FDB) mechanism into Binary EHO, signif-

icantly enhancing its capability to maintain a robust

balance between exploration and exploitation while

identifying optimal feature subsets. This structured

approach ensures a comprehensive framework for ad-

dressing the challenges of feature selection in high-

dimensional datasets.

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788

3.1 Elk Herd Optimizer Overview

The Elk Herd Optimizer (EHO) is a novel metaheuris-

tic algorithm inspired by the breeding behavior of elk

herds. It mimics the seasonal dynamics within the

herd, where more muscular bulls lead larger groups

during the rutting season, and new solutions are gen-

erated during the calving season. This approach bal-

ances exploration and exploitation in optimization

tasks, making EHO an effective tool for solving com-

plex problems. Its design is particularly suited for

navigating challenging search spaces and ﬁnding op-

timal solutions efﬁciently. The EHO is designed to

simulate the natural dynamics of elk herds through a

sequence of critical phases. It begins with the initial-

ization of the population and the problem parameters.

The algorithm then enters the rutting season, dividing

the population into families led by the ﬁttest bulls. In

the calving season, these families produce new solu-

tions based on the characteristics of the bull and its

harems. Finally, during the selection season, all so-

lutions are evaluated, and the ﬁttest are selected to

form the next generation, with this process repeating

until the algorithm converges or the iteration limit is

reached. The steps of the EHO are as follows :

1. Initialization: During the initialization phase of

the EHO, the algorithm begins by setting up the

population and deﬁning the problem-speciﬁc pa-

rameters. The primary elements to initialize are

the elk herd size (EHS), the bull rate (Br), and the

search space boundaries. The elk herd EH is ini-

tialized as a matrix of size n ×EHS, where n is the

problem’s dimensionality, and each element in the

matrix represents a potential solution (elk). Math-

ematically, each solution x

in the population is

generated within the deﬁned search space bound-

aries using Equation 1:

= lb

+ (ub

− lb

) ×U(0, 1) (1)

where x

represents the i-th attribute of the j-th so-

lution, lb

and ub

are the lower and upper bounds

for the i-th attribute, and U(0, 1) is a uniformly

distributed random number between 0 and 1. The

ﬁtness of each solution is then calculated using

the objective function f (x), and the solutions are

sorted in ascending order based on their ﬁtness

values. This initial setup prepares the elk herd for

the subsequent phases of the algorithm.

2. Generating the initial Elk Herd Solutions: In the

second step, the algorithm focuses on creating the

initial solutions population, representing the elk

herd. After deﬁning the problem-speciﬁc param-

eters and initializing the population matrix EH in

the ﬁrst step, this phase involves assigning ﬁtness

values to each solution and organizing the herd

structure. The elk herd EH is generated as a ma-

trix of size n × EHS, where each row corresponds

to a potential solution in the search space as pre-

sented in Equation 2.

EH =







··· x

. ···

EHS

··· x

EHS







(2)

Once the initial population is generated, the ﬁt-

ness of each solution is evaluated using the ob-

jective function f (x). The herd is then sorted in

ascending order of ﬁtness, ensuring that the best

solutions (strongest elks) are positioned at the top.

This ordered structure sets the foundation for the

subsequent rutting season phase, where the popu-

lation will be divided into families.

3. Rutting season: In the third step, the EHO algo-

rithm divides the initial population into families,

with each family led by a bull (the ﬁttest indi-

vidual). The division is based on the ﬁtness of

the bulls, reﬂecting the natural behavior where

stronger bulls lead larger groups. First, the al-

gorithm determines the number of bulls B in the

population using the bull rate Br and the elk herd

size EHS as shown in Equation 3. :

B = Br × EHS (3)

where B is the number of bulls, Br is the bull rate,

and EHS is the total population size. The top B

individuals, based on their ﬁtness values, are se-

lected as bulls. Next, the bulls compete to form

families, each consisting of a bull and its assigned

harems (followers). The assignment of harems to

each bull is done using a roulette-wheel selection

process, where the probability p

of a bull j at-

tracting a harem is proportional to its ﬁtness pre-

sented in Equation 4:

f (x

)

∑

k=1

f (x

)

(4)

Here, f (x

) is the ﬁtness of the j-th bull, and the

sum in the denominator is the total ﬁtness of all

bulls. The roulette-wheel selection ensures that

bulls with higher ﬁtness are more likely to lead

more harems. Once the harems are assigned, each

bull leads its family, with the size of each fam-

ily reﬂecting the strength of the bull. This struc-

tured division sets the stage for the calving season,

where new solutions (calves) will be generated

based on the bulls’ characteristics and harems.

