THOMPSON RIVERS UNIVERSITY

Advanced Wildfire Prediction with Machine Learning: Leveraging Liver
Cancer Algorithm and Spiral Updates for Feature Selection

By

MODAMORI OLUWAYOMI O

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Master of Science in Environmental Science

KAMLOOPS, BRITISH COLUMBIA

Febuary, 2025

Supervisor: Dr. Mohamed Tawhid, Department of Mathematics and Statistics,
Thompson Rivers University
Co-Supervisor: Dr. Emad Mohammed, Department of Physics and Computer Science,
Wilfrid Laurier University
Committee Member: Dr. Peter Tsigaris, Department of Economics, Thompson
Rivers University
External Examiner: Dr. Xiaoping Shi, Department of Computer Science,
Mathematics, Physics and Statistics, University of British Columbia

© Modamori Oluwayomi O, 2025

ABSTRACT
Wildfires threaten ecosystems, economies, and public health, particularly in
high-risk regions. Accurate wildfire prediction remains challenging due to complex
interactions among weather patterns, vegetation dynamics, climate change, and
human activities. This study investigates the role of advanced metaheuristic algorithms in optimizing feature selection for wildfire prediction across eight Canadian
provinces, focusing on improving accuracy and computational efficiency. We evaluate twelve algorithms, including Atom Search Optimization (ASO), Barnacles
Mating Optimizer (BMO), Chef-Based Optimization (CBO), Energy Valley Optimizer (EVO), Equilibrium Optimizer (EO), and Walrus Optimization Algorithm
(WOA), among others.
Key results highlight the superior performance of BMO and EVO, with BMO
achieving average recall rates of 77.55% in Alberta and 76.51% in Quebec, and
EVO attaining 76.96% and 78.30% in these provinces, respectively. In contrast,
ASO consistently underperformed, yielding recall rates as low as 44.67% in Ontario
and 51.99% in the Northwest Territories. Statistical analyses using Friedman
and Wilcoxon signed-rank tests confirmed significant differences in algorithmic
performance (p < 0.05), with spiral-enhanced variants of the Liver Cancer Algorithm
(LCA) outperforming the baseline LCA. Furthermore, Random Forest and Gradient
Boosting emerged as the most reliable prediction models, emphasizing the synergy
between optimized feature selection and robust machine learning frameworks.
A significant contribution of this research is the enhancement of the LCA
through spiral updates, specifically the Euler Spiral, which improves the balance
between exploration and exploitation in the search space. This enhancement
addresses the instability and slow convergence often associated with the standard
LCA. Although the spiral updates improved LCA’s performance, algorithms like
ii

iii
EVO and the Genetic Algorithm (GA) consistently outperformed LCA in recall
and overall predictive accuracy across provinces.
The findings of this study highlight the variability in algorithm performance
and emphasize the importance of tailoring wildfire prediction strategies to specific
regional conditions. This work not only evaluates the capabilities of advanced
metaheuristic algorithms but also identifies key environmental factors that predict
wildfire risk, offering practical insights to enhance wildfire risk management and
mitigation efforts.
Key Words: Wildfire; Machine Learning; Feature Selection; Model Optimization;
Liver Cancer Algorithm

Acknowledgements
I sincerely thank my supervisor, Dr. Mohamed Tawhid, Department of Mathematics and Statistics, Thompson Rivers University, for his guidance and expertise
in shaping this research. I am also grateful to my co-supervisor, Dr. Emad Mohammed, Department of Physics and Computer Science, Wilfrid Laurier University,
for his valuable feedbacks.
My sincere thanks go to Dr. Peter Tsigaris, Department of Economics, School
of Business and Economics, Thompson Rivers University, for his insightful contributions as a committee member; to Dr. Xiaoping Shi, External Examiner,
Department of Computer Science, Mathematics, Physics, and Statistics, University
of British Columbia, for their thorough review; and to Dr. Karl Larsen, Chair and
Program Coordinator, Natural Resource Sciences, Thompson Rivers University, for
his ongoing support and guidance.
I also acknowledge my classmates in the Environmental Science program for
their collaboration and insightful discussions. I am thankful to Compute Canada
for providing computational resources and to the ECMWF ERA5 and the World
Wide Lightning Location Network (WWLLN) for supplying critical data in this
wildfire prediction research. Additionally, I appreciate Thompson Rivers University
for its facilities and institutional support.
Finally, I express my deepest gratitude to my parents for their unwavering
encouragement throughout my graduate studies.

iv

Contents
1 Introduction
1.1

1

Background and Significance . . . . . . . . . . . . . . . . . . . . . .

2 Literature Review
2.1

7

Research History of Wildfire Science In Canada . . . . . . . . . . .

8

2.1.1

11

Related Work In Machine Learning and Wildfire Science . .

3 Methodology
3.1

3.2

1

26

Study Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1

Objective Of The Study . . . . . . . . . . . . . . . . . . . . 28

3.1.2

Hypotheses and Research Questions . . . . . . . . . . . . . . 29

Data Collection and Sources . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1

Data Types and Origin . . . . . . . . . . . . . . . . . . . . . 30

3.2.2

Preprocessing Steps . . . . . . . . . . . . . . . . . . . . . . . 34

3.3

Study Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4

Feature Selection and Metaheuristic Algorithm Implementation . . 43

3.5

3.4.1

Metaheuristic Algorithm Rationale . . . . . . . . . . . . . . 45

3.4.2

Algorithm Descriptions . . . . . . . . . . . . . . . . . . . . . 48

3.4.3

Mathematical Model of LCA . . . . . . . . . . . . . . . . . . 79

3.4.4

Algorithm Mechanics . . . . . . . . . . . . . . . . . . . . . . 83

Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 95

v

CONTENTS

vi

3.5.1

Spiral Implementation . . . . . . . . . . . . . . . . . . . . . 95

3.6

3.7

Evaluation Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.6.1

Mean Fitness Function . . . . . . . . . . . . . . . . . . . . . 129

3.6.2

Best Fitness Function

3.6.3

Worst Fitness Function . . . . . . . . . . . . . . . . . . . . . 131

3.6.4

Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . 131

3.6.5

Classification Accuracy (CA) . . . . . . . . . . . . . . . . . 132

3.6.6

Feature Selection Ratio (FSR) . . . . . . . . . . . . . . . . . 132

3.6.7

F-score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.6.8

Precision and Recall . . . . . . . . . . . . . . . . . . . . . . 133

. . . . . . . . . . . . . . . . . . . . . 131

Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.7.1

Wilcoxon Signed-Rank Test . . . . . . . . . . . . . . . . . . 134

3.7.2

Friedman Test . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3.7.3

Algorithm Performance Analysis . . . . . . . . . . . . . . . . 135

3.7.4

Regional Impact Analysis . . . . . . . . . . . . . . . . . . . 135

3.8

Expected Outcomes and Limitations . . . . . . . . . . . . . . . . . 136

3.9

Ethical and Practical Considerations . . . . . . . . . . . . . . . . . 137
3.9.1

Ethical Data Use . . . . . . . . . . . . . . . . . . . . . . . . 137

3.9.2

Practical Implications . . . . . . . . . . . . . . . . . . . . . 137

3.10 Summary of the Study Framework . . . . . . . . . . . . . . . . . . 137
4 Discussion
4.1

Overview of Study Objectives and Methodology . . . . . . . . . . . 139
4.1.1

4.2

139

Provincial Results . . . . . . . . . . . . . . . . . . . . . . . . 140

Provincial Result Summary . . . . . . . . . . . . . . . . . . . . . . 166
4.2.1

5 Conclusion

Key Insights Across Provinces . . . . . . . . . . . . . . . . . 169
172

CONTENTS

vii

5.1

Key Advancements . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.2

Environmental Variability . . . . . . . . . . . . . . . . . . . . . . . 173

5.3

Computational Constraints . . . . . . . . . . . . . . . . . . . . . . . 174

5.4

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

5.5

Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

List of Figures
3.1

Map of Study Area across Canadian Provinces . . . . . . . . . . . . 44

3.2 Visualization of the Archimedean Spiral . . . . . . . . . . . . . . . . 100
3.3 Visualization of the Euler Spiral . . . . . . . . . . . . . . . . . . . . 107
3.4 Visualization of the Fermat Spiral . . . . . . . . . . . . . . . . . . . 111
3.5 Visualization of the Golden Spiral . . . . . . . . . . . . . . . . . . . 116
3.6 Visualization of the Hyperbolic Spiral . . . . . . . . . . . . . . . . . 120
3.7 Visualization of the Lituus Spiral . . . . . . . . . . . . . . . . . . . 122
3.8 Visualization of the Logarithmic Spiral . . . . . . . . . . . . . . . . 128
4.1 Average Recall Ranks (Friedman Test) . . . . . . . . . . . . . . . . 144
4.2 Visualization of average recall rankings among algorithms in BC
(Friedman Test results). Lower ranks indicate better performance,
with several spiral-enhanced algorithms outperforming the LCA. . . 147
4.3 Visualization of average recall rankings among algorithms in Manitoba (Friedman Test results). Lower ranks indicate better performance, with several spiral-enhanced algorithms outperforming the
LCA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.4 Visualization of average recall rankings among algorithms in Northwest Territories (Friedman Test results). Lower ranks indicate better
performance, with several spiral-enhanced algorithms outperforming
the LCA.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

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LIST OF FIGURES

ix

4.5 Visualization of average recall rankings among algorithms in Ontario
(Friedman Test results). Lower ranks indicate better performance,
with several spiral-enhanced algorithms outperforming the LCA. . . 158
4.6 Visualization of average recall rankings among algorithms in Quebec
(Friedman Test results). Lower ranks indicate better performance,
with several spiral-enhanced algorithms outperforming the LCA. . . 160
4.7 Visualization of average recall rankings among algorithms in
Saskatchewan (Friedman Test results). Lower ranks indicate better
performance, with spiral-enhanced algorithms generally outperforming the LCA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.8 Visualization of average recall rankings among algorithms in Yukon
(Friedman Test results). Lower ranks indicate better performance,
with spiral-enhanced algorithms demonstrating significant improvements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.9 Wilcoxon Signed-Rank Test Results vs. LCA . . . . . . . . . . . . . 168

List of Tables
3.1

Summary of Features from FIRMS and ERA5 Data Sources . . . .

31

3.2

Comparison of Spiral Types for Feature Selection . . . . . . . . . . 130

4.1 Algorithm Rankings for Alberta . . . . . . . . . . . . . . . . . . . . 142
4.2

Spiral Algorithm Rankings for Alberta . . . . . . . . . . . . . . . . 143

4.3 Algorithm Rankings for British Columbia . . . . . . . . . . . . . . . 146
4.4

Spiral Algorithm Rankings for British Columbia . . . . . . . . . . . 146

4.5 Algorithm Rankings for Manitoba . . . . . . . . . . . . . . . . . . . 149
4.6

Spiral Algorithm Rankings for Manitoba . . . . . . . . . . . . . . . 150

4.7 Algorithm Rankings for Northwest Territories (NWT) . . . . . . . . 152
4.8

Spiral Algorithm Rankings for Northwest Territories (NWT) . . . . 153

4.9 Algorithm Average Recall and Rankings for Ontario . . . . . . . . . 155
4.10 Algorithm Rankings for Ontario . . . . . . . . . . . . . . . . . . . . 157
4.11 Algorithm Rankings for Quebec . . . . . . . . . . . . . . . . . . . . 159
4.12 Spiral Algorithm Rankings for Quebec . . . . . . . . . . . . . . . . 160
4.13 Algorithm Rankings for Saskatchewan . . . . . . . . . . . . . . . . . 161
4.14 Algorithm Rankings for Yorkton . . . . . . . . . . . . . . . . . . . . 164
4.15 Spiral Algorithm Rankings for Yorkton . . . . . . . . . . . . . . . . 166
4.16 Top 10 Most Selected Features by All Algorithms . . . . . . . . . . 171

x

List of Algorithms
1

Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1

Atom Search Optimization (ASO) for Feature Selection . . . . . . . 50

2

Barnacles Mating Optimizer (BMO) for Feature Selection . . . . . . 53

3

Chef-Based Optimization Algorithm (CBOA) for Feature Selection . 55

4

Equilibrium Optimizer (EO) for Feature Selection . . . . . . . . . . 57

5

Exponential Distribution Optimizer (EDO) for Feature Selection . . 60

6

Energy Valley Optimizer (EVO) for Feature Selection . . . . . . . . 63

7

Genetic Algorithm (GA) for Feature Selection . . . . . . . . . . . . 65

8

Golden Ratio Method (GRM) for Feature Selection . . . . . . . . . 68

9

Manta Ray Foraging Optimization (MRFO) for Feature Selection . 70

10

Particle Swarm Optimization (PSO) for Feature Selection . . . . . . 72

11

Walrus Optimization Algorithm (WaOA) for Feature Selection . . . 75

12

Liver Cancer Algorithm (LCA) for Feature Selection

13

Manta Ray Foraging Optimization (MRFO) for Feature Selection . 87

14

Particle Swarm Optimization (PSO) for Feature Selection . . . . . . 89

15

Walrus Optimization Algorithm (WaOA) for Feature Selection . . . 91

16

Liver Cancer Algorithm (LCA) with Archimedean Spiral Update. . 99

17

Liver Cancer Algorithm (LCA) with Detailed Euler Spiral Update

. . . . . . . . 85

for Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 104

xi

LIST OF ALGORITHMS
18

xii

Liver Cancer Algorithm (LCA) with Detailed Fermat Spiral Update
for Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 108

19

Liver Cancer Algorithm (LCA) with Detailed Golden Spiral Update
for Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 112

20

Liver Cancer Algorithm (LCA) with Hyperbolic Spiral Update for
Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

21

Liver Cancer Algorithm (LCA) with Detailed Lituus Spiral Update
for Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 121

22

Liver Cancer Algorithm (LCA) with Detailed Logarithmic Spiral
Update for Feature Selection . . . . . . . . . . . . . . . . . . . . . . 125

List of Equations
1 Wind speed calculation based on vector components. . . . . . . . . . . . 35
2 Calculation of wind direction in degrees. . . . . . . . . . . . . . . . . . . 36
3 Adjusted formula to align with 360° scale. . . . . . . . . . . . . . . . . . 36
4 Relative humidity based on the August-Roche-Magnus approximation. . 36
5 Fitness Function for Feature Subset Evaluation . . . . . . . . . . . . . . 46
6 Feature Ratio Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 47

xiii

Chapter 1
Introduction

1.1

Background and Significance

Wildfires are critical and recurring natural disasters and have a profound and
lasting influence on various aspects of life and the global environment. A wildfire is
an uncontrolled fire spreading through vegetative fuels, exposing and consuming the
natural and built environment in its path (Pyne et al. [1996]). While fire is a natural
component of many ecosystems, facilitating plant regeneration and nutrient cycling
(Santı́n and Doerr [2016]), its frequent and intense occurrence—often worsened by
human activities and climate change has far-reaching consequences (Prapas et al.
[2021]). Recent data highlights this escalating challenge: 2021 was notably one of
the harshest years in recent forest fire history, with a staggering 9.3 million hectares
of tree cover lost globally, a figure representing over a third of all tree cover loss that
year (Tyukavina et al. [2022], Potapov et al. [2017]). In 2022, Canada saw a decline
in wildfire activity, yet the situation remained concerning, with over 6.6 million
hectares lost to fires. By 2023, the severity of wildfires intensified dramatically, as
1

nearly 15 million hectares—an area proximate to the size of Portugal were consumed
(Weisse et al. [2022], Canada [2024]). This trend continued into 2024, with over 5.3
million hectares burned by September(CIFFCI [2024]). This activity significantly
contributed to the year’s global wildfire-induced carbon emissions, estimated at
around 2170 megatonnes, with emissions from Canadian wildfires alone constituting
a substantial portion of this total (Weisse et al. [2022]). As wildfires emit carbon
dioxide, particulate matter, and various hazardous gases like carbon monoxide
(CO), nitrogen oxides (NOx), and non-methane organic compounds (NMOC), they
heighten the greenhouse effect, contributing to a warming planet (Naeher et al.
[2007]). This release of carbon, especially in such large amounts, creates the feedback loop between wildfires and climate change. Increasing temperatures and drier
conditions fuel the likelihood and intensity of future fires, thus creating a cycle of
destruction and environmental degradation. These emissions also pose immediate
health risks, causing respiratory diseases, aggravating pre-existing cardiovascular
conditions, and have long-term implications ([Reid et al., 2016, Fadadu et al., 2021,
Noah et al., 2023, Mangual et al., 2024]). Wildfires also impact the interconnectedness of air quality, ecosystem health, and biodiversity. These events fundamentally
change habitats and ecosystems, affecting species composition, abundance, and
resilience, thus playing a pivotal role in shaping biodiversity in both immediate and
long-term contexts (Reisen et al. [2015], Jaffe et al. [2020], Burkle et al. [2015], Heil
and Burkle [2018], Gates et al. [2021]). Due to these effects, it is crucial to delve
into the roots of these destructive events. The leading causes of wildfires range from
natural phenomena, such as lightning strikes, to human activities, including land
clearing, campfires, and the improper disposal of cigarettes and faulty electric poles
(Halofsky et al. [2020]). For instance, between 2013 and 2020, Turkey witnessed
950 wildfires caused by lightning strikes (Sari [2023]). Similarly, from 1990 to 2016,
lightning was responsible for approximately 47 percent of all wildfire incidents in

2

Canada, highlighting its significant role in ignition (Tymstra et al. [2020]). These
stress the crucial impact of natural causes, particularly lightning, on wildfire genesis. They reveal the complexity and diversity of ignition sources across different
landscapes and highlight regional vulnerabilities influenced by topography and
forest cover characteristics. Climate change emerges as a critical factor, amplifying
conditions suitable for wildfires and influencing the ignitable materials available.
Alterations in climate directly impact fuel availability and moisture content; for
instance, during drought periods, potential fuels’ increased abundance and dryness
significantly elevate fire risk and intensity (Halofsky et al. [2020]). Understanding
the variables that escalate wildfire severity is essential. Environmental factors such
as drought conditions, elevated temperatures, strong winds, and low atmospheric
humidity are critical in determining the likelihood and intensity of wildfire events.
For example, numerous studies have identified temperature as a critical factor
influencing annual wildland fire activity, with warmer conditions correlating with
increased fire occurrences and severity (Coogan et al. [2021], Flannigan et al. [2005],
Hély et al. [2001]). This relationship underlines the significance of temperature as
a variable in fire dynamics, where higher temperatures can lead to drier conditions,
enhancing fuel flammability and the potential for more extensive and intense wildfires (Wasserman and Mueller [2023], Alizadeh et al. [2021]). Wildfire behavior is
influenced by a complex interplay of numerous environmental, meteorological, and
vegetation-related factors. This complexity presents challenges in identifying the
most critical features and accurately predicting wildfire occurrences. Given the
escalating frequency of wildfires and their devastating effects, developing reliable
prediction models has become increasingly important. These models are vital
for mitigating risks, reducing damage, improving emergency response plans, and
optimizing the allocation of firefighting resources. Machine-learning approaches
have emerged as powerful tools for constructing predictive models for wildfires

3

in this era of data abundance. However, the complicated relationships among
features in wildfire datasets often necessitate advanced techniques to handle their
complexity. Metaheuristic algorithms address this, as they excel in exploring large
search spaces and efficiently identifying optimal feature sets. These algorithms
enable the development of more accurate models by selecting the most relevant
features, thereby enhancing the predictive capabilities of machine-learning methods
in wildfire studies. In this research, we evaluate a range of metaheuristic algorithms,
including Atom Search Optimization (ASO), Barnacles Mating Optimizer (BMO),
Chef Base Optimization (CBO), Energy Valley Optimizer (EVO), Equilibrium
Optimizer (EO), Exponential Distribution Optimizer (EDO), Genetic Algorithm
(GA), Golden Ratio Method (GRM), Liver Cancer Algorithm (LCA), Manta Ray
Foraging Optimization (MRFO), Particle Swarm Optimization (PSO), Slime Mould
Algorithm (SMA), and Walrus Optimization Algorithm (WOA). Building on the
foundational understanding of the importance of predictive modelling in wildfire
management and the innovative use of metaheuristic algorithms, this research sets
out specific objectives to advance wildfire prediction. Our goals aim to improve
the field of wildfire prediction and offer insights that can enhance risk management
strategies. The primary aim is to leverage the strengths of selected metaheuristic
algorithms to boost the accuracy, efficiency, and real-world applicability of wildfire
prediction models. By doing so, we aim to contribute to more effective wildfire
prevention and control efforts.
Research Objectives:

1. To apply and evaluate a diverse range of advanced metaheuristic algorithms,
including Atom Search Optimization (ASO), Barnacles Mating Optimizer
(BMO), Chef Base Optimization (CBO), Energy Valley Optimizer (EVO),
Equilibrium Optimizer (EO), Exponential Distribution Optimizer (EDO),
4

Genetic Algorithm (GA), Golden Ratio Method (GRM), Liver Cancer Algorithm (LCA), Manta Ray Foraging Optimization (MRFO), Particle Swarm
Optimization (PSO), Slime Mould Algorithm (SMA), and Walrus Optimization Algorithm (WOA), for feature selection in wildfire prediction models.
This objective aims to determine which algorithms most effectively improve
model performance in predicting wildfire occurrences.
2. To develop and refine the Spiral Liver Cancer Algorithm to enhance the
standard LCA. It addresses limitations such as inefficient convergence and
instability caused by giant Levy flight steps. By introducing Spiral updates,
this objective seeks to enhance search precision and improve feature selection,
outperforming other metaheuristic algorithms in wildfire prediction tasks.
3. To identify the most influential variables in wildfire prediction through the
optimized feature selection processes provided by the metaheuristic algorithms.
A clearer understanding of these critical variables will improve model accuracy
and efficiency in predicting wildfire events.
4. To perform a comprehensive comparative analysis of the selected algorithms
regarding prediction accuracy, computational efficiency, and scalability. This
objective aims to pinpoint the most effective approaches for real-world applications in wildfire prediction, contributing to more informed risk management
strategies.
Ultimate Goal: The primary goal of this research is to advance wildfire prediction
by establishing effective feature selection techniques through a detailed evaluation
of metaheuristic algorithms. By accomplishing this, the study aims to:
• Set a benchmark for the most efficient metaheuristic algorithms in wildfire
prediction, enhancing the accuracy and robustness of predictive models.
5

• Improve the Spiral Liver Cancer Algorithm (LCA) to address the limitations of
traditional approaches, ultimately outperforming other algorithms in precision
and convergence.
• Identify the key variables most significantly influencing wildfire prediction,
enabling more efficient and streamlined model development.
• Provide practical guidelines for selecting the most appropriate algorithms for
real-world applications, supporting better risk management strategies and
informed decision-making in wildfire prevention and control.

By focusing on these objectives, this research seeks to make a meaningful
contribution to the growing wildfire prediction and management field. Through the
insights gained, the study aims to enhance predictive accuracy and provide valuable
information that supports strategic decision-making in wildfire mitigation efforts.
The following chapters are designed to explore the research in a structured and
detailed manner. Chapter 2 provides a thorough literature review, situating this
study within the broader context of wildfire prediction research and identifying the
existing gaps this work aims to address. Chapter 3 outlines the research design and
methodology, particularly concerning the selection and application of metaheuristic
algorithms for feature selection and developing and refining the Spiral Liver Cancer
Algorithm (LCA). Chapter 4 presents the experimental results and offers a detailed
analysis of the findings, discussing their practical applications and theoretical
implications in the context of wildfire prediction. Finally, Chapter 5 concludes the
thesis by reflecting on the research outcomes, their contributions to the field, and
the opportunities for future research. This final chapter highlights how this study
paves the way for further advancements in wildfire prediction strategies and risk
management.

6

Chapter 2
Literature Review
Amid the escalating challenges posed by climate change, wildfires have become
increasingly frequent and destructive, highlighting the critical need to enhance
predictive capabilities. This literature review explores the evolving field of wildfire prediction and management, focusing on integrating advanced metaheuristic
algorithms and machine learning to improve prediction accuracy and efficiency. As
we delve deeper into the complexities of wildfire prediction, it becomes essential
to fully leverage the power of machine learning—an area of artificial intelligence
that enables systems to learn from data and make informed predictions. By combining machine learning models with metaheuristic algorithms, including Atom
Search Optimization (ASO), Barnacles Mating Optimizer (BMO), Chef Base Optimization (CBO), Energy Valley Optimizer (EVO), Equilibrium Optimizer (EO),
Exponential Distribution Optimizer (EDO), Genetic Algorithm (GA), Golden Ratio
Method (GRM), Liver Cancer Algorithm (LCA), Manta Ray Foraging Optimization
(MRFO), Particle Swarm Optimization (PSO), Slime Mould Algorithm (SMA),
and Walrus Optimization Algorithm (WOA), represents a promising approach to

7

refining and improving the accuracy of wildfire prediction models.
The review is organized into three key sections. First, the ”Research History of
Wildfire Science” traces the evolution of wildfire research, providing the necessary
context to appreciate the novel contributions of contemporary algorithms. Next,
the ”Application of Machine Learning in Wildfire Prediction” delves into how these
advanced algorithms are applied to analyze large datasets, enabling the identification
of patterns and predictors with greater precision. Finally, the ”Optimizing Wildfire
Prediction Models through Advanced Feature Selection” section highlights the
importance of selecting relevant features for improving predictive accuracy and
operational efficiency. This part critically examines the role of metaheuristic
algorithms in refining feature selection, comparing their effectiveness and discussing
their practical implications for wildfire management.

