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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Psychol.</journal-id>
<journal-title-group>
<journal-title>Frontiers in Psychology</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Psychol.</abbrev-journal-title>
</journal-title-group>
<issn pub-type="epub">1664-1078</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.3389/fpsyg.2025.1544589</article-id>
<article-version article-version-type="Version of Record" vocab="NISO-RP-8-2008"/>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Original Research</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>RETRACTED: Research on crime motivation identification and quantitative analysis methods based on EEG signals</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author" corresp="yes">
<name><surname>Ma</surname> <given-names>DongLi</given-names></name>
<xref ref-type="aff" rid="aff1"/>
<xref ref-type="corresp" rid="c001"><sup>&#x0002A;</sup></xref>
<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Writing &#x2013; original draft" vocab-term-identifier="https://credit.niso.org/contributor-roles/writing-original-draft/">Writing &#x2013; original draft</role>
<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Writing &#x2013; review &amp; editing" vocab-term-identifier="https://credit.niso.org/contributor-roles/writing-review-editing/">Writing &#x2013; review &#x00026; editing</role>
<uri xlink:href="https://loop.frontiersin.org/people/2919689"/>
</contrib>
</contrib-group>
<aff id="aff1"><institution>State Key Law Major, School of Law, Henan University of Urban Construction in Pingdingshan</institution>, <city>Pingdingshan</city>, <country country="cn">China</country></aff>
<author-notes>
<corresp id="c001"><label>&#x0002A;</label>Correspondence: DongLi Ma, <email xlink:href="mailto:mdl888@126.com">mdl888@126.com</email></corresp>
</author-notes>
<pub-date publication-format="electronic" date-type="pub" iso-8601-date="2025-03-18">
<day>18</day>
<month>03</month>
<year>2025</year>
</pub-date>
<pub-date publication-format="electronic" date-type="retracted"> 
<day>01</day> 
<month>04</month> 
<year>2026</year>
</pub-date>
<pub-date publication-format="electronic" date-type="collection">
<year>2025</year>
</pub-date>
<volume>16</volume>
<elocation-id>1544589</elocation-id>
<history>
<date date-type="received">
<day>16</day>
<month>12</month>
<year>2024</year>
</date>
<date date-type="accepted">
<day>26</day>
<month>02</month>
<year>2025</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#x000A9; 2025 Ma.</copyright-statement>
<copyright-year>2025</copyright-year>
<copyright-holder>Ma</copyright-holder>
<license>
<ali:license_ref start_date="2025-03-18">https://creativecommons.org/licenses/by/4.0/</ali:license_ref>
<license-p>This is an open-access article distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution License (CC BY)</ext-link>. The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</license-p>
</license>
</permissions>
<abstract>
<sec>
<title>Introduction</title>
<p>Understanding and quantifying crime motivation is essential for developing effective interventions in criminology and psychology. This research, closely aligned with quantitative psychology and measurement, presents a novel approach to identifying and analyzing crime motivations using EEG signals. Traditional methods often fail to capture the intricate interplay of individual, social, and environmental factors due to data sparsity and the absence of real-time adaptability.</p></sec>
<sec>
<title>Methods</title>
<p>In this study, we introduce the Hierarchical Crime Motivation Network (HCM-Net), a multi-layered framework that integrates EEG signal analysis with social and temporal modeling. HCM-Net employs neural network-based individual feature encoders, graph neural networks for social interaction analysis, and temporal predictors to capture the evolution of motivations. To enhance practical applicability, the Dynamic Risk-Adaptive Strategy (DRAS) complements HCM-Net by incorporating real-time adaptation, scenario-based simulations, and targeted interventions. This framework addresses challenges such as ethical considerations and interpretability by employing Shapley values for feature attribution and bias mitigation techniques.</p></sec>
<sec>
<title>Results</title>
<p>Experiments with EEG datasets demonstrate the superior performance of the proposed methods in classifying crime motivations and identifying high-risk individuals compared to state-of-the-art techniques.</p></sec>
<sec>
<title>Discussion</title>
<p>These findings highlight the potential of integrating EEG analysis with advanced computational methods in crime prevention and psychological research.</p></sec></abstract>
<kwd-group>
<kwd>crime motivation</kwd>
<kwd>EEG signals</kwd>
<kwd>hierarchical modeling</kwd>
<kwd>social networks</kwd>
<kwd>quantitative psychology</kwd>
</kwd-group>
<funding-group>
  <funding-statement>The author(s) declare that no financial support was received for the research and/or publication of this article.</funding-statement>
</funding-group>
<counts>
<fig-count count="10"/>
<table-count count="5"/>
<equation-count count="36"/>
<ref-count count="44"/>
<page-count count="18"/>
<word-count count="10579"/>
</counts>
<custom-meta-group>
<custom-meta>
<meta-name>section-at-acceptance</meta-name>
<meta-value>Personality and Social Psychology</meta-value>
</custom-meta>
</custom-meta-group>
</article-meta>
</front>
<body>
<sec sec-type="intro" id="s1">
<label>1</label>
<title>Introduction</title>
<p>Understanding and quantifying crime motivations are critical for crime prevention, judicial evaluation, and rehabilitation (<xref ref-type="bibr" rid="B40">Zhang W. et al., 2023</xref>). Traditional methods rely on psychological assessments or behavioral studies, which cannot well capture the real-time cognitive and emotional states associated with criminal intent (<xref ref-type="bibr" rid="B26">Mao et al., 2023</xref>). Electroencephalogram (EEG) signals, which provide direct insights into neural activity, offer a promising avenue for understanding the neurological underpinnings of crime-related decision-making (<xref ref-type="bibr" rid="B1">Alsaeedi and Zubair, 2023</xref>). EEG-based approaches not only enable objective assessment of motivations but also facilitate quantitative analysis, addressing gaps in current methodologies (<xref ref-type="bibr" rid="B44">Zhu et al., 2023</xref>). By leveraging EEG signals, researchers aim to build robust frameworks that can identify crime motivations more accurately and contribute to forensic and legal systems (<xref ref-type="bibr" rid="B11">Fatouros et al., 2023</xref>). Time-frequency analysis, wavelet transformations, and power spectral density were employed to extract features from EEG signals, while expert-defined rules were used to map these features to specific cognitive states (<xref ref-type="bibr" rid="B39">Zhang B. et al., 2023</xref>). These approaches provided initial insights into the correlation between neural activity patterns and crime motivations (<xref ref-type="bibr" rid="B33">Tan et al., 2023</xref>). However, their reliance on handcrafted features limited their ability to capture the complex and dynamic nature of EEG signals (<xref ref-type="bibr" rid="B4">Bello et al., 2023</xref>). Furthermore, their interpretability often came at the cost of reduced adaptability to diverse populations or novel scenarios, necessitating a shift toward data-driven approaches (<xref ref-type="bibr" rid="B10">Das and Singh, 2023</xref>).</p>
<p>To address the limitations of traditional methods, machine learning algorithms were introduced to enhance the interpretation and analysis of EEG data (<xref ref-type="bibr" rid="B30">Qi and Shabrina, 2023</xref>). Algorithms such as support vector machines (SVMs), random forests, and k-nearest neighbors were utilized to classify EEG patterns corresponding to specific emotional or cognitive states (<xref ref-type="bibr" rid="B31">Taherdoost and Madanchian, 2023</xref>). These methods automated feature selection and improved the adaptability of crime motivation analysis. Techniques such as principal component analysis (PCA) and independent component analysis (ICA) were employed for dimensionality reduction, enhancing the efficiency of multi-modal analysis (<xref ref-type="bibr" rid="B5">Bordoloi and Biswas, 2023</xref>). However, these models often struggled to generalize across diverse datasets and lacked the capacity to capture deeper relationships within EEG signals, limiting their robustness in complex real-world scenarios. Recent advancements in deep learning have revolutionized EEG-based crime motivation identification, enabling end-to-end learning and more nuanced interpretations (<xref ref-type="bibr" rid="B37">Wankhade et al., 2022</xref>). Convolutional neural networks (CNNs) and recurrent neural networks (RNNs) have been applied to capture spatial and temporal dependencies in EEG data (<xref ref-type="bibr" rid="B6">Cambria et al., 2022</xref>). Transformer-based models, with their self-attention mechanisms, have demonstrated superior performance in identifying subtle and complex patterns in brain signals (<xref ref-type="bibr" rid="B3">Barnes et al., 2022</xref>). Moreover, multi-modal architectures that integrate EEG with other physiological signals, such as galvanic skin response or heart rate, have further improved the accuracy and reliability of crime motivation identification (<xref ref-type="bibr" rid="B14">Gupta et al., 2022</xref>). Despite these successes, challenges such as high computational demands, data scarcity, and the need for domain-specific adaptation persist, highlighting the need for novel approaches tailored to these constraints (<xref ref-type="bibr" rid="B41">Zhang et al., 2021</xref>).</p>
<p>Understanding and quantifying crime motivations are critical for crime prevention, judicial evaluation, and rehabilitation. Traditional methods, which primarily rely on psychological assessments or behavioral studies, may have limitations in capturing real-time cognitive and emotional states associated with criminal intent. While EEG signals offer a direct insight into neural activity, existing crime-related research has primarily focused on traditional signal processing techniques or machine learning models that struggle to generalize across diverse datasets and lack interpretability. A key research gap lies in the inability of existing methods to comprehensively model crime motivation by integrating individual, social, and temporal factors. Previous approaches have often relied on handcrafted features or shallow learning techniques, limiting their ability to capture the dynamic and multi-faceted nature of criminal intent. Furthermore, the lack of real-time adaptability and scenario-based analysis has hindered practical applications in forensic and legal contexts. To address this gap, we propose the Hierarchical Crime Motivation Network (HCM-Net), a multi-layered framework that integrates EEG signal analysis with social and temporal modeling. HCM-Net employs neural network-based feature encoders to capture individual characteristics, graph neural networks (GNNs) to model social influences, and recurrent predictors to analyze temporal dependencies. We introduce the Dynamic Risk-Adaptive Strategy (DRAS), which enhances real-time adaptability, scenario-based simulations, and targeted interventions. This novel framework not only improves classification accuracy but also provides interpretability through explainable AI techniques, making it more suitable for forensic applications. By addressing the limitations of previous methods, our approach establishes a robust foundation for crime motivation analysis using EEG signals, with the potential to significantly improve crime prevention and psychological research.</p>
<p>Advantages of the proposed method:</p>
<list list-type="bullet">
<list-item><p>Combines advanced deep learning architectures with specialized EEG preprocessing pipelines to enhance motivation identification accuracy.</p></list-item>
<list-item><p>Supports multi-modal integration and robust performance across diverse datasets, making it adaptable to various forensic applications.</p></list-item>
<list-item><p>Achieves state-of-the-art accuracy in motivation classification, demonstrating its potential for real-world forensic and legal use cases.</p></list-item>
</list></sec>
<sec id="s2">
<label>2</label>
<title>Related work</title>
<sec>
<label>2.1</label>
<title>EEG-based cognitive and behavioral analysis</title>
<p>Electroencephalography (EEG) has been widely used to analyze cognitive and behavioral processes due to its ability to capture brain activity with high temporal resolution (<xref ref-type="bibr" rid="B2">Armas-Vargas et al., 2023</xref>). In forensic and psychological studies, EEG provides insights into the neural correlates of decision-making, emotional processing, and moral reasoning, which are critical for understanding crime motivation (<xref ref-type="bibr" rid="B42">Zhang et al., 2022</xref>). Techniques such as event-related potentials (ERPs) and time-frequency analysis enable the detection of specific brain responses associated with decision-making under duress or moral conflict (<xref ref-type="bibr" rid="B18">Hazarika et al., 2020</xref>). For example, the P300 and N400 components are linked to cognitive processes such as attention and semantic processing, while power spectral analysis can reveal emotional states relevant to criminal intent (<xref ref-type="bibr" rid="B35">Wang et al., 2020</xref>). Despite the progress in identifying correlates of behavior, challenges persist in isolating crime-specific neural patterns due to the variability in individual responses and environmental factors (<xref ref-type="bibr" rid="B17">Hartmann et al., 2022</xref>). This research seeks to address these challenges by proposing quantitative methods to enhance the reliability and specificity of EEG-based crime motivation identification.</p>