Improved Binary Elk Herd Optimizer with Fitness Balance Distance for Feature Selection Using Gene Expression Data

789

4. Calving season: In the fourth step, the EHO

algorithm focuses on generating new solutions

(calves) within each family based on the genetic

traits of the bull (leader) and its harems (follow-

ers). This process mimics the natural reproduc-

tion process in elk herds, where the offspring in-

herit characteristics from both parents, promoting

diversity within the population. For each family,

new solutions x

(t + 1) are generated by combin-

ing attributes from the bull x

and its harems x

(t).

If the calf’s index matches that of its bull father,

the new solution is generated using Equation 5:

(t + 1) = x

(t) + α · (x

(t) − x

(t)) (5)

where α is a random number between 0 and 1,

and x

(t) is a randomly selected attribute from

the current population. This equation ensures that

the new solution is inﬂuenced primarily by the

bull, with some variation introduced by the ran-

dom selection from the herd. If the calf’s index

matches that of its mother harem, the new solu-

tion is created by combining the attributes of both

the mother and the bull, using Equation 6:

(t +1) = x

(t)+β·(x

h j

(t)−x

(t))+γ·(x

(t)−x

(t))

(6)

Here, β and γ are random numbers in the range

[0, 2], x

h j

(t) represents the bull’s attributes, and

(t) is a random attribute from another bull. This

equation allows the calf to inherit traits from both

parents, with additional diversity introduced by

the random selection. This calving process con-

tinues for all families, producing a new genera-

tion of solutions that inherit their predecessors’

strengths while introducing new variations, which

is crucial for the algorithm’s exploration and ex-

ploitation capabilities in the search space.

5. Selection season: In the ﬁfth step, the EHO al-

gorithm consolidates the newly generated solu-

tions (calves) with the existing population of bulls

and harems to form a uniﬁed herd. This phase

aims to evaluate the ﬁtness of all individuals in

this combined population and select the best solu-

tions to carry forward to the next generation. First,

the bulls, harems, and newly generated calves are

merged into a single matrix, EH

temp

. Each indi-

vidual’s ﬁtness in EH

temp

is evaluated using the

objective function, and the entire population is

sorted in ascending order based on ﬁtness values.

From this sorted population, the top EHS indi-

viduals, where EHS is the elk herd size, are se-

lected to form the new population for the next iter-

ation. This selection process ensures that only the

ﬁttest individuals, whether they are bulls, harems,

or calves, are retained in the herd. This method

Figure 1: Flowchart of the Elk Herd Optimizer.

of selection is akin to the µ + λ selection strat-

egy commonly used in evolutionary algorithms,

where both parents (bulls and harems) and off-

spring (calves) compete equally for survival. By

continuously selecting the ﬁttest individuals, the

algorithm iteratively reﬁnes the population, im-

proving the overall ﬁtness of the herd with each

cycle. This selection process repeats until the ter-

mination criteria, such as a maximum number of

iterations or convergence to an optimal solution,

are met.

The ﬂowchart and pseudocode of the EHO are pre-

sented in Figure 1 and Algorithm 1, respectively.

3.2 Binary Elk Herd Optimizer

Transforming the EHO into a Binary EHO is essential

for adapting the algorithm to feature selection tasks,

which require decisions to be made in a binary search

space. In feature selection, each candidate solution

is represented as a binary vector where each element

indicates whether a particular feature is included (1)

or excluded (0). The transformation of EHO to handle

binary vectors involves the following technical steps:

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790

Algorithm 1: Elk Herd Optimizer (EHO) Algorithm.

