
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39232022
70766
10.1038/s41598-024-70766-z
Article
Advancing sentiment classification through a population game model approach
http://orcid.org/0000-0001-8173-4003
Punetha Neha
http://orcid.org/0000-0001-8499-4433
Jain Goonjan goonjan_jain@hotmail.com

https://ror.org/01ztcvt22 grid.440678.9 0000 0001 0674 5044 Department of Applied Mathematics, Delhi Technological University, New Delhi, India
4 9 2024
4 9 2024
2024
14 205407 1 2024
21 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Computational Sentiment Analysis involves the automation of human emotion comprehension by categorizing sentiments as positive, negative, or neutral. In the contemporary digital environment, the extensive volume of social media content presents significant challenges for manual analysis, thereby necessitating the development and implementation of automated analytical tools. To address the limitations of existing techniques, which heavily rely on machine learning and extensive dataset pre-training, we propose an innovative unsupervised approach for sentiment classification. This novel methodology is grounded in game theory concepts, particularly the population game model, offering a promising solution by circumventing the need for extensive training procedures. We extract two textual features from review comments, namely context score and emotion score. Leveraging lexicon databases and numeric scores, this cognitive mathematical framework is language-independent. Competitive results are demonstrated across various domains (hotels, restaurants, electronic devices, etc.), and the efficacy of the proposed work is validated in two languages (English and Hindi). The highest accuracy recorded for the English domain dataset is 89%, while electronic Hindi reviews attain an 84% accuracy rate. The proposed model exhibits domain and language independence, validated through statistical analyses confirming the significance of the findings. The framework demonstrates noteworthy rationality and coherence in its outcomes.

Keywords

Sentiment classification
Context score
Emotion score
Population game model
Subject terms

Computer science
Information technology
Scientific data
Mathematics and computing
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

In the expanding digital ecosystem, where text is continuously generated across diverse platforms and devices, the task of accurately interpreting the sentiments embedded within the text has become a critical area of study. Computational Sentiment Analysis, an integral subfield of Natural Language Processing (NLP), addresses this challenge by automating the complex process of analyzing human sentiment conveyed through language. This task seeks to elucidate the intricate layers of emotions, opinions, and attitudes present in textual data, which has wide-ranging applications in numerous fields1. The vast and growing volume of online content—from consumer reviews to social media posts—renders manual analysis impractical and often infeasible. As a result, there is an increasing demand for automated sentiment analysis tools that can handle extensive datasets efficiently. Various sophisticated algorithms have been developed to fulfill this need, playing a crucial role in deriving actionable insights from textual data. Such advancements are essential for enabling organizations to make data-driven decisions and sustain a competitive edge in today’s digital landscape2.

For the extraction of sentiment from text, three types of approaches are most commonly used in the literature: lexicon-based approaches, machine-learning approaches, and hybrid approaches. Lexicons are predefined dictionaries of words or phrases with associated sentiment scores. However, these lexicons are not exhaustive and may not include all the words or expressions in a given domain or language3. This can lead to inaccurate sentiment analysis results, particularly for newer or domain-specific terms. In machine learning approaches, one major drawback is their dependence on high-quality, labeled training data. Acquiring and annotating large amounts of data can be time-consuming, expensive, and subjective2. Additionally, machine learning models can suffer from overfitting and fail to generalize well to new or unseen data. Furthermore, adapting models to new domains or handling noisy and unstructured data can pose significant challenges. Hybrid approaches in sentiment analysis, which combine multiple techniques such as lexicons, machine learning, and rule-based systems, also have certain drawbacks4. One drawback is increased complexity and computational overhead. Integrating different approaches and coordinating their interactions can be challenging, requiring additional effort in feature extraction, model design, and optimization5. Another drawback is the potential for conflicting results or errors when different techniques yield contrasting sentiment predictions. Combining disparate methods may introduce additional sources of noise and ambiguity, leading to decreased accuracy and reliability. Furthermore, hybrid approaches may require domain-specific knowledge and expertise to effectively select, combine, and fine-tune the various components. Overall, while hybrid approaches have the potential to leverage the strengths of multiple techniques, careful consideration, and rigorous evaluation are necessary to ensure their effectiveness and mitigate their drawbacks6. To overcome the limitations associated with traditional methods, we present a novel approach for sentiment analysis at the sentence level in review datasets. The primary goal is to develop a decision-making system capable of effectively determining the sentiment of online reviews. The main contributions of this study are as follows:Innovative mathematical sentiment analysis approach: This study presents a groundbreaking unsupervised sentiment classification model based on population game theory. Unlike traditional models, it does not require labelled datasets, making it highly adaptable and reducing the dependency on extensive pre-existing data. This innovation addresses a significant limitation in sentiment analysis by offering a more flexible and scalable solution.