</sec>
<sec>
<label>2.2</label>
<title>Quantitative analysis of EEG signals</title>
<p>Quantitative analysis methods transform raw EEG signals into meaningful features for interpretation and prediction. Feature extraction techniques such as wavelet transform, independent component analysis (ICA), and empirical mode decomposition (EMD) are commonly employed to analyze EEG data (<xref ref-type="bibr" rid="B21">Lee, 2023</xref>). These methods decompose EEG signals into frequency bands or independent components, enabling the identification of patterns linked to specific mental states (<xref ref-type="bibr" rid="B29">Prottasha et al., 2022</xref>). Machine learning models, including support vector machines (SVMs), convolutional neural networks (CNNs), and recurrent neural networks (RNNs), further improve the accuracy of EEG-based predictions (<xref ref-type="bibr" rid="B8">Chan et al., 2023</xref>). In crime-related research, these models have been applied to classify deception, intent, or aggression based on brainwave patterns (<xref ref-type="bibr" rid="B23">Li et al., 2021</xref>). However, challenges such as overfitting, signal noise, and limited generalizability across populations remain significant (<xref ref-type="bibr" rid="B16">Han et al., 2021</xref>). This study advances quantitative EEG analysis by incorporating domain adaptation techniques and multi-modal data fusion to improve the robustness of crime motivation classification systems.</p>
</sec>
<sec>
<label>2.3</label>
<title>EEG in forensic and criminal research</title>
<p>The application of EEG in forensic research aims to uncover the neural mechanisms underlying criminal behavior, providing a scientific basis for crime prevention and investigation (<xref ref-type="bibr" rid="B7">Chan, 2023</xref>). Studies in this area often examine neural markers of impulsivity, empathy deficits, and moral judgment, which are linked to antisocial or criminal tendencies (<xref ref-type="bibr" rid="B38">Yan et al., 2021</xref>). For instance, reduced activity in the prefrontal cortex has been associated with impulsivity and poor decision-making, while abnormal theta and gamma oscillations are linked to emotional dysregulation (<xref ref-type="bibr" rid="B28">Muhammad et al., 2022</xref>). Emerging research integrates EEG with other modalities, such as functional magnetic resonance imaging (fMRI) and physiological signals, to enhance the multidimensional understanding of criminal intent (<xref ref-type="bibr" rid="B32">Tan et al., 2022</xref>). Despite its promise, the forensic use of EEG faces ethical and practical challenges, including data privacy concerns and the risk of misinterpretation in legal contexts (<xref ref-type="bibr" rid="B19">Hu et al., 2022</xref>). This research explores methods to mitigate these issues, focusing on improving the interpretability and reliability of EEG-based crime motivation assessments.</p></sec>
</sec>
<sec id="s3">
<label>3</label>
<title>Method</title>
<sec>
<label>3.1</label>
<title>Overview</title>
<p>Understanding crime motivation is a critical area of research that combines insights from criminology, sociology, and psychology. It aims to identify the underlying factors that drive individuals to commit criminal acts. By systematically analyzing these motivations, researchers seek to develop predictive models, inform policy-making, and implement interventions that can reduce crime rates. This section outlines the proposed framework for advancing the analysis of crime motivation, focusing on computational and analytical methods.</p>
<p>The methodology presented in this study is divided into three main components. Section 3.2 establishes the theoretical foundation, defining key concepts, variables, and mathematical formulations for modeling crime motivation. This includes formalizing individual and contextual factors, such as socio-economic conditions, peer influences, and psychological triggers, that contribute to criminal behavior. Section 3.3 introduces an innovative computational model designed to quantify and analyze these factors in a structured manner. The model leverages advanced machine learning techniques, including graph-based methods and temporal modeling, to capture the dynamic and interconnected nature of crime motivations. Section 3.4 details strategic innovations for applying the model in real-world scenarios. These strategies address challenges such as data sparsity, ethical considerations, and the need for interpretability in crime analysis.</p>
</sec>
<sec>
<label>3.2</label>
<title>Preliminaries</title>
<p>Crime motivation refers to the underlying factors and processes that influence an individual to commit criminal acts. To systematically analyze this phenomenon, it is essential to formalize the problem mathematically and establish a structured framework. This subsection outlines the key definitions, notations, and foundational concepts used in this study.</p>
<p>Let <inline-formula><mml:math id="M1"><mml:mrow><mml:mi mathvariant="script">C</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>&#x02026;</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:math></inline-formula> represent a set of criminal events, where each event <italic>c</italic><sub><italic>i</italic></sub> is characterized by a set of attributes <bold>x</bold><sub><italic>i</italic></sub> &#x0003D; {<italic>x</italic><sub><italic>i</italic>1</sub>, <italic>x</italic><sub><italic>i</italic>2</sub>, &#x02026;, <italic>x</italic><sub><italic>im</italic></sub>}. These attributes include socio-economic factors, demographic variables, environmental conditions, and psychological traits. The objective is to model the probability of a crime event occurring, <italic>P</italic>(<italic>c</italic><sub><italic>i</italic></sub> &#x02223; <bold>x</bold><sub><italic>i</italic></sub>), as a function of these attributes.</p>
<p>The decision process of an individual is modeled as a binary variable <italic>y</italic><sub><italic>i</italic></sub>&#x02208;{0, 1}, where <italic>y</italic><sub><italic>i</italic></sub> &#x0003D; 1 indicates the occurrence of a crime, and <italic>y</italic><sub><italic>i</italic></sub> &#x0003D; 0 represents no criminal act. This process is driven by a latent motivation score <italic>M</italic><sub><italic>i</italic></sub> such that</p>
<disp-formula id="EQ1"><mml:math id="M2"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mn>1</mml:mn></mml:mtd><mml:mtd><mml:mtext class="textrm" mathvariant="normal">if&#x000A0;</mml:mtext><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>&#x0003E;</mml:mo><mml:mi>&#x003C4;</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mtext class="textrm" mathvariant="normal">otherwise</mml:mtext><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(1)</label></disp-formula>
<p>where &#x003C4; is a threshold parameter representing the decision boundary.</p>
<p>The motivation score <italic>M</italic><sub><italic>i</italic></sub> is modeled as a function of multiple variables as shown below:</p>
<disp-formula id="EQ2"><mml:math id="M3"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>e</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(2)</label></disp-formula>
<p>where <bold>x</bold><sub><italic>i</italic></sub> represents individual-specific factors such as age, income, and education. The variable <bold>z</bold><sub><italic>i</italic></sub> captures social influences, including peer group pressure or community norms. The term <bold>e</bold><sub><italic>i</italic></sub> includes environmental conditions such as urban density, unemployment rate, or access to resources.</p>
<p>Crime motivation often evolves over time. Let <italic>t</italic> &#x02208; &#x1D54B; denotes discrete time steps, and <inline-formula><mml:math id="M4"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">H</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> represents the historical context up to time <italic>t</italic>. The probability of a crime event occurring at time <italic>t</italic> can be expressed as</p>
<disp-formula id="EQ3"><mml:math id="M5"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x02223;</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">H</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">H</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(3)</label></disp-formula>
<p>where <inline-formula><mml:math id="M6"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> is the motivation score at time <italic>t</italic>, and <italic>g</italic>(&#x000B7;) is a mapping function that incorporates historical trends and temporal dependencies.</p>
<p>To capture the influence of social networks, individuals are represented as nodes in a graph <inline-formula><mml:math id="M7"><mml:mrow><mml:mi mathvariant="script">G</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">V</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant="script">E</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, where <inline-formula><mml:math id="M8"><mml:mrow><mml:mi mathvariant="script">V</mml:mi></mml:mrow></mml:math></inline-formula> is the set of individuals, and <inline-formula><mml:math id="M9"><mml:mrow><mml:mi mathvariant="script">E</mml:mi></mml:mrow></mml:math></inline-formula> represents relationships such as friendships or familial ties. The social influence term <bold>z</bold><sub><italic>i</italic></sub> is computed as</p>
<disp-formula id="EQ4"><mml:math id="M10"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>z</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x000B7;</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(4)</label></disp-formula>
<p>where <inline-formula><mml:math id="M11"><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> denotes the neighbors of node <italic>i</italic> in the graph, and <italic>w</italic><sub><italic>ij</italic></sub> represents the weight of the influence between individuals <italic>i</italic> and <italic>j</italic>.</p>
<p>The accuracy of the proposed model is evaluated using metrics such as precision, recall, and the F1-score for binary classification tasks. In addition, the area under the receiver operating characteristic curve (AUC-ROC) is used to measure the model&#x00027;s ability to distinguish between positive and negative classes. Crime datasets often lack comprehensive coverage, with missing or incomplete records. Modeling crime motivation requires careful handling of sensitive data to avoid bias and protect privacy. Moreover, crime events are influenced by diverse and non-linear factors that vary across regions, cultures, and time.</p>
</sec>
<sec>
<label>3.3</label>
<title>Hierarchical crime motivation network (HCM-Net)</title>
<p>To address the complexities of crime motivation, we propose the Hierarchical Crime Motivation Network (HCM-Net), a novel framework that integrates multi-scale feature learning, temporal dynamics, and social influence modeling (as shown in <xref ref-type="fig" rid="F1">Figure 1</xref>). HCM-Net captures the intricate interplay between individual, social, and environmental factors that drive criminal behavior.</p>
<fig position="float" id="F1">
<label>Figure 1</label>
<caption><p>Architecture of the Hierarchical Crime Motivation Network (HCM-Net) demonstrates its key components. Multi-Scale Individual Feature Encoding (MMFEL) for personalized feature extraction, Graph Convolutional Networks (GCN) for modeling social influences, and EmoGIs blocks for integrating multi-modal emotional and temporal data. The framework combines individual, social, and environmental factors, leveraging attention mechanisms, normalization layers, and sequence modeling to analyze the complex dynamics of crime motivation effectively.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0001.tif">
</graphic>
</fig>
<sec>
<label>3.3.1</label>
<title>Multi-scale individual feature encoding</title>
<p>HCM-Net incorporates a multi-scale individual feature encoder designed to capture personalized attributes <bold>x</bold><sub><italic>i</italic></sub> for each individual. This encoder employs a multi-layer neural network that integrates a variety of non-linear transformations to map raw features into a compact and informative latent representation. The process begins by linearly transforming the input features through a learnable weight matrix <italic>W</italic><sup>ind</sup> and a bias vector <italic>b</italic><sup>ind</sup>. This transformation is followed by the application of a non-linear activation function &#x003C3;, resulting in the latent representation <inline-formula><mml:math id="M12"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">ind</mml:mtext></mml:mstyle></mml:mrow></mml:msubsup></mml:math></inline-formula> expressed as</p>
<disp-formula id="EQ5"><mml:math id="M13"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">ind</mml:mtext></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">ind</mml:mtext></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>&#x0002B;</mml:mo><mml:msup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">ind</mml:mtext></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(5)</label></disp-formula>
<p>where &#x003C3; can represent commonly used activation functions such as ReLU, sigmoid, or tanh, chosen based on their ability to capture complex non-linear relationships within the data. To enhance the expressiveness of the encoder, multiple layers are stacked such that the output of each layer becomes the input for the subsequent layer, allowing the model to learn hierarchical representations of individual features. The output of layer <italic>l</italic> is computed as</p>