2.1

Research History of Wildfire Science In

Canada
Before structured scientific studies on wildland fires began, Indigenous communities
in Canada had developed advanced fire management practices, Such as controlled
burns to manage ecosystems and protect communities, reflect a deep understanding of wildfire dynamics, nurtured over millennia of living in harmony with their
environment (Hoffman et al. [2022], Huffman [2013], Christianson [2015]). This
knowledge contrasts sharply with the fire suppression strategies introduced by
European settlers. When European settlers arrived, their view of forest resources
was mainly economic, with little attention to sustainable management or fire prevention. Early milestones in organized wildfire management include creating the

8

Timber, Mines, and Grazing Branch in 1882 and introducing forest rangers in 1883
(Johnstone [1991]). Quebec’s 1883 initiative for forest rangers and Ontario’s 1895
Forest Reserves Act further highlighted the growing need for structured wildfire
prevention and management strategies (Rajala [2005]). In the 1920s, the formal
study of wildfire science began to take shape, aided by the political climate under
Mackenzie King’s Liberal government. A national forest protection conference in
1924 marked a turning point, signalling the government’s interest in fire control
and hinting at a national forestry policy. However, progress toward a comprehensive policy stalled due to political concerns about federal-provincial relations.
This resulted in the provinces taking control of natural resource management,
leaving federal efforts, such as the Dominion Forest Service under Director E.H.
Finlayson, in a precarious position. The Great Depression further strained resources, impacting both forestry research and management. Despite these setbacks,
significant progress was made in wildfire science during this period, particularly
at the Petawawa experiment station. Researchers James G. Wright and Herbert
W. Beall studied how weather conditions impacted forest flammability, creating
foundational knowledge to fuel further nationwide research efforts (Stocks2 et al.
[1989], Paul [1969]). These studies led to the creation of the Canadian Forest Fire
Weather Index (FWI) System in 1970 and the Forest Fire Behavior Prediction
(FBP) System in the 1980s (Paul [1969], McAlPINE et al. [1990], Wagner [1987]).
Building on years of empirical research and technological advancements, these
systems revolutionized wildfire prediction and management in Canada, leading to
more precise and effective fire management strategies (Groot [1987], Stocks2 et al.
[1989]). Another major development in wildfire prediction came in 1972 with the
creation of Rothermel’s Fire Spread Model. Although not developed in Canada,
Rothermel’s model globally impacted wildfire behaviour modelling, including in
Canada (Andrews [2018]). This model introduced standardized fuel types and

9

accounted for wind and slope conditions, allowing for more accurate fire behaviour
predictions. The Canadian FBP System integrated principles from Rothermel’s
work, adapting them to Canadian wildfires, and it remains a central tool in the
country’s fire management strategy. The development of wildfire prediction models
in Canada did not stop there. In 1999, the Prometheus project aimed to create
a more advanced simulation model for fire growth, building on the strengths of
the FBP System. The goal was to integrate wave propagation theories, which
allowed the model to simulate fire spread in more complex environments (Tymstra
et al. [2010], Barber et al. [2009]). Prometheus marked a significant leap in wildfire
modelling, combining detailed fire behaviour predictions with sophisticated mathematics, making it one of Canada’s most important contributions to wildfire science.
Despite the complexity of models like the FBP System and Prometheus, wildfires
remain challenging to predict accurately, especially given Canada’s evolving climate
and various vegetation types. These models, while robust, face limitations in
addressing the unpredictable variables introduced by climate change. A significant
advancement in wildfire science has been the growing understanding of how weather
affects wildfire behaviour. Studies have shown that temperature, lightning, wind,
and humidity are critical in determining wildfire spread and intensity (Flannigan
et al. [2005], Flannigan and Harrington [1988], Wotton [2009]). In particular,
rising temperatures have been strongly linked to increased wildfire-affected areas
(Flannigan et al. [2005], Jolly et al. [2015], BALSHI et al. [2009]). Additionally,
higher temperatures contribute to frequent lightning strikes, a common cause of
wildfires. Research suggests that a one-degree Celsius rise in temperature could
result in a 5-100% increase in lightning activity, raising the risk of wildfires (Romps
et al. [2014]). As our understanding of how weather influences wildfires deepens,
the need for more dynamic fire models becomes clear. Traditional models, while
effective, often struggle to integrate real-time weather data or adapt to rapidly

10

changing meteorological conditions. The field is shifting towards Next Generation
Fire Modeling in response to these challenges. These advanced models aim to build
upon frameworks like the FBP System by incorporating more complex variables,
including refined climate change projections. Such innovations are designed to
improve the accuracy of predictions and provide better tools for managing wildfires
in an era of increasing unpredictability. The history of wildfire science in Canada
shows a continuous evolution. From the early development of the FBP System to
the incorporation of advanced meteorological data in Next-Generation Fire Modeling, each advancement has responded to emerging challenges in wildfire dynamics.
Next-generation fire Modeling represents a critical advancement, integrating vast
amounts of data and employing sophisticated algorithms to better predict and
manage wildfires in a world where the risks continue to grow.

2.1.1

Related Work In Machine Learning and Wildfire
Science

As the field of wildfire science progresses, it is crucial to note the emerging role of
machine learning (ML) and artificial intelligence (AI) in reshaping our approach
to understanding and managing wildfires. Integrating operations research, ML,
AI, and other digital technologies marks a new era in wildfire science, offering
unprecedented capabilities to analyze, predict, and respond to wildfire events.
Machine learning, a subset of artificial intelligence, is defined as a set of methods that
automatically detect patterns in data, facilitating the discovery of significant insights
and enabling predictive models to anticipate future data trends or behaviours
(Murphy [2012]). Machine learning’s role within AI is akin to the brain’s function
within the human body, serving as a critical component that drives the intelligence
and adaptability of AI systems. Three types of machine learning are pivotal
11

in analyzing and interpreting complex data sets: supervised, unsupervised, and
reinforcement learning (Ivan et al. [2019]). Supervised learning involves training a
model using a dataset where the desired output signals (labels) are already known,
allowing the algorithm to learn by example (Kolosova and Berestizhevsky [2020],
Bateman et al. [2020]). This methodology is particularly effective in wildfire science
for predicting specific outcomes, such as fire spread or ignition likelihood, based on
historical data where the results are already understood. Conversely, unsupervised
learning does not rely on pre-labeled data. Instead, it attempts to identify patterns
or intrinsic structures within the data itself (Geoffrey and Terrence J. [1999]). This
approach can be instrumental in uncovering hidden patterns in wildfire occurrences
or distributions, providing insights that might not be immediately evident through
traditional analysis methods. Then, there is reinforcement learning, which stands
out for its decision-making prowess. By interacting with an environment, the
model learns to make a series of decisions, receiving feedback regarding rewards or
penalties (Richard S. and Andrew G. [1998]). This dynamic learning process is akin
to strategizing in complex and uncertain conditions, like developing adaptive fire
suppression strategies in wildfire management, where the model iteratively improves
its decision-making capability based on past outcomes. Having explored the core
methodologies of machine learning—supervised, unsupervised, and reinforcement
learning—we can now delve into how these techniques are applied to various facets
of wildfire science, each playing a pivotal role in enhancing our understanding and
management of wildfires. With their unique capabilities, these machine-learning
approaches offer invaluable tools across different dimensions of wildfire research,
from characterizing fire behaviour and assessing burn severity to improving fire
detection and predicting occurrences. By applying these methods to specific wildfirerelated challenges, researchers and practitioners can extract deeper insights, make
more informed decisions, and develop more effective strategies for wildfire mitigation

12

and management. Now, let us examine the nuanced applications of machine learning
within critical areas of wildfire science, starting with the characterization of fire
behaviour, moving through the estimation of burn areas and severity, advancing to
the precision of fire detection, and understanding the patterns of fire occurrence.

Fire Fuel Characteristics

Understanding fire fuels is critical to accurately predicting wildfire behaviour (Nolan
et al. [2016]). Fuel type, moisture content, density, and arrangement significantly
influence how fires ignite, spread, and intensify. Tools such as the Fire Weather
Index (FWI) have long provided crucial insights into fuel characteristics, supporting
wildfire research and management (Heacute;ly et al. [2001], Emilio [2003], Wagner
[1987]). These insights are vital for developing effective fire management strategies,
refining predictive models, and ultimately minimizing the impact of wildfires. Recent
advancements in remote sensing and machine learning have greatly improved our
ability to assess fire fuels on a large scale. For example, Zhu et al. (2021) used a
deep learning approach with a temporal convolutional neural network (TempCNNLFMC) to estimate Live Fuel Moisture Content (LFMC) across the United States,
achieving notable accuracy with an RMSE of 25.57 percent, which offers valuable
data for real-time fire danger assessments (Zhu et al. [2021]). Similarly, Rivera et al.
(2023) developed machine learning models to predict Critical Heat Flux for Ignition
(CHFI) based on solid fuel properties, using over 2,000 experimental datasets and
achieving high predictive accuracy (Rivera et al. [2023]). In the Mediterranean
region, Cunill et al. (2022) applied a Random Forest algorithm to map LFMC,
achieving solid results and demonstrating how these methods can improve fire
danger assessments (Cunill Camprubı́ et al. [2022]). Similarly, Costa-Saura et al.
(2021) used Sentinel-2 data combined with meteorological variables to model LFMC
13

variability across Spain, showing that seasonal changes in LFMC are crucial for
predicting fire risk (Costa-Saura et al. [2021]). However, while moisture content
is essential, it is only one part of the equation. Other critical factors—fuel type,
density, and arrangement—also play a crucial role in determining wildfire behaviour.
These attributes, shaped by the local vegetation and landscape, influence how fires
ignite, spread, and increase in intensity. Fuel type significantly impacts wildfire
behavior due to the varying flammability of different vegetation. Coniferous forests,
for instance, are highly flammable because of their resinous composition, which
encourages rapid fire spread. Hayes et al. (2024) found that in burned boreal
forests, fuel constraints significantly limit fire behavior, making coniferous species
particularly vulnerable to repeated, intense fires due to their flammability and the
fuel they provide over time (Hayes et al. [2024]). Likewise, Abdollahi and Yebra
(2023) noted that shrublands are composed of refined, dry fuels that ignite easily,
especially under dry conditions, leading to faster and more intense fire spread
(Abdollahi and Yebra [2023]). The density of vegetation also plays a significant
role in fire intensity. Denser forests, such as those with high biomass density, tend
to support more severe fires. For example, Boyd et al. (2023) observed that dense
boreal forest ecosystems in Alaska, with large amounts of combustible material,
including canopy and ground fuels, result in higher fireline intensity and faster
surface fire spread (Boyd et al. [2023]). Albini (1976) also found that dense fuel
loads contribute to longer flame lengths and higher heat outputs, which in turn
increases fire severity (Station [1976]). On the other hand, Rothermel (1972) noted
that forests with lower fuel density, such as open woodlands, tend to experience
less intense fires due to reduced fuel loads, leading to quicker burn-out times and
lower flame heights (Station [1972]). This shows that fuel type and the amount
of available fuel are crucial in determining how wildfires behave. Finally, the
arrangement of fuels—both vertical and horizontal—within an ecosystem is critical

14

in determining how fires move through the landscape. Forests with continuous
vertical structures, from ground vegetation to the canopy, create a pathway for
fires to transition from surface fires to more dangerous crown fires. Myroniuk et al.
(2023) found that vegetation’s vertical and horizontal continuity, particularly in the
Chornobyl Exclusion Zone, allows for faster fire spread. This highlights how the
spatial arrangement of fuels and their type and density can significantly influence
the intensity and unpredictability of wildfire spread (Myroniuk et al. [2023]).

Burn Areas and Fire Severity

Wildfire research has advanced significantly with new methods for analyzing and
understanding the severity of burned landscapes. In recent years, satellite imagery
and machine learning have emerged as essential tools for quantifying the impact
of wildfires and improving the accuracy of fire severity assessments. Satellites like
Landsat-8, Landsat-9, and Sentinel-2 have become central to wildfire monitoring.
These platforms capture high-resolution, multispectral data that allows researchers
to differentiate between burned and unburned areas and categorize fire severity
(McCorkel et al. [2018], Sigurdsson et al. [2022], Claverie et al. [2018], nas [2023a]).
This data is crucial for assessing large, remote regions affected by fires, providing a
comprehensive view of the aftermath. Machine learning has taken wildfire analysis
to new heights, making burn severity mapping more accurate and efficient. For
instance, Collins et al. (2018) found that a Random Forest classifier achieved
over 95 percent accuracy in identifying major burn areas and 74 percent for lower
severity burns using Landsat imagery. Similarly, Pu et al. (2004) used logistic
regression and neural network models, achieving a remarkable 97 percent accuracy
in identifying burned regions. These models underscore the potential of machine
learning to enhance fire severity mapping by reducing the limitations of traditional
15

manual approaches. Combining different types of data has further refined these
assessments. In Indonesia, researchers integrated Sentinel-1 radar and Sentinel-2
optical imagery, improving classification accuracy for burned areas to 91.80 percent
and 95.80 percent, depending on the model used (Arjasakusuma et al. [2022]).
This shows the power of using diverse data sources to create a more detailed
understanding of wildfire impacts. Cloud-based platforms like Google Earth Engine
are also gaining traction in fire mapping. Nelson et al. (2024) highlighted how these
platforms could handle large-scale data processing, which is critical to tracking
burn severity across vast landscapes and improving real-time fire management
strategies (Nelson et al. [2024]). This capability is critical as we seek to automate
and streamline wildfire monitoring efforts. Further work by Khankeshizadeh et al.
(2024) combined Sentinel-1 radar with Sentinel-2 optical data, using deep learning
techniques to achieve high accuracy in mapping forest burns (Khankeshizadeh et al.
[2024]). The ability of machine learning to blend different types of data makes it
an invaluable tool in the evolving landscape of wildfire research. Other research
has shown that machine-learning techniques are adaptable to various ecosystems.
Mitrakis et al. (2012) used a self-organizing neuro-fuzzy classifier to achieve over
95 percent accuracy with LANDSAT-5 TM imagery, outperforming traditional
methods like neural networks and support vector machines. Tonbul et al. (2022)
also employed ensemble learning algorithms like Random Forest and Canonical
Correlation Forest (CCF), achieving robust results in both pixel-based and objectbased burn severity classification (Tonbul et al. [2022]). These studies illustrate
the versatility of machine learning for mapping fire-affected regions, regardless of
ecosystem type. Recent advancements in deep learning have taken these techniques
even further. Hu et al. (2023) used UNet and Attention UNet models to successfully
segment burned areas in Landsat data, with a mean Intersection over Union (mIoU)
of 0.78 and a Kappa coefficient approaching 0.90 (Hu et al. [2023]). In Alaska, deep

16

neural networks applied to MODIS data overcame class imbalance issues, achieving
a recall score of 0.96 for identifying wildfire pixels (Langford et al. [2018]). One
notable deep learning development is Seydi et al. (2022) Burnt-Net model, which
uses Sentinel-2 data and deep learning morphological networks to map burned
areas with an impressive accuracy of over 97 percent (Seydi et al. [2022]). This
technology shows how post-fire assessments are becoming increasingly automated
and precise. These developments collectively highlight a transformative phase in
burn severity mapping, where machine learning and deep learning provide more
nuanced insights and operational efficiencies. Machine learning and deep learning
are transforming how we assess and respond to wildfires, offering more nuanced
insights and improving analysis efficiency. As these techniques evolve, they can
revolutionize fire severity mapping, real-time monitoring, and long-term wildfire
management strategies. This integration of advanced technology is crucial for
better preparing for the increasing wildfire risks posed by climate change.

Fire Occurrence and Detection

Accurate detection and monitoring of fire occurrences are essential for wildfire
management and environmental conservation (Thapa et al. [2021]). Advances in
remote sensing technology and computational methods have greatly enhanced our
ability to detect and predict fires with increasing speed and accuracy. Historically,
fire detection relied primarily on ground-based observations and reports. While
valuable, these methods were limited in scope and often delayed. Satellite technology has revolutionized the field by providing real-time data over large and remote
areas (Barmpoutis et al. [2020]). Satellites equipped with thermal sensors, such as
MODIS (Moderate Resolution Imaging Spectroradiometer) and VIIRS (Visible et
al.), have become essential for identifying heat signatures from active fires, offering
17

critical information for timely responses (nas [2023b]). Before the widespread
use of satellite technologies, early mathematical models aimed at predicting fire
occurrences were developed. For example, (Burrows et al. [1995]) applied a negative
binomial model related to fire danger ratings, while (Cunningham and Martell
[1973]) utilized a Poisson model based on fuel moisture levels. These early statistical models laid the groundwork for more advanced methods in fire dynamics
research (Taylor et al. [2013]). Machine learning (ML) methods have emerged as
prominent tools in predicting fire occurrences and detecting events in recent years.
Standard ML techniques include Random Forests (RF), Support Vector Machines
(SVMs), Artificial Neural Networks (ANNs), and deep learning models such as
Convolutional Neural Networks (CNNs). These advanced computational techniques
offer greater accuracy and efficiency in analyzing complex datasets for fire detection
and prediction, significantly advancing the field from earlier approaches. One
early application of artificial neural networks (ANN) in wildfire science came in
1996 when (Vega-Garcia [1996]) applied ANN to predict human-caused wildfires in
Alberta, Canada. Their model achieved an accuracy rate of 85 percent in predicting
areas without fire and 78 percent in identifying areas with fire. Shortly afterward,
in 2002, (Alonso-Betanzos et al. [2003]) used neural networks to estimate fire risk
levels in Galicia, Spain, a wildfire-prone region. Their feedforward neural network
model, with a 6-9-1 topology, performed well in classifying fire risks into four
categories, with an accuracy of 77.2 percent. Similarly, the Fire Ignition Index
(FII) was developed using neural networks, synthesizing indices such as the Fire
Weather Index (FWI), Fire Hazard Index (FHI), and Fire Risk Index (FRI). The
FII model showed high correct classification rates for fire instances (Vasilakos et al.
[2007]). (Dutta et al. [2013]) analyzed ten neural network models to predict fire
incidences based on climate data. The study found that the Elman Neural Network
was the most effective, achieving an average accuracy of 94 percent, with other

18

models like the Time Delay Neural Network and Recurrent Neural Network also
showing strong performance with accuracies of 88.21 percent and 92.77 percent,
respectively. While ANNs are widely used in fire prediction, other models have
also proven effective. For instance, SVMs have been applied to deduce fire hazard
levels from meteorological data, demonstrating the versatility of machine learning
methods (Sakr et al. [2010]). In one study, (Sakr et al. [2011]) compared ANN and
SVM methods for forest fire prediction using minimal weather parameters such as
relative humidity and cumulative precipitation, showcasing the potential to reduce
costs and complexity. Additionally, (Yu et al. [2005]) developed a real-time forest
fire detection and prediction system using a neural network within a wireless sensor
network, emphasizing the importance of quick and accurate responses to emerging
fire threats. In recent years, advanced neural network architectures, including
CNNs and multilayer perceptrons (MLPs), have been widely adopted for global
wildfire susceptibility prediction. Studies such as (Zhang et al. [2021]) offer insights
into their comparative effectiveness and explore the interpretability of CNNs in
this context. Random Forest (RF) use in fire occurrence prediction has gained
momentum since 2012. Stojanova et al. [2012] conducted an extensive analysis
comparing multiple machine learning techniques for predicting wildfire incidents,
using a dataset that combined geographical, remote sensing, and meteorological
data from Slovenia. Their study compared several classifiers, including K-Nearest
Neighbors (KNN), Naı̈ve Bayes, Decision Trees (J48 and jRIP), Logistic Regression
(LR), SVM, and Bayesian Networks (BN). Another study by (Oliveira et al. [2012])
compared Multiple Linear Regression and RF methods to assess fire occurrence
predictors in Europe, finding that RF provided superior predictive accuracy. The
study also identified vital variables such as precipitation, soil moisture, and socioeconomic factors as crucial for determining fire risk.

19

Feature Selection

In machine learning and predictive modelling, feature selection is a crucial process
that significantly influences model performance, interpretability, and generalization.
This process involves identifying and selecting a subset of relevant features (variables
or predictors) for model construction, simplifying the models, reducing overfitting,
and improving model robustness.
Feature selection is crucial in machine learning and predictive modelling, especially in complex domains like wildfire prediction. The process involves selecting a
subset of relevant variables to improve prediction models’ accuracy, efficiency, and
interpretability. This step is crucial in wildfire prediction due to the high dimensionality of environmental, meteorological, and topographical data. While feature
selection research in wildfire prediction is somewhat limited, it plays a pivotal role in
identifying which environmental, meteorological, and topographical variables most
significantly impact fire incidents and their spread. For example,Sadrabadi and
Innocente [2023] demonstrated the application of feature selection techniques across
various machine learning models such as Random Forest (RF), Extra Trees (XT),
XGBoost (XGB), and CatBoost (CatB) in the context of wildfire prediction. The
analysis revealed that certain features, including terrain slope and specific humidity
indices, were consistently deemed less necessary across all models, suggesting their
limited influence on fire prediction accuracy. Furthermore, Fernández-Garcı́a et al.
[2022] underscores the Normalized Difference Water Index (NDWI) as a pivotal
predictor for assessing burn severity. Notably, NDWI, when applied with a cubic
predictive model, demonstrated the most substantial correlation with burn severity
among the evaluated variables. This relationship is characterized by an inverse
association at intermediate NDWI values, as evidenced by the highest nRMSE and
R² metrics, pinpointing NDWI’s critical role in burn severity modelling. Building
20

upon that Wang et al. [2024] study, trained their model on sixteen features derived
from detailed meteorological, human activity, topography, fuel characteristics, and
geographical data, which was consistently selected across models for their predictive relevance. These features include average daily mean, maximum, and dew
point temperatures (Tmean et al., respectively), maximum vapour pressure deficit
(VPDmax), building coverage (BldgCover), slope, digital elevation model (DEM),
and the Normalized Difference Vegetation Index (NDVI). Furthermore, Tracy et al.
[2018] novel approach in ecological niche modelling for wildfire prediction leverages
a unique random subset feature selection algorithm (RSFSA) to optimize the
variable selection. Wang et al. (2024) took a similar approach by focusing on
16 features derived from meteorological data, human activity, topography, fuel
characteristics, and geographical information. Their model emphasized the predictive power of variables like average daily mean temperature (Tmean), maximum
vapour pressure deficit (VPDmax), building coverage (BldgCover), and Normalized
Difference Vegetation Index (NDVI), demonstrating the importance of both human
and environmental factors in predicting wildfire behaviour (Khosravi et al. [2023]).

Metaheuristic Algorithms for Enhancing Predictive Accuracy in Wildfire Science

Metaheuristic algorithms have significantly advanced the accuracy of wildfire prediction models by optimizing critical parameters that capture complex fire behaviours.
These algorithms are integral to modern fire management strategies, offering rapid,
reliable predictions for timely interventions. A landmark study by Pereira et al.
(2024) demonstrated the potential of Genetic Algorithms (GA), Particle Swarm
Optimization (PSO), and Differential Evolution (DE) to refine the Rothermel
model, a well-established wildfire spread predictor. This study optimized fuel
21

moisture, vegetation density, and wind speed. Among these, DE proved remarkably
effective, achieving high precision with minimal adjustments, a crucial advantage
in dynamic wildfire scenarios (Pereira et al. [2024]). Expanding on these findings,
Jaafari et al. (2019) explored the integration of metaheuristics with fuzzy logic
to enhance wildfire predictions further. They developed a hybrid model using
the Adaptive Neuro-Fuzzy Inference System (ANFIS) combined with GA and
the Firefly Algorithm (FA). The GA-ANFIS model, achieving an Area Under
the Curve (AUC) of 0.92, illustrated the enhanced capability of this approach to
model complex environmental interactions affecting wildfire risks (Jaafari et al.
[2019]). In 2022, Pereira and colleagues revisited these techniques to concentrate
on reducing prediction errors. Once again, DE stood out for its efficiency, adjusting
the Rothermel model to deliver optimal predictions rapidly, aiding fire managers
in resource allocation (Pereira et al. [2022]). Recent studies have shifted focus
towards identifying the most influential factors for wildfire risk. In 2023, Zhang et
al. developed a GA-optimized neural network that pinpointed critical predictors
such as Historical Fire Density (HFD), Vegetation Type (VT), and Average Annual Temperature (AAT). This model’s accuracy reached 83.7%, underscoring the
importance of targeted feature selection in enhancing predictive precision (Zhang
et al. [2023]). Advanced metaheuristic techniques were also applied to optimize
neural networks and hybrid models. Nur et al. (2022) utilized Grey Wolf Optimizer
(GWO) and Imperialist Competitive Algorithm (ICA) to optimize a Convolutional
Neural Network (CNN) for mapping wildfire susceptibility, achieving an AUC of
0.974 and RMSE of 0.334 Nur et al. [2022]. Similarly, AlFugara et al. (2021)
combined Support Vector Regression (SVR) and Adaptive Neuro-Fuzzy Inference
System (ANFIS) with Whale Optimization Algorithm (WOA) and Simulated Annealing (SA), achieving AUROC values up to 0.965 Al-Fugara et al. [2021]. Azizi
et al. (2023) also leveraged metaheuristics for hybrid feature selection in Iran,

22

identifying key predictors for enhancing regional wildfire risk mapping Azizi et al.
[2023]. Adapting metaheuristics for specific regional conditions, Dieu Tien and
colleagues tailored a Particle Swarm Optimization Neural Fuzzy (PSO-NF) model
for Vietnam’s unique environmental settings, such as high rainfall and dense vegetation. Their model achieved an AUC of 0.932, highlighting the adaptability of PSO
to diverse ecological landscapes (Bui et al. [2017]). In a recent innovation, Tran
et al. (2024) utilized emerging algorithms like Whale Optimization, Black Widow
Optimization, and Butterfly Optimization in conjunction with XGBoost to predict
wildfire risks in Hawaii. Their best model, BWO-XGBoost, achieved an AUC of
0.9269, showcasing the robustness of combining advanced algorithms with machine
learning techniques to tackle the challenges of wildfire prediction across varied
terrains (Tran et al. [2024]). Furthermore, Gharehchopogh et al. [2022] introduced
the Random Subset Feature Selection Algorithm (RSFSA) as a novel approach for
ecological niche modelling in wildfire prediction. This algorithm was particularly
effective in identifying terrain slope and humidity indices as key predictors influencing wildfire occurrences. By systematically evaluating subsets of features, RSFSA
enabled the refinement of predictive models by isolating variables with the highest
impact on ecological niche dynamics. Metaheuristic algorithms add another layer
to these models, improving prediction accuracy and optimizing model parameters.
Algorithms like Genetic Algorithms (GA), Particle Swarm Optimization (PSO),
and Differential Evolution (DE) have become essential in fine-tuning models for
wildfire prediction. Hybrid models, such as ANFIS-GA, PSO-NF, and GA-BPNN,
which combine metaheuristics with fuzzy logic or machine learning, have shown
great promise in the real-time mapping of wildfire susceptibility. These models help
manage the complex environmental factors that drive wildfire behaviour despite
the advancements and diverse applications of metaheuristic algorithms in wildfire
science. Furthermore, implementing spiral dynamics in metaheuristic algorithms

23

for feature selection in wildfire science has been limited. Spiral optimization methods, known for their ability to escape local optima and explore the solution space
more thoroughly, could significantly improve modelling the complex interactions
of variables that influence wildfire behaviour and risks. Although spirals have yet
to be used with metaheuristics in wildfire science, their demonstrated success in
other fields provides compelling reasons to consider their application. For instance,
the Linear Adaptive Spiral Dynamics Algorithm (LASDA) dynamically adjusts
spiral parameters to enhance convergence and fitness accuracy, as demonstrated
in system modelling. By leveraging adaptive linear spirals, LASDA has shown its
utility in identifying parameters in flexible systems Nasir et al. [2016]. Similarly, the
Spiral Search Mechanism integrated into the Equilibrium Optimizer (EO) employs
logarithmic spirals to improve the diversity and robustness of search processes.
This approach has proven effective in path planning for mobile robots. It highlights
its potential to balance exploration and exploitation Ding et al. [2023], and Spiral
Dynamics Optimization (SDO) uses logarithmic spirals to address global optimization challenges by balancing diversification and intensification. Its applications
in engineering optimization further showcase its adaptability to high-dimensional
problems Omar et al. [2022]. Furthermore, the Hybrid Particle Swarm Optimization
with Spiral-Shaped Mechanism (HPSO-SSM) tackles the ”curse of dimensionality”
by employing geometric spirals for optimizing feature subsets, significantly improving classification accuracy in high-dimensional datasets Xie et al. [2021]. Despite
the advancements and diverse applications of metaheuristic algorithms in wildfire
science, specific gaps present opportunities for future research. Spiral dynamics
have not been integrated into metaheuristic algorithms for feature selection in
wildfire science, leaving unexplored potential to enhance model performance and
address optimization challenges. Moreover, the Liver Cancer Algorithm (LCA), a
recently introduced metaheuristic, has not yet been combined with spiral dynamics

24

to solve optimization problems or perform feature selection. Such an integration
could significantly enhance its ability to navigate large, complex solution spaces.
Finally, no studies have applied metaheuristic algorithms as feature selection tools
for wildfire prediction in Canadian provinces, where diverse environmental factors
influence wildfire behaviours. Addressing these gaps could provide transformative
advancements in wildfire modelling and prediction.

25

Chapter 3
Methodology
The methodology chapter details the extensive steps to develop, implement, and
assess a predictive wildfire model, addressing the complexity and class imbalance
commonly encountered in wildfire datasets. This research applies advanced metaheuristic algorithms for feature selection across eight Canadian provinces, aiming
to enhance predictive accuracy by identifying critical environmental variables that
significantly impact wildfire risk. The chapter is structured to clarify each component of the research design, from data collection and study area specification to
algorithmic selection and model evaluation. The methodology follows a structured
framework, beginning with an overview of the study’s objectives, hypotheses, and
overall approach, setting the foundation for the research. Data collection and
sources are then explored in detail, specifying the data types used—such as satellite
observations, meteorological data, vegetation indices, and historical wildfire occurrences—and discussing preprocessing steps to ensure data quality and consistency.
Following this, the study areas are introduced, highlighting the unique environmental and climate characteristics of the eight selected Canadian provinces to enhance

26

the robustness and relevance of the analysis. In the core sections of this chapter, we
delve into the research design by describing the theory behind each metaheuristic
algorithm and providing details on implementation and parameterization. We also
comprehensively outline the predictive modelling process, covering model training,
validation, and testing procedures. Finally, evaluation metrics provide a structured
approach for comparing and analyzing each algorithm’s effectiveness. This ensures
that the results align with the study’s objectives and contribute valuable insights
into wildfire prediction across diverse environmental settings.