Dynamic and real-time sentiment tagging: The population game model dynamically adapts to changing language patterns and sentiment expressions over time. By framing sentiment categories as players and sentiment labels as strategies, the model achieves a Nash equilibrium that ensures stable and accurate sentiment classification. This dynamic adaptation is crucial for maintaining model effectiveness in real-world applications where language and context continuously evolve.

Cross-language and cross-domain robustness: The model’s design emphasizes language and domain independence, proven effective across various datasets, including English and Hindi reviews from multiple domains like hotels, restaurants, and electronic devices. This broad applicability highlights the model’s robustness and potential for wide-scale deployment in diverse sentiment analysis tasks.

Enhanced accuracy through strategic interaction modelling: By capturing the strategic interactions between context and emotion within an asymmetric game framework, the model provides a more nuanced understanding of sentiment dynamics. This leads to enhanced accuracy, with the study achieving high-performance metrics (89% for English and 84% for Hindi), demonstrating the model’s superiority over existing sentiment analysis techniques.

The study is structured as follows: Section “Related work” offers a concise review of previous scientific work in the realm of sentiment analysis. Section “Methodology” elaborates on the core tenets of the population game theory model and the proposed algorithmic approach for sentiment analysis. In Section “Result and evaluation”, we assess the efficacy of the proposed model by juxtaposing it against alternative methods across different review datasets. This evaluation is substantiated through adherence to specified guidelines and statistical significance tests, accompanied by a thorough analysis of the algorithm's strengths and limitations. Section “Discussion.” encapsulates the error rate, significance, real-life implementation, and limitations of the study and delineates potential avenues for future research. Finally, Section “Conclusion” presents the conclusion and the future work.

Related work

Textual sentiment analysis methods include Topic-based approaches, machine learning approaches, deep learning approaches, and mathematical modeling approaches. Each of these methods has its strengths and limitations, and understanding these can help identify the gap that our proposed work aims to fill as illustrated in Table 1.Table 1 Comparative evaluation of advanced sentiment analysis methodologies.