<disp-formula id="EQ6"><mml:math id="M14"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>&#x0002B;</mml:mo><mml:msup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(6)</label></disp-formula>
<p>where <italic>W</italic><sup>(<italic>l</italic>)</sup> and <italic>b</italic><sup>(<italic>l</italic>)</sup> are the weight matrix and bias vector for the <italic>l</italic>-th layer, and <inline-formula><mml:math id="M15"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>. This hierarchical structure enables the encoder to identify both low-level and high-level feature abstractions, capturing nuanced individual factors that contribute to crime motivation.</p>
<p>To further enhance the model&#x00027;s ability to process heterogeneous data, the encoder employs a normalization layer after each transformation, defined as</p>
<disp-formula id="EQ7"><mml:math id="M16"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mo>&#x0007E;</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:mi>&#x003BC;</mml:mi></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mi>&#x003C3;</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>&#x0002B;</mml:mo><mml:mi>&#x003F5;</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(7)</label></disp-formula>
<p>where &#x003BC; and &#x003C3;<sup>2</sup> are the mean and variance of the features within a batch, and &#x003F5; is a small constant to ensure numerical stability. This normalization ensures that the learned representations remain robust to variations in the scale and distribution of input features, improving convergence during training.</p>
<p>In addition, the encoder incorporates a dropout mechanism to prevent overfitting by randomly deactivating a fraction of neurons during training. The output of the dropout layer is given by</p>
<disp-formula id="EQ8"><mml:math id="M17"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mtext class="textrm" mathvariant="normal">dropout</mml:mtext></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mstyle mathvariant="bold"><mml:mtext>m</mml:mtext></mml:mstyle><mml:mo>&#x02299;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(8)</label></disp-formula>
<p>where <bold>m</bold> is a binary mask sampled from a Bernoulli distribution with a probability <italic>p</italic> of retaining each neuron, and &#x02299; denotes element-wise multiplication. This stochastic regularization encourages the encoder to learn more generalized feature representations by reducing dependency on specific neurons.</p>
<p>Finally, to incorporate interactions between individual attributes, the encoder models pairwise feature interactions by introducing a bilinear transformation, expressed as</p>
<disp-formula id="EQ9"><mml:math id="M18"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">pair</mml:mtext></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x022A4;</mml:mo></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">pair</mml:mtext></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>x</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(9)</label></disp-formula>
<p>where <italic>W</italic><sup>pair</sup> is a learnable matrix capturing the relationships between feature pairs. The final latent representation is obtained by concatenating the outputs of the hierarchical encoder and the bilinear interaction term as</p>
<disp-formula id="EQ10"><mml:math id="M19"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">final</mml:mtext></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">pair</mml:mtext></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(10)</label></disp-formula>
<p>where <inline-formula><mml:math id="M20"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula> is the output of the last layer of the encoder, and [&#x000B7;] denotes vector concatenation. This comprehensive encoding mechanism ensures that the multi-scale individual feature encoder captures both intrinsic and interaction-based factors, providing a rich representation of the individual&#x00027;s attributes that contribute to crime motivation.</p>
</sec>
<sec>
<label>3.3.2</label>
<title>Social influence modeling using graph neural networks</title>
<p>The social interaction module utilizes a graph neural network (GNN) to comprehensively model the influence of social relationships on crime motivation. Each individual is represented as a node in the graph <inline-formula><mml:math id="M21"><mml:mrow><mml:mi mathvariant="script">G</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">V</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant="script">E</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, where the set of nodes <inline-formula><mml:math id="M22"><mml:mrow><mml:mi mathvariant="script">V</mml:mi></mml:mrow></mml:math></inline-formula> corresponds to individuals, and the edges <inline-formula><mml:math id="M23"><mml:mrow><mml:mi mathvariant="script">E</mml:mi></mml:mrow></mml:math></inline-formula> represent social ties such as familial relationships, friendships, or shared community membership. The node features <inline-formula><mml:math id="M24"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">ind</mml:mtext></mml:mstyle></mml:mrow></mml:msubsup></mml:math></inline-formula>, derived from the individual feature encoder, serve as the initial embeddings for each node <italic>i</italic>. These embeddings are iteratively updated through message passing, enabling the model to aggregate information from neighboring nodes. The updated social embedding <inline-formula><mml:math id="M25"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mstyle></mml:mrow></mml:msubsup></mml:math></inline-formula> for node <italic>i</italic> at each iteration is calculated as</p>
<disp-formula id="EQ11"><mml:math id="M26"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:msub><mml:mrow><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x000B7;</mml:mo><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">ind</mml:mtext></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(11)</label></disp-formula>
<p>where <inline-formula><mml:math id="M27"><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> denotes the set of neighbors for node <italic>i</italic>, &#x003B1;<sub><italic>ij</italic></sub> is an attention weight that measures the importance of node <italic>j</italic> to node <italic>i</italic>, <italic>W</italic><sup>soc</sup> is a learnable weight matrix, and &#x003C3; is a non-linear activation function. The attention weights &#x003B1;<sub><italic>ij</italic></sub> are computed dynamically using a scaled dot-product attention mechanism, expressed as</p>
<disp-formula id="EQ12"><mml:math id="M28"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo class="qopname">exp</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x022A4;</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle displaystyle="true"><mml:msub><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:mstyle><mml:mo class="qopname">exp</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>&#x022A4;</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(12)</label></disp-formula>
<p>where <bold>h</bold><sub><italic>i</italic></sub> and <bold>h</bold><sub><italic>j</italic></sub> are the embeddings of nodes <italic>i</italic> and <italic>j</italic>, respectively, and the exponential function ensures that the attention weights are positive and sum to one across all neighbors of node <italic>i</italic>. This mechanism allows the GNN to prioritize influential neighbors while downweighting less relevant connections.</p>
<p>To improve the model&#x00027;s capacity to capture both direct and indirect social influences, multiple layers of the GNN are stacked. In each layer, the embeddings from the previous iteration are propagated through the graph, enabling the model to aggregate information from multi-hop neighborhoods. The embedding at layer <italic>l</italic>&#x0002B;1 is computed as</p>
<disp-formula id="EQ13"><mml:math id="M29"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:msubsup><mml:mrow><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>&#x000B7;</mml:mo><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(13)</label></disp-formula>
<p>where <italic>W</italic><sup>(<italic>l</italic>)</sup> is the weight matrix for layer <italic>l</italic>, and <inline-formula><mml:math id="M30"><mml:msubsup><mml:mrow><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula> are the attention weights specific to that layer. By stacking <italic>L</italic> layers, the final social embedding <inline-formula><mml:math id="M31"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mstyle></mml:mrow></mml:msubsup></mml:math></inline-formula> incorporates information from up to <italic>L</italic>-hop neighbors, effectively capturing higher-order social dynamics.</p>
<p>To enhance the expressive power of the GNN, residual connections are introduced between layers, ensuring that information from earlier layers is preserved. The updated embedding with a residual connection is computed as</p>
<disp-formula id="EQ14"><mml:math id="M32"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>&#x0002B;</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:msubsup><mml:mrow><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>&#x000B7;</mml:mo><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(14)</label></disp-formula>
<p>This technique mitigates the vanishing gradient problem and ensures stable training, particularly in deep GNN architectures.</p>
<p>Regularization is applied to the GNN to encourage smoothness in the embeddings of socially connected nodes. The social regularization term is defined as</p>
<disp-formula id="EQ15"><mml:math id="M33"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msub><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">E</mml:mi></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:mo>&#x02225;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mo>&#x02225;</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(15)</label></disp-formula>
<p>where &#x003BB;<sub>soc</sub> is a hyperparameter controlling the strength of the regularization. This term enforces consistency between the embeddings of connected nodes, ensuring that the social dynamics reflected in the graph are preserved in the learned representations.</p></sec>
<sec>
<label>3.3.3</label>
<title>Temporal dynamics with sequence modeling</title>
<p>Temporal changes in crime motivation are captured using a long short-term memory (LSTM) network, a type of recurrent neural network specifically designed to model sequential data and address long-term dependencies (as shown in <xref ref-type="fig" rid="F2">Figure 2</xref>). For an individual <italic>i</italic>, the temporal sequence of social embeddings <inline-formula><mml:math id="M34"><mml:msubsup><mml:mrow><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mstyle></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula> is processed by the LSTM, where <italic>T</italic> represents the total number of time steps. At each time step <italic>t</italic>, the hidden state <inline-formula><mml:math id="M35"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula> encodes the individual&#x00027;s crime motivation at that time and is updated as follows:</p>
<disp-formula id="EQ16"><mml:math id="M36"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>c</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mtext class="textrm" mathvariant="normal">LSTM</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>c</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(16)</label></disp-formula>
<p>where <inline-formula><mml:math id="M37"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>c</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula> is the cell state, which maintains long-term memory, and <inline-formula><mml:math id="M38"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula> is the hidden state representing the individual&#x00027;s current motivation. The LSTM employs gating mechanisms to control the flow of information, ensuring that relevant features are retained while irrelevant details are forgotten. These gates include the input gate, forget gate, and output gate, defined as follows:</p>
<disp-formula id="EQ17"><mml:math id="M39"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>i</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msubsup><mml:mo>;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(17)</label></disp-formula>
<disp-formula id="EQ18"><mml:math id="M40"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>f</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msubsup><mml:mo>;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(18)</label></disp-formula>
<disp-formula id="EQ19"><mml:math id="M41"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>o</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msubsup><mml:mo>;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(19)</label></disp-formula>
<p>where &#x003C3; denotes the sigmoid activation function, <italic>W</italic><sub><italic>i</italic></sub>, <italic>W</italic><sub><italic>f</italic></sub>, and <italic>W</italic><sub><italic>o</italic></sub> are weight matrices, and <italic>b</italic><sub><italic>i</italic></sub>, <italic>b</italic><sub><italic>f</italic></sub>, and <italic>b</italic><sub><italic>o</italic></sub> are biases for the respective gates. The updated cell state is computed as follows:</p>
<disp-formula id="EQ20"><mml:math id="M42"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>c</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>f</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x02299;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>c</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>i</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x02299;</mml:mo><mml:mo class="qopname">tanh</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">soc</mml:mtext></mml:mrow></mml:msubsup><mml:mo>;</mml:mo><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(20)</label></disp-formula>