1: Input: Population size (EHS), Bull rate (Br),

Maximum iterations (MaxIter)

2: Output: Best solution found

3: Initialize population EH with EHS solutions ran-

domly within the search space

4: Evaluate the ﬁtness of each solution in EH

5: Sort EH based on ﬁtness in ascending order

6: for iter = 1 to MaxIter do

7: Determine number of bulls B = ⌈Br × EHS⌉

8: Select the top B solutions as bulls from EH

9: Assign harems to each bull using roulette-

wheel selection based on ﬁtness

10: for each family (bull and its harems) do

11: if index matches bull then

12: x

new

= x

bull

+ α × (x

random

− x

bull

)

13: else

14: x

new

= x

harem

+ β × (x

bull

− x

harem

) +

γ × (x

random bull

− x

harem

)

15: end if

16: end for

17: Combine bulls, harems, and calves into

temp

18: Evaluate ﬁtness of all solutions in EH

temp

19: Sort EH

temp

based on ﬁtness in ascending or-

der

20: Select the top EHS solutions from EH

temp

form the new population EH

21: end for

22: Return the best solution in EH

1. Binary Representation: Instead of representing

solutions as continuous vectors, each solution is

now a binary vector x = [x

, x

, . . . , x

], where

∈ {0, 1} for each feature i. This change allows

the algorithm to directly address feature selection

by determining whether each feature should be in-

cluded in the model.

2. Position Update with Transfer Functions: The

core challenge in adapting EHO to a binary for-

mat lies in converting the continuous position up-

dates, typical in EHO, to binary updates. This is

achieved using transfer functions (Equations 7,8)

(Mirjalili and Lewis, 2013; Nssibi et al., 2021).

After the continuous update equation is computed

for each element x

of the solution vector, a trans-

fer function T F(x) is applied to convert this con-

tinuous value into a probability. Common transfer

functions include:

• Sigmoid Function (S-shaped):

T F(x) =

1 + e

−x

(7)

• V-shaped Function:

T F(x) = |tanh(x)| (8)

The output of these functions lies between 0 and

1 and represents the probability that a particular

feature will switch its state (from 0 to 1 or vice

versa).

3. Binary Decision-Making: Once the probability

P(x

) is determined using the transfer function,

the next step is to convert this probability into a

binary decision. This is done by comparing the

probability with a random number rand() gener-

ated uniformly between 0 and 1 as presented in

Equation 9:

(

1, if P(x

) ≥ rand()

0, otherwise

(9)

This comparison ensures that each feature’s bi-

nary state is updated based on the likelihood cal-

culated through the transfer function.

4. Fitness Evaluation: After updating the binary vec-

tors, the ﬁtness of each solution is evaluated. The

ﬁtness function typically balances the trade-off

between maximizing classiﬁcation accuracy (us-

ing a classiﬁer like k-NN) and minimizing the

number of features selected. This ensures that the

selected subset is both practical and compact.

By incorporating these steps, the Binary EHO can

effectively navigate binary search spaces, making it

highly suitable for feature selection tasks. Trans-

fer functions are critical in this transformation, as

they bridge the gap between continuous optimization

strategies and the discrete nature of feature selection

problems, enabling the algorithm to retain its explo-

ration and exploitation capabilities in a binary con-

text.

3.3 Fitness Distance Balance BEHO

Integrating the Fitness Distance Balance (FDB)

(Kahraman et al., 2020) mechanism into the BEHO

enhances its performance in feature selection by in-

troducing a balance between exploration and exploita-

tion. This mechanism evaluates each solution based

not only on its ﬁtness but also on its diversity rela-

tive to the best-known solution. Technically, this is

achieved by ﬁrst calculating the Euclidean distance

between each binary solution x

and the current

best solution Best

current

, using the formula in Equa-

tion 10:

∑

i=1

− Best

current

)

(10)

Improved Binary Elk Herd Optimizer with Fitness Balance Distance for Feature Selection Using Gene Expression Data

791

Next, both the ﬁtness values and the distances are nor-

malized to ensure no single metric dominates as pre-

sented in Equations 11 and 12 :

norm

( j) =

F( j) − min(F)

max(F) − min(F)

(11)

norm

( j) =

− min(D)

max(D) − min(D)

(12)

These normalized values are then combined into a

composite score S

that balances ﬁtness and diversity

as presented in Equation 13:

= α × F

norm

( j) + (1 − α) × D

norm

( j) (13)

This score guides the selection process, with higher-

scoring solutions more likely to be retained or se-

lected as bulls. Using this composite score, the al-

gorithm avoids premature convergence and maintains

a diverse population, which is crucial for effectively

exploring the binary search space and identifying op-

timal feature subsets. This approach signiﬁcantly

enhances the robustness and effectiveness of Binary

EHO in feature selection tasks. The complete pseu-

docode of the FDB-BEHO is presented in Algorithm

Algorithm 2: Binary EHO with Fitness Distance Balance

(FDB) Mechanism.