Sr. no.	References	Category	Approach	Strengths	Limitations	Research gap filled by the proposed method	
1	García-Pablos et al.7	Topic modelling	Word2Vector-Latent Dirichlet Allocation (W2VLDA)	Almost unsupervised, versatile across domains	Requires extensive tuning, and variable performance across domains	Consistent performance across domains with less tuning required	
2	Yiran and Srivastava8	Deep Learning	Linear Discriminant Analysis (LDA)	Robust for topic clustering and sentiment analysis	Challenges in interpretability, limited practical applicability	Improved interpretability and practical applicability	
3	Song et al.9	Deep Learning	Attention-based Long Short-Term Memory (LSTM)	Enhances aspect-level sentiment analysis with sentiment lexicon embedding	Deteriorates with longer sequences, impacting effectiveness	Maintains effectiveness with long sequences without attention mechanism limitations	
4	Mee et al.10	Machine Learning	Hybrid approach (regression and sentiment analysis)	Versatile, combines multiple methods	Relies heavily on TF-IDF, and can overlook contextual nuances	Better capture of contextual nuances without relying solely on TF-IDF	
5	Fiok et al.11	Deep Learning	Unsupervised sentiment analysis for Twitter accounts	Scalable, handles large volumes of data	Struggles with evolving language trends and slang usage	Adapts more effectively to evolving language trends and slang	
6	Daniel and Meena12	Machine Learning	Lagrangian Support Vector Machine (LSVM)	High accuracy	Struggles with high-dimensional data require careful feature selection	Handles high-dimensional data more efficiently without extensive feature selection	
7	Basiri et al.13	Machine Learning	Bidirectional neural network	Addresses LSTM and GRU limitations	Suffers from vanishing gradient problems in deep architectures	Avoids vanishing gradient problems through different modelling techniques	
8	Perikos et al.2	Machine Learning	Recurrent neural network (RNN)-based attention models	Effective in sentiment analysis tasks	Prone to overfitting with limited training data	Reduces overfitting risk by not relying on large labeled datasets	
9	Nguyen et al.14	Machine Learning	Ontology embedding	Enhances aspect capturing	Faces difficulties with polysemy and ambiguity in domain-specific languages	Training independent	
10	Dai et al.15	Machine Learning	Selective Domain Adaptation (SDA)	Feature-level transfer of knowledge	Challenges in handling domain shifts and maintaining model interpretability	Maintains interpretability and adapts better to domain shifts	
11	Vashishtha and Susan4	Machine Learning	Keyphrase extraction using fuzzy entropy and k-means clustering	Offers interpretability	Relies on predefined clustering algorithms, and might overlook emerging sentiment patterns	Dynamically captures emerging sentiment patterns without relying on predefined algorithms	
12	Peng et al.1	Machine Learning	BERT Post Training (BERT-PT)	Adaptability, state-of-the-art performance	Requires substantial computational resources and labeled data for fine-tuning	Requires fewer computational resources and works well without extensive labeled data	
13	Chiha et al.16	Machine Learning	Sentiment Analysis Based on Sentiment Clustering (SASC)	Combines deep learning and fuzzy logic	Sensitive to choice of clustering algorithms and sentiment cluster granularity	Less sensitive to clustering algorithm choices and granularity	
14	Hu et al. 17	Deep Learning	Attention-based Point Network (AttPNet)	Innovative attention mechanisms	Complexity in training and deployment, especially with limited computational resources	Simplifies training and deployment while maintaining effectiveness	
15	Liu and Shen18	Deep Learning	Recurrent Memory Neural Network (ReMemNN)	Enhances sentiment analysis with memory components	Performance degrades with noisy or sparse data	Maintains performance even with noisy or sparse data	
16	Zhao and Yu19	Deep Learning	BERT with knowledge incorporation	Enhanced aspect-based sentiment analysis	High computational demands and pre-training requirements	Reduces computational demands and pre-training requirements	
17	Zhang et al.20	Deep Learning	Syntactic and semantic analysis	Comprehensive understanding	High computational demands may limit scalability	Improves scalability while maintaining comprehensive understanding	
18	Atandoh et al.21	Deep Learning	BERT-MultiLayer Convolutional Neural Network (B-MLCNN)	Computational efficiency	Overlooks nuanced sentiment expressions within sentences or phrases	Better captures nuanced sentiment expressions within sentences or phrases	
18	Zuheros et al.22	Mathematical model	SA-MpMcDM for TripAdvisor	Effective in specific domains	Effectiveness varies across different review platforms, requires extensive customization	More adaptable across different review platforms without extensive customization	
20	Punetha and Jain23	Mathematical model	Bayesian game model	Suitable for sentiment classification	Relies on weights allocated to textual features criteria	Rating independent, adaptability over time	
21	Various (SOTM)3, (GRA-ST)24, (TOGT-ST)3, (COPGT-ST)25)	Mathematical model	MCDM-based frameworks for sentiment classification	Different approaches for sentiment orientation and tagging	Reliance on weights and extensive customization requirements	Independent of weights assigned to the criteria	

Addressing research gap

Below are some of the research gaps that we identified by analyzing different approaches, mentioned in Table 1.Topic modeling: Topic modeling methods, such as (W2VLDA), often lack contextual understanding, relying on word co-occurrence without grasping the meaning, which can result in incoherent topics. The need to predefine the number of topics can be challenging, and the models are computationally intensive and sensitive to hyperparameters. They also ignore word order, affect meaning, and may struggle with short texts. Additionally, interpretability and pre-processing quality can impact their effectiveness.