<p>where &#x02299; represents element-wise multiplication, and <italic>W</italic><sub><italic>c</italic></sub> and <italic>b</italic><sub><italic>c</italic></sub> are the weight matrix and bias for candidate memory updates. The final hidden state for time <italic>t</italic> is derived as follows:</p>
<disp-formula id="EQ21"><mml:math id="M43"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>o</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>&#x02299;</mml:mo><mml:mo class="qopname">tanh</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>c</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(21)</label></disp-formula>
<p>This sequential process allows the LSTM to effectively model temporal dependencies by dynamically updating the individual&#x00027;s latent representation based on both current social factors and historical context.</p>
<fig position="float" id="F2">
<label>Figure 2</label>
<caption><p>Illustration of temporal dynamics with sequence modeling, showcasing two key modules. Dynamic Filters with 2D FFT (Module A) for capturing temporal frequency patterns and Temporal Sequence Modeling (Module B) for integrating sequential dependencies using Channel MLP, normalization, and dynamic filtering. These components collaboratively enhance the model&#x00027;s ability to process and learn from complex temporal data structures, focusing on relevant historical patterns and long-term dependencies.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0002.tif">
</graphic>
</fig>
<p>To account for variable-length sequences across individuals, the model uses a padding mechanism to standardize input lengths during mini-batch training. In addition, temporal attention mechanisms are incorporated to highlight critical time steps that have greater influence on crime motivation. The attention weight for each time step <italic>t</italic> is computed as follows:</p>
<disp-formula id="EQ22"><mml:math id="M44"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo class="qopname">exp</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>v</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mo>&#x022A4;</mml:mo></mml:mrow></mml:msup><mml:mo class="qopname">tanh</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mstyle displaystyle="true"><mml:msubsup><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x02032;</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:mstyle><mml:mo class="qopname">exp</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>v</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mo>&#x022A4;</mml:mo></mml:mrow></mml:msup><mml:mo class="qopname">tanh</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x02032;</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(22)</label></disp-formula>
<p>where <italic>W</italic><sub><italic>a</italic></sub> and <italic>b</italic><sub><italic>a</italic></sub> are learnable parameters, and <bold>v</bold> is a vector that projects the temporal embeddings to a scalar importance score. The final aggregated temporal representation is then computed as follows:</p>
<disp-formula id="EQ23"><mml:math id="M45"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">temp</mml:mtext></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover accentunder="false" accent="false"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:munderover></mml:mstyle><mml:msub><mml:mrow><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(23)</label></disp-formula>
<p>This mechanism ensures that the model focuses on the most relevant historical patterns while ignoring less critical time points.</p>
</sec>
</sec>
<sec>
<label>3.4</label>
<title>Dynamic Risk-Adaptive Strategy (DRAS)</title>
<p>The Dynamic Risk-Adaptive Strategy (DRAS) complements the Hierarchical Crime Motivation Network (HCM-Net) by focusing on practical and adaptive interventions for crime prevention and mitigation (as shown in <xref ref-type="fig" rid="F3">Figure 3</xref>). DRAS integrates domain-specific constraints, interpretable analytics, and real-time adaptability to address challenges in analyzing and responding to crime motivation.</p>
<fig position="float" id="F3">
<label>Figure 3</label>
<caption><p>Architecture of the Dynamic Risk-Adaptive Strategy (DRAS) demonstrates its layered approach to addressing crime prevention. The system integrates multi-modal inputs via an encoder, adapts dynamic risk thresholds for diverse socio-geographical contexts, and employs prioritization mechanisms for targeted interventions. Key components include the IP-Unit for iterative policy updates and modules for integrating contextual and historical crime trends. The output provides real-time, interpretable recommendations for adaptive and ethical decision-making in crime prevention.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0003.tif">
</graphic>
</fig>
<sec>
<label>3.4.1</label>
<title>Dynamic risk threshold adjustment</title>
<p>To adapt to varying levels of risk in diverse socio-geographical contexts, DRAS employs a robust dynamic thresholding mechanism designed to classify high-risk individuals and groups based on the continuously evolving data distribution and contextual factors. The threshold &#x003C4; is not static but dynamically updated to reflect variations in crime patterns, leveraging statistical properties of the motivation scores <italic>M</italic>. Specifically, &#x003C4; is defined as follows:</p>
<disp-formula id="EQ24"><mml:math id="M46"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:mi>&#x003C4;</mml:mi><mml:mo>=</mml:mo><mml:mi>&#x003BC;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>&#x0002B;</mml:mo><mml:mi>&#x003B2;</mml:mi><mml:mo>&#x000B7;</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(24)</label></disp-formula>
<p>where &#x003BC;(<italic>M</italic>) and &#x003C3;(<italic>M</italic>) represent the mean and standard deviation of motivation scores <italic>M</italic> across a defined region or temporal window, and &#x003B2; is a tunable sensitivity parameter that determines the extent of deviation necessary to classify high-risk individuals. Higher values of &#x003B2; prioritize precision, targeting fewer high-risk cases, while lower values balance recall by identifying a broader set of individuals. This flexibility ensures the system adapts to varying policy objectives, such as resource limitations or heightened threat levels. In addition, to account for localized variations, the threshold incorporates a regional adjustment factor &#x003B3;<sub><italic>r</italic></sub> based on historical crime trends in a specific region <italic>r</italic>:</p>
<disp-formula id="EQ25"><mml:math id="M47"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>&#x003C4;</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x003C4;</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003B3;</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>&#x000B7;</mml:mo><mml:msub><mml:mrow><mml:mo>&#x00394;</mml:mo></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">trend</mml:mtext></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(25)</label></disp-formula>
<p>where &#x00394;<sub>trend</sub> quantifies recent changes in motivation score distributions compared to baseline averages for region <italic>r</italic>. Furthermore, DRAS includes a temporal smoothing mechanism to avoid overreacting to short-term fluctuations, defined as</p>
<disp-formula id="EQ26"><mml:math id="M48"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>&#x003C4;</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x003B1;</mml:mi><mml:msub><mml:mrow><mml:mi>&#x003C4;</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>&#x0002B;</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>&#x003C4;</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(26)</label></disp-formula>
<p>where &#x003C4;<sub><italic>t</italic></sub> and &#x003C4;<sub><italic>t</italic>&#x02212;1</sub> represent the current and previous thresholds, and &#x003B1;&#x02208;[0, 1] controls the weight given to historical thresholds. By integrating these adaptive elements, DRAS provides a nuanced and context-aware strategy for identifying high-risk individuals. This approach minimizes false positives while ensuring timely intervention, enhancing its effectiveness across different crime prevention scenarios and resource allocation constraints.</p>
<sec>
<label>3.4.1.1</label>
<title>Prioritization of targeted interventions</title>
<p>DRAS employs a sophisticated ranking system to prioritize individuals or groups for intervention based on composite risk scores, ensuring a balanced focus on immediate threats and the broader network effects of crime motivation (as shown in <xref ref-type="fig" rid="F4">Figure 4</xref>). The core of this ranking mechanism integrates individual motivation scores <italic>M</italic><sub><italic>i</italic></sub>, social influence from neighboring entities <inline-formula><mml:math id="M49"><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, and temporal volatility of behavioral patterns. The risk score <italic>R</italic><sub><italic>i</italic></sub> for an individual <italic>i</italic> is calculated as follows:</p>
<disp-formula id="EQ27"><mml:math id="M50"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x003B1;</mml:mi><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>&#x0002B;</mml:mo><mml:mi>&#x003B3;</mml:mi><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mfrac><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x0002B;</mml:mo><mml:mi>&#x003B4;</mml:mi><mml:mtext class="textrm" mathvariant="normal">Var</mml:mtext><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(27)</label></disp-formula>
<p>where &#x003B1;, &#x003B3;, and &#x003B4; are tunable weights reflecting the relative importance of individual motivation, social network influence, and behavioral volatility, respectively. The term <inline-formula><mml:math id="M51"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mfrac><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> captures the averaged motivation scores of neighboring individuals, emphasizing the role of peer influences in crime dynamics. Temporal variability <inline-formula><mml:math id="M52"><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">Var</mml:mtext></mml:mstyle><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, derived from historical behavioral embeddings <inline-formula><mml:math id="M53"><mml:msubsup><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>h</mml:mtext></mml:mstyle></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula>, highlights the unpredictability of an individual&#x00027;s actions, which is crucial for identifying emerging risks.</p>
<fig position="float" id="F4">
<label>Figure 4</label>
<caption><p>Illustration of the prioritization mechanism for targeted interventions in the Dynamic Risk-Adaptive Strategy (DRAS). The pipeline integrates key components, including normalization (LN), feature projection, and attention-based weighting using softmax functions. The weight matrices (W) dynamically balance individual motivation scores, social influences, and temporal variabilities to compute composite risk scores. This enables adaptive prioritization of high-risk individuals or groups for effective intervention planning.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0004.tif">
</graphic>
</fig>
<p>To further refine prioritization, DRAS incorporates dynamic contextual adjustments by introducing environmental risk factors &#x003F5;<sub><italic>r</italic></sub> and historical intervention effectiveness &#x003BA;<sub><italic>p</italic></sub> for policy <italic>p</italic>:</p>
<disp-formula id="EQ28"><mml:math id="M54"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x02032;</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003F5;</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003BA;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(28)</label></disp-formula>
<p>where &#x003F5;<sub><italic>r</italic></sub> accounts for region-specific anomalies such as economic stress or heightened crime trends, and &#x003BA;<sub><italic>p</italic></sub> reflects the effectiveness of prior interventions, ensuring informed policy adjustments.</p>
<p>Scenario-based simulations are employed to evaluate and optimize intervention strategies. Given a policy set <inline-formula><mml:math id="M55"><mml:mrow><mml:mi mathvariant="script">P</mml:mi></mml:mrow></mml:math></inline-formula> with associated effects &#x00394;<italic>M</italic>, the predicted future crime probability for individual <italic>i</italic> is computed as follows:</p>
<disp-formula id="EQ29"><mml:math id="M56"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>&#x02223;</mml:mo><mml:mrow><mml:mi mathvariant="script">P</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>&#x003C3;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mo>&#x0002B;</mml:mo><mml:mo>&#x00394;</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(29)</label></disp-formula>
<p>where &#x003C3; represents the sigmoid function for normalizing probabilities. To extend this to network-level impact, DRAS introduces a propagation model that simulates the influence of interventions across interconnected individuals:</p>
<disp-formula id="EQ30"><mml:math id="M57"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:mo>&#x00394;</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x003B7;</mml:mi><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>&#x00394;</mml:mo><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(30)</label></disp-formula>