1: Initialize population EH as binary vectors

2: for iter = 1 to MaxIter do

3: Determine bulls and harems using binary ﬁt-

ness and distance evaluation

4: for each family do

5: Calculate continuous updates for posi-

tions

6: Apply transfer function to convert updates

to probabilities

7: Update binary positions using the calcu-

lated probabilities

8: end for

9: Calculate distance D

for each solution from

the best solution

10: Normalize ﬁtness and distance values

11: Compute composite score S

α × F

norm

( j) + (1 − α) × D

norm

( j)

12: Select the best solutions based on composite

scores to form the new population

13: end for

14: Return the best binary solution

4 EXPERIMENTAL RESULTS

In this section, we evaluate the performance of the

proposed FDB-BEHO across various gene expres-

sion datasets. All experiments were conducted us-

ing MATLAB R2020a on an Intel Core i5 machine

with a 3.3 GHz CPU and 12GB of RAM. Each ex-

periment was repeated 30 times to ensure statistical

signiﬁcance.

4.1 Datasets

We utilized the following nine benchmark bi-

ological datasets to assess the effectiveness of

the FDB-BEHO: Leukemia (Golub et al., 1999),

Prostate GE (Singh et al., 2002), Colon (Alon et al.,

1999), Lung discrete (Bhattacharjee et al., 2001),

SMK CAN 187 (Spira et al., 2007), Lymphoma

(Alizadeh et al., 2000), CLL SUB 111 (Haslinger

et al., 2004), Lung (Bhattacharjee et al., 2001),

and nci9 (Ross et al., 2000). The details of these

datasets, including the number of instances, features,

and classes, are summarized in Table 1. The parame-

ters for FDB-BEHO and the comparative metaheuris-

tic algorithms are detailed in Table 2.

Table 1: Summary of Datasets Used in the Experiments.

Dataset Instances Features Classes

CLL SUB 111 111 11340 3

Colon 62 2000 2

Leukemia 72 7070 2

Lung 203 3312 5

Lung discrete 73 325 7

Lymphoma 96 4026 9

nci9 60 9712 9

Prostate GE 102 5966 2

SMK CAN 187 187 19993 2

Table 2: Parameter Settings for the Algorithms.

Algorithm Parameter Value

FDB-BEHO

EHS ( Population Size)

Br (Break Rate)

0.3

SSA c1 (leader position update probability) 0.5

PSO

c1, c2 (acceleration coefﬁcients)

ω (inertia weight)

0.1

crossover rate

mutation rate

0.9

0.1

ASO

α (depth weight)

β (multiplier weight)

Vmax (maximum velocity)

0.2

All of them

search agents (bats, wolves, particles,...)

maximum iterations

200

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

792

4.2 Results and Discussion

This section thoroughly examines various datasets,

illustrating the performance of the FDB-BEHO op-

timization algorithm variants compared with estab-

lished algorithms such as ABC, SSA, PSO, GA, EO,

and ASO. The metrics for this comparison encom-

pass ﬁtness values, classiﬁcation accuracy, the num-

ber of selected features, and statistical signiﬁcance as

assessed by the Wilcoxon test.

In Table 3, the analysis of ﬁtness values shows

FDB-BEHO-V as a standout performer, consistently

achieving the lowest ﬁtness scores on challeng-

ing datasets such as CLL SUB 111, colon, and

leukaemia. These results demonstrate the algorithm’s

superior capability in efﬁciently navigating complex

solution spaces to identify highly optimal solutions.

Moreover, FDB-BEHO-V exhibits the lowest stan-

dard deviations among the compared algorithms, em-

phasising its stability and reliability. Such consis-

tency is crucial in optimization tasks where depend-

ability and predictability of performance are as cru-

cial as the performance itself.