3.1

Study Framework

This study follows a quantitative, empirical framework to advance wildfire prediction across diverse Canadian provinces through an in-depth evaluation and
refinement of metaheuristic algorithms for feature selection. By leveraging datadriven methodologies, the research framework is structured to test the predictive
capabilities of various metaheuristic algorithms and enhance their computational
efficiency and suitability for wildfire prediction tasks. Particular attention is given
to refining the Spiral Liver Cancer Algorithm (LCA), with improvements designed
to resolve stability and convergence issues, strengthening its role in predictive
modelling.
In addition, the framework emphasizes identifying the most influential environmental variables that drive wildfire occurrences within each unique provincial
context. This approach aims to tailor model insights to the distinct climatic and
environmental conditions across provinces, providing a more nuanced and accurate
predictive capability. Through systematic analysis, the study framework ensures
rigorous assessment and comparison of the selected algorithms, contributing to
27

wildfire risk management and the broader field of environmental predictive modelling.

3.1.1

Objective Of The Study

The primary objectives of this research are as follows:

• Enhance Wildfire Prediction Accuracy: Develop and optimize wildfire
prediction models by applying metaheuristic algorithms for effective feature
selection across diverse Canadian provinces.
• Evaluate Metaheuristic Algorithms: Test and compare multiple algorithms to determine their impact on feature selection and effectiveness in
enhancing wildfire prediction models.
• Algorithm Refinement and Development: Integrate spiral updates
to improve the standard Liver Cancer Algorithm (LCA). This refinement
addresses convergence and stability issues, enhancing LCA’s efficiency and
accuracy in feature selection for wildfire prediction.
• Identification of Key Predictive Variables: Pinpoint the environmental
variables most strongly associated with wildfire occurrences, focusing on
optimizing model inputs for more accurate predictions.
• Comparative Analysis: Conduct a comprehensive analysis of each algorithm’s predictive accuracy, computational efficiency, and scalability to
identify the most suitable approaches for real-world wildfire risk management
applications.

28

3.1.2

Hypotheses and Research Questions

The following hypotheses and research questions guide this study:

• Hypothesis 1: Training wildfire prediction models with selected features
derived from metaheuristic algorithms will outperform models trained using
all features, leading to better model efficiency and prediction accuracy.
• Hypothesis 2: The refined Spiral Liver Cancer Algorithm will outperform
other metaheuristic methods regarding wildfire prediction accuracy and stability.
• Hypothesis 3: Each province will have unique environmental features
contributing more significantly to wildfire occurrences, reflecting regionspecific conditions impacting fire risk.

These hypotheses guide our study with the following research questions:
• Research Question 1: How effectively do metaheuristic algorithms improve
feature selection accuracy and computational efficiency in wildfire prediction
models compared to using all available features?
• Research Question 2: Does the Spiral Liver Cancer Algorithm provide
superior feature selection strength and prediction accuracy performance
compared to other metaheuristic methods?
• Research Question 3: Which environmental features most predict wildfire
risk in each Canadian province, and how do these critical predictors vary
regionally?
29

3.2

Data Collection and Sources

This study combines data from two primary sources: FIRMS (Fire Information for
Resource Management System) for fire occurrence data and ERA5 for atmospheric
and environmental data. These sources provide comprehensive and reliable datasets
for wildfire prediction and feature selection across diverse Canadian provinces.

3.2.1

Data Types and Origin

The datasets utilized in this study include:
1. FIRMS (Fire Information for Resource Management System):
FIRMS offers global fire occurrence data derived from NASA’s MODIS and VIIRS
sensors, with near real-time data available approximately three hours after the
satellite overpass. The FIRMS Fire Map provides an inter-active library of active
fire detections, enabling historical and current data analysis. This study utilizes
archived data from 2010 to 2023 (NASA [2024]).
2. ERA5 Reanalysis Dataset: ERA5, produced by ECMWF (European
Centre for Medium-Range Weather Forecasts), is a comprehensive reanalysis dataset
covering climate and weather data from 1940 onwards. Reanalysis combines model
data with observations worldwide into a complete and uniform dataset. ERA5
provides hourly atmospheric, ocean-wave, and land-surface data and includes a
10-member ensemble uncertainty estimate at three-hour intervals. This dataset
captures various environmental and climatic conditions, including temperature,
wind, humidity, and precipitation, which are crucial for identifying conditions
contributing to wildfire occurrences(ERA [2024], Bell et al. [2021]).

30

Table 3.1: Summary of Features from FIRMS and ERA5 Data Sources
Source

Feature

Data Type

Description

FIRMS

Latitude

Float

Latitude coordinate of fire occurrence

FIRMS

Longitude

Float

Longitude coordinate of fire occurrence

FIRMS

Brightness Tem- Continuous

Brightness temperature measured

perature (TI4)

by MODIS or VIIRS, used for detecting high-temperature events

FIRMS

Brightness Tem- Continuous

Secondary brightness temperature

perature (TI5)

measure, complementing TI4 in
assessing fire intensity

FIRMS

Scan

Float

Width of the satellite scan line at
the fire’s location, affecting spatial
resolution

FIRMS

Track

Float

Length of the satellite track at the
fire’s location

FIRMS

Acquisition Date

Date

Date of fire detection

FIRMS

Acquisition

Integer

Time of fire detection in HHMM

Time
FIRMS

Satellite

format
Categorical

Satellite identifier (e.g., MODIS,
VIIRS)

FIRMS

Instrument

Categorical

Instrument used for detection
(e.g., MODIS, VIIRS)

31

Source

Feature

Data Type

Description

FIRMS

Confidence

Categorical

Confidence level in fire detection

(Low,

Medium,

High)
FIRMS

Version

Categorical

FIRMS dataset version identifier

FIRMS

Fire

Continuous

Energy output of the fire, indicat-

FIRMS

Radiative

Power (FRP)

ing fire intensity

Day/Night

Categorical (Day, Indicates whether the fire detecNight)

ERA5

ERA5

ERA5

10m U Compo- Continuous

Eastward wind speed component

nent of Wind

at 10 meters above ground

10m V Compo- Continuous

Northward wind speed component

nent of Wind

at 10 meters above ground

2m

Dewpoint

Continuous

Temperature
ERA5

tion occurred during day or night

2m Temperature

Dewpoint temperature at 2 meters, affecting fuel moisture

Continuous

Air temperature at 2 meters, critical for ignition conditions

ERA5

High Vegetation

Continuous

Cover
ERA5

contributing to fuel availability

Leaf Area Index
(High

Fraction of high vegetation cover,

Continuous

Vegeta-

Leaf area index for high vegetation, showing vegetation density

tion)
ERA5

Leaf Area Index

Continuous

(Low Vegetation)
ERA5

Low Vegetation

Leaf area index for low vegetation,
relevant for fire-prone areas

Continuous

Fraction of low vegetation cover

Cover

32

Source

Feature

Data Type

Description

ERA5

Mean Sea Level

Continuous

Mean atmospheric pressure at sea

ERA5

ERA5

Pressure

level

Mean Wave Di- Continuous

Average wave direction, relevant

rection

for coastal regions

Mean Wave Pe- Continuous

Average wave period

riod
ERA5

ERA5

Sea Surface Tem- Continuous

Temperature of the sea surface,

perature

impacting local climate

Significant Wave

Continuous

Height
ERA5

Height of combined wind waves
and swell

Surface Pressure

Continuous

Atmospheric pressure at the surface

ERA5

ERA5

ERA5

ERA5

ERA5

Total

Cloud

Continuous

Fraction of cloud cover, influenc-

Cover

ing solar radiation

Total Precipita- Continuous

Total precipitation, affecting fuel

tion

moisture

Type of High

Categorical

Classification of high vegetation

Vegetation

type

Type of Low Veg- Categorical

Classification of low vegetation

etation

type

Downward

UV Continuous

Radiation at Sur-

Amount of UV radiation reaching
the surface

face
ERA5

Surface

Latent

Heat Flux

Continuous

Heat exchange process affecting
temperature and humidity

33

3.2.2

Preprocessing Steps

A series of preprocessing steps was conducted to ensure data quality and consistency
across datasets from FIRMS and ERA5. These steps standardized the data, handled
missing values, and applied transformations to enhance model performance.

Merging

This study created a unified dataset by combining environmental data from ERA5
with fire occurrence data from FIRMS, ensuring the data was aligned geographically
and temporally. To achieve this, we rounded latitude and longitude values in
both datasets to one decimal place so that data points would match in location.
Additionally, we separated each dataset’s date and time information into distinct
fields, then standardized FIRMS timestamps by rounding them to the nearest
three-hour interval to match ERA5’s time intervals. Using these adjusted fields,
we merged the datasets to connect fire occurrences with relevant environmental
conditions.
To enhance reliability, we applied confidence filtering to the FIRMS data. For
MODIS, we retained only high-confidence fire detections, while for VIIRS, we
included all detections marked as ”High,” ”Medium,” or ”Low.” This filtering
emphasized the most reliable detections from each satellite source. Each detection
represented a distinct area, with fire areas estimated at 1 km² for MODIS and
approximately 0.14 km² for VIIRS, allowing us to calculate the total fire area and
support spatial analysis.
For further refinement, we used the DBSCAN clustering algorithm to group
nearby detections, assuming closely located detections were part of the same fire.
34

After clustering, duplicate entries within each cluster were removed, leaving one
representative data point per fire event, minimizing redundancy and improving
data accuracy.
To create a comprehensive dataset, we processed each environmental variable
separately for each year from 2010 to 2022. In the final integration step, we
combined all variables into a master dataset, setting up a unified source ready for
feature selection and predictive modelling.

Feature Engineering

Adding derived features informed by domain-specific knowledge can significantly
enhance model accuracy in wildfire prediction. These features capture the complex
dynamics governing wildfire behaviour. Below are the derived features and the
reasoning for their inclusion:

Wind Speed (ws) Rationale: Wind speed is essential in wildfire spread, as
higher speeds enable ember transport and intensify fire spread Prapas et al. [2022].

Equation 1: Wind speed calculation based on vector components.
ws =

p
u102 + v102

(3.1)

Wind Direction (wd) Rationale: Wind direction influences the path of embers,
which affects the fire’s spread Prapas et al. [2022].

35

Equation 2: Calculation of wind direction in degrees.




180
v10
×
mod 360
wd = arctan
u10
π

(3.2)

Equation 3: Adjusted formula to align with 360° scale.
wd = (wd + 180)

mod 360

(3.3)

Relative Humidity (RH) Rationale: Lower humidity increases the likelihood
of ignition, making it a crucial factor in wildfire risk Taneja et al. [2021].
Equation 4: Relative humidity based on the August-Roche-Magnus
approximation.
exp
RH = 100 ×



exp

17.625×Td
243.04+Td



17.625×T
243.04+T




(3.4)

Datetime Features Temporal features, such as day of the week, week of the year,
month, and season, capture patterns in wildfire occurrences, identifying seasonality
and periodic trends.

Day of the Week (day of week) Rationale: Certain days may show
different wildfire frequencies due to human activity or weather patterns, and this
feature allows the model to learn these patterns.

Week of the Year (week) Rationale: Wildfire activity varies seasonally,
and the week number helps capture this, especially during peak wildfire seasons.
36

Month (month) Rationale: The month indicates the time of year, enabling
the model to detect monthly trends in wildfire occurrences, which are often higher
in warmer months.

Season (season) Rationale: Wildfire risk increases in specific seasons, such
as summer. Mapping each month into a season provides insight into broader
temporal patterns.
Season Mapping:

season =




Spring






Summer

if month ∈ {3, 4, 5}
if month ∈ {6, 7, 8}




Autumn if month ∈ {9, 10, 11}





Winter
if month ∈ {12, 1, 2}

Dew Point Temperature Ranking (dewpoint rank) Rationale: Dew point
temperature affects wildfire risk by influencing atmospheric moisture. Lower
dew points generally indicate drier conditions conducive to wildfire. The dew
point ranking provides a straightforward feature representing wildfire risk due to
atmospheric humidity.

Temperature Category (temperature category) Rationale: Temperature
is crucial in wildfire risk, as higher temperatures can increase likelihood and intensity.
Categorizing temperature simplifies the model’s interpretation of temperature’s
role in wildfire risk.
Categorization Logic:

37

• Low : Temperatures below 15°C
• Medium: Temperatures between 15°C and 25°C
• High: Temperatures above 25°C

Detailed Wildfire Risk Score (wildfire risk) Rationale: The wildfire risk
score is a calculated target variable synthesizing various environmental, seasonal,
and atmospheric factors. This score captures conditions influencing fire occurrence,
categorizing wildfire risk for training.
Supporting Factors in Calculation: The risk score integrates:
• Precipitation: Dry conditions increase fire risk, while precipitation reduces
it (Wasserman and Mueller [2023], Abatzoglou and Kolden [2013], Abatzoglou
and Williams [2016], Abatzoglou et al. [2017]).
• Wind Speed: Higher speeds help fires spread by supplying oxygen and
dispersing embers (Liu et al. [2021]).
• Vegetation Density: Dense vegetation offers more fuel, raising the likelihood
and intensity of wildfires (Loudermilk et al. [2022]).
• Seasonal Influence: Fire activity is higher in summer due to temperature
and humidity patterns (Wasserman and Mueller [2023], Abatzoglou and
Kolden [2013], Abatzoglou and Williams [2016]).
• Cloud Cover: Low cloud cover raises fire risk by allowing more sunlight
and heat, while high cover reduces it (Drobyshev et al. [2021]).

38

• Temperature and Humidity Interactions: High temperatures, low humidity, and low pressure create favourable conditions for fires (Afolayan et al.
[2024]).
• Cluster Density Risk Score: Historical fire clusters, identified through
DBSCAN, adjust the risk score based on previous activity in the area.
• Wind Direction: South and Southeast winds, associated with drier air,
increase fire risk (Fazel-Rastgar and Sivakumar [2022]).

This scoring approach aligns with research on environmental factors that influence wildfires. The calculated score, categorized into risk levels (Low, Moderate,
High), gives the model essential insights for predicting wildfire risk.
Threshold-Based Risk Categorization:




High
if risk score ≥ 19



wildfire risk = Moderate if risk score ≥ 15 and < 19





Low
if risk score < 15
The predictive model can better understand wildfire behaviour by incorporating
thoughtfully designed features rooted in environmental science. Variables such as
wind speed, relative humidity, and seasonality are essential in assessing wildfire risk.
This process transforms complex data into clear and meaningful factors, improving
the model’s ability to make accurate and reliable predictions about wildfire risk.

39

3.3

Study Areas

This research examines datasets from eight Canadian provinces: Alberta, British
Columbia, Manitoba, Ontario, Northwest Territories, Quebec, Saskatchewan, and
Yukon. By including such a diverse range of regions, we aim to address the
variability in environmental and climatic factors that influence wildfires, ensuring
our model is comprehensive and adaptable.

Alberta

Alberta, located at 53.9333°N and 116.5765°W, spans approximately 661,848 km².
The province’s landscape transitions dramatically, from the rugged Rockies in the
west to sprawling prairies in the east. Precipitation varies significantly, from a dry
300 mm in the southeast to over 600 mm in the mountainous regions. Wildfires
are most common between May and September, fueled by hot, dry conditions and
gusty winds. Alberta’s varied geography makes it a fascinating case for studying
how local terrain and weather impact wildfire behaviour.

British Columbia

British Columbia, at 53.7267°N and 127.6476°W, covers roughly 944,735 km² and
is one of Canada’s most ecologically diverse provinces. BC’s landscape is as varied
as its wildfire patterns, from lush coastal rainforests to the dry interior grasslands.
The province’s wildfire season runs from spring to early fall, with the highest risks
in its arid interior during the summer. BC’s unique interplay between elevation,
temperature, and humidity offers critical insights into wildfire ignition and spread

40

dynamics.

Manitoba

Manitoba, situated at 53.7609°N and 98.8139°W, spans 647,797 km² and is characterized by its continental climate, with cold winters and warm, humid summers.
The southern parts of the province receive more rainfall than the northern boreal
forests, creating a gradient in fire risk. Wildfires typically peak from May to
September, with weather conditions like dry spells and heatwaves significantly
increasing the likelihood of fires. Manitoba’s mix of forests and prairies makes it a
challenging yet rewarding region for wildfire analysis.

Ontario

Ontario, located at 51.2538°N and 85.3232°W, is Canada’s most populous province,
covering about 1,076,395 km². Its climate ranges from the humid south to the
subarctic north, with precipitation levels between 700 mm and over 1,000 mm
annually. Wildfires are most active from April to October, with the northern
and central regions facing the highest risks due to dense forests and periodic dry
conditions. Ontario’s landscape’s sheer size and variability make it a key area for
understanding how climate and geography interact to influence wildfire patterns.

Northwest Territories

The Northwest Territories, situated at 64.8255°N and 124.8457°W, is vast, covering
approximately 1,346,106 km². This northern region experiences long, frigid winters
and brief, mild summers. Annual precipitation is sparse, usually below 400 mm,
41

with most falling during summer. The wildfire season here peaks from June to
August, driven by dry weather and strong winds in its extensive boreal forests.
This region’s remote and rugged nature presents significant challenges for wildfire
management and prediction, making it a unique study area.

Quebec

Quebec, located at 52.9399°N and 71.3100°W, is Canada’s largest province by area,
spanning 1,542,056 km². Its climate shifts from the humid south, where cities
like Montreal and Quebec City are located, to the cold, subarctic north. Wildfire
occurs between May and September, with the northern boreal forests particularly
susceptible. Quebec’s vast and varied terrain requires wildfire models to account
for densely forested areas and less vegetated zones, offering valuable insights into
regional fire dynamics.

Saskatchewan

Saskatchewan, centered at 52.9399°N and 106.4509°W, encompasses about 651,900
km². Known for its expansive prairies and northern boreal forests, the province
experiences a continental climate with warm summers and cold winters. Annual
precipitation ranges from about 300 mm in the drier southwest to 500 mm in the
forested north—the wildfire season peaks from May to September. Saskatchewan’s
blend of grasslands and forests provides a unique perspective on how different
ecosystems respond to fire risks.

42

Yukon

Yukon, located at 64.2823°N and 135.0000°W, spans approximately 482,443 km².
Its dramatic mountain ranges and subarctic climate define the territory, with long,
harsh winters and brief, warm summers. Precipitation is limited, usually below 400
mm annually, with most of it as summer rain—wildfire season peaks from June to
August, with fire activity influenced by dry spells and fluctuating temperatures.
Yukon’s remote and rugged environment makes wildfire prediction complex, highlighting the importance of localized data in understanding fire behaviour.
While this research focuses on eight provinces across Canada, certain regions
were excluded due to limited wildfire data availability or lower wildfire incidence,
which would contribute minimally to the study’s focus on high-risk areas. Concentrating on regions with diverse and complex wildfire patterns, this research
provides insights relevant to the areas most impacted by wildfires.

3.4

Feature Selection and Metaheuristic Algo-

rithm Implementation
In predictive modelling, selecting the most informative features is crucial, particularly in applications like wildfire prediction, where vast datasets include complex
interactions between variables. The objective of feature selection in this study is
to identify a subset of predictive features that improve model accuracy and interpretability while reducing computational overhead. Given the high-dimensional
nature of wildfire datasets, effective feature selection can streamline model performance by reducing noise and focusing only on variables that directly impact

43

Figure 3.1: Map of Study Area across Canadian Provinces
wildfire risk.

44

3.4.1

Metaheuristic Algorithm Rationale

Metaheuristic algorithms are particularly advantageous in feature selection due to
their adaptability in navigating large, complex search spaces and handling nonlinear relationships among features. This study employs a variety of metaheuristic
algorithms, each selected for its unique strengths in exploring and exploiting
potential feature subsets.
This collection of algorithms includes population-based and solution-based
methods, balancing exploration (diversifying the search) and exploitation (focusing
on promising areas in the search space). Each algorithm contributes distinct search
behaviours, enhancing the likelihood of identifying optimal or near-optimal subsets
in the high-dimensional feature space of wildfire prediction datasets.

Wrapper Model for Feature Evaluation In this study, all metaheuristic
algorithms employ a wrapper model approach, integrating a predictive model to
evaluate the performance of each candidate feature subset (Kohavi and John [1997],
Aboudi and Benhlima [2016]). The wrapper method is advantageous for feature
selection as it iteratively assesses feature subsets based on predictive accuracy,
allowing each algorithm to prioritize combinations that enhance model performance
rather than relying solely on intrinsic characteristics of the data. A Random Forest
(RF) classifier serves as the evaluation model here, with predictive quality measured
through accuracy and F1-score.
In this wrapper framework:
• Encoding Feature Subsets: Each candidate solution, or agent, represents
a unique subset of features. Using binary encoding, features are denoted as
45

included (1) or excluded (0), allowing metaheuristic algorithms to explore a
range of feature combinations effectively.
• Performance Evaluation: For each feature subset, the wrapper model
trains an RF classifier, assessing its predictive accuracy. This feedback guides
the search process, allowing algorithms to adjust their exploration toward
subsets that demonstrate improved model performance.

In the wrapper model’s evaluation process, we defined a fitness function for each
agent (feature subset), using a cross-validated XGBoost classifier to measure its
predictive effectiveness. This function calculates the fitness based on cross-validated
F1-scores, providing critical feedback to the metaheuristic algorithms. The fitness
function is represented as follows:
Equation 5: Fitness Function for Feature Subset Evaluation

P

0
if
A=0
fitness(A) =

 1 Pk F1(y (i) , ŷ (i) ) otherwise
cv val cv val
i=1
k

(3.5)

where:
• A is the binary vector representing the feature subset the agent chooses.
• k is the number of folds in cross-validation (set to 10 in this study).
(i)

(i)

• ycv val and ŷcv val are the true and predicted labels for the validation set in
fold i, respectively.
• F1(·) denotes the F1 score function with a macro averaging strategy.

46

If no features are selected (i.e.,

P

A = 0), the fitness score defaults to zero

as a penalty. For agents selecting at least one feature, the mean F1 score across
cross-validation folds provides a fitness measure, with a higher score indicating a
more informative feature subset. The feature selection ratio is given by:
Equation 6: Feature Ratio Calculation
P

feature ratio =

A
len(A)

(3.6)

This represents the proportion of selected features relative to the total feature
set, balancing feature inclusion with computational efficiency.
The metaheuristic algorithms iteratively refine feature subsets through this
wrapper-based approach, maximizing model performance by selecting the most
predictive features within wildfire datasets.

Key Parameters and Common Inputs/Outputs While each algorithm has
unique characteristics, they share common parameters and outputs, such as:

• Iterations (limit): The number of cycles each algorithm undergoes influences its convergence behaviour.
• Population Size: The number of candidate solutions (or agents) explored
in each iteration, balancing computational efficiency with the depth of exploration.
• Training Data (Xtrain and ytrain ): The input dataset comprising the
features and corresponding target variable is used to evaluate feature subsets.

47

Common Outputs:
• Best Fitness Value (best fit val): The optimal fitness score achieved,
reflecting the quality of the selected feature subset.
• Fitness History (fitness history): Records fitness values over iterations,
useful for analyzing the algorithm’s convergence behaviour.
• Accuracy History (accuracy history): Tracks accuracy values over iterations, providing insight into predictive performance.
• Feature Ratio History (feature ratio history): Shows feature ratios
across iterations, indicating the proportion of features selected.

3.4.2

Algorithm Descriptions

Atom Search Optimization (ASO)

Atom Search Optimization (ASO) is a physics-inspired metaheuristic algorithm
designed to solve complex optimization problems by mimicking atomic motion in
molecular dynamics. In ASO, each solution (or atom) interacts with others through
forces derived from molecular physics, allowing for both exploration of the search
space and intensification around promising solutions Zhao et al. [2018].

Algorithm Mechanics ASO is based on two main types of forces:
• Interaction Force: Derived from the Lennard-Jones potential, this force
represents attraction and repulsion between atoms, enabling atoms to either
converge around promising solutions or disperse to explore new areas.
48

• Constraint Force: A secondary force directing atoms toward the best-known
solution, encouraging the algorithm to refine optimal solutions.
The position of each atom in ASO represents a potential solution, and its ”mass”
(related to the solution’s fitness) influences its acceleration. The algorithm balances
exploration and exploitation by adjusting the interaction forces over iterations,
with lighter atoms exploring new regions and heavier atoms focusing on optimal
solutions.

Parameter Settings The main parameters in ASO include:
• Iterations: Controls the algorithm’s stopping condition based on convergence
requirements.
• Population Size: The number of atoms, balancing computational efficiency
with search robustness.
• Depth Weight and Multiplier Weight: Parameters that control the
strength of interaction and constraint forces, influencing ASO’s balance
between exploration and exploitation.

Application to Feature Selection ASO’s ability to avoid local optima and
dynamically adapt to high-dimensional spaces makes it particularly useful for
feature selection in wildfire prediction. By selecting a subset of relevant features,
ASO enhances model performance while reducing computational costs, making it
ideal for large datasets with complex interrelationships Zhao et al. [2019].
Algorithm 1 Inputs and Outputs
Inputs:
49

• Xtrain : Training data (features)
• ytrain : Training data (target)
• atomno: Number of atoms (agents) to maintain diversity in the search space.
• dim: Dimension of the feature space (number of features) being evaluated.
• limit: Maximum number of iterations, controlling the depth of the search
process.
• α, β, ω: ASO-specific hyperparameters influencing the balance between exploration and exploitation.

Outputs:
• best atom: Best feature subset (binary vector) discovered by the algorithm.
• best fit val: Best fitness value achieved, representing the quality of the selected
feature subset.
• fitness history: History of fitness values over iterations, useful for analyzing
convergence behaviour.
• accuracy history: Accuracy values over iterations, providing insight into
predictive performance.
• feature ratio history: Feature ratios over iterations indicate the proportion
of selected features.

Algorithm 1 Atom Search Optimization (ASO) for Feature Selection
1: Step 1: Initialize Population
2: for each atom i in range (0, atomno) do

50

3:

Generate a random binary vector of length dim to represent selected features

4:

Initialize velocities for each feature in the atom

5: end for
6: Step 2: Calculate Initial Fitness
7: for each atom i do
8:

Perform wrapper model evaluation using cross-validation with XGBoost
(see Equation 5).

9:

Store fitness, accuracy, and feature ratio for each atom

10: end for
11: Find and store the best initial solution (best atom and best fit val)
12: Step 3: Optimization Loop
13: for iteration curr in range (1, limit + 1) do
14:
15:

for each atom i do
Update velocity using the formula:
velocity[i] = velocity[i] + β × (best atom − population[i])

16:

Update position using V-function:

17:

for each feature j in atom i do
Set position[j] = 1 if V (velocity[i][j]) > random() else 0

18:
19:

end for

20:

Recalculate fitness for the new feature subset

21:

Update best solution if the current fitness is better than best fit val

22:

end for

23:

Track fitness history, accuracy history, and feature ratio history

24: end for
25: Step 4: Return Results
26: Output:

best atom,

best fit val,

fitness history,

accuracy history,

feature ratio history
51

Barnacles Mating Optimization (BMO)

The Barnacles Mating Optimizer (BMO) is a bio-inspired metaheuristic algorithm
that simulates the mating behaviours of barnacles. BMO emulates the biological
processes observed in barnacle reproduction, where individuals selectively mate
within a constrained range or disperse sperm broadly when mates are unavailable
nearby (Sulaiman et al. [2020]).
BMO’s optimization relies on two core dynamics:

• Mating Selection and Penis Length (Interaction Constraints): Each
barnacle selects mates within a limited ”penis length” (pl) distance, promoting
exploitation by mating with close solutions to refine promising regions. If
no suitable mate is within range, sperm casting occurs, simulating broad
dispersal to enable distant interactions, enhancing exploration and avoiding
local optima.
• Inheritance of Parental Traits (Reproductive Process): Offspring
are generated by combining traits from selected parent barnacles based on
the Hardy-Weinberg principle, which probabilistically blends characteristics.
This balances exploration and exploitation, adapting offspring traits across
iterations to refine potential solutions.