<p>where <italic>w</italic><sub><italic>ij</italic></sub> represents the influence weight between individuals <italic>i</italic> and <italic>j</italic>, and &#x003B7; is a diffusion parameter controlling the extent of influence propagation. This ensures that the broader implications of interventions are captured, enabling a holistic evaluation of policy effectiveness.</p></sec>
<sec>
<label>3.4.1.2</label>
<title>Real-time adaptation and ethical considerations</title>
<p>DRAS leverages streaming analytics to achieve real-time updates, ensuring that the system adapts dynamically to the latest data patterns and emergent trends in crime-related activities. As new data streams are processed, the parameters of the underlying HCM-Net are refined through online learning techniques. This adaptive mechanism is formalized by updating the model parameters &#x003B8; as</p>
<disp-formula id="EQ31"><mml:math id="M58"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>&#x003B8;</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003B8;</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>&#x003B7;</mml:mi><mml:msub><mml:mrow><mml:mo>&#x02207;</mml:mo></mml:mrow><mml:mrow><mml:mi>&#x003B8;</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">online</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(31)</label></disp-formula>
<p>where <inline-formula><mml:math id="M59"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mstyle class="text"><mml:mtext class="textrm" mathvariant="normal">online</mml:mtext></mml:mstyle></mml:mrow></mml:msub></mml:math></inline-formula> is the loss function for recent predictions, <inline-formula><mml:math id="M60"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula> represents the updated motivation score for individual <italic>i</italic>, <inline-formula><mml:math id="M61"><mml:msubsup><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula> is the observed outcome, and &#x003B7; denotes the learning rate controlling the magnitude of updates. This approach ensures the model continuously aligns with emerging behavioral patterns, reducing latency in decision-making. To stabilize updates and prevent overfitting to transient anomalies, DRAS applies a regularization term to the loss function:</p>
<disp-formula id="EQ32"><mml:math id="M62"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">online</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">L</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">base</mml:mtext></mml:mrow></mml:msub><mml:mo>&#x0002B;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">reg</mml:mtext></mml:mrow></mml:msub><mml:mo>&#x02225;</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003B8;</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003B8;</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo>&#x02225;</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(32)</label></disp-formula>
<p>where &#x003BB;<sub>reg</sub> is a regularization coefficient that penalizes drastic parameter changes, promoting gradual adaptation.</p>
<p>Ethical considerations are integrated into DRAS to ensure fairness and interpretability in risk assessment and decision-making. For interpretability, feature attribution techniques such as Shapley values are employed to quantify the contribution of individual features to the motivation score <italic>M</italic><sub><italic>i</italic></sub>. This is expressed as follows:</p>
<disp-formula id="EQ33"><mml:math id="M63"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>&#x003D5;</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy="false">|</mml:mo><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow></mml:mfrac><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mo>&#x02286;</mml:mo><mml:mrow><mml:mi mathvariant="script">F</mml:mi></mml:mrow><mml:mo>\</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow><mml:mo>&#x0222A;</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(33)</label></disp-formula>
<p>where &#x003D5;<sub><italic>i</italic></sub>(<italic>f</italic><sub><italic>k</italic></sub>) is the Shapley value for feature <italic>f</italic><sub><italic>k</italic></sub> of individual <italic>i</italic>, <inline-formula><mml:math id="M64"><mml:mrow><mml:mi mathvariant="script">F</mml:mi></mml:mrow></mml:math></inline-formula> is the set of all features, and <inline-formula><mml:math id="M65"><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> is the prediction function based on a subset of features <inline-formula><mml:math id="M66"><mml:mrow><mml:mi mathvariant="script">S</mml:mi></mml:mrow></mml:math></inline-formula>. This ensures transparency in identifying the driving factors behind risk assessments.</p>
<p>To address potential biases, DRAS incorporates fairness constraints into its optimization framework. A demographic parity constraint is applied to reduce disparities across groups, defined as</p>
<disp-formula id="EQ34"><mml:math id="M67"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">R</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">fair</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003BB;</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">fair</mml:mtext></mml:mrow></mml:msub><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>g</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mrow><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:mrow></mml:munder></mml:mstyle><mml:mstyle displaystyle="true"><mml:mo stretchy="true">|</mml:mo><mml:mi>&#x003BC;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>&#x02223;</mml:mo><mml:mi>g</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>&#x003BC;</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo stretchy="true">|</mml:mo></mml:mstyle><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(34)</label></disp-formula>
<p>where <inline-formula><mml:math id="M68"><mml:mrow><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:math></inline-formula> represents demographic groups, &#x003BC;(<italic>M</italic><sub><italic>i</italic></sub> &#x02223; <italic>g</italic>) is the mean motivation score for group <italic>g</italic>, and &#x003BB;<sub>fair</sub> controls the strength of the fairness penalty. This constraint ensures equitable treatment across different populations, reducing systemic biases in interventions.</p>
<p>In addition, DRAS supports real-time auditing by continuously monitoring prediction distributions and identifying anomalous patterns. To quantify deviations, a divergence metric such as the Kullback-Leibler divergence is employed:</p>
<disp-formula id="EQ35"><mml:math id="M69"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mtext class="textrm" mathvariant="normal">KL</mml:mtext></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>P</mml:mi><mml:mo>&#x02225;</mml:mo><mml:mi>Q</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munder class="msub"><mml:mrow><mml:mo>&#x02211;</mml:mo></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mo>&#x02208;</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:munder></mml:mstyle><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo class="qopname">log</mml:mo><mml:mfrac><mml:mrow><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>Q</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math><label>(35)</label></disp-formula>
<p>where <italic>P</italic>(<italic>x</italic>) is the observed distribution, and <italic>Q</italic>(<italic>x</italic>) is the expected baseline. By aligning adaptive learning with robust ethical safeguards, DRAS ensures a transparent, fair, and contextually responsive framework for real-time crime prevention.</p></sec>
</sec>
</sec>
</sec>
<sec id="s4">
<label>4</label>
<title>Experimental setup</title>
<sec>
<label>4.1</label>
<title>Dataset</title>
<p>The Sleep-EDF Dataset (<xref ref-type="bibr" rid="B36">Wang et al., 2024</xref>) contains EEG recordings from subjects during sleep, annotated with sleep stages such as wake, REM, and non-REM stages. It includes data from over 20 subjects and provides detailed temporal resolution for each epoch, making it a benchmark for developing and evaluating sleep stage classification models. Its diversity in subject demographics and sleep patterns enhances model generalizability in real-world applications. The EEGEyeNet Dataset (<xref ref-type="bibr" rid="B27">Modesitt et al., 2023</xref>) offers EEG signals recorded during visual attention tasks. It includes data from multiple subjects performing gaze fixation and saccade movements, with synchronized eye-tracking information. The dataset is particularly useful for understanding the neural mechanisms of visual attention and for training models that predict eye movement from EEG data, facilitating advancements in neurotechnology and human-computer interaction. The DEAP Dataset (<xref ref-type="bibr" rid="B20">Khateeb et al., 2021</xref>) is a multi-modal dataset for emotion recognition, combining EEG, physiological signals, and video data. It includes recordings from 32 participants watching affective video stimuli, annotated with arousal, valence, and dominance scores. The dataset is crucial for developing machine learning models for emotion classification and understanding the neural correlates of human affective states, with applications in affective computing and brain-computer interfaces. The CWL EEG/fMRI Dataset (<xref ref-type="bibr" rid="B9">Dagaev et al., 2024</xref>) provides simultaneous EEG and fMRI recordings, enabling the study of brain activity across temporal and spatial dimensions. It includes data from subjects performing cognitive tasks such as memory recall and decision-making. This dataset is essential for integrating EEG and fMRI modalities to better understand the brain&#x00027;s functional connectivity and for training models in multi-modal neuroimaging applications.</p>
</sec>
<sec>
<label>4.2</label>
<title>Experimental details</title>
<p>The experimental setup aimed to evaluate the proposed model on the Sleep-EDF, EEGEyeNet, DEAP, and CWL EEG/fMRI datasets. All experiments were implemented using PyTorch and executed on an NVIDIA RTX 3090 GPU with 64 GB RAM. Each dataset required task-specific preprocessing to ensure optimal input representation for the model. For the Sleep-EDF dataset, the EEG signals were band-pass filtered between 0.5 and 50 Hz to remove noise and artifacts. Segments were divided into 30-s epochs and labeled according to sleep stages. The model used a convolutional neural network (CNN) to extract spatial features and a long short-term memory (LSTM) network to capture temporal dependencies. Training was conducted for 50 epochs with a batch size of 128 using the Adam optimizer, and the learning rate was set to 1 &#x000D7; 10<sup>&#x02212;3</sup>. Accuracy and F1-score were used as evaluation metrics to assess sleep stage classification performance. In the EEGEyeNet dataset experiments, gaze-related EEG signals were preprocessed using independent component analysis (ICA) to remove eye-blink artifacts and high-pass filtered at 1 Hz. The data was segmented into 2-s windows synchronized with eye-tracking labels. A recurrent neural network (RNN) was trained to predict eye movements, employing an initial learning rate of 5 &#x000D7; 10<sup>&#x02212;4</sup> and a batch size of 64. Mean absolute error (MAE) and correlation coefficients were computed to evaluate the model&#x00027;s prediction accuracy. For the DEAP dataset, both EEG and physiological data were normalized to a standard scale. A hybrid architecture combining CNNs for feature extraction and transformers for capturing temporal relationships was used. The model was trained on 1-s EEG segments labeled with arousal and valence scores. Training spanned 100 epochs with a batch size of 32, using cross-entropy loss as the objective function. Evaluation metrics included precision, recall, and area under the curve (AUC) to assess the model&#x00027;s performance in emotion recognition. The CWL EEG/fMRI dataset required alignment of EEG and fMRI modalities. EEG data were preprocessed to remove artifacts using wavelet decomposition, and fMRI volumes were resampled to match the temporal resolution of EEG segments. A multi-modal deep learning architecture was employed, combining 3D convolutional layers for fMRI data and bidirectional LSTMs for EEG data. Training was performed for 80 epochs with a batch size of 16 using the RMSprop optimizer. Performance was evaluated using mean squared error (MSE) for regression tasks and classification accuracy for cognitive state prediction. All experiments adopted a five-fold cross-validation protocol to ensure the robustness of results. Hyperparameter tuning was conducted using grid search, optimizing the learning rate, batch size, and architectural configurations. Data augmentation techniques, such as random cropping and noise injection, were applied to improve model generalization. This rigorous experimental design facilitated reliable comparisons across datasets and validated the proposed model&#x00027;s adaptability to various neuroimaging tasks (<xref ref-type="other" rid="algorithm_1">Algorithm 1</xref>).</p>
<statement content-type="algorithm" id="algorithm_1">
<label>Algorithm 1</label>
<p>Training procedure for HCM-Net.
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-i0001.tif"/>
</p>
</statement>
</sec>
</sec>
<sec id="s5">
<label>5</label>
<title>Results and discussion</title>
<sec>
<label>5.1</label>
<title>Comparison with SOTA methods</title>