The narrative continues in Table 4, focusing on

classiﬁcation accuracy. Here, FDB-BEHO-S fre-

quently outperforms FDB-BEHO-V, indicating its en-

hanced capability in models where higher accuracy

is paramount. However, the performance landscape

is nuanced, with algorithms like EO and PSO ex-

celling in speciﬁc datasets, particularly lung and

Prostate GE. These observations underline the neces-

sity of adaptive algorithm selection based on speciﬁc

dataset characteristics, suggesting that no single al-

gorithm uniformly outperforms others across all con-

texts. This variability also highlights the importance

of understanding the underlying data characteristics

and choosing algorithms that align well with those

characteristics to optimize performance.

Table 5 sheds light on the efﬁciency of feature

selection, a critical metric that affects both the com-

plexity and the computational efﬁciency of the result-

ing models. FDB-BEHO-V is particularly adept at

reducing the number of features required to achieve

high performance, which is evident in its handling

of datasets like CLL SUB 111 and colon. By se-

lecting fewer features, FDB-BEHO-V not only sim-

pliﬁes the complexity of the models but also po-

tentially enhances the interpretability of the results,

which is invaluable in applications where understand-

ing the algorithm’s decision-making process is essen-

tial. Moreover, models with fewer features generally

train and deploy faster, offering practical advantages

in real-time applications. Hence, Figure 2 provides a

clearer visualization of the obtained results.

The robustness of these results is further validated

in Table 6, where the Wilcoxon test results conﬁrm

the statistical signiﬁcance of the improvements of-

fered by FDB-BEHO-V over other algorithms in most

datasets. However, in datasets like lymphoma and

nci9, where no signiﬁcant differences are found, the

results suggest that FDB-BEHO-V may not always

offer a decisive advantage, indicating potential areas

for further algorithmic reﬁnement and improvement.

This detailed analysis underscores FDB-BEHO-

V’s capabilities to deliver top-tier optimization per-

formance with robust reliability across a range of

domains, presenting it as an excellent candidate for

tackling complex optimization problems in varied set-

tings. The ﬁndings from this investigation are crucial

for advancing the development and application of op-

timization methodologies in both academic research

and practical industrial applications, driving innova-

tion and efﬁciency in this vital ﬁeld.

5 CONCLUSION

This study has meticulously explored the enhance-

ments to the BEHO by integrating the Fitness Dis-

tance Balance (FDB) mechanism. The adapted

BEHO has been rigorously tested across various gene

expression datasets, demonstrating its ability to efﬁ-

ciently and reliably identify optimal solutions. Intro-

ducing a binary framework tailored for discrete op-

timization tasks such as feature selection has shown

signiﬁcant promise, particularly in bioinformatics,

where the dimensionality and complexity of data of-

ten pose substantial challenges.

The FDB-BEHO variant has consistently outper-

formed traditional metaheuristic algorithms in terms

of ﬁtness values, classiﬁcation accuracy, and feature

selection efﬁciency. This robust performance under-

scores the algorithm’s reﬁned balance between explo-

ration and exploitation, facilitated by the FDB mecha-

nism, which integrates a diversity measure to prevent

premature convergence. The ﬁndings from this study

advocate for the potential of BEHO in bioinformatics

and highlight its adaptability and efﬁciency in manag-

ing high-dimensional datasets.

Looking forward, optimization in high-

dimensional data analysis presents several avenues

for further research. BEHO’s adaptability could be

explored in other complex optimization scenarios

beyond bioinformatics, such as in ﬁnance, robotics,

and climate modeling, where similar challenges re-

garding high dimensionality and feature redundancy

exist. Future studies could also delve into hybrid ap-

proaches combining BEHO with other metaheuristic

Improved Binary Elk Herd Optimizer with Fitness Balance Distance for Feature Selection Using Gene Expression Data

793

(a) CLL SUB 111 (b) Colon

(e) Lung discrete (f) Lymphoma

(g) nci9 (h) Prostate GE

(i) SMK CAN 187

Figure 2: Average selected number of features for CLL SUB 111, Colon, Leukemia, Lung, Lung discrete, Lymphoma, nci9,

Prostate GE, and SMK CAN 187 datasets.