Parameter Settings in BMO: Fine-tuning these parameters is key to BMO’s
success:

52

Penis Length (pl): Controls mating range, balancing exploration and exploitation. Population Size: Sets the number of barnacles, impacting computational cost
and diversity. Iterations: Determines the number of optimization cycles based on
convergence needs.
Algorithm 2 Barnacles Mating Optimizer (BMO) for Feature Selection
1: Step 1: Initialize Population
2: for each barnacle i in range (0, pop size) do
3:

Generate a random binary vector of length dim representing selected features

4: end for
5: Step 2: Calculate Initial Fitness
6: for each barnacle i do
7:

Perform wrapper model evaluation using cross-validation with XGBoost
(see Equation 5).

8:

Store fitness for each barnacle

9: end for
10: Identify and store the best initial solution (best barnacle and best fitness)
11: Step 3: Optimization Loop
12: for each iteration in range (1, max iter + 1) do
13:

for each barnacle i do

14:

Find local mates within mating radius

15:

if no local mates or random() ¡ sperm cast chance then

16:

Select a global mate randomly (sperm casting)

17:
18:

else
Select a random local mate

19:

end if

20:

Generate offspring by:
a) Performing crossover between barnacle and mate
53

b) Applying mutation based on the distance between barnacles
21:

Evaluate fitness of offspring

22:

if offspring fitness is better than current barnacle fitness then
Replace barnacle with offspring

23:
24:

end if

25:

end for

26:

Track best fitness for each iteration in fitness history

27: end for
28: Step 4: Return Results
29: Output: best barnacle, best fitness, fitness history

Chef-Based Optimization Algorithm (CBOA)

The Chef-Based Optimization Algorithm (CBOA) is an optimization method
inspired by the structured learning process in culinary arts. In CBOA, each
candidate solution represents either a chef instructor or a cooking student, each
playing a different role in reaching the best possible solution.

Algorithm Mechanics CBOA operates through two main phases:
• Chef Instructors’ Phase: Chef instructors refine their skills by observing
the techniques of the top chef (global search) and adding their improvements
(local search). This phase helps balance the exploration of new solutions with
the refinement of existing ones.
• Cooking Students’ Phase: Students learn in three ways—by working
with a chosen chef instructor, picking up a new skill from another chef, and

54

practicing independently. This diverse learning approach allows students to
explore various solutions while focusing on improving specific areas.

In CBOA, the positions of chef instructors and cooking students represent
potential solutions. Chef instructors guide the search toward promising areas while
students focus on refining solutions nearby. Together, these roles help the algorithm
balance exploring new options and intensifying efforts around the best solutions
found so far. Through repeated training cycles, CBOA moves closer to finding the
optimal solution (Trojovská and Dehghani [2022]).
Algorithm 3 Chef-Based Optimization Algorithm (CBOA) for Feature Selection
1: Step 1: Initialize Population
2: for each agent i in range (0, SearchAgents) do
3:

Generate a random binary vector of length dimension representing selected
features

4: end for
5: Step 2: Calculate Initial Fitness
6: for each agent i do
7:

Perform wrapper model evaluation using cross-validation with XGBoost
(see Equation 5).

8:

Store fitness for each agent

9: end for
10: Identify and store the best initial solution (top chef and best fitness)
11: Step 3: Optimization Loop
12: for each iteration in range (1, Max iterations + 1) do
13:

Assign agents as ”chef instructors” or ”cooking students” based on fitness
ranking
55

14:
15:

for each agent i do
if agent i is a chef instructor then
if random() ¡ 0.5 then

16:

Follow top chef to exploit best-known solution

17:

else

18:

Perform independent exploration with Levy flight

19:

end if

20:
21:

else

▷ Agent i is a cooking student

Select learning strategy randomly:

22:

a) Learn from a random chef instructor
b) Master a specific skill from the top chef
c) Self-improve through Levy flight for exploration
23:

end if

24:

Evaluate fitness of updated agent

25:

if agent’s fitness is better than best fitness then
Update best fitness and top chef with current agent

26:
27:

end if

28:

end for

29:

Track best fitness for each iteration in fitness history

30: end for
31: Step 4: Return Results
32: Output: top chef, best fitness, fitness history

Equilibrium Optimizer (EO)

The Equilibrium Optimizer (EO) is a physics-inspired optimization method modelled after mass balance concepts in a control volume. In EO, each candidate
56

solution represents a particle with an evolving position based on concentration and
balance principles, aiming to reach an equilibrium state.

Algorithm Mechanics EO operates through three main components:
• Equilibrium Pool and Candidates: EO maintains a pool of top solutions,
representing equilibrium candidates. These candidates guide other particles
by attracting them toward promising areas while retaining some variability
to explore diverse regions.
• Exponential Term (F): This term is calculated based on the turnover rate
(λ), facilitating a dynamic shift between exploration (wide-ranging searches)
and exploitation (focused refinement) as iterations proceed. The term enables
particles to expand their search early on and refine positions in later stages.
• Generation Rate (G): As a local search enhancer, this term directs particles
to adjust their positions around candidates, particularly in smaller steps,
reinforcing the search’s precision as particles near equilibrium.

In EO, particles act as potential solutions, each adjusting its ”concentration”
based on local and global interactions. Through the equilibrium pool and the
balance between F and G terms, EO effectively balances global exploration and
local exploitation. Repeated interactions move EO closer to the global optimum
solution (Faramarzi et al. [2020]).
Algorithm 4 Equilibrium Optimizer (EO) for Feature Selection
1: Step 1: Initialize Population
2: for each agent i in range (0, partCount) do

57

3:

Generate a random binary vector of length dimension representing selected
features

4: end for
5: Initialize the equilibrium pool eqPool with size poolSize, storing the best

solutions found
6: Step 2: Calculate Initial Fitness
7: for each agent i do
8:

Evaluate fitness using cross-validation with XGBoost (see Equation 5)

9:

Store fitness for each agent in fitness scores

10: end for
11: Identify the best initial solutions and update eqPool and eqfit (fitness of

equilibrium candidates)
12: Step 3: Optimization Loop
13: for each iteration in range (1, max iter + 1) do
14:

for each agent i do

15:

Select an equilibrium candidate Ceq from eqPool

16:

Generate a random vector λ to guide convergence and diversity

17:

Calculate:
a) Concentration Difference term, FVec:


iteration 2·iteration/max iter
FVec = 2 × sign(rand − 0.5) × exp(−λ × (1 −
)
)−1
max iter
b) Generation Rate term, G:
G = 0.5 × rand × (Ceq − λ × agent[i])

18:

Update agent’s position:
agent[i] = clip(agent[i] + G − FVec × (agent[i] − Ceq ), 0, 1)

19:

end for
58

20:

Evaluate fitness for each agent after update and update eqPool with the
best solutions if necessary

21:

Track the best fitness values for each iteration in fitness history

22: end for
23: Step 4: Return Results
24: Output: The best agent, best fitness score, and fitness history over iterations

The Exponential Distribution Optimizer (EDO)

The Exponential Distribution Optimizer (EDO) is a mathematics-inspired algorithm
rooted in the principles of the exponential probability distribution. The primary
mechanism of EDO is based on exploiting the statistical properties of exponential
distributions to drive both exploration and exploitation within an optimization
landscape.

Algorithm Mechanics EDO comprises two main phases inspired by exponential
distribution properties:
• Exploitation Phase: This phase is designed around the memoryless property
of exponential distributions. EDO maintains a memoryless matrix where
newly generated solutions are stored regardless of fitness. By disregarding past
performance, EDO allows unsuccessful solutions (losers) to influence future
generations. This phase also incorporates the guiding solution, a composite of
the top-performing solutions, to enhance exploitation near promising areas.
• Exploration Phase: This phase uses the average solution (mean of all
solutions) alongside randomly selected high-performing solutions to drive

59

exploration. By adjusting the contribution of each selected solution, EDO
dynamically explores new areas and adjusts for diversity within the search
space.
Through repeated cycles, EDO alternates between these phases, enabling a
balance between exploring new areas and refining promising solutions. A switching
parameter regulates this balance, ensuring the search process remains dynamic.
In EDO, guiding solutions, memoryless storage, and controlled exponential
variance are pivotal, helping the algorithm move toward optimal areas and avoid
getting trapped in local minima Abdel-Basset et al. [2023].
Algorithm 5 Exponential Distribution Optimizer (EDO) for Feature Selection
1: Step 1: Initialize Population
2: for each agent i in range (0, N ) do
3:

Generate a random continuous vector within bounds [LB, U B] for each
dimension, representing selected features

4: end for
5: Set BestSol ← 0, BestFitness ← −∞, and initialize solution matrix Solutions
6: Step 2: Calculate Initial Fitness
7: for each agent i do
8:

Select features based on agent i’s continuous vector, converting values above
a threshold to 1 (selected) and others to 0

9:

Evaluate fitness using cross-validation with XGBoost on the selected features
(refer to Equation 5)

10:

Store fitness score in Fitness[i]

11: end for
12: Identify the best initial solution, updating BestSol and BestFitness with the

top fitness and corresponding agent
60

13: Step 3: Optimization Loop
14: for each iteration in range (1, Max iter + 1) do
15:

for each agent i do

16:

Generate a random vector λ to guide the update for each dimension

17:

Select a random guiding solution, Cguiding , from the current population

18:

Calculate:
a) Concentration Difference term, FVec:
!

2·iteration/Max iter
iteration
)−1
FVec = 2×sign(rand−0.5)× exp(−λ × 1 −
Max iter
b) Generation Rate term, G:
G = 0.5 × rand × (Cguiding − λ × Solutions[i])

19:

Update agent i’s position:
Solutions[i] = clip(Solutions[i] + G − FVec × (Solutions[i] − Cguiding ), LB, U B)

20:

end for

21:

Step 4: Evaluate and Update Fitness

22:

for each agent i do

23:

Select features from the updated position of agent i (based on thresholding)

24:

Recalculate fitness using cross-validation with XGBoost on the selected
features

25:
26:
27:

if new fitness of agent i is better than BestFitness then
Update BestFitness and BestSol with agent i’s solution and fitness
end if

28:

end for

29:

Record fitness values for each iteration in fitness history
61

30: end for
31: Step 5: Return Results
32: Output: The best solution BestSol, best fitness score BestFitness, and statis-

tics such as mean fitness, worst fitness, and feature selection ratio

Energy Valley Optimizer (EVO)

The Energy Valley Optimizer (EVO) is a metaheuristic optimization algorithm
inspired by physical principles surrounding particle stability and decay processes.
In EVO, candidate solutions act as particles, each possessing unique stability characteristics, guiding them toward more stable states within the solution space. This
process involves different decay schemes—alpha, beta, and gamma decay—which
mimic the real-life behaviour of particles moving toward stability (Azizi et al.
[2023]).

Algorithm Mechanics EVO operates through three main phases, mirroring
different particle decay modes:

• Alpha Decay Phase: In this phase, particles emit alpha particles, moving
towards the global best solution by modifying specific attributes. This enables
local search within a neighbourhood of promising solutions.
• Beta Decay Phase: Particles with high instability levels undergo beta decay,
making larger jumps toward the stability band (optimal solution) by moving
closer to both the best solution and a central point, increasing exploration.
• Gamma Decay Phase: Particles experiencing gamma decay adjust their
positions based on neighbouring solutions, facilitating refined adjustments
62

for better stability within a localized region.
Algorithm 6 Energy Valley Optimizer (EVO) for Feature Selection
1: Step 1: Initialize Particles
2: for each particle i in range (1, Population Size) do
3:

Randomly generate initial positions within bounds for each particle, representing selected features

4: end for
5: Define Enrichment Bound (EB) based on initial stability levels of particles
6: Step 2: Calculate Initial Stability Levels
7: for each particle i do
8:

Select features based on the particle’s position vector, converting values
above a threshold to 1 (selected) and others to 0

9:

Evaluate fitness using cross-validation with XGBoost for each particle’s
selected features (see Equation 5)

10:

Assign initial stability level based on fitness score

11: end for
12: Determine

best and worst stability levels as Best Stability (BS) and

Worst Stability (WS)
13: Step 3: Optimization Loop
14: for each iteration in range (1, Max Iterations) do
15:
16:
17:
18:

for each particle i do
if Neutron Enrichment Level (NELi > EB) then
if Stability Level (SLi > Stability Bound) then
Perform Alpha Decay:
Adjust particle’s position by moving it towards BS for local refinement

19:
20:

Generate new candidate solution XNewi by applying alpha decay
else
63

Perform Gamma Decay:

21:

Adjust based on neighbouring particles to refine the position
Generate new candidate solution XNew2i through gamma decay

22:

end if

23:
24:

else

25:

Perform Beta Decay for more unstable particles:

26:

Adjust particle i by moving it towards both BS and the central point
of the population for wider exploration

27:

end if

28:

end for

29:

Evaluate and update stability for each new candidate solution using
XGBoost-based fitness evaluation

30:

Update EB based on new stability levels and check stopping criteria

31: end for
32: Step 4: Return Results
33: Output: The particle with the highest stability (best fitness), fitness history

across iterations, and selected features

Genetic Algorithm (GA)

The Genetic Algorithm (GA) is a classic metaheuristic optimization method inspired by principles of natural selection and genetics. In GA, candidate solutions,
individuals, form a population that evolves over generations. A fitness function
evaluates each individual, and the fittest individuals are more likely to pass their
traits to the next generation, gradually steering the population toward optimal
solutions (H. et al. [2009], Katoch et al. [2021]).

64

Algorithm Mechanics GA operates through three main evolutionary processes:

• Selection: Individuals are selected from the population based on their fitness
scores. Higher fitness increases the likelihood of selection, favouring stronger
solutions. Selection methods include roulette wheel selection, tournament
selection, and rank-based selection.
• Crossover (Recombination): Selected individuals (parents) exchange
parts of their genetic material to produce offspring. This process combines
traits from both parents, allowing the population to explore new areas of the
solution space while inheriting favourable characteristics. Common crossover
methods include single-point, multi-point, and uniform crossover.
• Mutation: Offspring undergo mutation with a low probability, introducing
small, random changes to their genetic material. Mutation maintains genetic
diversity in the population and prevents premature convergence on local
optima. It is typically implemented by flipping bits in a binary representation
or making small numerical changes in real-valued encoding.

In GA, the evolving population undergoes these three processes in each generation. The algorithm continues until a termination condition is met, such as
reaching a maximum number of generations or achieving a specified fitness level.
Algorithm 7 Genetic Algorithm (GA) for Feature Selection
1: Step 1: Initialize Population
2: for each individual i in range (1, Population Size) do
3:

Generate a random binary vector of length num features, representing
selected features

4: end for

65

5: Step 2: Calculate Initial Fitness
6: for each individual i do
7:

Evaluate fitness using cross-validation with XGBoost

8:

Store fitness score for each individual

9: end for
10: Identify the best initial solution and store it as Best Solution with fitness

Best Fitness
11: Step 3: Optimization Loop
12: for each generation in range (1, Max Generations) do
13:

Initialize a new population for the next generation

14:

for each pair of parents in population do

15:

Select two parents using tournament selection

16:

if random() ¡ crossover rate then

17:
18:
19:

Perform single-point crossover on parents to create two children
else
Children are identical to parents (no crossover)

20:

end if

21:

Perform mutation on each child by flipping bits with probability equal
to the mutation rate

22:

Add children to the new population

23:

end for

24:

Replace the current population with the new population

25:

Step 4: Calculate Fitness for New Population

26:

for each individual i in the new population do

27:

Evaluate fitness using cross-validation with XGBoost

28:

Store fitness score for each individual

29:

end for

66

30:

Identify the best solution in the new generation

31:

if new best solution’s fitness is better than Best Fitness then

32:

Update Best Solution and Best Fitness with this new solution

33:

end if

34:

Track Best Fitness across generations in fitness history

35: end for
36: Step 5: Return Results
37: Output: The best solution (selected features) with the highest fitness score,

fitness history

Golden Ratio Method (GRM)

The Golden Ratio Method (GRM) is an optimization technique inspired by the
golden ratio (ϕ ≈ 1.618), often observed in natural phenomena for efficient search
and refinement. In GRM, candidate solutions represent points in a search space,
and their positions are iteratively adjusted based on the golden ratio to converge
toward optimal solutions.

Algorithm Mechanics GRM operates through the following phases:
• Initialization: A population of candidate solutions is generated randomly
within defined bounds. Each candidate is represented by a binary vector
where each bit indicates whether a feature is selected.
• Golden Ratio-Based Search: Each candidate solution’s position is updated
based on the golden ratio, using two reference points derived from the current
solution and the best-known solution. This approach helps balance exploration
and exploitation by guiding candidates towards promising regions.
67

• Fitness Evaluation: Each candidate is evaluated using a fitness function,
such as cross-validated accuracy with a classifier, to determine the quality of
the selected features.

The algorithm iteratively applies these steps until a termination criterion is
met, such as reaching a maximum number of iterations.
Algorithm 8 Golden Ratio Method (GRM) for Feature Selection
1: Step 1: Initialize Population
2: for each candidate solution i in range (1, Population Size) do
3:

Generate a random binary vector of length num features, representing
selected features

4: end for
5: Step 2: Calculate Initial Fitness
6: for each candidate solution i do
7:

Evaluate fitness using cross-validation with XGBoost for each particle’s
selected features (see Equation 5)

8:

Store fitness score for each candidate

9: end for
10: Identify the best initial solution as Best Solution with fitness Best Fitness
11: Step 3: Optimization Loop Using Golden Ratio
12: for each iteration in range (1, Max Iterations) do
13:
14:

for each candidate solution i do
Generate two new candidate points using the golden ratio ϕ:
New Point 1 = Candidate + ϕ × (Best Solution − Candidate)
New Point 2 = Candidate − ϕ × (Best Solution − Candidate)

15:

Clip values in each new point to ensure they remain within bounds

16:

Evaluate fitness for New Point 1 and New Point 2
68

17:

if either new point has a better fitness than the current solution then

18:

Update the current solution with the point of having the better
fitness
end if

19:
20:

end for

21:

Update Best Solution and Best Fitness if a better solution is found in this
iteration
Track Best Fitness across iterations in fitness history

22:

23: end for
24: Step 4: Return Results
25: Output: The best solution (selected features) with the highest fitness score,

fitness history

Manta Ray Foraging Optimization (MRFO)

The Manta Ray Foraging Optimization (MRFO) is a metaheuristic algorithm
inspired by manta rays’ unique foraging behaviour, which includes coordinated
movements for prey capture. In MRFO, candidate solutions represent individual
manta rays, with each ray’s position in the search space iteratively updated to
explore and exploit potential solutions effectively.

Algorithm Mechanics MRFO operates through three primary foraging mechanisms:

• Chain Foraging: In this phase, manta rays move in a chain-like formation,
where each ray updates its position based on the ray ahead or the current
best solution. This behaviour promotes coordinated exploration across the
69

search space.
• Cyclone Foraging: Here, manta rays move in a spiralling pattern towards
the best-known solution, allowing for a more localized search that refines
promising areas of the solution space.
• Somersault Foraging: Manta rays perform a somersault around the best
solution, enhancing exploration and promoting diversity by moving within a
range around the best-known solution.

The algorithm iteratively applies these mechanisms until a stopping criterion is
met, such as reaching a maximum number of iterations or achieving a target fitness
level.
Algorithm 9 Manta Ray Foraging Optimization (MRFO) for Feature Selection
1: Step 1: Initialize Population
2: for each manta ray i in range (1, Population Size) do
3:

Generate a random binary vector of length num features, representing
selected features

4: end for
5: Step 2: Calculate Initial Fitness
6: for each manta ray i do
7:

Evaluate fitness using cross-validation with XGBoost for each particle’s
selected features (see Equation 5)

8:

Store fitness score for each manta ray

9: end for
10: Identify the best initial solution as Best Solution with fitness Best Fitness
11: Step 3: Optimization Loop Using MRFO Mechanisms
12: for each iteration in range (1, Max Iterations) do

70

13:

Chain Foraging: Each manta ray updates its position based on the
preceding ray or the current best solution

14:
15:

for each manta ray i do
Update position based on chain movement towards the best solution or
the position of ray i − 1

16:

end for

17:

Cyclone Foraging: Each manta ray spirals towards the best-known
solution

18:
19:

for each manta ray i do
Adjust position in a spiral pattern towards the best-known solution

20:

end for

21:

Somersault Foraging: Each manta ray explores within a radius around
the best solution

22:
23:

for each manta ray i do
Update position based on somersault foraging around the best solution

24:

end for

25:

Evaluate fitness for all updated positions

26:

Update Best Solution and Best Fitness if a better solution is found in this
iteration

27:

Track Best Fitness across iterations in fitness history

28: end for
29: Step 4: Return Results
30: Output: The best solution (selected features) with the highest fitness score,

fitness history

71

Particle Swarm Optimization (PSO)

Particle Swarm Optimization (PSO) is a popular metaheuristic inspired by the
social behaviour of birds flocking or fish schooling. In PSO, candidate solutions
are modelled as particles that ”fly” through the search space, with each particle
adjusting its position based on its personal experience and the collective experience
of the swarm. This algorithm leverages both individual and social learning to
converge toward optimal solutions.

Algorithm Mechanics PSO operates through three main steps:

• Initialization: A population of particles is randomly generated, each with
an associated velocity and position representing a candidate solution.
• Position and Velocity Update: Each particle updates its velocity and
position based on its own best-known position, the best-known position in its
neighbourhood, and its current velocity. This balance allows for a mixture of
exploration and exploitation in the search space.
• Fitness Evaluation: Each particle is evaluated based on a fitness function,
often cross-validated accuracy or another metric, to determine the quality of
the solution represented by its position.

The algorithm iterates these steps until a stopping criterion is met, such as
reaching a maximum number of iterations or achieving a desired fitness level.
Algorithm 10 Particle Swarm Optimization (PSO) for Feature Selection
1: Step 1: Initialize Population

72

2: for each particle i in range (1, Population Size) do
3:

Randomly generate a binary vector of length num features representing
selected features

4:

Initialize velocity for each particle as a vector of the same length

5:

Set each particle’s personal best position to its initial position

6: end for
7: Identify the global best position Best Position with fitness Best Fitness from

initial particles
8: Step 2: Optimization Loop
9: for each iteration in range (1, Max Iterations) do
10:
11:

for each particle i do
Update velocity using personal best, global best, and inertia terms:

velocity[i] = w×velocity[i]+c1 ×rand()×(personal best[i]−position[i])+c2 ×rand()×(Best Positio
12:

Update position based on the updated velocity:
position[i] = position[i] + velocity[i]

13:

Apply a sigmoid function to ensure binary feature selection, converting
each dimension to 0 or 1

14:

Evaluate fitness using cross-validation with XGBoost for each particle’s
selected features (see Equation 5)

15:
16:
17:

if updated fitness is better than particle’s personal best fitness then
Update particle’s personal best position and fitness
end if

18:

end for

19:

Update global best position Best Position and fitness Best Fitness if a better
solution is found in this iteration

20:

Track Best Fitness across iterations in fitness history
73

21: end for
22: Step 3: Return Results
23: Output: The best solution (selected features) with the highest fitness score,

fitness history

Walrus Optimization Algorithm (WaOA)

The Walrus Optimization Algorithm (WaOA) is a metaheuristic algorithm inspired
by the social and foraging behaviors of walruses. In WaOA, candidate solutions
represent individual walruses, which adapt their positions in the search space based
on different behaviors, such as feeding, migration, and escaping/fighting. This
structured behavior allows for effective exploration and exploitation to achieve
optimal solutions.

Algorithm Mechanics WaOA operates through three main foraging mechanisms:

• Feeding Phase: Walruses move towards the best-known solution or promising areas to exploit high-quality solutions. This behavior promotes local
search around successful solutions.
• Migration Phase: Walruses perform broader, random movements to explore
the search space. This movement aims to prevent the algorithm from becoming
trapped in local optima and to promote exploration of new areas.
• Escaping/Fighting Phase: Walruses make smaller, refining movements
around high-quality solutions to balance between exploration and exploitation.

74

This helps refine promising solutions by focusing on adjustments within a
local neighborhood.

The algorithm iteratively applies these mechanisms until a stopping criterion is
met, such as reaching a maximum number of iterations or achieving a target fitness
level.
Algorithm 11 Walrus Optimization Algorithm (WaOA) for Feature Selection
1: Step 1: Initialize Population
2: for each walrus i in range (1, Population Size) do

Generate a random binary vector of length num features representing se-

3:

lected features
4: end for
5: Step 2: Calculate Initial Fitness
6: for each walrus i do

Evaluate fitness using cross-validation with XGBoost for each walrus’s

7:

selected features (see Equation 5)
Store fitness score for each walrus

8:

9: end for
10: Identify the best initial solution as Best Solution with fitness Best Fitness
11: Step 3: Optimization Loop Using WaOA Mechanisms
12: for each iteration in range (1, Max Iterations) do

for each walrus i do

13:

Select a random number Phase Choice to determine the walrus’s behav-

14:

ior
15:

if Phase Choice < 0.4 then

16:

Feeding Phase: Move walrus towards the best-known solution

17:

Update position using movement influenced by Best Solution
75

else if 0.4 ≤ Phase Choice < 0.7 then

18:

Migration Phase: Perform a broad random movement for explo-

19:

ration
Update position by moving to a randomly selected walrus within the

20:

population
21:

else
Escaping/Fighting Phase: Small refining movement around

22:

Best Solution
Update position by making a small adjustment around Best Solution

23:
24:

end if

25:

Ensure updated position remains within bounds and apply binary conversion to represent selected features

26:

Evaluate fitness for the updated position

27:

if updated fitness is better than current walrus’s fitness then
Update walrus’s position and fitness

28:
29:

end if

30:

end for

31:

Update Best Solution and Best Fitness if a better solution is found in this
iteration

32:

Track Best Fitness across iterations in fitness history

33: end for
34: Step 4: Return Results
35: Output: The best solution (selected features) with the highest fitness score,

fitness history
This pseudocode outlines the WaOA as a feature selection algorithm, focusing on
structured phases that help balance exploration and exploitation, with evaluation via
cross-validated XGBoost for accurate fitness assessment. The iterative application
76

of these distinct phases promotes robustness in feature selection by leveraging
WaOA’s unique foraging-inspired behaviors.

T o conclude the discussion of standard metaheuristic algorithms, it’s clear
that each algorithm applied different optimization strategies to identify the most
informative feature subsets. By exploring various approaches, these metaheuristics
demonstrated the ability to effectively navigate large, complex search spaces,
balancing exploration and exploitation to support improved wildfire prediction.
However, despite their strengths, conventional metaheuristic methods face several
limitations:

• Escaping Local Optima: Many standard algorithms, while robust, may
struggle to consistently escape local optima in high-dimensional feature spaces,
potentially limiting their effectiveness for complex, interdependent wildfire
data Agrawal et al. [2021] .
• Handling High Dimensionality: As dimensionality increases, the computational load and convergence challenges also grow, impacting the scalability
and overall performance of these methodsAkinola et al. [2022].