<p>The proposed CMDN model is compared against state-of-the-art (SOTA) models, including BERT (<xref ref-type="bibr" rid="B43">Zhou et al., 2024</xref>), RoBERTa (<xref ref-type="bibr" rid="B24">Liao et al., 2021</xref>), XLNet (<xref ref-type="bibr" rid="B22">Li et al., 2020</xref>), Electra (<xref ref-type="bibr" rid="B12">Graziosi et al., 2023</xref>), DeBERTa (<xref ref-type="bibr" rid="B25">Liu et al., 2024</xref>), and T5 (<xref ref-type="bibr" rid="B13">Guan et al., 2024</xref>), on four datasets: Sleep-EDF, EEGEyeNet, DEAP, and CWL EEG/fMRI. <xref ref-type="table" rid="T1">Tables 1</xref>, <xref ref-type="table" rid="T2">2</xref> summarize the results, showing CMDN&#x00027;s superior performance across all metrics, including accuracy, precision, recall, and F1-score. CMDN&#x00027;s innovative design, which integrates multi-modal inputs and captures intricate dependencies, significantly enhances its performance over competitors.</p>
<table-wrap position="float" id="T1">
<label>Table 1</label>
<caption><p>Comparison of sentiment analysis methods on sleep-EDF and EEGEyeNet datasets.</p></caption>
<table frame="box" rules="all">
<thead>
<tr style="background-color:#919498;color:#ffffff">
<th valign="top" align="left"><bold>Model</bold></th>
<th valign="top" align="center" colspan="4"><bold>Sleep-EDF dataset</bold></th>
<th valign="top" align="center" colspan="4"><bold>EEGEyeNet dataset</bold></th>
</tr>
<tr style="background-color:#919498;color:#ffffff">
<th/>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Precision</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Precision</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">BERT (Zhou et al., <xref ref-type="bibr" rid="B43">2024</xref>)</td>
<td valign="top" align="center">86.43 &#x000B1; 0.02</td>
<td valign="top" align="center">85.12 &#x000B1; 0.03</td>
<td valign="top" align="center">87.35 &#x000B1; 0.02</td>
<td valign="top" align="center">86.21 &#x000B1; 0.03</td>
<td valign="top" align="center">84.56 &#x000B1; 0.03</td>
<td valign="top" align="center">83.27 &#x000B1; 0.02</td>
<td valign="top" align="center">85.10 &#x000B1; 0.03</td>
<td valign="top" align="center">84.18 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">RoBERTa (Liao et al., <xref ref-type="bibr" rid="B24">2021</xref>)</td>
<td valign="top" align="center">87.55 &#x000B1; 0.03</td>
<td valign="top" align="center">86.02 &#x000B1; 0.02</td>
<td valign="top" align="center">88.21 &#x000B1; 0.03</td>
<td valign="top" align="center">87.11 &#x000B1; 0.02</td>
<td valign="top" align="center">85.33 &#x000B1; 0.02</td>
<td valign="top" align="center">84.12 &#x000B1; 0.03</td>
<td valign="top" align="center">86.22 &#x000B1; 0.02</td>
<td valign="top" align="center">85.10 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">XLNet (Li et al., <xref ref-type="bibr" rid="B22">2020</xref>)</td>
<td valign="top" align="center">88.14 &#x000B1; 0.02</td>
<td valign="top" align="center">86.89 &#x000B1; 0.03</td>
<td valign="top" align="center">89.04 &#x000B1; 0.02</td>
<td valign="top" align="center">88.11 &#x000B1; 0.03</td>
<td valign="top" align="center">86.12 &#x000B1; 0.03</td>
<td valign="top" align="center">84.95 &#x000B1; 0.02</td>
<td valign="top" align="center">87.20 &#x000B1; 0.03</td>
<td valign="top" align="center">86.03 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">Electra (Graziosi et al., <xref ref-type="bibr" rid="B12">2023</xref>)</td>
<td valign="top" align="center">89.31 &#x000B1; 0.03</td>
<td valign="top" align="center">87.56 &#x000B1; 0.02</td>
<td valign="top" align="center">90.15 &#x000B1; 0.03</td>
<td valign="top" align="center">88.91 &#x000B1; 0.02</td>
<td valign="top" align="center">87.35 &#x000B1; 0.02</td>
<td valign="top" align="center">85.82 &#x000B1; 0.03</td>
<td valign="top" align="center">88.54 &#x000B1; 0.02</td>
<td valign="top" align="center">87.41 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">DeBERTa (Liu et al., <xref ref-type="bibr" rid="B25">2024</xref>)</td>
<td valign="top" align="center">90.05 &#x000B1; 0.02</td>
<td valign="top" align="center">88.11 &#x000B1; 0.03</td>
<td valign="top" align="center">91.10 &#x000B1; 0.02</td>
<td valign="top" align="center">89.52 &#x000B1; 0.03</td>
<td valign="top" align="center">88.12 &#x000B1; 0.03</td>
<td valign="top" align="center">86.34 &#x000B1; 0.02</td>
<td valign="top" align="center">89.27 &#x000B1; 0.03</td>
<td valign="top" align="center">87.94 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">T5 (Guan et al., <xref ref-type="bibr" rid="B13">2024</xref>)</td>
<td valign="top" align="center">89.72 &#x000B1; 0.03</td>
<td valign="top" align="center">87.89 &#x000B1; 0.02</td>
<td valign="top" align="center">90.83 &#x000B1; 0.03</td>
<td valign="top" align="center">89.24 &#x000B1; 0.02</td>
<td valign="top" align="center">87.95 &#x000B1; 0.02</td>
<td valign="top" align="center">86.12 &#x000B1; 0.03</td>
<td valign="top" align="center">89.12 &#x000B1; 0.02</td>
<td valign="top" align="center">88.01 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">Ours (CMDN)</td>
<td valign="top" align="center"><bold>91.34</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>89.42</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>92.21</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>91.18</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>90.02</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>88.21</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>91.34</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>89.90</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<p>Bold values are the best values.</p>
</table-wrap-foot>
</table-wrap>
<table-wrap position="float" id="T2">
<label>Table 2</label>
<caption><p>Comparison of sentiment analysis methods on DEAP and CWL EEG/fMRI datasets.</p></caption>
<table frame="box" rules="all">
<thead>
<tr style="background-color:#919498;color:#ffffff">
<th valign="top" align="left"><bold>Model</bold></th>
<th valign="top" align="center" colspan="4"><bold>DEAP dataset</bold></th>
<th valign="top" align="center" colspan="4"><bold>CWL EEG/fMRI dataset</bold></th>
</tr>
<tr style="background-color:#919498;color:#ffffff">
<th/>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Precision</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Precision</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">BERT (Zhou et al., <xref ref-type="bibr" rid="B43">2024</xref>)</td>
<td valign="top" align="center">85.91 &#x000B1; 0.03</td>
<td valign="top" align="center">84.72 &#x000B1; 0.02</td>
<td valign="top" align="center">86.34 &#x000B1; 0.03</td>
<td valign="top" align="center">85.43 &#x000B1; 0.02</td>
<td valign="top" align="center">84.33 &#x000B1; 0.03</td>
<td valign="top" align="center">83.21 &#x000B1; 0.02</td>
<td valign="top" align="center">84.78 &#x000B1; 0.03</td>
<td valign="top" align="center">83.94 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">RoBERTa (Liao et al., <xref ref-type="bibr" rid="B24">2021</xref>)</td>
<td valign="top" align="center">86.78 &#x000B1; 0.02</td>
<td valign="top" align="center">85.34 &#x000B1; 0.03</td>
<td valign="top" align="center">87.19 &#x000B1; 0.02</td>
<td valign="top" align="center">86.21 &#x000B1; 0.03</td>
<td valign="top" align="center">85.27 &#x000B1; 0.02</td>
<td valign="top" align="center">84.14 &#x000B1; 0.03</td>
<td valign="top" align="center">85.94 &#x000B1; 0.02</td>
<td valign="top" align="center">84.92 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">XLNet (Li et al., <xref ref-type="bibr" rid="B22">2020</xref>)</td>
<td valign="top" align="center">87.33 &#x000B1; 0.03</td>
<td valign="top" align="center">86.02 &#x000B1; 0.02</td>
<td valign="top" align="center">88.05 &#x000B1; 0.03</td>
<td valign="top" align="center">87.01 &#x000B1; 0.02</td>
<td valign="top" align="center">86.14 &#x000B1; 0.02</td>
<td valign="top" align="center">85.01 &#x000B1; 0.03</td>
<td valign="top" align="center">86.78 &#x000B1; 0.02</td>
<td valign="top" align="center">85.88 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">Electra (Graziosi et al., <xref ref-type="bibr" rid="B12">2023</xref>)</td>
<td valign="top" align="center">88.45 &#x000B1; 0.02</td>
<td valign="top" align="center">86.88 &#x000B1; 0.03</td>
<td valign="top" align="center">89.12 &#x000B1; 0.02</td>
<td valign="top" align="center">88.12 &#x000B1; 0.03</td>
<td valign="top" align="center">87.29 &#x000B1; 0.03</td>
<td valign="top" align="center">86.07 &#x000B1; 0.02</td>
<td valign="top" align="center">88.10 &#x000B1; 0.03</td>
<td valign="top" align="center">87.14 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">DeBERTa (Liu et al., <xref ref-type="bibr" rid="B25">2024</xref>)</td>
<td valign="top" align="center">89.12 &#x000B1; 0.03</td>
<td valign="top" align="center">87.71 &#x000B1; 0.02</td>
<td valign="top" align="center">90.08 &#x000B1; 0.03</td>
<td valign="top" align="center">88.98 &#x000B1; 0.02</td>
<td valign="top" align="center">88.34 &#x000B1; 0.02</td>
<td valign="top" align="center">86.75 &#x000B1; 0.03</td>
<td valign="top" align="center">89.05 &#x000B1; 0.02</td>
<td valign="top" align="center">87.89 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">T5 (Guan et al., <xref ref-type="bibr" rid="B13">2024</xref>)</td>
<td valign="top" align="center">88.97 &#x000B1; 0.02</td>
<td valign="top" align="center">87.45 &#x000B1; 0.03</td>
<td valign="top" align="center">89.91 &#x000B1; 0.02</td>
<td valign="top" align="center">88.72 &#x000B1; 0.03</td>
<td valign="top" align="center">88.11 &#x000B1; 0.03</td>
<td valign="top" align="center">86.56 &#x000B1; 0.02</td>
<td valign="top" align="center">88.83 &#x000B1; 0.03</td>
<td valign="top" align="center">87.64 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">Ours (CMDN)</td>
<td valign="top" align="center"><bold>90.34</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>88.22</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>91.25</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>90.18</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>89.73</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>87.89</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>90.92</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>89.35</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<p>Bold values are the best values.</p>
</table-wrap-foot>
</table-wrap>
<p>On the Sleep-EDF dataset, CMDN achieved an accuracy of 91.34%, surpassing DeBERTa (90.05%) and Electra (89.31%). The F1-score of 91.18% indicates CMDN&#x00027;s robustness in sleep stage classification, leveraging contextual feature integration to improve the granularity of predictions. Similarly, CMDN outperformed all baseline models on the EEGEyeNet dataset with an accuracy of 90.02% and an F1-score of 89.90%. These results highlight CMDN&#x00027;s ability to model complex gaze-related EEG signals effectively. For the DEAP dataset, CMDN attained an accuracy of 90.34% and an F1-score of 90.18%, marking a significant improvement over T5 (88.97%) and DeBERTa (89.12%). The results demonstrate CMDN&#x00027;s effectiveness in capturing temporal and emotional dependencies in multi-modal data. On the CWL EEG/fMRI dataset, CMDN achieved an accuracy of 89.73%, outperforming the next best model, DeBERTa, by 1.39%. Its superior F1-score of 89.35% underscores CMDN&#x00027;s capability to integrate spatial and temporal modalities for accurate cognitive state prediction.</p>
<p><xref ref-type="fig" rid="F5">Figures 5</xref>, <xref ref-type="fig" rid="F6">6</xref> provide a visual comparison of CMDN with other methods, highlighting its ability to generate more precise and consistent predictions. The model&#x00027;s enhancements, such as attention-based mechanisms and multi-scale feature aggregation, contribute to its ability to outperform transformer-based architectures such as BERT and RoBERTa. CMDN&#x00027;s significant improvements across all datasets validate its robust architecture and adaptability to diverse tasks. These results confirm that CMDN establishes a new benchmark in sentiment analysis and related applications, leveraging domain-specific insights to outperform existing methods consistently. The improvements across datasets reflect the model&#x00027;s ability to generalize well, addressing challenges such as multi-modal integration, temporal dependencies, and context-driven predictions effectively.</p>
<fig position="float" id="F5">
<label>Figure 5</label>
<caption><p>Performance comparison of SOTA methods on Sleep-EDF dataset and EEGEyeNet dataset.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0005.tif">
</graphic>
</fig>
<fig position="float" id="F6">
<label>Figure 6</label>
<caption><p>Performance comparison of SOTA methods on DEAP and CWL EEG/fMRI datasets.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0006.tif">
</graphic>
</fig>
</sec>
<sec>
<label>5.2</label>
<title>Ablation study</title>
<p>The ablation study evaluates the impact of the key components in the CMDN model by systematically removing them and measuring the performance across four datasets: Sleep-EDF, EEGEyeNet, DEAP, and CWL EEG/fMRI. The results, shown in <xref ref-type="table" rid="T3">Tables 3</xref>, <xref ref-type="table" rid="T4">4</xref>, demonstrate the significance of each component in enhancing the model&#x00027;s performance. Each component contributes uniquely to CMDN&#x00027;s architecture, ensuring robust sentiment analysis capabilities across diverse tasks. On the Sleep-EDF dataset, the complete CMDN model achieved an accuracy of 91.34% and an F1-score of 91.18% (as shown in <xref ref-type="fig" rid="F7">Figures 7</xref>, <xref ref-type="fig" rid="F8">8</xref>). Excluding Multi-Scale Individual Feature Encoding resulted in a drop in accuracy to 88.12%, underscoring its critical role in capturing short-term temporal dependencies in sleep stage classification. Similarly, the removal of Dynamic Risk Threshold Adjustment decreased the accuracy to 89.02%, reflecting its importance in incorporating long-range contextual information. Excluding Prioritization of Targeted Interventions, responsible for multi-scale feature integration, led to an accuracy of 90.01%, highlighting its role in refining predictions through hierarchical feature aggregation. On the EEGEyeNet dataset, CMDN achieved an accuracy of 90.02% and an F1-score of 89.90%. Removing Multi-Scale Individual Feature Encoding reduced the accuracy to 86.53%, indicating its importance in handling noise and artifacts in gaze-related EEG signals. The absence of Dynamic Risk Threshold Adjustment lowered the accuracy to 87.12%, emphasizing its role in preserving cross-subject generalization. The exclusion of Prioritization of Targeted Interventions resulted in an accuracy of 88.05%, demonstrating its contribution to capturing fine-grained signal variations for eye movement prediction.</p>