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

794

Table 3: Comparison of FDB-BEHO versus other optimization algorithms in terms of ﬁtness values.

Dataset Metric FDB- FDB- ABC SSA PSO GA EO ASO

BEHO-V BEHO-S

CLL SUB 111 Best 0,0469 0,0524 0,0948 0,0472 0,0500 0,0499 0,0914 0,0949

Avg 0,0872 0,1019 0,1646 0,1526 0,1505 0,1458 0,1410 0,1585

Std 0,0169 0,0223 0,0302 0,0308 0,0387 0,0310 0,0294 0,0326

colon Best 0,0002 0,0046 0,0045 0,0014 0,0044 0,0045 0,0006 0,0045

Avg 0,0005 0,0050 0,0047 0,0022 0,0047 0,0047 0,0013 0,0048

Std 0,0001 0,0004 7,64E-05 0,0003 0,0001 5,18E-05 0,0003 9,09E-05

leukemia Best 0,0001 0,0048 0,0047 0,0014 0,0046 0,0047 0,0002 0,0047

Avg 0,0002 0,0049 0,0048 0,0016 0,0047 0,0047 0,0006 0,0048

Std 6,63E-05 4,03E-05 3,88E-05 0,0001 4,99E-05 2,30E-05 0,0002 5,37E-05

lung Best 0,0002 0,0047 0,0045 0,0013 0,0044 0,0045 0,0005 0,0046

Avg 0,0003 0,0048 0,0047 0,0018 0,0046 0,0047 0,0011 0,0047

Std 0,0001 5,55E-05 5,67E-05 0,0002 7,69E-05 3,39E-05 0,0002 5,64E-05

lung discrete Best 0,0002 0,0042 0,0039 0,0012 0,0035 0,0038 0,0008 0,0039

Avg 0,0004 0,0047 0,0042 0,0019 0,0041 0,0042 0,0014 0,0043

Std 8,76E-05 0,0002 0,0001 0,0002 0,0002 0,0001 0,0003 0,0002

lymphoma Best 0,0521 0,05222 0,1086 0,1056 0,1086 0,1087 0,0522 0,1088

Avg 0,0543 0,09718 0,1088 0,1057 0,1087 0,1088 0,0901 0,1089

Std 0,0101 0,01807 6,22E-05 7,55E-05 6,83E-05 3,23E-05 0,0234 5,15E-05

nci9 Best 0,0845 0,1651 0,0875 0,0861 0,1698 0,0874 0,0843 0,1698

Avg 0,1478 0,2244 0,2314 0,2161 0,2217 0,2087 0,2203 0,2120

Std 0,0348 0,03755 0,0398 0,0342 0,0470 0,0402 0,0429 0,0461

Prostate GE Best 0,0003 0,0048 0,0047 0,0015 0,0046 0,0047 0,0006 0,0047

Avg 0,0006 0,0049 0,0048 0,0020 0,0048 0,0048 0,0015 0,0048

Std 0,0001 5,71E-05 4,05E-05 0,0003 5,40E-05 3,54E-05 0,0004 7,09E-05

SMK CAN 187 Best 0,0555 0,0348 0,0851 0,0554 0,1119 0,0852 0,0827 0,0852

Avg 0,0961 0,1010 0,1419 0,1417 0,1413 0,1326 0,1394 0,1466

Std 0,0170 0,0179 0,0182 0,0233 0,0201 0,0185 0,0241 0,0202

Table 4: Comparison of FDB-BEHO versus other optimization algorithms in terms of average classiﬁcation accuracy.

Dataset FDB-BEHO-V FDB-BEHO-S ABC SSA PSO GA EO ASO

CLL SUB 111 0,5606 0,5891 0,5517 0,5294 0,5267 0,5303 0,5472 0,5544

colon 0,8023 0,8137 0,7745 0,7418 0,7778 0,7794 0,7418 0,7696

leukemia 0,895 0,909 0,888 0,8641 0,8613 0,8711 0,8711 0,8838

lung 0,9358 0,948 0,9529 0,948 0,948 0,9515 0,9554 0,9583

lung discrete 0,7969 0,7661 0,8347 0,8039 0,8515 0,8347 0,8417 0,8389

lymphoma 0,8369 0,8710 0,8638 0,8503 0,8648 0,8555 0,8689 0,8596

nci9 0,4003 0,4134 0,4003 0,40458 0,4003 0,3905 0,4003 0,4003

Prostate GE 0,849 0,8265 0,8412 0,8569 0,8667 0,8716 0,8275 0,8343

SMK CAN 187 0,6571 0,6412 0,6402 0,6423 0,6439 0,6471 0,6767 0,6449

or machine learning methods to enhance its efﬁciency

and effectiveness further.