Given these limitations, a specialized approach for feature selection is advantageous. The Liver Cancer Algorithm (LCA) was selected for targeted improvement
due to its adaptability to high-dimensional data and robust search capabilities.
However, LCA has unique challenges, as highlighted in recent studies. Despite its
strengths, the algorithm can exhibit limitations in precise refinement near optimal
solutions, given its reliance on Levy Flight and Random Opposition-Based Learning
(ROBL). These mechanisms, while enhancing exploration and diversity, may lead

77

to overshooting and can occasionally slow down convergence near optimal solutions
due to their inherent randomness.
To address these issues, integrating a Spiral Update mechanism into LCA offers
a systematic approach to enhance search precision. The Spiral Update guides the
algorithm around promising solutions in a controlled path, providing more accurate,
fine-tuned adjustments as it nears the optimal solution. This improvement enables
LCA to more effectively balance exploration and exploitation, reducing the tendency
to overshoot while maintaining a steady convergence path. In high-dimensional
contexts such as wildfire prediction, the refined search offered by Spiral Update
increases the precision and stability of the algorithm, positioning it to outperform
conventional methods by reliably converging toward optimal solutions.

Liver Cancer Algorithm (LCA)

The Liver Cancer Algorithm (LCA) is a bio-inspired metaheuristic optimization
algorithm that simulates the progression and mutation patterns observed in liver
cancer cells to explore potential solutions in an optimization search space. In LCA,
candidate solutions act as individual cells, with each cell representing a unique
feature subset. The algorithm iteratively refines these cells, allowing them to
”mutate” and explore different areas of the search space.

78

3.4.3

Mathematical Model of LCA

Tumor Growth

The mathematical model of the Liver Cancer Algorithm (LCA) is centered around
simulating the tumor growth and metastasis process. The initial stage involves
calculating the tumor’s size, which is critical for setting up the algorithm’s population. This process is modeled by:

Positionji =





π
lengthj · widthj · heightj − lb + (ub − lb) − rd · Positionji (3.7)
6

where Positionji represents the estimated volume of the tumor for the i-th agent
in the j-th dimension, lb and ub are the lower and upper bounds of the search
space, respectively, and rd is a random number between 0 and 1 that facilitates
exploration by introducing stochastic variation.
To model the growth of the tumor, the position size is updated dynamically
using:

Position =

π
· f · (l · w)3/2
6

(3.8)

where f = 1 is a constant representing a specific tumor type, and l and w
denote the length and width of the tumor, respectively. This formulation captures
the geometric expansion behavior of the tumor, allowing the algorithm to balance
exploration and exploitation effectively.
The initial size calculation ensures a diverse and well-distributed population,
79

enabling the algorithm to adaptively explore the search space during optimization.

Optimization Stages

Following the initial tumor size calculation, the next phase simulates the replication
process observed in malignant tumors. This stage models the tumor’s ability to
duplicate itself in different regions within the liver, reflecting its invasive behavior.
The replication phase relies on the exponential growth model, which is commonly
used to describe cellular division in cancers, including hepatocellular carcinoma.
The tumor growth process is expressed mathematically as:
P Gji =

dV
= r × P ositiont ,
dt

t ∈ [1 . . . T ],

i ∈ [1 . . . N ],

(3.9)

where P G represents the tumor growth rate, r denotes the radius of the tumor
modeled as an ellipsoidal shape, and P ositiont is derived from the earlier initialization step. The parameters include: - T : Maximum number of iterations, - N :
Population size or number of agents.

Advanced Search Dynamics

To improve exploration and avoid local optima, the replication phase integrates a
Lévy flight-based search mechanism. The random walk process follows:
Lv(D) = 0.01 ×

rand(1, D) · σ
1

,

(3.10)

|rand(1, D)| β
where D is the dimension size, β controls the step distribution, and σ determines
the scale factor. The scale σ is calculated as:
σ=

Γ(1 + β) sin πβ
2

 β−1
1+β
2
Γ 2 β2


 ! β1
,

(3.11)
80

where Γ denotes the Gamma function.

Exploration and Decision Mechanism. Based on the growth dynamics, each
tumor evaluates its surroundings and determines its next position through the
following steps:
y = P osition + P G,

(3.12)

Z = Y + S × LF (D),

(3.13)

where S represents random scaling factors to enhance exploration.
Finally, positions are updated based on fitness evaluations:


y, if f it(y) < f it(P ositioni ),
P ositioni,t+1 =

z, if f it(z) < f it(P ositioni ),

(3.14)

where f it(·) represents the fitness function used to evaluate solution quality.
In the final stage of the LCA, tumor progression mimics the metastatic behavior
of cancer by applying genetic operators such as mutation and crossover.
Mutation: Mutation introduces diversity by randomly altering components of
the tumor position. The mutation operator is defined as:

yMut =



P osition if rand1 ≥ ζ

y

zMut =

else



P osition if rand2 ≥ ζ

z

(3.15)

(3.16)

else

where:
81

ζ=

1
,
T

y = |P osition − P ositionj |,

z =y−S

(3.17)

Crossover: Crossover combines elements from two parents to generate offspring.
The new position after crossover is:

wCross = τ · yMut + (1 − τ ′ ) · zMut ,

τ ̸= τ ′

(3.18)

Selection: A greedy selection process evaluates the fitness of each position,
retaining the best solutions for the next iteration:





yMut



P ositioni,t+1 = zMut





wCross

if f it(yMut ) < f it(P ositioni )
if f it(zMut ) < f it(P ositioni )

(3.19)

if f it(wCross ) < f it(P ositioni )

The mutation and crossover operations ensure sufficient exploration and exploitation within the search space, enabling the LCA algorithm to converge effectively
towards optimal solutions.
Following the initialization, the LCA simulates tumor replication and spreading
within the host organ, reflecting stages of disease progression. This adaptive search
process is governed by:
Position = 6 × f × (ℓ · w)3/2

(3.20)

where f is a constant reflecting specific tumor characteristics, and ℓ and w represent
the evolving dimensions of the tumor within the search space.

82

The replication of the tumor, which signifies a critical phase of disease progression, is mathematically described by:
dV
= r · Positiont ,
dt

t ∈ [1, T ],

i ∈ [1, N ]

(3.21)

where V is the volume of the tumor position, r defines the growth rate, T is the
maximum number of iterations, and N represents the number of search agents.
To effectively simulate the metastatic spread, LCA utilizes Lévy flight dynamics,
enhancing its exploratory capabilities across the problem landscape:
L(D) = 0.01 × rand(1, D) × σ

σ=

Γ(1 + β) × sin(πβ/2)
Γ((1 + β)/2) × β × 2(β−1)/2

(3.22)

1/β
(3.23)

These equations govern the stochastic and strategic dispersal of search agents,
mimicking the unpredictable and aggressive spread of liver cancer cells across the
liver landscape, optimizing the algorithm’s capability to locate and refine the best
solutions for complex optimization problems.

3.4.4

Algorithm Mechanics

The Liver Cancer Algorithm (LCA) employs mutation-based processes inspired by
tumor growth, replication, and metastasis to perform optimization. Its mechanics
are as follows:

• Primary Mutation: This process introduces random changes in the position
of candidate solutions, simulating the unpredictable growth of tumors. It
promotes diversity within the population, ensuring broad exploration of the
search space to avoid premature convergence.
83

• Opposition-Based Mutation: Inspired by opposition-based learning
(ROBL), this mechanism generates complementary solutions by reflecting
positions across their opposite values within the search bounds. This process
expands the search area, allowing the algorithm to escape local optima
and improve solution diversity.
• Crossover Mutation: Borrowing principles from genetic algorithms, LCA
combines features of high-fitness solutions (parents) to produce offspring with
enhanced traits. This crossover mechanism allows exploitation of promising
areas while retaining diversity in the population.

The algorithm iteratively applies these mutation processes until a termination
criterion, such as the maximum number of iterations, is met.

84

Algorithm 12 Liver Cancer Algorithm (LCA) for Feature Selection
1: Step 1: Initialization
2: Initialize population size N and maximum iterations T
3: for each agent i in range (1, N ) do
4:

Generate random positions Positioni using Eq. (3.7)

5: end for
6: Step 2: Fitness Evaluation
7: for each agent i do
8:

Evaluate fitness using cross-validation (e.g., XGBoost) for selected features

9:

Store fitness scores and identify Best Position and Best Fitness

10: end for
11: Step 3: Iterative Optimization Process
12: for each iteration t in range (1, T ) do
13:

for each agent i do

14:

Compute fitness for current position

15:

if Position < 0.8 or Position > 0.8 then
Update tumor position using Eq. (3.20)

16:
17:

else if 0.8 < Position < 1.96 then

▷ Replication Phase

Update position using Eq. (3.20) (Exponential Growth)

18:
19:

▷ Exploitation Phase

else

▷ Exploration Phase

20:

Apply mutation using Eq. (3.15) and Eq. (3.16)

21:

Apply crossover using Eq. (3.18)

22:

Evaluate new positions using fitness and select the best

23:

end if

24:

end for

25:

Step 4: Update Population and Best Fitness

26:

Compare updated fitness values with Best Fitness

27:

if new fitness improves then

28:
29:

Update Best Position and Best Fitness
end if

30: end for
31: Step 5: Output Results
32: Return Best Position and Best Fitness

85

Manta Ray Foraging Optimization (MRFO)

The Manta Ray Foraging Optimization (MRFO) is a metaheuristic algorithm
inspired by manta rays’ unique foraging behaviour, which includes coordinated
movements for prey capture. In MRFO, candidate solutions represent individual
manta rays, with each ray’s position in the search space iteratively updated to
explore and exploit potential solutions effectively.

Algorithm Mechanics MRFO operates through three primary foraging mechanisms:

• Chain Foraging: In this phase, manta rays move in a chain-like formation,
where each ray updates its position based on the ray ahead or the current
best solution. This behaviour promotes coordinated exploration across the
search space.
• Cyclone Foraging: Here, manta rays move in a spiralling pattern towards
the best-known solution, allowing for a more localized search that refines
promising areas of the solution space.
• Somersault Foraging: Manta rays perform a somersault around the best
solution, enhancing exploration and promoting diversity by moving within a
range around the best-known solution.

The algorithm iteratively applies these mechanisms until a stopping criterion is
met, such as reaching a maximum number of iterations or achieving a target fitness
level.

86

Algorithm 13 Manta Ray Foraging Optimization (MRFO) for Feature Selection
1: Step 1: Initialize Population
2: for each manta ray i in range (1, Population Size) do
3:

Generate a random binary vector of length num features, representing
selected features

4: end for
5: Step 2: Calculate Initial Fitness
6: for each manta ray i do
7:

Evaluate fitness using cross-validation with XGBoost for each particle’s
selected features (see Equation 5)

8:

Store fitness score for each manta ray

9: end for
10: Identify the best initial solution as Best Solution with fitness Best Fitness
11: Step 3: Optimization Loop Using MRFO Mechanisms
12: for each iteration in range (1, Max Iterations) do
13:

Chain Foraging: Each manta ray updates its position based on the
preceding ray or the current best solution

14:
15:

for each manta ray i do
Update position based on chain movement towards the best solution or
the position of ray i − 1

16:

end for

17:

Cyclone Foraging: Each manta ray spirals towards the best-known
solution

18:
19:
20:

for each manta ray i do
Adjust position in a spiral pattern towards the best-known solution
end for

87

21:

Somersault Foraging: Each manta ray explores within a radius around
the best solution

22:
23:

for each manta ray i do
Update position based on somersault foraging around the best solution

24:

end for

25:

Evaluate fitness for all updated positions

26:

Update Best Solution and Best Fitness if a better solution is found in this
iteration

27:

Track Best Fitness across iterations in fitness history

28: end for
29: Step 4: Return Results
30: Output: The best solution (selected features) with the highest fitness score,

fitness history

Particle Swarm Optimization (PSO)

Particle Swarm Optimization (PSO) is a popular metaheuristic inspired by the
social behaviour of birds flocking or fish schooling. In PSO, candidate solutions
are modelled as particles that ”fly” through the search space, with each particle
adjusting its position based on its personal experience and the collective experience
of the swarm. This algorithm leverages both individual and social learning to
converge toward optimal solutions.

Algorithm Mechanics PSO operates through three main steps:

• Initialization: A population of particles is randomly generated, each with
an associated velocity and position representing a candidate solution.
88

• Position and Velocity Update: Each particle updates its velocity and
position based on its own best-known position, the best-known position in its
neighbourhood, and its current velocity. This balance allows for a mixture of
exploration and exploitation in the search space.
• Fitness Evaluation: Each particle is evaluated based on a fitness function,
often cross-validated accuracy or another metric, to determine the quality of
the solution represented by its position.

The algorithm iterates these steps until a stopping criterion is met, such as
reaching a maximum number of iterations or achieving a desired fitness level.
Algorithm 14 Particle Swarm Optimization (PSO) for Feature Selection
1: Step 1: Initialize Population
2: for each particle i in range (1, Population Size) do
3:

Randomly generate a binary vector of length num features representing
selected features

4:

Initialize velocity for each particle as a vector of the same length

5:

Set each particle’s personal best position to its initial position

6: end for
7: Identify the global best position Best Position with fitness Best Fitness from

initial particles
8: Step 2: Optimization Loop
9: for each iteration in range (1, Max Iterations) do
10:
11:

for each particle i do
Update velocity using personal best, global best, and inertia terms:

velocity[i] = w×velocity[i]+c1 ×rand()×(personal best[i]−position[i])+c2 ×rand()×(Best Positio

89

12:

Update position based on the updated velocity:
position[i] = position[i] + velocity[i]

13:

Apply a sigmoid function to ensure binary feature selection, converting
each dimension to 0 or 1

14:

Evaluate fitness using cross-validation with XGBoost for each particle’s
selected features (see Equation 5)

15:

if updated fitness is better than particle’s personal best fitness then
Update particle’s personal best position and fitness

16:
17:

end if

18:

end for

19:

Update global best position Best Position and fitness Best Fitness if a better
solution is found in this iteration

20:

Track Best Fitness across iterations in fitness history

21: end for
22: Step 3: Return Results
23: Output: The best solution (selected features) with the highest fitness score,

fitness history

Walrus Optimization Algorithm (WaOA)

The Walrus Optimization Algorithm (WaOA) is a metaheuristic algorithm inspired
by the social and foraging behaviors of walruses. In WaOA, candidate solutions
represent individual walruses, which adapt their positions in the search space based
on different behaviors, such as feeding, migration, and escaping/fighting. This
structured behavior allows for effective exploration and exploitation to achieve
optimal solutions.
90

Algorithm Mechanics WaOA operates through three main foraging mechanisms:

• Feeding Phase: Walruses move towards the best-known solution or promising areas to exploit high-quality solutions. This behavior promotes local
search around successful solutions.
• Migration Phase: Walruses perform broader, random movements to explore
the search space. This movement aims to prevent the algorithm from becoming
trapped in local optima and to promote exploration of new areas.
• Escaping/Fighting Phase: Walruses make smaller, refining movements
around high-quality solutions to balance between exploration and exploitation.
This helps refine promising solutions by focusing on adjustments within a
local neighborhood.

The algorithm iteratively applies these mechanisms until a stopping criterion is
met, such as reaching a maximum number of iterations or achieving a target fitness
level.
Algorithm 15 Walrus Optimization Algorithm (WaOA) for Feature Selection
1: Step 1: Initialize Population
2: for each walrus i in range (1, Population Size) do
3:

Generate a random binary vector of length num features representing selected features

4: end for
5: Step 2: Calculate Initial Fitness
6: for each walrus i do

91

Evaluate fitness using cross-validation with XGBoost for each walrus’s

7:

selected features (see Equation 5)
Store fitness score for each walrus

8:

9: end for
10: Identify the best initial solution as Best Solution with fitness Best Fitness
11: Step 3: Optimization Loop Using WaOA Mechanisms
12: for each iteration in range (1, Max Iterations) do

for each walrus i do

13:

Select a random number Phase Choice to determine the walrus’s behav-

14:

ior
if Phase Choice < 0.4 then

15:
16:

Feeding Phase: Move walrus towards the best-known solution

17:

Update position using movement influenced by Best Solution
else if 0.4 ≤ Phase Choice < 0.7 then

18:

Migration Phase: Perform a broad random movement for explo-

19:

ration
20:

Update position by moving to a randomly selected walrus within the
population

21:
22:

else
Escaping/Fighting Phase: Small refining movement around
Best Solution

23:

Update position by making a small adjustment around Best Solution

24:

end if

25:

Ensure updated position remains within bounds and apply binary conversion to represent selected features

26:

Evaluate fitness for the updated position

27:

if updated fitness is better than current walrus’s fitness then

92

Update walrus’s position and fitness

28:
29:

end if

30:

end for

31:

Update Best Solution and Best Fitness if a better solution is found in this
iteration

32:

Track Best Fitness across iterations in fitness history

33: end for
34: Step 4: Return Results
35: Output: The best solution (selected features) with the highest fitness score,

fitness history
This pseudocode outlines the WaOA as a feature selection algorithm, focusing on
structured phases that help balance exploration and exploitation, with evaluation via
cross-validated XGBoost for accurate fitness assessment. The iterative application
of these distinct phases promotes robustness in feature selection by leveraging
WaOA’s unique foraging-inspired behaviors.

T o conclude the discussion of standard metaheuristic algorithms, it’s clear
that each algorithm applied different optimization strategies to identify the most
informative feature subsets. By exploring various approaches, these metaheuristics
demonstrated the ability to effectively navigate large, complex search spaces,
balancing exploration and exploitation to support improved wildfire prediction.
However, despite their strengths, conventional metaheuristic methods face several
limitations:

• Escaping Local Optima: Many standard algorithms, while robust, may
struggle to consistently escape local optima in high-dimensional feature spaces,
93

potentially limiting their effectiveness for complex, interdependent wildfire
data Agrawal et al. [2021] .
• Handling High Dimensionality: As dimensionality increases, the computational load and convergence challenges also grow, impacting the scalability
and overall performance of these methodsAkinola et al. [2022].

Given these limitations, a specialized approach for feature selection is advantageous. The Liver Cancer Algorithm (LCA) was selected for targeted improvement
due to its adaptability to high-dimensional data and robust search capabilities.
However, LCA has unique challenges, as highlighted in recent studies. Despite its
strengths, the algorithm can exhibit limitations in precise refinement near optimal
solutions, given its reliance on Levy Flight and Random Opposition-Based Learning
(ROBL). These mechanisms, while enhancing exploration and diversity, may lead
to overshooting and can occasionally slow down convergence near optimal solutions
due to their inherent randomness.
To address these issues, integrating a Spiral Update mechanism into LCA offers
a systematic approach to enhance search precision. The Spiral Update guides the
algorithm around promising solutions in a controlled path, providing more accurate,
fine-tuned adjustments as it nears the optimal solution. This improvement enables
LCA to more effectively balance exploration and exploitation, reducing the tendency
to overshoot while maintaining a steady convergence path. In high-dimensional
contexts such as wildfire prediction, the refined search offered by Spiral Update
increases the precision and stability of the algorithm, positioning it to outperform
conventional methods by reliably converging toward optimal solutions.
Given these limitations, a specialized approach for feature selection is advantageous. The Liver Cancer Algorithm (LCA) was selected for targeted improvement
94

due to its adaptability to high-dimensional data and robust search capabilities.
However, LCA has unique challenges, as highlighted in recent studies. Despite its
strengths, the algorithm can exhibit limitations in precise refinement near optimal
solutions, given its reliance on Levy Flight and Random Opposition-Based Learning
(ROBL). These mechanisms, while enhancing exploration and diversity, may lead
to overshooting and can occasionally slow down convergence near optimal solutions
due to their inherent randomness.
To address these issues, integrating a Spiral Update mechanism into LCA offers
a systematic approach to enhance search precision. The Spiral Update guides the
algorithm around promising solutions in a controlled path, providing more accurate,
fine-tuned adjustments as it nears the optimal solution. This improvement enables
LCA to more effectively balance exploration and exploitation, reducing the tendency
to overshoot while maintaining a steady convergence path. In high-dimensional
contexts such as wildfire prediction, the refined search offered by Spiral Update
increases the precision and stability of the algorithm, positioning it to outperform
conventional methods by reliably converging toward optimal solutions.

3.5

Proposed Algorithm

3.5.1

Spiral Implementation

Spiral structures generally appear in forms such as galaxies, hurricanes, seashells,
and plant growth patterns. A consistent yet dynamic progression marks these
natural phenomena, inspiring optimization techniques that mimic their structured
navigation through complex environments. The convergent nature of spirals,

95

coupled with their orderly expansion, makes them highly effective in optimization.
They offer a path that balances thorough search space exploration with a steady
movement towards optimal solutions (Omar et al. [2022], Tamura and Yasuda
[2011]).
The spiral’s rotating and inward-curving trajectory enables smooth transitions
toward high-potential areas without the abrupt jumps often seen in random exploration methods Polezhaev [2019]. This structured progression supports controlled
convergence, particularly in high-dimensional and complex search spaces. Specifically, spiral dynamics provides the following advantages in optimization:

• Improved Search Precision: The gradual approach of the spiral path
allows algorithms to fine-tune adjustments as they get closer to the optimal
solution. This precise control helps reduce the risk of overshooting highpotential regions, addressing a common limitation in methods such as Levy
flights.
• Enhanced Exploration and Exploitation Balance: Through systematic
rotation throughout the search space, spiral dynamics strike a balance between
exploring new regions and exploiting promising ones. This balanced approach
is essential in optimization, preventing premature convergence and ensuring
a comprehensive search space exploration.
• Adaptability to Boundaries: Spiral paths can be adjusted to remain
within defined boundaries, making them particularly suitable for constrained
optimization tasks. This flexibility allows the spiral dynamics to operate
effectively in diverse problems where boundary conditions must be respected.

In optimization, the spiral dynamic’s predictable yet flexible path offers substan96

tial benefits for navigating high-dimensional spaces with complex interdependencies.
For algorithms such as the Liver Cancer Algorithm (LCA), integrating a Spiral
Update mechanism introduces a systematic search pattern that enhances precision
in convergence. This enhancement is especially valuable in complex domains such
as wildfire prediction, where efficient exploration and accurate targeting of highimpact feature subsets are critical to improve predictive performance.

Parameter Definitions

To formalize the implementation of various spiral types, the following parameters
are consistently used throughout their mathematical representations:
• a: A scaling constant determining the spiral’s initial size or growth rate.
• b: A constant controlling the spacing or rate of growth between successive
loops of the spiral.
• φ: The polar angle, measured in radians, represents the rotational component
of the spiral.
• θ: The angular increment, a step size determining the resolution of the spiral
path.
• r: The radius or distance from the origin to a point on the spiral.
• v: A constant velocity used in some formulations to integrate linear motion
with rotational dynamics.
• ω: The angular velocity defines the rate at which the spiral rotates about
the origin.

97

These parameters are adapted for each specific spiral type to achieve tailored
exploration and exploitation dynamics.
We utilized several spiral path types to implement the Spiral Update in the
Liver Cancer Algorithm (LCA), each tailored to different optimization strategies.
By incorporating these specific spiral dynamics into the LCA, we developed a more
refined approach to feature selection, particularly well-suited for high-dimensional
and interdependent datasets. Below, we outline each spiral type, its mathematical
formulation, and its unique contributions to enhancing the LCA’s performance.

The Archimedean Spiral
• Archimedean Spiral: The Archimedean spiral, named after the Greek
mathematician Archimedes, describes a path in which a point moves outward
from a fixed origin at a constant speed while rotating with a constant angular
velocity. This type of spiral can be mathematically represented in polar
coordinates by the formula:

r =b·θ

(3.24)

where r is the radius (distance from the origin), θ is the angle, and b is a
constant controlling the spacing between loops. A higher value of b increases
the distance between successive spiral turns.
The Archimedean spiral is beneficial in optimization, especially for systematically exploring search spaces. Because the radial distance r grows
proportionally with θ, the spiral expands evenly from the origin. This steady
spacing enables controlled and incremental searches, avoiding large jumps
and ensuring consistent coverage across the search space.
98

Derivation of the Archimedean Spiral Formula To derive this formula,
consider a point moving at constant velocity v along the x-axis in Cartesian
coordinates. If the xy-plane rotates around the z-axis with a constant angular
velocity ω, then the position of the point at any time t is given by:

x = (vt + c) cos(ωt) and y = (vt + c) sin(ωt)

(3.25)

where c is an initial radial offset. This formulation integrates linear motion
with rotational movement, creating a spiral path over time. In polar coordinates, this can be expressed as:

r=

v
·θ+c
ω

(3.26)

Key Characteristics
– Constant Separation Distance: Each loop of the Archimedean spiral
is separated by a constant distance along any ray from the origin. For θ
measured in radians, this spacing is 2πb.
– Uniform Expansion: For large θ, movement along the spiral approximates uniform acceleration, providing a stable path ideal for controlled,
predictable searches.
– Symmetry: The Archimedean spiral is symmetrical, with two mirrored
arms—one for θ > 0 and one for θ < 0—supporting balanced search
patterns.

Algorithm 16 Liver Cancer Algorithm (LCA) with Archimedean Spiral Update.
99

Figure 3.2: Visualization of the Archimedean Spiral
1: Step 1: Initialize Population
2: for each agent i in range (0, SearchAgents) do
3:

Generate a binary vector of length dimension for feature selection

4:

Generate opposite agents to enhance diversity and combine with initial
agents

5: end for
6: Step 2: Calculate Initial Fitness
7: for each agent i do
8:

Select features based on agent i’s binary vector

9:

Evaluate fitness using cross-validation with XGBoost on the selected features

10:

Store fitness score in fit[i]

11: end for
12: Identify the best initial solution, updating best score and best features

100

13: Step 3: Optimization Loop
14: for each iteration t in range (1, Max iterations) do
15:
16:

for each agent i do
Exploration with Levy Flight: With probability 0.5, apply Levy
flight to diversify the agent’s position

17:
18:

if random() ¡ 0.5 then
Generate a step using Levy flight and adjust the agent’s position

19:

end if

20:

**Archimedean Spiral Update**: With probability 0.5, apply

21:

if random() ¡ 0.5 then

22:

Set best agent ← X[argmax(fit)]

23:

Compute the direction vector direction = best agent − X[i]

24:

Calculate the Euclidean norm of the direction vector:
norm = ∥direction∥

25:
26:

if norm = 0 then
Set norm = 1 to avoid division by zero

27:

end if

28:

Normalize the direction vector:
unit direction =

direction
norm

29:

Set the angular increment θ (e.g., θ = 0.1)

30:

Calculate the spiral step size based on constant b:
step = b · θ · unit direction

31:

Update agent’s position in the direction of the spiral step:
new position = X[i] + step
101

Convert updated position to binary by setting values above 0.5 to 1

32:

and others to 0:
X[i] = (new position > 0.5)
33:

end if

34:

Crossover and Mutation:

35:

Select a partner agent at random

36:

Apply crossover to generate a child:
child ← crossover(X[i], X[partner idx])

37:

Apply mutation to the child:
mutated child ← mutation(child, mutation rate)

38:

Calculate the fitness of the mutated child

39:

if mutated child fitness ¿ current agent fitness then

40:

Replace the current agent with the mutated child

41:

end if

42:

end for

43:

Update Best Solution:

44:

if Current best fitness is greater than previous best then

45:
46:

Update best score and best features
end if

47: end for
48: Step 4: Return Results
49: Output: The best solution best features, best fitness score best score, and

fitness history over iterations

102

The Euler Spiral
• Euler Spiral: The Euler spiral, also known as the Clothoid or Cornu spiral,
is defined by a variable curvature that increases linearly with the arc length.
As the spiral progresses, its curvature continuously changes, allowing for
smooth transitions in direction Levien [2008], Seggern [1994], Cherchi et al.
[2013]. In polar coordinates, the Euler spiral does not have a straightforward
formula like the Archimedean spiral but can be described parametrically
using Fresnel integrals:
Z t
x(t) = C(t) =

cos
0

Z t


sin

y(t) = S(t) =
0

π u2
2



π u2
2





du

(3.27)

du

(3.28)

where C(t) and S(t) are the Fresnel integrals, which describe the coordinates
of the point on the spiral.
The Euler spiral is handy for optimizing smooth exploration. Its linearly
changing curvature provides a gradual transition in direction, preventing
abrupt movements. This helps navigate complex search spaces while maintaining a controlled exploration path.