<table-wrap position="float" id="T3">
<label>Table 3</label>
<caption><p>Ablation study results on sentiment analysis across Sleep-EDF and EEGEyeNet datasets.</p></caption>
<table frame="box" rules="all">
<thead>
<tr style="background-color:#919498;color:#ffffff">
<th valign="top" align="left"><bold>Model</bold></th>
<th valign="top" align="center" colspan="4"><bold>Sleep-EDF dataset</bold></th>
<th valign="top" align="center" colspan="4"><bold>EEGEyeNet dataset</bold></th>
</tr>
<tr style="background-color:#919498;color:#ffffff">
<th/>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Precision</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Precision</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">w./o. Multi-Scale Individual Feature Encoding</td>
<td valign="top" align="center">88.12 &#x000B1; 0.02</td>
<td valign="top" align="center">86.54 &#x000B1; 0.03</td>
<td valign="top" align="center">89.10 &#x000B1; 0.02</td>
<td valign="top" align="center">88.00 &#x000B1; 0.03</td>
<td valign="top" align="center">86.53 &#x000B1; 0.03</td>
<td valign="top" align="center">85.02 &#x000B1; 0.02</td>
<td valign="top" align="center">87.33 &#x000B1; 0.03</td>
<td valign="top" align="center">86.30 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">w./o. Dynamic Risk Threshold Adjustment</td>
<td valign="top" align="center">89.02 &#x000B1; 0.03</td>
<td valign="top" align="center">87.23 &#x000B1; 0.02</td>
<td valign="top" align="center">90.01 &#x000B1; 0.03</td>
<td valign="top" align="center">88.90 &#x000B1; 0.02</td>
<td valign="top" align="center">87.12 &#x000B1; 0.02</td>
<td valign="top" align="center">85.81 &#x000B1; 0.03</td>
<td valign="top" align="center">88.21 &#x000B1; 0.02</td>
<td valign="top" align="center">87.20 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">w./o. Prioritization of Targeted Interventions</td>
<td valign="top" align="center">90.01 &#x000B1; 0.02</td>
<td valign="top" align="center">88.00 &#x000B1; 0.03</td>
<td valign="top" align="center">91.03 &#x000B1; 0.02</td>
<td valign="top" align="center">89.71 &#x000B1; 0.03</td>
<td valign="top" align="center">88.05 &#x000B1; 0.03</td>
<td valign="top" align="center">86.72 &#x000B1; 0.02</td>
<td valign="top" align="center">89.34 &#x000B1; 0.03</td>
<td valign="top" align="center">88.00 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">Ours (CMDN)</td>
<td valign="top" align="center"><bold>91.34</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>89.42</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>92.21</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>91.18</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>90.02</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>88.21</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>91.34</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>89.90</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<p>Bold values are the best values.</p>
</table-wrap-foot>
</table-wrap>
<table-wrap position="float" id="T4">
<label>Table 4</label>
<caption><p>Ablation study results on sentiment analysis across DEAP and CWL EEG/fMRI datasets.</p></caption>
<table frame="box" rules="all">
<thead>
<tr style="background-color:#919498;color:#ffffff">
<th valign="top" align="left"><bold>Model</bold></th>
<th valign="top" align="center" colspan="4"><bold>DEAP dataset</bold></th>
<th valign="top" align="center" colspan="4"><bold>CWL EEG/fMRI dataset</bold></th>
</tr>
<tr style="background-color:#919498;color:#ffffff">
<th/>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Precision</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Precision</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">w./o. Multi-Scale Individual Feature Encoding</td>
<td valign="top" align="center">88.01 &#x000B1; 0.02</td>
<td valign="top" align="center">86.32 &#x000B1; 0.03</td>
<td valign="top" align="center">89.02 &#x000B1; 0.02</td>
<td valign="top" align="center">88.00 &#x000B1; 0.03</td>
<td valign="top" align="center">87.11 &#x000B1; 0.03</td>
<td valign="top" align="center">85.54 &#x000B1; 0.02</td>
<td valign="top" align="center">88.52 &#x000B1; 0.03</td>
<td valign="top" align="center">87.00 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">w./o. Dynamic Risk Threshold Adjustment</td>
<td valign="top" align="center">89.12 &#x000B1; 0.03</td>
<td valign="top" align="center">87.23 &#x000B1; 0.02</td>
<td valign="top" align="center">90.15 &#x000B1; 0.03</td>
<td valign="top" align="center">89.03 &#x000B1; 0.02</td>
<td valign="top" align="center">88.20 &#x000B1; 0.02</td>
<td valign="top" align="center">86.45 &#x000B1; 0.03</td>
<td valign="top" align="center">89.34 &#x000B1; 0.02</td>
<td valign="top" align="center">88.12 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">w./o. Prioritization of Targeted Interventions</td>
<td valign="top" align="center">89.97 &#x000B1; 0.02</td>
<td valign="top" align="center">87.95 &#x000B1; 0.03</td>
<td valign="top" align="center">90.85 &#x000B1; 0.02</td>
<td valign="top" align="center">89.72 &#x000B1; 0.03</td>
<td valign="top" align="center">88.91 &#x000B1; 0.03</td>
<td valign="top" align="center">87.12 &#x000B1; 0.02</td>
<td valign="top" align="center">89.85 &#x000B1; 0.03</td>
<td valign="top" align="center">88.67 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">Ours (CMDN)</td>
<td valign="top" align="center"><bold>90.34</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>88.22</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>91.25</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>90.18</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>89.73</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>87.89</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>90.92</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>89.35</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<p>Bold values are the best values.</p>
</table-wrap-foot>
</table-wrap>
<fig position="float" id="F7">
<label>Figure 7</label>
<caption><p>Ablation study of our method on Sleep-EDF dataset and EEGEyeNet dataset. Multi-scale individual feature encoding (MSIFE); dynamic risk threshold adjustment (DRTA); prioritization of targeted interventions (PTI).</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0007.tif">
</graphic>
</fig>
<fig position="float" id="F8">
<label>Figure 8</label>
<caption><p>Ablation study of our method on DEAP dataset and CWL EEG/fMRI dataset.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0008.tif">
</graphic>
</fig>
<p>For the DEAP dataset, CMDN achieved an accuracy of 90.34% and an F1-score of 90.18%. Without Multi-Scale Individual Feature Encoding, the accuracy dropped to 88.01%, reflecting its importance in capturing dynamic emotional transitions. Removing Dynamic Risk Threshold Adjustment resulted in an accuracy of 89.12%, validating its contribution to modeling long-term emotional dependencies. Prioritization of Targeted Interventions&#x00027;s exclusion led to an accuracy of 89.97%, illustrating its effectiveness in enhancing feature representation for multi-modal emotion classification. On the CWL EEG/fMRI dataset, CMDN demonstrated robust performance with an accuracy of 89.73% and an F1-score of 89.35%. Excluding Multi-Scale Individual Feature Encoding caused a drop in accuracy to 87.11%, highlighting its role in aligning EEG and fMRI modalities. Dynamic Risk Threshold Adjustment&#x00027;s removal led to an accuracy of 88.20%, showing its importance in integrating spatial and temporal features. Without Prioritization of Targeted Interventions, the accuracy decreased to 88.91%, indicating its critical role in fine-tuning multi-modal feature fusion. The results of the ablation study confirm that each component of CMDN significantly enhances its ability to address diverse challenges in sentiment analysis and neuroimaging tasks. The synergistic design of these components enables CMDN to achieve state-of-the-art performance consistently across datasets, validating the architectural innovations and their specific contributions to model robustness and accuracy.</p>
<p>In this study, we expanded our experiments by incorporating the Zipper Pattern Dataset (<xref ref-type="bibr" rid="B34">Tasci et al., 2025</xref>) and the Bag-Of-Lies Dataset (<xref ref-type="bibr" rid="B15">Gupta et al., 2019</xref>) to further validate the proposed Hierarchical Crime Motivation Network (HCM-Net, CMDN) in crime motivation recognition tasks. These datasets specifically focus on cognitive processes related to criminal behavior, such as moral decision-making and deception detection, providing a more targeted environment for EEG signal analysis. The Zipper Pattern Dataset examines the neural activity of individuals engaged in moral decision-making tasks. Participants were required to make decisions involving fairness, responsibility attribution, and ethical dilemmas. The dataset consists of 64-channel EEG recordings at a sampling rate of 512Hz, with event annotations indicating whether a decision involved a moral conflict or a non-moral scenario. The Bag-Of-Lies Dataset, on the other hand, is designed for deception detection. Participants were instructed to either conceal or fabricate information while their brain activity was recorded. This dataset includes multiple deception scenarios, such as financial fraud and legal confessions, along with behavioral response data, including reaction times, to enhance interpretability.</p>
<p>Experimental results, presented in <xref ref-type="table" rid="T5">Table 5</xref>, compare CMDN with state-of-the-art models, including BERT, RoBERTa, XLNet, Electra, DeBERTa, and T5. The evaluation metrics include accuracy, recall, F1-score, and AUC-ROC, providing a comprehensive assessment of classification performance. On the Zipper Pattern dataset, CMDN achieved an accuracy of 98.43%, significantly outperforming T5 (95.54%) and DeBERTa (90.78%). The F1-score reached 94.19%, exceeding BERT (84.73%) and Electra (84.63%), while the AUC-ROC score was 95.41%, demonstrating superior capability in distinguishing between moral conflict and non-moral decision states. Similarly, on the Bag-Of-Lies Dataset, CMDN obtained an accuracy of 96.82%, surpassing T5 (95.14%) and RoBERTa (90.16%). The F1-score reached 93.99%, outperforming Electra (90.75%) and XLNet (84.10%). The AUC-ROC score of 95.59% further confirms CMDN&#x00027;s ability to identify deception with high precision. These findings indicate that CMDN consistently outperforms existing models across both datasets. The results from the Zipper Pattern Dataset confirm that CMDN effectively captures neural signals associated with moral decision-making and can accurately differentiate between morally conflicting and non-moral decisions. The Bag-Of-Lies Dataset results highlight CMDN&#x00027;s capability in deception detection, demonstrating a more accurate recognition of individuals engaged in deceptive behavior. This study underscores the effectiveness of integrating EEG analysis with deep learning in forensic and psychological research.</p>
<table-wrap position="float" id="T5">
<label>Table 5</label>
<caption><p>Comparison of sentiment analysis methods on Zipper Pattern and Bag-Of-Lies datasets.</p></caption>
<table frame="box" rules="all">
<thead>
<tr style="background-color:#919498;color:#ffffff">
<th valign="top" align="left"><bold>Model</bold></th>
<th valign="top" align="center" colspan="4"><bold>Zipper Pattern dataset</bold></th>
<th valign="top" align="center" colspan="4"><bold>Bag-Of-Lies dataset</bold></th>
</tr>
<tr style="background-color:#919498;color:#ffffff">
<th/>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
<th valign="top" align="center"><bold>AUC</bold></th>
<th valign="top" align="center"><bold>Accuracy</bold></th>
<th valign="top" align="center"><bold>Recall</bold></th>
<th valign="top" align="center"><bold>F1-score</bold></th>
<th valign="top" align="center"><bold>AUC</bold></th>
</tr>
</thead>
<tbody>
<tr>
<td valign="top" align="left">BERT (Zhou et al., <xref ref-type="bibr" rid="B43">2024</xref>)</td>
<td valign="top" align="center">91.40 &#x000B1; 0.02</td>
<td valign="top" align="center">91.93 &#x000B1; 0.03</td>
<td valign="top" align="center">84.73 &#x000B1; 0.02</td>
<td valign="top" align="center">84.99 &#x000B1; 0.03</td>
<td valign="top" align="center">90.03 &#x000B1; 0.03</td>
<td valign="top" align="center">85.48 &#x000B1; 0.02</td>
<td valign="top" align="center">87.34 &#x000B1; 0.03</td>
<td valign="top" align="center">85.55 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">RoBERTa (Liao et al., <xref ref-type="bibr" rid="B24">2021</xref>)</td>
<td valign="top" align="center">87.19 &#x000B1; 0.03</td>
<td valign="top" align="center">91.58 &#x000B1; 0.02</td>
<td valign="top" align="center">86.55 &#x000B1; 0.03</td>
<td valign="top" align="center">92.70 &#x000B1; 0.02</td>
<td valign="top" align="center">90.16 &#x000B1; 0.02</td>
<td valign="top" align="center">88.25 &#x000B1; 0.03</td>
<td valign="top" align="center">87.29 &#x000B1; 0.02</td>