Machine learning: Machine learning methods, such as Lagrangian Support Vector Machine (LSVM) and Bidirectional Neural Networks, face major challenges while handling high-dimensional data, particularly, overfitting, especially with limited training data. Recurrent neural network (RNN)-based attention models are particularly prone to overfitting. Ontology embeddings struggle with polysemy and ambiguity in domain-specific languages. Selective Domain Adaptation (SDA) encounters difficulties in managing domain shifts while maintaining model interpretability. Keyphrase extraction methods using fuzzy entropy and k-means clustering, which rely on predefined algorithms, may miss emerging sentiment patterns. Furthermore, methods like BERT Post Training (BERT-PT) have high demands for computational resources and labeled data for fine-tuning.

Deep learning: Deep learning models, including BERT and Attention-Based Point Networks (AttPNet), have significant drawbacks due to their substantial computational resource requirements and the complexity involved in their deployment. Techniques such as Recurrent Memory Neural Networks (ReMemNN) experience performance degradation when dealing with noisy or sparse data. BERT with knowledge incorporation demands extensive computational power and pre-training. Syntactic and semantic analysis approaches struggle with scalability due to high computational demands. BERT-Multi Layer Convolutional Neural Network (B-MLCNN) often overlooks nuanced sentiment expressions within sentences or phrases.

Mathematical approaches: Existing mathematical models for sentiment analysis, such as SA-MpMcDM, face drawbacks due to the need for extensive customization across different review platforms, which limits their adaptability. Bayesian game models based approaches are dependent on the probability of the occurrence of textual features. Various MCDM-based frameworks for sentiment classification are challenged by their reliance on weights and the extensive customization required.

In summary, the proposed method fills these research gaps by offering consistent performance across domains, effective handling of high-dimensional data, simplified deployment, and adaptability to language trends and domain shifts, thus ensuring scalability and robustness in sentiment analysis. These contributions aim to enhance the reliability, applicability, and practicality of sentiment analysis methodologies in real-world scenarios.

Methodology

The utilization of the population game models for sentiment analysis is given in Section “Population game model”. Section “Proposed methodology” contains the proposed methodology for sentiment analysis. We give a detailed illustrative example of the proposed model in Section “Numerical illustration”.

Population game model

The population game model is a mathematical framework that integrates concepts from game theory and evolutionary biology to analyze the dynamics of strategic interactions within a population26. The population game model offers a compelling framework for sentiment analysis due to its alignment with the dynamics of strategic interactions inherent in sentiment classification.

Sentiment analysis inherently involves strategic interactions between different sentiment categories or strategies. By framing sentiment categories as players and sentiment labels as strategies within the population game model, we accurately capture the strategic nature of sentiment classification. This representation allows for a nuanced understanding of how different sentiment strategies compete and evolve. As a first step, the model defines the “players” as various elements contributing to sentiment, such as context and emotion, while sentiment categories like positive and negative sentiments are regarded as distinct “strategies.” The performance of each strategy is then quantified through a payoff matrix, indicating how well the strategies predict sentiment in given text data. From there, the model calculates the fitness of each sentiment category based on its predictive capability in the given text data. This fitness assessment forms the basis for determining the dynamic changes in strategy proportions over time. Utilizing replicator dynamics, the model describes how the proportions of different sentiment strategies evolve. This dynamic process ensures that the sentiment analysis model adapts and optimizes its strategies over time in response to changing language patterns, sentiment expressions, and user preferences. As the proportions of sentiment categories evolve, the model reaches a steady state, representing a Nash equilibrium where the distribution of sentiment categories stabilizes. At this point, no single sentiment category can unilaterally improve its performance, indicating a convergence to a stable state where the sentiment analysis model maintains a consistent distribution of sentiment categories. Throughout this iterative process, the population game model enables continuous adaptation and optimization of sentiment analysis algorithms. By updating strategy proportions according to replicator dynamics, the model ensures adaptability to changes in sentiment trends and linguistic patterns, ultimately leading to improved performance and adaptability in real-world applications. Evaluation and validation of the model's performance at this steady state help ensure its effectiveness in accurately capturing sentiment in text data.