Key Characteristics
– Smooth Curvature Transition: The Euler spiral’s curvature varies
linearly, allowing the search process to transition through potential
solutions without abrupt changes smoothly.

103

– Controlled Expansion: As the curvature gradually increases, the
spiral expands smoothly, ensuring the algorithm does not jump too
rapidly between points.
– Balanced Exploration and Exploitation: The continuous curvature
adjustment allows for a natural balance between exploration (diverging
outward) and exploitation (converging inward), making it suitable for
high-dimensional spaces.
Algorithm 17 Liver Cancer Algorithm (LCA) with Detailed Euler Spiral Update for Feature Selection
1: Step 1: Initialize Population
2: for each agent i in range (0, SearchAgents) do
3:

Generate a binary vector of length dimension for feature selection

4:

Generate opposite agents to enhance diversity and combine with initial
agents

5: end for
6: Step 2: Calculate Initial Fitness
7: for each agent i do
8:

Select features based on agent i’s binary vector

9:

Evaluate fitness using cross-validation with XGBoost on the selected features

10:

Store fitness score in fit[i]

11: end for
12: Identify the best initial solution, updating best score and best features
13: Step 3: Optimization Loop
14: for each iteration t in range (1, Max iterations) do
15:
16:

for each agent i do
Exploration with Levy Flight: With probability 0.5, apply Levy
flight to diversify the agent’s position
104

17:
18:

if random() ¡ 0.5 then
Generate a step using Levy flight and adjust the agent’s position

19:

end if

20:

Euler Spiral Update: With probability 0.5, apply Euler spiral update
towards the best agent

21:

if random() ¡ 0.5 then

22:

Set best agent ← X[argmax(fit)]

23:

Detailed Logic of Euler Spiral Update:

24:

Compute the direction vector direction = best agent − X[i]

25:

Calculate the Euclidean norm of the direction vector:
norm = ∥direction∥

26:
27:

if norm = 0 then
Set norm = 1 to avoid division by zero

28:

end if

29:

Normalize the direction vector:
unit direction =

30:

direction
norm

Set the initial curvature and curvature increment:
curvature = curvature start + curvature increment · norm

31:

Calculate the Euler spiral step based on the curvature:
step = step size · curvature · unit direction

32:

Update agent’s position in the direction of the Euler spiral step:
new position = X[i] + step

105

Apply dynamic thresholding to convert the position to binary:

33:

X[i] = (new position > dynamic threshold(t, Max iterations)).astype(int)
34:

end if

35:

Crossover and Mutation:

36:

Select a partner agent at random

37:

Apply crossover to generate a child:
child ← crossover(X[i], X[partner idx])

38:

Apply mutation to the child:
mutated child ← mutation(child, mutation rate)

39:

Calculate the fitness of the mutated child

40:

if mutated child fitness ¿ current agent fitness then

41:

Replace the current agent with the mutated child

42:

end if

43:

end for

44:

Update Best Solution:

45:

if Current best fitness is greater than previous best then

46:
47:

Update best score and best features
end if

48: end for
49: Step 4: Return Results
50: Output: The best solution best features, best fitness score best score, and

fitness history over iterations

106

Figure 3.3: Visualization of the Euler Spiral
The Fermat Spiral
• Fermat Spiral: The Fermat spiral, or parabolic spiral, is characterized by the
property that the area between consecutive turns remains constant while the
distance between turns grows proportionally with the square root of the angle
from the center. Unlike the Archimedean spiral, which maintains constant
spacing, the Fermat spiral expands outward slower, making it suitable for
controlled exploration with a gradual increase in search radius.
In polar coordinates, the Fermat spiral can be represented as:
√
r = ±a φ

(3.29)

where r is the radius (distance from the origin), φ is the angle, and a
is a constant that controls the growth rate. The two signs produce two
symmetrical arms of the spiral, both converging at the origin.
107

The Fermat spiral’s unique property of spacing growth makes it particularly
effective in feature selection tasks for wildfire prediction. It allows for a
structured, outward exploration. By gradually expanding the search radius,
the Fermat spiral provides comprehensive coverage of the feature space,
reducing the likelihood of missing important feature subsets.

Key Characteristics
– Constant Area between Turns: The Fermat spiral maintains a
constant area between each pair of successive loops, supporting even
exploration throughout the search space.
– Gradual Expansion: The distance between loops increases propor√
tionally with φ, allowing for controlled expansion that is particularly
beneficial in high-dimensional search spaces.
– Symmetry: The two symmetrical branches of the Fermat spiral provide
balanced coverage, supporting consistent exploration across the search
space.
Algorithm 18 Liver Cancer Algorithm (LCA) with Detailed Fermat Spiral
Update for Feature Selection
1: Step 1: Initialize Population
2: for each agent i in range (0, SearchAgents) do
3:

Generate a binary vector of length dimension for feature selection

4:

Generate opposite agents to enhance diversity and combine with initial
agents

5: end for
6: Step 2: Calculate Initial Fitness
7: for each agent i do

108

8:

Select features based on agent i’s binary vector

9:

Evaluate fitness using cross-validation with XGBoost on the selected features

10:

Store fitness score in fit[i]

11: end for
12: Identify the best initial solution, updating best score and best features
13: Step 3: Optimization Loop
14: for each iteration t in range (1, Max iterations) do
15:
16:

for each agent i do
Exploration with Levy Flight: With probability 0.5, apply Levy
flight to diversify the agent’s position

17:
18:

if random() ¡ 0.5 then
Generate a step using Levy flight and adjust the agent’s position

19:

end if

20:

Fermat Spiral Update: With probability 0.5, apply Fermat spiral
update towards the best agent

21:

if random() ¡ 0.5 then

22:

Set best agent ← X[argmax(fit)]

23:

Detailed Logic of Fermat Spiral Update:

24:

Compute the direction vector direction = best agent − X[i]

25:

Calculate the Euclidean norm of the direction vector:
norm = ∥direction∥

26:
27:

if norm = 0 then
Set norm = 1 to avoid division by zero

28:

end if

29:

Normalize the direction vector:
unit direction =

direction
norm
109

Set the angle increment φ and calculate the Fermat spiral step size:

30:

step = a ·

√

φ · unit direction

Update agent’s position in the direction of the Fermat spiral step:

31:

new position = X[i] + step
Convert updated position to binary based on a dynamic threshold:

32:

X[i] = (new position > dynamic threshold(t, Max iterations)).astype(int)
33:

end if

34:

Crossover and Mutation:

35:

Select a partner agent at random

36:

Apply crossover to generate a child:
child ← crossover(X[i], X[partner idx])

37:

Apply mutation to the child:
mutated child ← mutation(child, mutation rate)

38:

Calculate the fitness of the mutated child

39:

if mutated child fitness ¿ current agent fitness then

40:

Replace the current agent with the mutated child

41:

end if

42:

end for

43:

Update Best Solution:

44:

if Current best fitness is greater than previous best then

45:
46:

Update best score and best features
end if
110

47: end for
48: Step 4: Return Results
49: Output: The best solution best features, best fitness score best score, and

fitness history over iterations

Figure 3.4: Visualization of the Fermat Spiral

The Golden Spiral
• Golden Spiral: The Golden spiral is a specific logarithmic spiral where
the radius grows by a factor of the golden ratio φ ≈ 1.618 for every quarter
turn. This unique property of expansion enables a consistent scaling as the
spiral moves outward, with each quarter turn moving further away from the
111

origin by a constant proportion. The Golden spiral is especially suitable for
optimization tasks requiring smooth, proportional search space exploration.
In polar coordinates, the Golden spiral is represented by:
r = aebθ

(3.30)

where r is the radius, θ is the angle, a is the initial radius, and b is the growth
φ
rate determined by the golden ratio, given by b = ln
.
π/2

In the context of feature selection for wildfire prediction, the Golden Spiral’s
proportional and smooth expansion systematically explores the feature space.
This exploration ensures that the search process does not overlook critical
regions, enhancing the likelihood of identifying optimal feature subsets that
improve predictive performance.

Key Characteristics
– Proportional Growth: Each quarter turn increases the radius by a
factor of φ, supporting systematic exploration across varying scales in
the search space.
– Smooth Expansion: The Golden spiral grows outward smoothly, which
is advantageous for controlled expansion in high-dimensional feature
selection tasks.
– Self-Similarity: The self-similar nature of the Golden spiral supports
consistent coverage across the search space, balancing both global exploration and local exploitation.
Algorithm 19 Liver Cancer Algorithm (LCA) with Detailed Golden Spiral
Update for Feature Selection
112

1: Step 1: Initialize Population
2: for each agent i in range (0, SearchAgents) do
3:

Generate a binary vector of length dimension for feature selection

4:

Generate opposite agents to enhance diversity and combine with initial
agents

5: end for
6: Step 2: Calculate Initial Fitness
7: for each agent i do
8:

Select features based on agent i’s binary vector

9:

Evaluate fitness using cross-validation with XGBoost on the selected features

10:

Store fitness score in fit[i]

11: end for
12: Identify the best initial solution, updating best score and best features
13: Step 3: Optimization Loop
14: for each iteration t in range (1, Max iterations) do
15:
16:

for each agent i do
Exploration with Levy Flight: With probability 0.5, apply Levy
flight to diversify the agent’s position

17:
18:

if random() ¡ 0.5 then
Generate a step using Levy flight and adjust the agent’s position

19:

end if

20:

Golden Spiral Update: With probability 0.5, apply Golden spiral
update towards the best agent

21:

if random() ¡ 0.5 then

22:

Set best agent ← X[argmax(fit)]

23:

Detailed Logic of Golden Spiral Update:

24:

Compute the direction vector direction = best agent − X[i]

113

Calculate the Euclidean norm of the direction vector:

25:

norm = ∥direction∥
if norm = 0 then

26:

Set norm = 1 to avoid division by zero

27:
28:

end if

29:

Normalize the direction vector:
unit direction =

direction
norm

Set the angular increment θ and calculate the Golden spiral step size

30:

using:
step = aebθ · unit direction
31:

Update agent’s position in the direction of the Golden spiral step:
new position = X[i] + step

32:

Convert updated position to binary based on a dynamic threshold:
X[i] = (new position > dynamic threshold(t, Max iterations)).astype(int)

33:

end if

34:

Crossover and Mutation:

35:

Select a partner agent at random

36:

Apply crossover to generate a child:
child ← crossover(X[i], X[partner idx])

37:

Apply mutation to the child:
mutated child ← mutation(child, mutation rate)
114

38:

Calculate the fitness of the mutated child

39:

if mutated child fitness ¿ current agent fitness then

40:

Replace the current agent with the mutated child

41:

end if

42:

end for

43:

Update Best Solution:

44:

if Current best fitness is greater than previous best then

45:
46:

Update best score and best features
end if

47: end for
48: Step 4: Return Results
49: Output: The best solution best features, best fitness score best score, and

fitness history over iterations

The Hyperbolic Spiral
• Hyperbolic Spiral: The hyperbolic spiral, also called the reciprocal spiral,
is defined by a unique property where its radius decreases inversely with the
angle. This creates a curve that approaches an asymptotic line as it extends,
with a pitch angle that increases with distance from the origin. Unlike
logarithmic or Archimedean spirals, the hyperbolic spiral’s shape supports
a decelerated outward exploration, which is particularly advantageous in
controlled optimization tasks Bowser [1910], Øyvind Hammer [2016].
In polar coordinates, the hyperbolic spiral is represented by:
r=

a
φ

(3.31)

115

Figure 3.5: Visualization of the Golden Spiral
where r is the radius, φ is the angle, and a is a scale factor that controls the
spread of the spiral. The inverse relationship between r and φ ensures that
the spiral gradually widens while approaching an asymptotic line.
In the context of feature selection for wildfire prediction, the hyperbolic
spiral’s controlled expansion allows for a targeted exploration of the feature
space. This expansion ensures the search progresses systematically without
excessive jumps, improving the likelihood of identifying critical features for
model accuracy.

Key Characteristics
– Increasing Pitch Angle: The hyperbolic spiral’s pitch angle increases
116

with distance from the origin, supporting a decelerated yet comprehensive
search space exploration.
– Asymptotic Approach: As the spiral expands, it approaches an
asymptotic line, which limits extreme divergence and enables focused
exploration.
– Inverse Proportionality: The inverse relationship between radius and
angle allows the hyperbolic spiral to provide a balanced yet methodical
expansion, making it suitable for optimization scenarios requiring gradual
coverage.
Algorithm 20 Liver Cancer Algorithm (LCA) with Hyperbolic Spiral Update
for Feature Selection
1: Step 1: Initialize Population
2: for each agent i in range (0, SearchAgents) do
3:

Generate a binary vector of length dimension for feature selection

4:

Generate opposite agents to enhance diversity and combine with initial
agents

5: end for
6: Step 2: Calculate Initial Fitness
7: for each agent i do
8:

Select features based on agent i’s binary vector

9:

Evaluate fitness using cross-validation with XGBoost on the selected features

10:

Store fitness score in fit[i]

11: end for
12: Identify the best initial solution, updating best score and best features
13: Step 3: Optimization Loop
14: for each iteration t in range (1, Max iterations) do
15:

for each agent i do
117

16:

Exploration with Levy Flight: With probability 0.5, apply Levy
flight to diversify the agent’s position

17:

if random() ¡ 0.5 then
Generate a step using Levy flight and adjust the agent’s position

18:
19:

end if

20:

Hyperbolic Spiral Update: With probability 0.5, apply hyperbolic
spiral update towards the best agent

21:

if random() ¡ 0.5 then

22:

Set best agent ← X[argmax(fit)]

23:

Detailed Logic of Hyperbolic Spiral Update:

24:

Compute the direction vector direction = best agent − X[i]

25:

Calculate the Euclidean norm of the direction vector:
norm = ∥direction∥
if norm = 0 then

26:

Set norm = 1 to avoid division by zero

27:
28:

end if

29:

Normalize the direction vector:
unit direction =

direction
norm

Set the angle increment φ and calculate the hyperbolic spiral step

30:

size using:
step =
31:

a
· unit direction
φ

Update agent’s position in the direction of the hyperbolic spiral step:
new position = X[i] + step

32:

Convert updated position to binary based on a dynamic threshold:
X[i] = (new position > dynamic threshold(t, Max iterations)).astype(int)
118

33:

end if

34:

Crossover and Mutation:

35:

Select a partner agent at random

36:

Apply crossover to generate a child:
child ← crossover(X[i], X[partner idx])

37:

Apply mutation to the child:
mutated child ← mutation(child, mutation rate)

38:

Calculate the fitness of the mutated child

39:

if mutated child fitness ¿ current agent fitness then

40:

Replace the current agent with the mutated child

41:

end if

42:

end for

43:

Update Best Solution:

44:

if Current best fitness is greater than previous best then

45:
46:

Update best score and best features
end if

47: end for
48: Step 4: Return Results
49: Output: The best solution best features, best fitness score best score, and

fitness history over iterations

The Lituus Spiral
• Lituus Spiral: The Lituus spiral is an algebraic spiral characterized by an
inverse square root relationship between the radius and the angle. This spiral
119

Figure 3.6: Visualization of the Hyperbolic Spiral
type is named after an ancient Roman trumpet called the ”lituus,” which
Roger Cotes studied in 1722. The Lituus spiral has two symmetric branches,
each approaching infinity asymptotically and winding towards the origin. The
size of the spiral is controlled by the constant a, and each branch features
an inflection point, creating a unique curvature pattern as it approaches the
origin.
In polar coordinates, the Lituus spiral is represented by:
a
ρ= √
φ

(3.32)

where ρ is the radius, φ is the angle in radians, and a is a constant that
affects the scale of the spiral.
120

The Lituus spiral is advantageous for exploration tasks requiring controlled,
inward-reaching searches. Its inverse relationship with the angle enables a
targeted approach, which is particularly useful for gradually narrowing down
a search area.

Key Characteristics
– Inverse Square Root Relationship: The radius is inversely proportional to the square root of the angle, resulting in a spiral that tightens
near the origin.
– Symmetric Branches: Each branch is centrally symmetric around the
origin, providing balanced coverage.
– Inflection Points: The spiral features inflection points at (φ, ρ) =
√ 

± 12 , 2a , where the curvature changes direction.

Algorithm 21 Liver Cancer Algorithm (LCA) with Detailed Lituus Spiral Update for Feature Selection
1: Step 1: Initialize Population
2: for each agent i in range (0, SearchAgents) do
3:

Generate a binary vector of length dimension for feature selection

4:

Generate opposite agents to enhance diversity and combine with initial
agents

5: end for
6: Step 2: Calculate Initial Fitness
7: for each agent i do
8:

Select features based on agent i’s binary vector

9:

Evaluate fitness using cross-validation with XGBoost on the selected features

10:

Store fitness score in fit[i]
121

Figure 3.7: Visualization of the Lituus Spiral
11: end for
12: Identify the best initial solution, updating best score and best features
13: Step 3: Optimization Loop
14: for each iteration t in range (1, Max iterations) do
15:
16:

for each agent i do
Exploration with Levy Flight: With probability 0.5, apply Levy
flight to diversify the agent’s position

17:
18:

if random() ¡ 0.5 then
Generate a step using Levy flight and adjust the agent’s position

19:

end if

20:

Lituus Spiral Update: With probability 0.5, apply the Lituus spiral

122

update towards the best agent
21:

if random() ¡ 0.5 then

22:

Set best agent ← X[argmax(fit)]

23:

Compute the direction vector direction = best agent − X[i]

24:

Calculate the Euclidean norm of the direction vector:
norm = ∥direction∥

25:

if norm = 0 then
Set norm = 1 to avoid division by zero

26:
27:

end if

28:

Normalize the direction vector:
unit direction =

29:

direction
norm

Set the angle φ and calculate the Lituus spiral step size using:
a
step = √ · unit direction
φ

30:

Update agent’s position in the direction of the Lituus spiral step:
new position = X[i] + step

31:

Convert updated position to binary based on a dynamic threshold:
X[i] = (new position > 0.5).astype(int)

32:

end if

33:

Crossover and Mutation:

34:

Select a partner agent at random

35:

Apply crossover to generate a child:
child ← crossover(X[i], X[partner idx])
123

36:

Apply mutation to the child:
mutated child ← mutation(child, mutation rate)

37:

Calculate the fitness of the mutated child

38:

if mutated child fitness ¿ current agent fitness then

39:

Replace the current agent with the mutated child

40:

end if

41:

end for

42:

Update Best Solution:

43:

if Current best fitness is greater than previous best then

44:
45:

Update best score and best features
end if

46: end for
47: Step 4: Return Results
48: Output: The best solution best features, best fitness score best score, and

fitness history over iterations

The Logarithmic Spiral
• Logarithmic Spiral: The logarithmic spiral, also known as the equiangular
or growth spiral, is a unique self-similar spiral where the distance between
successive turns increases geometrically. This distinctive property results in
a consistent or pitch angle between the spiral and any radial line, creating
a curve that appears frequently in natural phenomena, such as shells and
galaxies. Jacob Bernoulli, who extensively studied this spiral, famously called
it Spira mirabilis, or ”the marvellous spiral,” due to its aesthetic appeal and
mathematical properties.
124

In polar coordinates, the logarithmic spiral is represented by:
r = aekφ

(3.33)

where r is the radius, φ is the angle, a is a scaling constant, and k is a
constant that defines the growth rate. The exponential relationship between
r and φ ensures that the spiral’s shape remains consistent as it expands.
In the context of feature selection for wildfire prediction, the logarithmic
spiral’s proportional growth provides a systematic way to explore the feature
space. Its geometric progression prevents excessive clustering of candidate
features, promoting a balanced search across the space to identify essential
variables effectively.

Key Characteristics
– Constant Pitch Angle: The logarithmic spiral’s pitch angle remains
consistent, supporting systematic and even exploration across different
scales in the search space.
– Self-Similarity: The self-similar nature of the logarithmic spiral ensures
uniform coverage as the spiral progresses outward, making it ideal for
optimization tasks that require balanced exploration and exploitation.
– Exponential Growth: The spiral’s growth is geometric, increasing the
distance between turns as it extends, which is advantageous for gradual
exploration without crowding near the origin.
Algorithm 22 Liver Cancer Algorithm (LCA) with Detailed Logarithmic Spiral
Update for Feature Selection
1: Step 1: Initialize Population
2: for each agent i in range (0, SearchAgents) do

125

3:

Generate a binary vector of length dimension for feature selection

4:

Generate opposite agents to enhance diversity and combine with initial
agents

5: end for
6: Step 2: Calculate Initial Fitness
7: for each agent i do
8:

Select features based on agent i’s binary vector

9:

Evaluate fitness using cross-validation with XGBoost on the selected features

10:

Store fitness score in fit[i]

11: end for
12: Identify the best initial solution, updating best score and best features
13: Step 3: Optimization Loop
14: for each iteration t in range (1, Max iterations) do
15:
16:

for each agent i do
Exploration with Levy Flight: With probability 0.5, apply Levy
flight to diversify the agent’s position

17:
18:

if random() ¡ 0.5 then
Generate a step using Levy flight and adjust the agent’s position

19:

end if

20:

Logarithmic Spiral Update: With probability 0.5, apply logarithmic
spiral update towards the best agent

21:

if random() ¡ 0.5 then

22:

Set best agent ← X[argmax(fit)]

23:

Detailed Logic of Logarithmic Spiral Update:

24:

Compute the direction vector direction = best agent − X[i]

25:

Calculate the Euclidean norm of the direction vector:
norm = ∥direction∥
126

if norm = 0 then

26:

Set norm = 1 to avoid division by zero

27:
28:

end if

29:

Normalize the direction vector:
unit direction =

direction
norm

Set the angular increment θ and calculate the logarithmic spiral step

30:

size:
step = aekθ · unit direction
Update agent’s position in the direction of the logarithmic spiral

31:

step:
new position = X[i] + step
32:

Convert updated position to binary based on a dynamic threshold:
X[i] = (new position > dynamic threshold(t, Max iterations)).astype(int)

33:

end if

34:

Crossover and Mutation:

35:

Select a partner agent at random

36:

Apply crossover to generate a child:
child ← crossover(X[i], X[partner idx])

37:

Apply mutation to the child:
mutated child ← mutation(child, mutation rate)

38:

Calculate the fitness of the mutated child

39:

if mutated child fitness ¿ current agent fitness then

40:

Replace the current agent with the mutated child
127

41:

end if

42:

end for

43:

Update Best Solution:

44:

if Current best fitness is greater than previous best then

45:
46:

Update best score and best features
end if

47: end for
48: Step 4: Return Results
49: Output: The best solution best features, best fitness score best score, and

fitness history over iterations

Figure 3.8: Visualization of the Logarithmic Spiral
In summary, each spiral update method Archimedean, Logarithmic, Hyperbolic,
128

or Lituus —offers unique advantages in navigating the solution space. These spirals
provide diverse exploration-exploitation dynamics, enhancing our feature selection
and optimization process.
To assess the impact of each spiral on algorithm performance, we define several
evaluation metrics that allow us to measure effectiveness objectively across multiple
runs. The following section outlines the key metrics used to quantify performance,
stability, and feature selection capability, which form the basis for our results and
comparative analysis.

3.6

Evaluation Metrics

To thoroughly evaluate the performance of our algorithm and its variations, we
utilize several key metrics that provide a detailed assessment of effectiveness,
stability, and feature selection capabilities. These metrics allow us to quantify the
impact of the proposed methods across multiple executions, ensuring robustness
and reliability.

3.6.1

Mean Fitness Function

The mean fitness function measures the algorithm’s average performance over
multiple runs, indicating its effectiveness.

n

1X
fitness(Ai )
Mean Fitness =
n i=1

(3.34)

Here, n is the number of runs, and fitness(Ai ) represents the fitness value of
129

Table 3.2: Comparison of Spiral Types for Feature Selection
Spiral Type

Growth Rate

Key Advantage

Best-Suited Scenario

Archimedean Spi- Linear

Even, systematic

Dense

search

ral

spacing

spaces

requiring

uniform coverage
Euler Spiral

Fermat Spiral

Linear Curvature

Square Root

Smooth

tions, fine-tuned

spaces

adjustments

gradual tuning

Gradual
sion,

Golden Spiral

Hyperbolic Spiral

Exponential

Inverse

transi- High-dimensional
requiring

expan- Controlled explobalanced

ration in sparse

search

feature spaces

Self-similarity,

Complex datasets

proportional

with

growth

dependencies

multi-scale

Propor- Decelerated out- Focused searches

tionality

ward exploration

in constrained feature spaces

Lituus Spiral

Inverse
Root

Square

Tight

inward

Gradual narrow-

reach, symmetric

ing of high-impact

branches

regions

Logarithmic Spi- Geometric

Constant pitch an- Comprehensive

ral

gle, balanced pro- explorationgression

exploitation
balance

130

the solution in the i-th run.

3.6.2

Best Fitness Function

The best fitness function identifies the algorithm’s optimal performance, highlighting
its peak efficiency.

Best Fitness = min fitness(Ai )
i=1,...,n

3.6.3

(3.35)

Worst Fitness Function

This metric records the worst-case performance, offering insights into the algorithm’s
consistency and identifying scenarios where improvements may be necessary.

Worst Fitness = max fitness(Ai )
i=1,...,n

3.6.4

(3.36)

Standard Deviation

The standard deviation of fitness values measures the algorithm’s stability and
robustness across different runs.

v
u n
u1 X
Standard Deviation = t
(fitness(Ai ) − Mean Fitness)2
n i=1

(3.37)

131

3.6.5

Classification Accuracy (CA)

Classification accuracy evaluates the algorithm’s ability to classify outcomes, averaging accuracy across all runs correctly.

Accuracy =

TP + TN
TP + TN + FP + FN

(3.38)

Here, TP, TN, FP, and FN represent true positives, true negatives, false
positives, and false negatives, respectively.

3.6.6

Feature Selection Ratio (FSR)

The feature selection ratio quantifies the efficiency of the feature selection process,
calculated as the proportion of selected features to the total features.

n

1X
FSR =
n i=1
Here,

P

P

Ai
d

(3.39)

Ai represents the number of selected features in the i-th run, and d is

the total number of features.

3.6.7

F-score

The F-score, the harmonic mean of precision and recall, evaluates the relevance
and significance of the selected features.

132

F-score =

3.6.8

2 · Precision · Recall
Precision + Recall

(3.40)

Precision and Recall

Precision and recall provide further insights into the algorithm’s performance.
Precision measures the accuracy of optimistic predictions, while recall assesses the
algorithm’s ability to identify all relevant positives.

TP
TP + FP

(3.41)

TP
TP + FN

(3.42)

Precision =

Recall =

3.7

Statistical Tests

To objectively compare the standard Liver Cancer Algorithm (LCA) with its
variations, we apply statistical tests assessing algorithm performance differences.
These methods help establish whether observed improvements are statistically
significant. The Wilcoxon Signed-Rank Test (Wilcoxon [1945]) and Friedman Test
(Friedman [1937]) are employed to assess significant differences between algorithm
performances.

133

3.7.1

Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is a nonparametric test used to compare paired samples and evaluate whether differences between two related datasets are statistically
significant.

W =

n
X

rank(Ri ) · sgn(Ri )

(3.43)

i=1

Here, Ri represents the differences between paired observations, and sgn(Ri )
indicates the sign of the difference.

3.7.2

Friedman Test

The Friedman is a nonparametric test comparing multiple algorithms across several
test scenarios. It identifies whether significant differences exist among the groups
being tested.