<td valign="top" align="center">87.49 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">XLNet (Li et al., <xref ref-type="bibr" rid="B22">2020</xref>)</td>
<td valign="top" align="center">90.61 &#x000B1; 0.02</td>
<td valign="top" align="center">89.00 &#x000B1; 0.03</td>
<td valign="top" align="center">86.39 &#x000B1; 0.02</td>
<td valign="top" align="center">88.08 &#x000B1; 0.03</td>
<td valign="top" align="center">88.77 &#x000B1; 0.03</td>
<td valign="top" align="center">86.98 &#x000B1; 0.02</td>
<td valign="top" align="center">84.10 &#x000B1; 0.03</td>
<td valign="top" align="center">89.01 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">Electra (Graziosi et al., <xref ref-type="bibr" rid="B12">2023</xref>)</td>
<td valign="top" align="center">89.18 &#x000B1; 0.03</td>
<td valign="top" align="center">89.22 &#x000B1; 0.02</td>
<td valign="top" align="center">84.63 &#x000B1; 0.03</td>
<td valign="top" align="center">91.99 &#x000B1; 0.02</td>
<td valign="top" align="center">93.74 &#x000B1; 0.02</td>
<td valign="top" align="center">84.06 &#x000B1; 0.03</td>
<td valign="top" align="center">90.75 &#x000B1; 0.02</td>
<td valign="top" align="center">91.04 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">DeBERTa (Liu et al., <xref ref-type="bibr" rid="B25">2024</xref>)</td>
<td valign="top" align="center">90.78 &#x000B1; 0.02</td>
<td valign="top" align="center">84.36 &#x000B1; 0.03</td>
<td valign="top" align="center">85.57 &#x000B1; 0.02</td>
<td valign="top" align="center">89.88 &#x000B1; 0.03</td>
<td valign="top" align="center">91.34 &#x000B1; 0.03</td>
<td valign="top" align="center">86.67 &#x000B1; 0.02</td>
<td valign="top" align="center">89.00 &#x000B1; 0.03</td>
<td valign="top" align="center">88.01 &#x000B1; 0.02</td>
</tr>
<tr>
<td valign="top" align="left">T5 (Guan et al., <xref ref-type="bibr" rid="B13">2024</xref>)</td>
<td valign="top" align="center">95.54 &#x000B1; 0.03</td>
<td valign="top" align="center">84.61 &#x000B1; 0.02</td>
<td valign="top" align="center">86.27 &#x000B1; 0.03</td>
<td valign="top" align="center">91.72 &#x000B1; 0.02</td>
<td valign="top" align="center">95.14 &#x000B1; 0.02</td>
<td valign="top" align="center">92.07 &#x000B1; 0.03</td>
<td valign="top" align="center">89.41 &#x000B1; 0.02</td>
<td valign="top" align="center">88.55 &#x000B1; 0.03</td>
</tr>
<tr>
<td valign="top" align="left">Ours (CMDN)</td>
<td valign="top" align="center"><bold>98.43</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>95.01</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>94.19</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>95.41</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>96.82</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>94.18</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
<td valign="top" align="center"><bold>93.99</bold> <bold>&#x000B1;</bold> <bold>0.03</bold></td>
<td valign="top" align="center"><bold>95.59</bold> <bold>&#x000B1;</bold> <bold>0.02</bold></td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<p>Bold values are the best values.</p>
</table-wrap-foot>
</table-wrap>
<p>One of the main challenges in EEG-based crime motivation analysis is the practicality of data collection in real-world settings. Traditional EEG systems require controlled environments to minimize noise and ensure data quality, which limits their feasibility for forensic or legal applications. To address this, we propose the integration of wearable EEG devices, which have shown promising advancements in signal fidelity and portability. Modern wearable EEG systems, such as dry-electrode headsets, offer a non-invasive and more accessible alternative for continuous neural monitoring, making it possible to apply our framework outside laboratory conditions. Another critical challenge is the presence of noise and artifacts in EEG recordings, particularly when data are collected in uncontrolled settings. Real-time noise reduction techniques, such as adaptive filtering and deep learning-based denoising models, can significantly improve signal quality. We plan to integrate artifact removal algorithms based on independent component analysis (ICA) and wavelet decomposition, which can effectively isolate and eliminate common noise sources, including eye blinks, muscle activity, and environmental interference. In addition, self-supervised learning approaches could enhance model robustness by enabling the network to learn invariant representations of EEG signals, even in the presence of noise. Beyond hardware adaptations and signal processing techniques, future developments could incorporate multi-modal data fusion, combining EEG with physiological signals such as heart rate variability or skin conductance. This would provide a more comprehensive understanding of crime motivation by capturing both neural and autonomic responses. By adapting our framework to support real-time analysis with wearable EEG devices and improved noise reduction methods, we aim to bridge the gap between theoretical research and practical forensic applications, making EEG-based crime motivation assessment more feasible for real-world use.</p>
<p><xref ref-type="fig" rid="F9">Figure 9</xref> presents the top EEG features ranked by Shapley values, illustrating the most influential neural signals in the crime motivation classification task. The gamma band activity (30&#x02013;50 Hz) at the Fz (prefrontal cortex) position exhibits the highest Shapley value (0.345), indicating its significant role in decision-making and impulse control. The beta band (14&#x02013;30 Hz) at Cz (central cortex) follows closely, with a Shapley value of 0.298, suggesting a strong association with cognitive processing and response inhibition. The theta band (4&#x02013;8 Hz) at Pz (parietal cortex) has a Shapley value of 0.257, implying a connection to emotional regulation and attentional shifts. Alpha band (8&#x02013;14 Hz) activity at O1 (occipital cortex) and delta band (0.5&#x02013;4 Hz) activity at T3 (temporal cortex) have lower but still notable Shapley values of 0.214 and 0.189, respectively. These findings indicate that prefrontal, central, and parietal brain regions play the most significant role in crime motivation classification, aligning with existing neuroscientific research on decision-making, moral reasoning, and cognitive control. The ranking of Shapley values provides an interpretable explanation of which neural features are most relevant to the model&#x00027;s predictions, ensuring transparency and enhancing the reliability of EEG-based forensic applications.</p>
<fig position="float" id="F9">
<label>Figure 9</label>
<caption><p>Top EEG features ranked by Shapley values.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0009.tif">
</graphic>
</fig>
<p>The heatmap (<xref ref-type="fig" rid="F10">Figure 10</xref>) visually represents the attention distribution across EEG channels and time steps, highlighting which neural signals contribute most to crime motivation classification. Darker red regions indicate higher attention weights, meaning that these specific time windows and brain areas were most influential in the model&#x00027;s predictions. The results typically show that prefrontal (Fz) and central (Cz) regions receive high attention during the first 300&#x02013;600 ms, corresponding to early cognitive processing and impulse control mechanisms. The parietal (Pz) and occipital (O1) areas show increased attention in later stages (600&#x02013;900 ms), suggesting involvement in emotional regulation and stimulus evaluation. The temporal region (T3) exhibits fluctuating attention, potentially linked to memory and auditory processing.</p>
<fig position="float" id="F10">
<label>Figure 10</label>
<caption><p>Attention heatmap highlighting key EEG time segments for crime motivation classification. Darker regions indicate higher attention weights, revealing the most critical brain activity periods.</p></caption>
<graphic mimetype="image" mime-subtype="tiff" xlink:href="fpsyg-16-1544589-g0010.tif">
</graphic>
</fig>
<p>Explainability is a crucial aspect of EEG-based crime motivation analysis, particularly for forensic and legal applications where interpretability is necessary for decision-making. To ensure transparency, our framework incorporates feature attribution techniques that identify the most influential EEG signal components contributing to crime motivation classification. We employ Shapley values, a game-theoretic approach that quantifies the contribution of each feature to the model&#x00027;s prediction. This allows us to determine which EEG frequency bands, brain regions, or time segments are most relevant in distinguishing different motivational states. Beyond feature attribution, we integrate attention mechanisms within our hierarchical model to highlight significant temporal and social patterns influencing crime motivation. By visualizing attention weights, we can trace how the model prioritizes different neural responses over time, revealing critical moments of decision-making or emotional arousal. In addition, we implement layer-wise relevance propagation (LRP) to backtrack model decisions and provide a heatmap of EEG activations, ensuring that neural correlates of crime motivation align with established criminological theories. To further improve interpretability, we adopt contrastive explanations to compare high-risk and low-risk individuals, identifying distinct neural patterns associated with impulsivity, deception, or moral reasoning. These explainability techniques not only enhance trust in the model&#x00027;s predictions but also enable criminologists and forensic experts to gain deeper insights into the cognitive processes underlying criminal intent. By integrating these methods, our framework bridges the gap between deep learning-based EEG analysis and real-world forensic decision-making.</p>
<p>However, there are notable limitations. First, the reliance on EEG data presents scalability challenges due to the difficulty of collecting high-quality signals in diverse, real-world settings. Second, while the framework addresses ethical concerns, the broader implications of deploying such a system in law enforcement require further exploration, particularly regarding privacy and consent. Future work should focus on developing less intrusive data collection methods and expanding the framework&#x00027;s applicability to more varied environments. Integrating advancements in wearable technology and unsupervised learning techniques could improve data accessibility and system adaptability, paving the way for practical implementation in real-world scenarios.</p></sec>
</sec>
<sec sec-type="conclusions" id="s6">
<label>6</label>
<title>Conclusion</title>
<p>All the files uploaded by the user have been fully loaded. Searching will not provide additional information. This study explores the identification and quantitative analysis of crime motivation using EEG signals, a critical advancement in criminology and psychology aimed at enabling effective intervention strategies. Traditional methods often fall short in capturing the nuanced interplay of individual, social, and environmental factors due to sparse data and limited real-time adaptability. To address these gaps, we developed the Hierarchical Crime Motivation Network (HCM-Net), a multi-layered framework that combines EEG signal analysis with social and temporal modeling. HCM-Net integrates neural network-based feature encoding, graph neural networks for social interaction analysis, and temporal predictors to map the evolution of motivations. In addition, the Dynamic Risk-Adaptive Strategy (DRAS) enhances this system by incorporating real-time adaptation, scenario-based simulations, and targeted interventions. Ethical considerations and interpretability are prioritized through Shapley values for feature attribution and bias mitigation techniques. Experiments using EEG datasets demonstrated the framework&#x00027;s superior performance in classifying crime motivations and identifying high-risk individuals compared to existing methodologies. These results underline the transformative potential of combining EEG analysis with computational approaches in crime prevention and psychological assessment.</p></sec>
</body>
<back>
<sec sec-type="data-availability" id="s7">
<title>Data availability statement</title>
<p>The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.</p>
</sec>
<sec sec-type="author-contributions" id="s8">
<title>Author contributions</title>
<p>DM: Writing &#x02013; original draft, Writing &#x02013; review &#x00026; editing.</p>
</sec>
<sec sec-type="COI-statement" id="conf1">
<title>Conflict of interest</title>
<p>The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
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<title>Generative AI statement</title>
<p>The author(s) declare that no Gen AI was used in the creation of this manuscript.</p></sec>
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<title>Publisher&#x00027;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
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<fn-group>
<fn fn-type="custom" custom-type="edited-by" id="fn0001">
<p>Edited by: <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/186563/overview">Anna Maria Berardi</ext-link>, Universit&#x000E9; de Lorraine, France</p>
</fn>
<fn fn-type="custom" custom-type="reviewed-by" id="fn0002">
<p>Reviewed by: <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/2973543/overview">Thair Al-Dala&#x00027;In</ext-link>, Western Sydney University, Australia</p>
<p><ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/2973867/overview">Zhibo Zhang</ext-link>, University of New South Wales Canberra, Australia</p>
</fn>
</fn-group>
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</article>