Overall, the population game model offers a suitable theoretical framework for sentiment classification due to its ability to capture the dynamic nature of sentiment analysis, optimize sentiment analysis algorithms, achieve stable equilibrium, and facilitate systematic evaluation and validation. By leveraging the strategic interactions inherent in sentiment classification, this model enhances the accuracy, adaptability, and performance of sentiment analysis systems in various real-world applications.

We employ the population game model to accurately reflect the diverse and varying influences of context and emotion on strategic interactions. Unlike symmetric games, where all players have identical strategies and payoffs, asymmetric games allow us to capture the inherent differences between the players' roles and payoffs in our model. This distinction is crucial because the strategies and outcomes for context and emotion are not identical and affect each other differently. By employing an asymmetric game framework, we can more precisely model and analyze the dynamics of how context and emotion interact, evolve, and reach equilibrium, leading to a more realistic and applicable understanding of the strategic behavior within the population.

A population game model is represented by a tuple G =  <N, S, A, B, p, q, F1, F2, E1, E2, p˙, q˙> where each component is defined as follows.Players (N): The model consists of N players. In the specific context of this task, there are two players: context (C) and emotion (E) for sentiment analysis.

Strategies (S): The strategies available to players. In the proposed work, we have two strategies viz., positive and negative.

Payoff matrices (A and B):

Matrix A represents the payoffs for player 1 (context). It is denoted by A = [aij], where aij represents the payoff for context playing strategy i against emotion playing strategy j.

Matrix B represents the payoffs for player 2 (emotion). It is denoted by B = [bij], where bij represents the payoff for emotion playing strategy j against context playing strategy i.

Strategy proportion (p and q):

p: A vector of proportions for Player 1 (context). Since we use only two strategies, thus this vector is of size two (02). The individual values are referred to as p1 and p2. p1 is the proportion of context choosing the positive strategy and p2 is the proportion of context choosing the negative strategy. The additional constraint is: p1 + p2 = 1 with p1, p2 ≥ 0.

q: A vector of proportions for Player 2 (emotion). Similarly, q1 is the proportion of emotion choosing the positive strategy and q2 is the proportion of emotion choosing the negative strategy. The constraint is q1 + q2 = 1 with q1, q2 ≥ 0.

Combined strategy proportion (π): A combined vector representing the strategy proportions of both players.π=π1π2=p1p2q1q2

where π1=p1p2 represents the strategy proportions of player 1 and π2=q1q2 represents the strategy proportion of player 2.

Fitness functions (F1 and F2):

The fitness of individual players in the population game model is calculated to understand and analyze the dynamics of the population and how different strategies contribute to the overall success of the individual players. It helps to understand which strategies are more effective in maximizing payoffs and providing individuals with a competitive advantage. For player 1 (context), the fitness (F1(π, i)) is calculated using Eq. (1). Equation (2) shows the equation to calculate the fitness F2π2,j of player 2 (emotion).1 F1π1,i=∑jaij∗qj∀i,j∈{1,2}

2 F2π2,j=∑ipi∗bij∀i,j∈{1,2}

The fitness of players helps us to understand which strategies are more effective in maximizing payoffs and providing players with a competitive advantage.