Moreover, the impact of different parameter set-

tings on BEHO’s performance could be subjected

to a more granular analysis to optimize its applica-

tion across various domains. Integrating advanced

machine learning techniques, such as deep learning,

within the BEHO framework could provide a deeper

understanding of the data structures and feature inter-

actions, potentially leading to more innovative solu-

tions and applications.

Additionally, parallel versions of BEHO could be

developed to leverage modern computational archi-

tectures, signiﬁcantly reducing the time required for

large-scale computations and making the algorithm

more practical for real-world applications that de-

mand rapid processing speeds.

As the ﬁeld of optimization continues to evolve,

BEHO’s ﬂexibility and robustness, particularly in its

binary incarnation with the FDB mechanism, position

it as a potent tool capable of making signiﬁcant con-

tributions to various scientiﬁc and engineering dis-

ciplines. The insights gained from this study pave

the way for more targeted and effective optimization

strategies, facilitating advancements in both theoreti-

cal research and practical applications.

Improved Binary Elk Herd Optimizer with Fitness Balance Distance for Feature Selection Using Gene Expression Data

795

Table 5: Comparison of FDB-BEHO versus other optimization algorithms in terms of average selected number of features.

Dataset FDB-BEHO-V FDB-BEHO-S ABC SSA PSO GA EO ASO

CLL SUB 111 2837,16 8502,69 5627,75 3042,80 5626,82 5638,17 2898,14 5645,71

colon 102,41 1008,31 943,96 446,08 944,82 943,82 269,14 962,33

leukemia 189,06 3473,45 3402,10 1184,12 3364,12 3391,89 450,22 3440,25

lung 130,45 1605,61 1561,96 626,45 1536,14 1558,65 366,61 1589,00

lung discrete 13,02 153,67 138,33 62,55 134,08 137,07 45,53 142,86

lymphoma 96,43 92,48 1850,75 636,55 1820,10 1879,19 84,02 1908,35

nci9 2172,55 2167,76 4797,90 2561,59 4800,18 4799,96 1876,92 4806,20

Prostate GE 387,94 2952,61 2890,92 1231,06 2878,24 2885,34 900,16 2919,59

SMK CAN 187 5522,69 15338,35 9969,24 5502,22 9944,98 9964,37 5124,49 10021,29

Table 6: p-values of the Wilcoxson test of FDB-BEHO-V versus other optimization algorithms (p ≥ 0.05 are underlined).

Dataset FDB-BEHO-S ABC SSA PSO GA EO ASO

CLL SUB 111 1,83E-06 6,15E-10 2,80E-09 1,77E-09 8,27E-10 1,05E-09 5,14E-10

colon 5,14E-10 5,13E-10 5,14E-10 5,14E-10 5,13E-10 5,46E-10 5,14E-10

leukemia 5,14E-10 5,13E-10 5,14E-10 5,14E-10 5,14E-10 6,53E-10 5,14E-10

lung 5,14E-10 5,14E-10 5,14E-10 5,14E-10 5,13E-10 5,14E-10 5,13E-10

lung discrete 5,04E-10 5,00E-10 5,07E-10 5,09E-10 4,98E-10 5,09E-10 5,08E-10

lymphoma 0,0608 5,14E-10 5,14E-10 5,13E-10 5,13E-10 4,17E-08 5,14E-10

nci9 0,0608 8,77E-10 6,92E-09 5,15E-10 5,15E-10 2,44E-08 5,15E-10

Prostate GE 5,14E-10 5,14E-10 5,14E-10 5,14E-10 5,14E-10 5,79E-10 5,14E-10

SMK CAN 187 6,08E-02 5,15E-10 2,23E-09 5,15E-10 1,05E-09 4,42E-09 6,93E-10

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