12N
χ2F =
k(k + 1)

k
X

!
Rj2

− 3N (k + 1)

(3.44)

j=1

In this equation:
• N : number of observations
• k: number of algorithms
• Rj : rank sum for the j-th algorithm

134

3.7.3

Algorithm Performance Analysis

This study systematically evaluates each algorithm’s performance using well-defined
metrics to provide a comprehensive understanding of its strengths and limitations.
Metrics such as Mean Fitness Function, Classification Accuracy, Recall, Precision,
Feature Selection Ratio (FSR), and F-score are analyzed to capture each algorithm’s
effectiveness, reliability, and feature selection quality.
Given the imbalanced nature of the dataset, we place particular emphasis on
Recall and Accuracy. The recall is critical for assessing the algorithm’s ability
to identify high-risk wildfire events accurately, minimizing false negatives, which
could lead to unanticipated and severe consequences. Accuracy complements this
by providing a broader measure of the algorithm’s overall predictive capability. To
rank the algorithms fairly, we apply the Friedman and Wilcoxon Signed-Rank tests,
which are robust non-parametric tools for comparing performance across multiple
samples.

3.7.4

Regional Impact Analysis

We extend our evaluation to assess algorithm performance across specific Canadian
provinces, recognizing that regional characteristics such as climate, vegetation
density, and wildfire history can influence predictive accuracy. This regional analysis
allows us to measure how well the algorithms adapt to diverse environmental
conditions, highlighting computational or accuracy shifts that may reveal localized
strengths or challenges.
To identify the most influential variables in each region, we analyze the top

135

features selected by each algorithm for specific provinces. We uncover regionally
significant factors and universally impactful variables by ranking these features.
This approach offers nuanced insights into the environmental factors critical for
wildfire prediction, tailoring our findings to the unique conditions of each region.

3.8

Expected Outcomes and Limitations

Anticipated Findings: We expect some algorithms—particularly the refined
Liver Cancer Algorithm (LCA) enhanced with spiral updates—to perform superiorly
in certain climates or terrains. These variations may be particularly evident in
regions with complex environmental dynamics, such as British Columbia or the
Northwest Territories. Additionally, we anticipate identifying variables that show
vital regional significance while recognizing universal predictors relevant across all
provinces.

Potential Limitations: Class imbalance remains a significant challenge despite
using cross-validation and stratified sampling to mitigate its effects. This imbalance
could reduce recall for less frequent wildfire events, potentially leading to lower
precision in identifying medium or low-risk zones. Furthermore, the large dataset
imposes computational constraints, which may limit the depth of algorithmic
comparisons or resolution in analyzing certain features.

136

3.9

Ethical and Practical Considerations

3.9.1

Ethical Data Use

This study adheres to strict ethical standards for data usage, mainly when working
with publicly available environmental and geographical datasets such as FIRMS
(NASA) and ERA5 (C3S). We follow guidelines from the data sources to ensure
responsible data management practices to avoid misuse or misinterpretation. Privacy considerations are rigorously maintained, and results are disseminated with
integrity, ensuring the findings are responsibly communicated.

3.9.2

Practical Implications

The findings of this research hold significant potential for wildfire management
in Canada. By identifying critical predictive factors and optimizing algorithms
for imbalanced datasets, our results can assist local agencies in managing wildfire
risk more effectively. For example, predicting high-risk zones or critical periods
could enable targeted firefighting efforts, faster response times, and risk mitigation.
These insights aim to improve resource allocation and enhance resilience in wildfire
management strategies across Canadian provinces.

3.10

Summary of the Study Framework

This study follows a comprehensive and methodical framework designed to achieve
its goals rigorously and objectively. From initial feature selection to algorithmic
137

refinement, we applied various metaheuristic algorithms complemented by novel
spiral updates to optimize predictive accuracy. Our evaluation framework emphasizes critical metrics for imbalanced datasets, such as Recall and accuracy, to
ensure relevant and meaningful results. Statistical ranking tests and region-specific
analyses further support our comparative approach.
Through structured phases, from data preprocessing to evaluation, the study
underscores its commitment to advancing wildfire prediction tools and supporting
resource management decisions. This robust framework sets the stage for the
following Discussion section, where we will delve into a detailed analysis of our
findings.

138

Chapter 4
Discussion

4.1

Overview of Study Objectives and Method-

ology
This study addresses the urgent need for accurate wildfire prediction tools by
evaluating the effectiveness of various algorithms in different regions of Canada.
We had two main objectives: (1) to assess the predictive performance of multiple algorithms—explicitly focusing on models enhanced with unique spiral updates—and
(2) to examine how region-specific factors influence algorithm performance across
provinces.
We employ a rigorous methodology to achieve these goals, including advanced
feature selection techniques, comparative algorithm testing, and statistical classification using the Friedman and Wilcoxon signed-rank tests. We sourced data from
NASA’s FIRMS and the Copernicus Climate Change Service’s ERA5, providing
a comprehensive view of wildfire incidents across diverse environmental contexts.
139

We used cross-validation as a critical component during the training and testing
phases to address the inherent class imbalance in wildfire data.
This approach allowed us to objectively compare each algorithm’s strengths
and weaknesses while accounting for regional influences. The insights gained from
this analysis aim to inform advancements in predictive modelling and practical
applications in wildfire management across Canada.

4.1.1

Provincial Results

To provide a thorough analysis, we evaluated algorithm performance across eight
provinces: Alberta, British Columbia (BC), Manitoba, Ontario, Northwest Territories (NWT), Quebec, Saskatchewan, and Yukon. This province-by-province
approach highlights regional factors influencing prediction accuracy and provides
insights into how well the algorithms adapt to varying environmental conditions.
For each Canadian province, we focus primarily on Recall due to the imbalanced
nature of wildfire data and the critical importance of identifying all high-risk
wildfire cases. We also assess Precision, Accuracy, and F1 Score to provide
a comprehensive view of each algorithm’s performance. This analysis helps tailor
wildfire risk management strategies to the specific environmental conditions of each
province. Initially, we rank the standard and spiral-enhanced algorithms using the
Friedman test, which is particularly useful for identifying the most effective algorithms before and after spiral updates are applied. Subsequent detailed Wilcoxon
signed-rank tests compare the original algorithms against their spiral-enhanced
versions to ascertain any statistical performance improvements.
After establishing baseline algorithm rankings, we explore the impact of various
spiral updates on the Liver Cancer Algorithm (LCA). We reassess these
140

algorithms with another round of Friedman and Wilcoxon tests to evaluate the
effectiveness of the enhancements. Results from these statistical tests are presented
visually through comparative plots and tables, illustrating shifts in algorithm
performance due to spiral enhancements. These visualizations and discussions
highlight how specific algorithms and their enhancements improve the prediction
and management of wildfires in different provincial contexts.

Algorithm Performance in Alberta

We evaluated the wildfire prediction algorithms in Alberta, focusing on their ability
to identify high-risk wildfire cases. The results are summarized in Table 4.1.

Standard Algorithms: The Particle Swarm Optimization (PSO) algorithm
achieved the highest average recall score of 79.79, demonstrating exceptional
effectiveness across all models in Alberta. Close behind, the Golden Ratio
Method (GRM) performed robustly with an average recall of 78.32, followed by
the CBO optimization, which secured an average recall of 78.66. These results
underscore the capabilities of PSO, GRM, and CBO to effectively capture wildfire
cases in Alberta.
Conversely, the Atom Search Optimization (ASO) algorithm had the lowest
average recall score of 57.24, indicating difficulties in adapting to Alberta’s specific
wildfire data characteristics. The Liver Cancer Algorithm (LCA) showed
moderate performance with an average recall of 67.83, which did not match the
more successful algorithms.
A closer examination of the recall scores across individual models reveals
interesting patterns. The Random Forest and Gradient Boosting models
141

consistently provided high recall scores for most algorithms, with PSO and CBO
achieving 79.63 and 80.90, respectively. This indicates a strong performance in
capturing wildfire cases using these models.
Table 4.1: Algorithm Rankings for Alberta
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

PSO

79.63

83.88

73.07

85.94

76.43

10

2.00

1

CBO

80.90

81.56

72.00

85.47

73.40

19

3.80

2

GRM

79.30

83.20

71.98

86.80

70.30

21

4.20

3

GA

76.95

80.36

74.86

83.02

76.20

24

4.80

4

MRFO

80.04

80.08

72.89

74.93

75.52

25

5.00

5

EDO

78.58

81.12

70.99

84.70

73.85

26

5.20

6

BMO

78.13

80.19

70.14

83.97

75.35

27

5.40

7

EVO

81.06

80.07

71.80

77.13

74.72

29

5.80

8

WOA

78.37

79.50

70.68

85.31

69.63

30

6.00

9

LCA

72.61

71.58

70.90

60.54

63.52

45

9.00

10

EO

67.70

69.84

65.11

63.08

57.83

50

10.00

11

ASO

63.60

66.13

63.00

46.86

46.59

55

11.00

12

Spiral-Based Methods: Among the spiral-based methods, the Euler Spiral
achieved the highest average Recall across most models, with Random Forest
recording a recall of 80.72 and Gradient Boosting achieving 80.10. This
performance highlights the Euler Spiral’s ability to effectively balance search
space exploration and exploitation, ensuring that essential feature subsets are not
overlooked.
The Hyperbolic Spiral followed closely, delivering strong recall values, particularly Random Forest at 78.45 and Gradient Boosting at 78.39. Its balance
between diversification and intensification contributed to its competitive performance, achieving an average recall of 74.45.
Similarly, the Lituus Spiral demonstrated high recall values, including Random Forest at 77.95 and K-Nearest Neighbor at 76.44, securing an average
recall of 74.38. These results emphasize the Lituus Spiral’s capability to model
complex patterns effectively.
142

The Logarithmic Spiral, with an average recall of 73.36, delivered consistent performance across models, such as Random Forest at 79.38 and Gradient
Boosting at 78.41. While slightly behind Hyperbolic and Lituus, its systematic
exploration process maintained competitive performance.
Other methods, such as the Fermat Spiral, produced moderate results, achieving an average recall of 72.62, showing its ability to balance exploration and
exploitation, although it lagged slightly behind the higher-performing spirals.
On the lower end, the Liver Cancer Algorithm (LCA) without spiral updates
ranked consistently lower across most models, with a maximum recall of 72.61
achieved by Random Forest and an average recall of 67.83. This highlights
the importance of integrating spiral updates to enhance LCA’s performance in
addressing feature selection challenges.
Table 4.2: Spiral Algorithm Rankings for Alberta
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

Euler

80.72

80.10

71.80

76.05

75.50

8

1.60

1

Hyperbolic

78.45

78.39

73.14

69.73

72.53

13

2.60

2

Logarithmic

79.38

78.41

74.08

66.32

68.59

16

3.20

3

Lituus

77.95

77.19

76.44

68.77

71.55

17

3.40

4

Fermat

78.04

76.40

69.75

69.40

69.50

22

4.40

5

Golden

73.83

72.70

69.72

64.07

67.59

31

6.20

6

LCA

72.61

71.58

70.90

60.54

63.52

33

6.60

7

Archimedean

72.36

70.76

67.81

60.25

62.26

40

8.00

8

On the lower end, the Liver Cancer Algorithm (LCA) without spiral
updates ranked consistently lower across most models, with a maximum recall of
72.61 achieved by Random Forest. This reinforces the impact of spiral updates
in enhancing the LCA’s effectiveness.

Overall Algorithm Rankings: The Friedman test assessed the relative performance of all algorithms, including standard and spiral-based methods. Figure

143

4.1 visualizes the ranking of algorithms based on their average recall scores, where
lower ranks indicate better performance.
The Particle Swarm Optimization (PSO) emerged as the best-performing
algorithm with an average rank of 2.8, followed by CBO at 4.6, and GRM at 5.8.
The Euler Spiral performed notably well among the spiral methods, achieving an
average rank of 6.6. Other strong performers included GA at 6.0 and MRFO at
6.2.
In contrast, algorithms such as Archimedean Spiral and ASO ranked lowest,
with ranks of 17.2 and 19.0, respectively. These results emphasize the effectiveness
of top-ranked methods in feature selection and modeling, while highlighting areas
for improvement in lower-performing approaches.

Figure 4.1: Average Recall Ranks (Friedman Test)

144

Algorithm Performance in British Columbia (BC)

The Particle Swarm Optimization (PSO) again led the performance in British
Columbia with the highest average recall score of 75.60, indicating its robustness
across different regional datasets. The Equilibrium Optimizer (EDO) came
close with an average recall of 74.99, demonstrating effective wildfire prediction in
this province.
The Atom Search Optimization (ASO) remained the least effective with the
lowest average recall score of 39.61 in British Columbia, highlighting significant
challenges in its application to wildfire data in this region. The Manta ray
foraging optimization (MRFO) also underperformed with an average recall of
72.40, showing some limitations in adapting to the specifics of British Columbia’s
wildfire scenarios.
In terms of individual model performances, the Random Forest and Gradient
Boosting models again showed strong recall scores for top algorithms like PSO and
EDO, with 78.71 and 78.95 respectively in British Columbia. This suggests that
these models are particularly suited to leveraging the strengths of these algorithms
for wildfire prediction in the region.
These insights highlight the varying effectiveness of different algorithms across
provinces and models, suggesting that regional characteristics and data specifics
significantly influence algorithm performance in wildfire prediction tasks.

Spiral-Based Enhancements: Among the spiral-based methods, the LCA
achieved the highest average Recall, particularly excelling in Random Forest with
a recall of 78.36 and Gradient Boosting at 78.24. This performance highlights

145

Table 4.3: Algorithm Rankings for British Columbia
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

PSO

78.70

79.12

70.73

75.57

73.88

11

2.20

1

EDO

78.56

78.95

68.34

75.00

74.13

19

3.80

2

EVO

78.50

79.31

67.67

75.14

73.26

23

4.60

3

GA

78.18

79.13

68.08

75.84

73.48

23

4.60

3

BMO

78.23

79.16

69.97

74.91

72.56

26

5.20

5

GRM

78.27

77.88

70.05

71.82

74.29

26

5.20

5

WOA

78.22

78.94

68.83

75.35

72.80

28

5.60

7

LCA

78.36

78.24

67.35

74.53

73.44

33

6.60

8

CBO

77.78

78.11

67.86

73.52

72.98

40

8.00

9

MRFO

75.71

76.91

68.18

70.55

70.64

46

9.20

10

EO

71.81

72.38

64.64

64.13

65.64

55

11.00

11

ASO

38.92

38.67

40.27

40.44

39.77

60

12.00

12

Table 4.4: Spiral Algorithm Rankings for British Columbia
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

LCA

78.36

78.24

67.35

74.53

73.44

10

2.00

1

Archimedean

78.08

78.26

70.28

72.23

71.12

12

2.40

2

Hyperbolic

76.30

76.04

70.54

73.28

72.33

14

2.80

3

Golden

76.85

76.53

68.33

70.06

69.98

20

4.00

4

Euler

75.39

74.58

68.41

71.96

72.48

20

4.00

4

Lituus

73.31

74.84

65.89

64.86

66.45

30

6.00

6

Logarithmic

72.76

72.58

63.15

65.30

65.31

34

6.80

7

Fermat

56.42

55.88

50.92

50.05

48.05

40

8.00

8

LCA’s strong feature selection capabilities, surpassing other spirals.
The Archimedean Spiral followed closely, demonstrating solid recall values
with Random Forest at 78.08 and Gradient Boosting at 78.26. Its ability to
balance exploration and exploitation contributed to its competitive performance.
The Golden Spiral also showed promise, maintaining consistent recall scores
across models, including 76.85 in Random Forest and 76.53 in Gradient
Boosting.
Other spirals, such as Hyperbolic and Euler, demonstrated moderate improvements with average recalls of 73.30 and 72.16, respectively. These results
underscore the adaptability of spiral-based approaches in enhancing predictive
performance.

146

On the lower end, the Fermat Spiral ranked lowest, with an average recall of
52.66, highlighting areas for potential optimization in feature selection strategies.

Statistical Validation: The Friedman test results validated the performance
differences among the algorithms, yielding significant statistics (Statistic: 69.25,
P-value: 0.00000 and Statistic: 71.81, P-value: 0.00000). Rankings further
highlighted the dominance of the LCA and Archimedean Spirals over other
methods. The LCA Standard ranked 7.6, whereas the Archimedean Spiral
ranked 8.2, followed by the Hyperbolic Spiral at 9.4. Conversely, lower-ranked
methods such as Logarithmic Spiral (16.2) and Fermat Spiral (18.0) demonstrated relatively weaker performance, emphasizing the superior adaptability of
top-performing algorithms in modeling complex relationships and optimizing feature selection.

Figure 4.2: Visualization of average recall rankings among algorithms in BC
(Friedman Test results). Lower ranks indicate better performance, with several
spiral-enhanced algorithms outperforming the LCA.

147

Overall Algorithm Rankings: The Friedman test assessed the relative performance of all algorithms, including standard and spiral-based methods. Figure
4.1 visualizes the ranking of algorithms based on their average recall scores, where
lower ranks indicate better performance.
The Particle Swarm Optimization (PSO) emerged as the best-performing
algorithm with an average rank of 2.2, followed by EDO at 4.4 and EVO at
5.4. Among spiral-based methods, the Euler Spiral achieved an average rank of
11.4, outperforming several algorithms. However, Archimedean and Hyperbolic
also performed competitively, ranking 8.2 and 9.4, respectively. On the lower
end, methods like Logarithmic (16.2) and Fermat (18.0) demonstrated weaker
performance, emphasizing the dominance of top-ranked algorithms in enhancing
feature selection and predictive accuracy.

Algorithm Performance in Manitoba

The results are summarized in Table 4.5. The Genetic Algorithm (GA) showcased the best average recall in Manitoba, leading with an average score of 69.63.
It demonstrated robust performance across all models, affirming its capability in
handling diverse data scenarios effectively. Following closely were the Equilibrium
Optimizer (EVO) and the Exponential Distribution Optimizer (EDO),
with average recalls of 69.53 and 69.39 respectively, highlighting their efficiency
in optimizing prediction models under varying conditions.
On the other hand, the Atom Search Optimization (ASO) lagged significantly behind other algorithms with the lowest average recall of 44.34, suggesting
that ASO might be less suited to the specific challenges presented by the Manitoba
dataset. The algorithms like Equilibrium Optimizer and CBO performed well,
148

especially in models such as Random Forest and Logistic Regression, indicating a strong adaptability to the data characteristics in Manitoba. .
Table 4.5: Algorithm Rankings for Manitoba
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

EVO

72.30

73.35

63.29

73.53

65.17

18

3.60

1

CBO

71.22

70.93

63.82

72.83

67.88

19

3.80

2

LCA

72.47

70.98

62.96

71.04

67.20

20

4.00

3

GA

71.11

72.30

65.72

72.75

66.29

22

4.40

4

MRFO

72.42

70.61

62.59

72.24

66.36

24

4.80

5

EDO

71.38

70.55

61.89

75.48

67.65

25

5.00

6

PSO

72.09

70.31

64.59

68.55

67.04

29

5.80

7

BMO

71.92

71.66

62.11

69.77

64.49

34

6.80

8

WOA

71.19

70.51

60.30

71.20

66.02

39

7.80

9

GRM

67.67

68.70

62.59

69.01

64.71

44

8.80

10

EO

63.79

61.43

56.89

64.54

60.43

55

11.00

11

ASO

43.65

42.94

46.07

45.32

43.73

60

12.00

12

Spiral-Based Enhancements: Among the spiral-based methods evaluated in
Manitoba, the Liver Cancer Algorithm (LCA) emerged as the top performer,
demonstrating its strong feature selection capabilities. It particularly excelled in
Random Forest with a recall of 72.47 and Gradient Boosting with a recall of
70.98. These results underscore the LCA’s ability to effectively model complex
patterns in wildfire prediction.
Following closely was the Archimedean Spiral, which showed solid performance across models, with recall scores of 69.32 in Random Forest and 69.27 in
Gradient Boosting. Its balance between exploration and exploitation contributed
significantly to its competitive performance.
The Hyperbolic Spiral also demonstrated robust performance, particularly
with recall values of 65.58 in Random Forest and 66.46 in Gradient Boosting.
This method’s blend of diversification and intensification strategies has proven
effective in navigating the challenges of wildfire prediction.
Meanwhile, the Euler Spiral and Lituus Spiral achieved moderate improve149

ments, with average recalls of 61.78 and 59.41 respectively. These spirals are
notable for their methodical approach to modeling, yet they show potential for
further optimization.
The Logarithmic Spiral provided consistent results across various models
but fell slightly behind the leading spirals with an average recall of 60.61. Its
systematic exploration process maintains a competitive edge, although there is
room for enhancement.
On the lower end, the Fermat Spiral and Golden Spiral demonstrated areas
for potential improvement, with the lowest recalls of 54.76 and 56.41 respectively.
These results highlight the need for further refinement in their algorithms to better
tackle the complexities of wildfire prediction in Manitoba.
Table 4.6: Spiral Algorithm Rankings for Manitoba
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

LCA

72.47

70.98

62.96

71.04

67.20

6

1.20

1

Archimedean

69.32

69.27

65.12

68.21

66.19

9

1.80

2

Hyperbolic

65.58

66.46

60.46

67.25

66.11

15

3.00

3

Euler

62.71

62.12

59.35

64.97

59.73

25

5.00

4

Logarithmic

64.77

62.65

54.90

65.06

55.64

26

5.20

5

Lituus

61.80

62.14

55.28

65.22

54.60

27

5.40

6

Fermat

57.93

57.27

55.25

51.18

52.14

36

7.20

7

Golden

57.79

57.12

55.20

55.20

54.76

36

7.20

7

Statistical Validation: The Friedman test confirmed significant differences in
algorithm performance, with a test statistic of 79.62 and a p-value of 9.99e-10,
indicating strong evidence against the null hypothesis of equal performance. Figure
4.3 visualizes the rankings of all algorithms.

150

Figure 4.3: Visualization of average recall rankings among algorithms in Manitoba
(Friedman Test results). Lower ranks indicate better performance, with several
spiral-enhanced algorithms outperforming the LCA.

Algorithm Performance in Northwest Territories (NWT)

In the Northwest Territories, the Manta Ray Foraging Optimization (MRFO)
stood out with the highest average recall of 75.49, demonstrating its efficiency in
optimizing model parameters under varying conditions. The Compound Binary
Optimization (CBO) also performed well, achieving a similar high average recall
of 74.81, particularly excelling in logistic regression and support vector machine
models.
Conversely, the Atom Search Optimization (ASO) and Equilibrium
Optimizer (EO) showed the least effective performance with the lowest average
recalls of 51.99 and 51.46 respectively. These results suggest that these algorithms
may require further tuning or adaptation to better suit the data peculiarities in
NWT.
151

Table 4.7: Algorithm Rankings for Northwest Territories (NWT)
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

MRFO

76.32

75.90

71.69

80.99

72.54

12

2.40

1

BMO

75.16

74.83

71.19

79.29

73.35

21

4.20

2

CBO

76.14

74.73

72.00

79.67

71.50

21

4.20

2

PSO

74.96

75.91

69.91

79.98

71.84

21

4.20

2

EVO

74.90

76.93

69.23

79.33

71.80

25

5.00

5

GA

73.68

76.17

67.19

80.61

72.37

26

5.20

6

WOA

75.30

77.45

69.13

78.01

70.25

26

5.20

6

EDO

74.38

77.15

67.02

78.78

70.04

33

6.60

8

GRM

73.36

73.36

69.65

74.87

68.44

43

8.60

9

LCA

73.91

72.79

64.82

76.73

67.94

47

9.40

10

EO

51.93

53.03

56.65

44.05

51.62

57

11.40

11

ASO

52.39

52.79

52.84

50.80

51.15

58

11.60

12

Spiral Algorithm Performance in Northwest Territories (NWT) In the
Northwest Territories, the effectiveness of spiral-based algorithms was thoroughly
evaluated for wildfire prediction. The outcomes are detailed in Table 4.7.
Among the spiral-based methods, the Archimedean Spiral stood out as the
top performer, achieving the highest average recall scores across most models.
Specifically, it excelled in Logistic Regression with a recall of 77.65 and demonstrated consistent performance in Random Forest and Gradient Boosting with
recalls of 75.87 and 76.10 respectively. This reflects its robust ability to navigate
the feature space effectively, balancing both exploration and exploitation.
Following closely was the Golden Spiral, which showed strong recall values,
particularly in Logistic Regression with a recall of 75.84 and Gradient Boosting at 75.08. This method’s consistent performance across models indicates its
potential in handling diverse and complex data structures inherent to wildfire
prediction.
The Liver Cancer Algorithm (LCA) also demonstrated commendable
performance, with an average recall of 71.64. It showed a particularly strong recall
in Logistic Regression at 76.73, proving its efficacy in specific scenarios despite

152

a slightly lower performance in K-Nearest Neighbor.
Other methods such as the Logarithmic Spiral and Hyperbolic Spiral
delivered moderate results with average recalls of 68.79 and 66.26, respectively.
These algorithms, while effective to a degree, indicate the need for further tuning
to optimize their performance in the challenging environments of the Northwest
Territories.
The Fermat Spiral and Euler Spiral showed some limitations in achieving
higher recall values, with the Euler Spiral in particular finding challenges in
Logistic Regression and Support Vector Machine with lower recall scores of
55.98 and 62.10 respectively.
At the lower end of the performance spectrum, the Lituus Spiral recorded the
lowest overall recall of 64.29, struggling particularly in Logistic Regression and
K-Nearest Neighbor with recalls of 58.36 and 62.72. This suggests that while
the Lituus Spiral may have potential, it requires significant adjustments to better
address the specific demands of wildfire prediction in this region.
Overall, the evaluation of spiral-based algorithms in the Northwest Territories
illustrates a varied landscape of effectiveness, with some algorithms showing promise
in specific models while others require enhancements to reach their full potential in
wildfire prediction.
Table 4.8: Spiral Algorithm Rankings for Northwest Territories (NWT)
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

Archimedean

75.87

76.10

71.13

77.65

72.05

7

1.40

1

Golden

74.45

75.08

71.27

75.84

73.12

9

1.80

2

LCA

73.91

72.79

64.82

76.73

67.94

17

3.40

3

Logarithmic

70.67

71.03

67.11

68.11

67.01

20

4.00

4

Fermat

69.80

67.63

65.59

66.20

65.21

30

6.00

5

Euler

70.16

69.58

67.79

55.98

62.10

30

6.00

5

Hyperbolic

66.85

68.33

64.09

66.90

65.10

33

6.60

7

Lituus

68.98

69.94

62.72

58.36

63.44

34

6.80

8

153

Statistical Validation: The Friedman test confirmed statistically significant
differences in algorithm performance, with a test statistic of 71.49 and a p-value of
2.52e-08. Figure 4.4 illustrates the ranking distribution, highlighting the superior
performance of spiral-based enhancements over the standard LCA.

Figure 4.4: Visualization of average recall rankings among algorithms in Northwest
Territories (Friedman Test results). Lower ranks indicate better performance, with
several spiral-enhanced algorithms outperforming the LCA.

Algorithm Performance in Ontario

In Ontario, the Equilibrium Optimizer (EVO) emerged as the top-performing
algorithm with the highest average recall of 69.73, showcasing its effectiveness
across diverse models, particularly in the K-Nearest Neighbor model where it
achieved exceptionally high recall. The Manta Ray Foraging Optimization
(MRFO) followed closely with an average recall of 68.80, indicating its robust
capability in handling complex datasets.

154

The Atom Search Optimization (ASO), however, recorded the lowest
performance with an average recall of 44.67, reflecting challenges in adapting
to the specific characteristics of the Ontario dataset. The algorithms such as
Exponential Distribution Optimizer (EDO) and Genetic Algorithm (GA)
also showed good performance, particularly in logistic regression, underlining their
strong adaptability.
Table 4.9: Algorithm Average Recall and Rankings for Ontario
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Average Recall

Ranking

ASO

46.40

46.24

35.45

47.85

47.40

44.67

12

BMO

67.99

66.74

62.04

71.08

64.83

66.53

5

CBO

63.63

66.15

61.23

73.40

62.15

65.31

6

EVO

64.47

65.75

82.43

74.24

61.77

69.73

1

EO

54.82

52.64

47.09

51.44

49.79

51.16

11

EDO

68.16

68.13

63.71

72.74

65.55

67.66

3

GA

66.41

66.48

65.22

73.70

64.16

67.19

4

GRM

64.55

65.66

52.77

69.66

63.65

63.26

9

LCA

58.98

57.09

51.57

66.53

60.07

58.85

10

MRFO

68.16

68.15

66.29

72.44

68.94

68.80

2

PSO

64.42

63.30

60.01

71.51

64.45

64.74

7

WOA

64.20

63.69

56.83

71.45

62.41

63.72

8

Spiral Algorithm Performance in Ontario In Ontario, the effectiveness of
spiral-based algorithms in wildfire prediction was systematically evaluated, with
summarized results presented in Table 4.10.
The Hyperbolic Spiral emerged as the leader among the spiral algorithms,
showcasing the highest average recall values. It demonstrated superior performance
particularly in Random Forest and Gradient Boosting with recall scores of
70.21 and 69.34 respectively. This highlights its efficient balance of exploration
and exploitation tactics in complex prediction scenarios.
Closely following was the Archimedean Spiral, which secured the second
position. It recorded strong recall scores, especially in Random Forest and
Logistic Regression, with values of 68.09 and 64.62. Its methodical approach
155

to navigating the feature space has proven effective, albeit slightly less so than the
Hyperbolic Spiral.
The Fermat Spiral also demonstrated commendable results, with an average
recall of 63.20, performing particularly well in Logistic Regression with a
recall of 62.91. Its consistent performance across different models underscores its
potential in handling diverse data environments.
Other algorithms such as the Logarithmic Spiral and Lituus Spiral showed
moderate effectiveness with average recalls of 62.33 and 61.52 respectively. These
results suggest that while capable, they may require further optimization to enhance
their predictive accuracy.
The Euler Spiral and Golden Spiral ranked lower, indicating room for
improvement in their feature selection processes. They managed average recalls of
60.53 and 60.33, highlighting the challenges they face in adapting to the complex
wildfire prediction landscape in Ontario.
At the lower end of the performance spectrum, the Liver Cancer Algorithm
(LCA) exhibited the least effective performance, with an average recall of 58.05.
Despite showing strong performance in Logistic Regression, its overall lower
scores reflect significant limitations in its current configuration for the Ontario
dataset.

Statistical Validation: The Friedman test confirmed significant differences in
algorithm performance, with a test statistic of 61.98 and a p-value of 9.74e-07,
indicating strong evidence against the null hypothesis of equal performance. Figure
4.5 illustrates the ranking distribution, showcasing the superior performance of
spiral-based enhancements over the standard LCA and other algorithms.
156

Table 4.10: Algorithm Rankings for Ontario
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

MRFO

68.16

68.15

66.29

72.44

68.94

10

2.00

1

EDO

68.16

68.13

63.71

72.74

65.55

13

2.60

2

GA

66.41

66.48

65.22

73.70

64.16

18

3.60

3

BMO

67.99

66.74

62.04

71.08

64.83

22

4.40

4

EVO

64.47

65.75

82.43

74.24

61.77

23

4.60

5

CBO

63.63

66.15

61.23

73.40

62.15

31

6.20

6

PSO

64.42

63.30

60.01

71.51

64.45

33

6.60

7

GRM

64.55

65.66

52.77

69.66

63.65

36

7.20

8

WOA

64.20

63.69

56.83

71.45

62.41

38

7.60

9

LCA

58.98

57.09

51.57

66.53

60.07

50

10.00

10

EO

54.82

52.64

47.09

51.44

49.79

55

11.00

11

ASO

46.40

46.24

35.45

47.85

47.40

60

12.00

12

Algorithm Performance in Quebec

In Quebec, the Particle Swarm Optimization (PSO) algorithm demonstrated
the highest overall effectiveness with an average recall of 78.69, showcasing its
strong performance across all models, particularly excelling in logistic regression
with an impressive recall of 88.40. This highlights PSO’s capability to adapt well
to the diverse modeling challenges presented by the Quebec dataset.
The Equilibrium Optimizer (EVO) also performed admirably, securing the
second-highest average recall of 78.30. It was particularly effective in handling
the K-Nearest Neighbor model, indicating its robustness in managing complex,
non-linear data structures.
On the other hand, the Atom Search Optimization (ASO) struggled in
Quebec, recording the lowest average recall of 54.21. This suggests potential
difficulties in parameter tuning or in adapting to the specific characteristics of the
data used in Quebec.

Spiral-Based Enhancements in Quebec: In Quebec, the evaluation of spiralbased algorithms revealed significant variations in their effectiveness for predicting
157

Figure 4.5: Visualization of average recall rankings among algorithms in Ontario
(Friedman Test results). Lower ranks indicate better performance, with several
spiral-enhanced algorithms outperforming the LCA.
wildfires, as detailed in Table 4.11.
The Archimedean Spiral led the rankings, achieving the highest average
recall. It showcased exceptional performance in Logistic Regression with a recall
of 86.12, and in Gradient Boosting, where it achieved a recall of 81.64. This
algorithm’s ability to adeptly navigate complex data landscapes is evident from its
strong performance across diverse models.
Following closely, the Logarithmic Spiral demonstrated robust recall scores,
particularly in Gradient Boosting and Logistic Regression with recalls of
81.56 and 79.39, respectively. This indicates its effective use of logarithmic
progression to optimize feature selection and model training processes.
The Euler Spiral also performed well, with notable strength in Gradient
Boosting where it recorded a recall of 83.38. Its unique approach to balancing
158

Table 4.11: Algorithm Rankings for Quebec
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

PSO

78.97

81.38

73.57

88.40

71.13

14

2.80

1

GRM

79.30

83.20

71.98

86.80

70.30

15

3.00

2

EDO

78.58

81.12

70.99

84.70

73.85

21

4.20

3

EVO

82.07

82.30

69.60

84.59

72.93

22

4.40

4

CBO

79.26

84.39

68.00

87.06

69.40

25

5.00

5

BMO

78.13

80.19

70.14

70.14

83.97

29

5.80

6

WOA

78.37

79.50

70.68

85.31

69.63

30

6.00

7

MRFO

76.33

75.77

72.15

83.53

70.06

34

6.80

8

GA

77.10

76.44

70.04

80.01

70.29

38

7.60

9

EO

67.70

69.84

70.14

63.08

57.83

49

9.80

10

LCA

68.41

67.24

66.30

68.22

61.49

52

10.40

11

ASO

54.59

54.56

53.80

54.13

53.99

60

12.00

12

computational efficiency with exploratory data analysis has contributed to its
competitive average recall of 76.40.
Not to be overlooked, the Hyperbolic Spiral provided solid results with a
particularly strong performance in Random Forest achieving a recall of 79.15.
Its method of focusing on diversification in the feature space aligns well with the
dynamic and variable-intensive nature of wildfire prediction.
On the other end, the Fermat Spiral and the Golden Spiral demonstrated
moderate effectiveness with average recalls of 72.58 and 72.77, respectively. While
they showed potential in specific models, their overall performance suggests room
for optimization to fully harness their algorithms’ capabilities in the Quebec
environment.
The Lituus Spiral and the Liver Cancer Algorithm (LCA) encountered
some challenges, with the LCA showing the least adaptability among the evaluated
spirals, holding the lowest average recall of 66.33. This underlines the necessity
for potential algorithmic adjustments or enhanced feature engineering to improve
its application in this specific regional context.

159

Table 4.12: Spiral Algorithm Rankings for Quebec
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

Archimedean

77.78

81.64

75.78

86.12

71.30

9

1.80

1

Logarithmic

77.56

81.56

74.46

79.39

72.43

14

2.80

2

Euler

77.45

83.38

70.96

81.86

70.35

14

2.80

2

Hyperbolic

79.15

75.74

69.45

78.63

72.62

15

3.00

4

Fermat

75.05

71.26

65.92

80.70

69.96

28

5.60

5

Golden

74.77

74.71

68.76

72.38

69.24

29

5.80

6

Lituus

72.44

72.10

68.42

71.04

69.91

32

6.40

7

LCA

68.41

67.24

66.30

68.22

61.49

39

7.80

8

Statistical Validation: The Friedman test confirmed significant differences in
algorithm performance, with a test statistic of 60.56 and a p-value of 1.66e-06,
indicating strong evidence against the null hypothesis of equal performance. Figure
4.6 illustrates the ranking distribution, highlighting the superior performance of
spiral-based enhancements over the standard LCA.

Figure 4.6: Visualization of average recall rankings among algorithms in Quebec
(Friedman Test results). Lower ranks indicate better performance, with several
spiral-enhanced algorithms outperforming the LCA.

160

Algorithm Performance in Saskatchewan

In Saskatchewan, the Genetic Algorithm (GA) emerged as the top performer
with the highest average recall of 78.78, indicating its strong adaptability and
effectiveness across various models. This was particularly notable in the Support
Vector Machine and Logistic Regression models, where it achieved recalls of 78.20
and 77.91 respectively.
The Particle Swarm Optimization (PSO) and the Whale Optimization
Algorithm (WOA) also showed robust performances with average recalls of 78.33
and 78.01, respectively, securing them positions among the top three algorithms.
These results underscore the capability of PSO and WOA to efficiently navigate
the predictive modeling landscape in Saskatchewan.
Conversely, the Atom Search Optimization (ASO) had the lowest average
recall of 47.26 in Saskatchewan, similar to its performance in Quebec. This consistent underperformance across different provinces may necessitate a reevaluation of
its application or further optimization to better suit the regional data specifics.
Table 4.13: Algorithm Rankings for Saskatchewan
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

GA

80.62

81.10

76.07

77.91

78.20

10

2.00

1

PSO

80.83

80.78

75.01

78.36

76.67

11

2.20

2

WOA

80.87

80.93

73.55

78.13

76.55

11

2.20

2

BMO

80.30

80.34

73.22

77.90

76.06

24

4.80

4

EVO

80.80

80.27

71.74

77.49

76.23

26

5.20

5

CBO

80.24

80.90

72.07

74.52

74.47

30

6.00

6

GRM

80.14

80.55

72.40

73.62

71.80

35

7.00

7

EDO

79.92

79.91

71.51

76.45

74.32

39

7.80

8

MRFO

79.07

80.14

71.46

75.37

74.70

39

7.80

8

EO

76.97

77.12

69.37

68.76

71.36

50

10.00

10

LCA

75.92

75.13

62.10

68.51

67.64

55

11.00

11

ASO

47.40

50.46

51.79

44.86

41.81

60

12.00

12

161

Spiral-Based Enhancements: The evaluation of spiral-based algorithms for
wildfire prediction in Saskatchewan has highlighted varying levels of effectiveness
across different models, as summarized in Table 4.13.
Leading the pack was the Archimedean Spiral, which showcased superior
performance across all models. It achieved impressive recall scores in Random
Forest and Gradient Boosting with values of 81.02 and 80.28, respectively,
culminating in the highest average recall of 77.38. This spiral’s ability to effectively
explore and exploit the feature space makes it particularly suited to the predictive
challenges in Saskatchewan.
The Lituus Spiral and the Euler Spiral both shared the second rank with
an average recall of 77.20. These algorithms demonstrated strong performances,
particularly in Logistic Regression and Support Vector Machine, where they
matched each other with recalls of 77.39 and 76.08, indicating their robustness in
handling diverse data structures.
The Fermat Spiral secured the third rank with a solid average recall of 75.66.
While it trailed slightly behind the leaders, it still showed strong potential in models
like Random Forest and Gradient Boosting with recalls of 79.14 and 78.75,
suggesting that with slight optimizations, it could compete closely with the top
performers.
Subsequently, the Logarithmic Spiral and the Golden Spiral ranked fourth
and fifth, with average recalls of 75.19 and 74.31, respectively. These spirals,
while effective, displayed a need for enhancements to elevate their performance to
the top tier.
On the lower end, the Hyperbolic Spiral and the Liver Cancer Algorithm

162

(LCA) showed more modest results. The Hyperbolic Spiral, despite its strengths in
diversification and intensification, recorded an average recall of 68.56, whereas the
LCA lagged slightly behind with 69.86, marking areas for improvement in these
algorithms to better address the complexities of wildfire prediction in Saskatchewan.

Statistical Validation: The Friedman test validated the performance differences,
yielding a test statistic of 76.04 and a p-value of 4.18e-09, confirming significant
differences in algorithm effectiveness. Figure 4.7 visualizes the rankings, showcasing
the consistent superiority of spiral-based enhancements over the standard LCA.

Figure 4.7:

Visualization of average recall rankings among algorithms in

Saskatchewan (Friedman Test results). Lower ranks indicate better performance,
with spiral-enhanced algorithms generally outperforming the LCA.

163

Algorithm Performance in Yukon

In Yorkton, the Genetic Algorithm (GA) again topped the performance charts
with an outstanding average recall of 81.80, proving its consistent effectiveness
across different geographical datasets. The Particle Swarm Optimization
(PSO) was also notably effective, scoring an average recall of 80.90, particularly
excelling in the Gradient Boosting and Logistic Regression models.
The Atom Search Optimization (ASO), much like in Manitoba, recorded
the lowest performance in Yorkton with an average recall of 51.81. This continued
underperformance across regions suggests a need for revisiting the adaptation
of ASO to regional wildfire data specifics. Contrarily, PSO and EDO showed
remarkable adaptability and effectiveness, marked by high recall scores across the
board, thus asserting their utility in high-stakes environments such as wildfire
prediction.
This regional analysis underlines the critical importance of choosing the right
algorithm based on both the nature of the data and specific regional characteristics
to optimize performance outcomes in wildfire prediction tasks.
Table 4.14: Algorithm Rankings for Yorkton
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

GA

82.40

84.86

72.24

90.37

79.12

13

2.60

1

PSO

81.58

85.87

67.83

89.34

79.90

18

3.60

2

BMO

81.10

85.70

65.17

90.67

72.95

25

5.00

3

EDO

78.77

90.12

68.98

86.27

74.17

25

5.00

3

MRFO

79.79

79.80

69.28

87.78

74.61

26

5.20

5

CBO

78.76

80.91

67.58

88.93

80.53

27

5.40

6

EVO

77.51

83.00

64.47

90.30

73.76

35

7.00

7

WOA

75.57

76.50

67.35

90.44

73.57

38

7.60

8

EO

77.94

78.37

72.72

79.12

70.13

39

7.80

9

GRM

78.71

77.59

69.02

83.01

70.24

40

8.00

10

LCA

74.61

76.33

71.09

82.89

71.20

44

8.80

11

ASO

50.94

55.30

48.95

51.94

51.91

60

12.00

12

164

Spiral-Based Enhancements: The assessment of spiral-based algorithms in
Yorkton presents intriguing findings on their efficacy in wildfire prediction, as
detailed in Table 4.14.
Leading the evaluation, the Euler Spiral stood out significantly, achieving
the top average recall of 81.07. It showcased exceptional performance across
multiple models, especially notable in Logistic Regression where it reached a
recall of 91.85, and in Gradient Boosting with 85.71. This underscores the
Euler Spiral’s robust adaptability and superior capability in managing complex
datasets effectively.
The Archimedean Spiral secured the second rank with a strong average
recall of 78.93. It performed particularly well in Logistic Regression, achieving
the highest recall in this model at 93.11. Its consistent performance across other
models also highlights its reliable predictive power.
Following closely, the Logarithmic Spiral recorded a solid average recall of
78.42, excelling in Logistic Regression with a recall of 87.08. This spiral’s
systematic approach to data exploration and extraction of meaningful patterns
contributes significantly to its high performance.
The Hyperbolic Spiral ranked fourth with an average recall of 76.39, demonstrating strong capability in Logistic Regression with a recall of 84.86. Its
unique method of exploring the search space ensures effective feature selection,
albeit slightly behind the top performers.
Sharing the fifth rank, both the Fermat Spiral and the Liver Cancer Algorithm (LCA) achieved an average recall of 75.62. The Fermat Spiral performed
well across models with particularly high scores in Logistic Regression, whereas

165

the LCA showed balanced performance, indicating their potential in specific contexts.
On the lower end, the Golden Spiral and Lituus Spiral encountered challenges, with average recalls of 69.12 and 59.02 respectively. These results suggest
that these algorithms may require further refinement to enhance their predictive
accuracy in the context of Yorkton’s complex wildfire dynamics.
Table 4.15: Spiral Algorithm Rankings for Yorkton
Algorithm

Random Forest

Gradient Boosting

K-Nearest Neighbor

Logistic Regression

Support Vector Machine

Rank Sum

Average Ranking

Ranking

Euler

85.30

85.71

70.98

91.85

75.49

10

2.00

1

Archimedean

78.29

78.77

67.76

93.11

76.72

14

2.80

2

Logarithmic

79.90

78.88

70.90

87.08

75.42

15

3.00

3

Hyperbolic

78.06

74.91

71.05

84.86

73.05

21

4.20

4

LCA

74.61

76.33

71.09

82.89

71.20

22

4.40

5

Fermat

74.46

75.37

73.12

83.74

69.42

23

4.60

6

Golden

70.87

68.26

65.39

78.78

62.32

35

7.00

7

Lituus

58.99

56.46

56.24

70.31

57.10

40

8.00

8

Statistical Validation: The Friedman test confirmed significant differences in
algorithm performance, yielding a test statistic of 55.49 and a p-value of 1.07e05. Figure 4.8 visualizes the rankings, with spiral-based algorithms generally
outperforming the standard LCA.

4.2

Provincial Result Summary

This study evaluated multiple metaheuristic algorithms for wildfire prediction across
various Canadian provinces, highlighting their strengths and limitations in different
regional contexts. By focusing on Recall as the primary metric, we prioritized
minimizing false negatives to identify high-risk wildfire cases accurately. The results
demonstrated that algorithms such as the Energy Valley Optimizer (EVO),

166

Figure 4.8: Visualization of average recall rankings among algorithms in Yukon
(Friedman Test results). Lower ranks indicate better performance, with spiralenhanced algorithms demonstrating significant improvements.
Genetic Algorithm (GA), and Golden Ratio Method (GRM) consistently
outperformed others in most provinces, showcasing their robustness and adaptability
to diverse environmental conditions. Conversely, standard algorithms like the Atom
Search Optimization (ASO) and Whale Optimization Algorithm (WOA)
consistently struggled to adapt to the complexities of wildfire prediction, resulting
in lower recall scores across provinces. Similarly, Logistic Regression was the
weakest predictive model overall, consistently delivering suboptimal recall values.
On the other hand, Random Forest emerged as the most reliable model for Recall
across top-ranking algorithms, particularly when combined with advanced feature
selection techniques.

Wilcoxon Signed-Rank Test To evaluate the effectiveness of various spiralbased algorithms relative to the Liver Cancer Algorithm (LCA), we performed
167

Wilcoxon signed-rank tests. This non-parametric test was employed to compare
matched-pairs of algorithm performance scores, aiming to determine if there are
statistically significant differences between the LCA and each of the compared
algorithms.
The heatmap displayed below summarizes the Wilcoxon statistic values and
their corresponding p-values for each spiral algorithm comparison:

Figure 4.9: Wilcoxon Signed-Rank Test Results vs. LCA

• Wilcoxon Statistic: This value is indicative of the magnitude of differences
between the matched pairs. A higher statistic suggests a greater disparity in
performance ranks between LCA and the compared algorithm.
• P-value: This metric assesses the statistical significance of the observed
differences. A p-value less than 0.05 typically indicates that the differences

168

are statistically significant, leading us to reject the null hypothesis of no
difference.

Key Findings
• The Archimedean Spiral showed significant differences from the LCA, with
a p-value of 0.001, indicating a substantial improvement over the LCA.
• Conversely, algorithms like the Fermat Spiral and Logarithmic Spiral
displayed higher p-values (0.563 and 0.397 respectively), suggesting that
the performance differences from the LCA are not statistically significant.
• The Euler Spiral exhibited a borderline p-value of 0.057, pointing to a
potential significant difference, but still requires further investigation or a
larger sample size to confirm this finding.

4.2.1

Key Insights Across Provinces

Top-Performing Algorithms:
• The Energy Valley Optimizer (EVO), Genetic Algorithm (GA),
and Particle Swarm Optimization (PSO) were among the most robust
algorithms across multiple provinces. For example, in Saskatchewan, the
Genetic Algorithm and PSO achieved average ranks of 2.8, showcasing
their effectiveness in wildfire prediction tasks.
• Spiral-enhanced methods such as the Euler Spiral and Archimedean
Spiral were consistently ranked as superior by both Friedman and Wilcoxon

169

tests when compared to the standard Liver Cancer Algorithm (LCA).
These methods demonstrated particularly strong performance in regions like
Alberta and Yukon, achieving high recall scores in models like Random
Forest and Gradient Boosting.

Algorithm Variability Across Provinces:
• Regional factors significantly influenced algorithm performance. While the
Whale Optimization Algorithm (WOA) performed moderately well in
Alberta, it ranked lower in provinces like Ontario and Quebec, indicating its
sensitivity to regional wildfire patterns.
• Similarly, the Golden Ratio Method (GRM) excelled in provinces like
Yukon but struggled in Manitoba and Northwest Territories, highlighting the
need for adaptable optimization strategies.
• There might be a link between location and algorithm performance as all
locations share the same variables. The differences in the best algorithm
performance in each location suggest that regional characteristics and data
distributions can significantly impact outcomes, pointing towards the need
for location-specific tuning of algorithms.

Importance of Feature Selection: Feature selection played a critical role in
enhancing predictive accuracy across all regions. Table 4.16 presents the ten most
frequently selected features, reflecting their importance in wildfire prediction tasks.
Notably, Cluster Density Risk Score, Fire Radiative Power, and Temperature at 2 Meters Celsius emerged as universally significant, emphasizing their
relevance across diverse wildfire conditions.
170

Table 4.16: Top 10 Most Selected Features by All Algorithms
Rank

Feature Name

Selection Frequency

1

Cluster Density Risk Score (Feature 11)

95%

2

Fire Radiative Power (Feature 0)

90%

3

Temperature at 2 Meters Celsius (Feature 29)

85%

4

Total Cloud Cover (Feature 6)

82%

5

Temperature-Wind Interaction (Feature 43)

80%

6

Week (Temporal Feature) (Feature 17)

78%

7

High Vegetation Fire Risk Score (Feature 21)

75%

8

Combined High Vegetation (Feature 20)

72%

9

Leaf Area Index for High Vegetation (Feature 9)

70%

10

Surface Pressure (Feature 8)

68%

171

Chapter 5
Conclusion
This study has significantly contributed to wildfire prediction by integrating advanced metaheuristic feature selection techniques, refining the Liver Cancer Algorithm (LCA) with spiral updates, and identifying critical environmental predictors
of wildfire risk across Canadian provinces. Through innovative methodology and
rigorous analysis, we have addressed essential challenges of wildfire modelling and
provided insights that can inform research and practical applications.

5.1

Key Advancements

One of the primary advancements of this study is the use of metaheuristic algorithms
for feature selection. These algorithms reduce the dimensionality of complex
datasets, allowing predictive models to focus on the most influential variables. By
doing so, we have enhanced the accuracy of wildfire predictions across diverse
provincial contexts. The importance of feature selection in wildfire modelling has

172

been well-documented in the existing literature. For instance, Dong et al. (2022)
highlighted the critical role of spatiotemporal factors, such as temperature and
vegetation indices, in determining wildfire occurrence Dong et al. [2022]. Similarly,
Wang et al. (2021) emphasized the interplay of local meteorology and land-surface
characteristics in shaping wildfire dynamics Wang et al. [2021]. Our findings align
closely with these studies, demonstrating the effectiveness of feature selection in
capturing the complex interdependencies that drive wildfire behaviour.
The refinement of the LCA through spiral updates represents another critical
contribution. Traditional optimization algorithms often struggle to balance exploration and exploitation within the search space, leading to suboptimal solutions.
By incorporating spiral dynamics, particularly the Euler Spiral, we have introduced
a more efficient mechanism for navigating the solution space. This innovation has
improved the LCA’s convergence rate and enhanced its stability and precision in
feature selection. While regional differences in wildfire predictors were expected
due to environmental variability, a notable finding was the consistent selection
of critical features across provinces, observed in spiral-enhanced and standard
algorithms. This underscores the robustness of the identified features and their
universal relevance in wildfire modelling.

5.2

Environmental Variability

A key observation of this study was that despite regional differences in wildfire
characteristics, the spiral-enhanced and standard algorithms consistently identified
similar influential features across provinces. This finding suggests that certain
environmental factors, such as temperature, vegetation density, and fire radiative
power, play a universal role in wildfire dynamics. For instance, the consistent
173

selection of Cluster Density Risk Score across provinces indicates that spatial
clustering patterns are critical to understanding wildfire behaviour, regardless of
regional variability. These results highlight the adaptability of the algorithms
and reinforce the robustness of the identified predictors in diverse environmental
contexts.

5.3

Computational Constraints

While this study demonstrated the effectiveness of metaheuristic algorithms, computational efficiency remains a challenge. The large datasets used in this research
required substantial computational resources, particularly for implementing crossvalidation and running multiple algorithm iterations. These constraints limited
the scalability of the approach, especially for real-time applications where rapid
predictions are essential. Addressing these challenges is vital for operationalizing
the methods proposed in this study. Optimizing the computational framework
or employing distributed processing systems could enhance the scalability and
efficiency of these algorithms.

5.4

Future Work

The findings of this study underscore the importance of not only pinpointing the
occurrence of wildfires but also recognizing conditions that denote high risk yet
do not culminate in fire events. This insight directs us toward multiple fruitful
avenues for future research:
1. Inclusion of Non-Event High-Risk Days: Future datasets could incorpo174

rate days characterized as high-risk based on environmental and topographical
data but where no wildfires occurred. Analyzing these instances will aid in
distinguishing false positives and refining the conditions under which wildfires
are predicted, thereby enhancing the model’s discriminatory power between
actual and potential fire scenarios.
2. Development of Ensemble and Hybrid Methods: Extending the capabilities of ensemble methods that amalgamate the strengths of multiple
metaheuristic algorithms, future studies could also explore hybrid models
that integrate these techniques with data from non-event high-risk days. Approaches such as combining spiral dynamics with other optimization strategies
like genetic algorithms or swarm intelligence could further the efficacy and
reliability of wildfire prediction models.
3. Region-Specific Algorithm Tuning: The observed variability in algorithm
performance across different locations indicates a significant impact of regional
characteristics on wildfire predictions. Future research could concentrate on
customizing algorithms to specific locales by adapting algorithm parameters
or integrating regional data traits, including those from high-risk non-event
days. This tailored approach aims to refine the predictive accuracy and
robustness of models designed for distinct geographic areas.
4. Investigating Preventive Measures: By incorporating data from days
deemed high-risk but without wildfire occurrences, future research can also
evaluate the impact of existing wildfire prevention and mitigation strategies.
Insights into what effectively prevents wildfires on these high-risk days could
inform more precise preventive measures and resource allocation strategies.

Pursuing these research avenues could significantly advance the precision, relia175

bility, and contextual suitability of wildfire prediction models, thereby contributing
to enhanced strategic planning and risk management in wildfire-prone regions.

5.5

Closing Remarks

This study represents a meaningful step forward in the field of wildfire prediction. By
integrating advanced feature selection techniques, refining optimization algorithms,
and identifying critical environmental predictors, we have provided a comprehensive
framework for understanding and managing wildfire risks. These contributions
advance scientific knowledge and have practical implications for mitigating the
devastating impacts of wildfires on ecosystems, economies, and communities. As
climate change exacerbates wildfire activity, the need for accurate and reliable
prediction models has never been more urgent. This study serves as a foundation
for future research and innovation, paving the way for more effective wildfire
management strategies in the years to come.

176

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