Expected fitness (E1 and E2):

Expected fitness is the average expected payoff of individual players in the population, considering the proportions of different strategies. It is calculated by taking the weighted average of expected payoffs for each individual, where the weights are given by the proportions of strategies in the population. Equations (3) and (4) show the formulas to calculate the expected fitness of the two players – context and emotion, respectively.3 E1π1,π2=∑ipi∗F1π1,i=∑ipi∗(∑jaij∗qj)

4 E2π1,π2=∑jqj∗F2π2,j=∑jqj∗(∑ipi∗bij)

Replicator dynamics (pi˙ and qi˙):

The rate of change of strategies concerning time t is known as replicator dynamics27. It can be represented as a set of differential equations where the rate of change of the proportion of strategy j is determined by the difference between the fitness of strategy i and the average fitness of all strategies in the population. The proportions of strategies in the population can change over time according to replicator dynamics, which describe how the proportions evolve based on the fitness of different strategies. The rate of change of strategy proportion over time for player 1 (context) is given by Eq. (5). For player 2 (emotion), the rate of change of strategy proportion over time is given by Eq. (6).5 pi˙=dpidt=pi∗F1π1,i-E1π1,π2∀i∈{1,2}

6 qj˙=dqjdt=qj∗F2π2,j-E2π1,π2∀j∈{1,2}

Steady-state:

The steady state is reached when the proportions of strategies no longer change over time. At this point, the model reaches a Nash equilibrium, where no player can improve their payoffs by unilaterally changing their strategy. In other words, the sentiment analysis model converges to a stable state where the distribution of sentiment categories remains constant. Mathematically, this is represented by Eqs. (7) and (8) for player 1 (context) and player 2 (emotion), respectively.7 dpidt=0⇒pi∗F1π1,i-E1π1,π2=0

8 dqjdt=0⇒qj∗F2π2,j-E2π1,π2=0

Additionally, the sum of all strategy proportions in the population must be 1 as given by Eqs. (9) and (10) for context and emotion, respectively.9 ∑ipi=1thissimplifiestop1+p2=1

10 ∑jqj=1thissimplifiestoq1+q2=1

Note on time dependence: The strategy proportions pi and qj are explicit functions of time t. This means that the proportions of strategies for both players are not static but can change over time. The dynamics of these proportions are governed by the replicator equations, reflecting how the fitness and relative success of strategies drive the evolution of these proportions in the population. As the game progresses, these proportions adjust until they reach a steady state where they remain constant, indicating that the population has reached a Nash equilibrium.

Proposed methodology

The proposed methodology comprises of two phases—Phase I and Phase II. Figure 1 outlines the flowchart delineating the two-phase framework. Phase I involves the extraction of textual features from the review comments, utilizing suitable algorithms. Subsequently, we generate a decision matrix, facilitating further analysis. In Phase II, a population game model is implemented, tailored to the task of sentiment classification. Further elaboration on the different phases is provided in subsequent subsections. Table 2 lists the notations employed in the study.Fig. 1 Flowchart illustrating the structure of the proposed framework.

Table 2 Comprehensive list of notations and their definitions utilized in the study.

Notations	Description	
F1π1,i	Fitness of player 1 (context)	
F2π2,j	Fitness of player 2 (emotion)	
E1π1,π2	Expected Fitness of Player 1	
E2π1,π2	Expected Fitness of Player 2	
p1, q1	Proportion of Positive strategy of both players	
p2, q2	Proportion of Negative strategy of both players	
pi˙	Replicator dynamics of player i	
qj˙	Replicator dynamics of player j	
(PCS⊗)	Normalized Positive context score	
(NCS⊗)	Normalized Negative context score	
(PES⊗)	Normalized Positive emotion score	
(NES⊗)	Normalized Negative emotion score	
(c+, c−, e+, e−)	Payoff values of players	

Phase I: features extraction

Phase I of the methodology is initiated by following a cleaning process to enhance the quality and uniformity of the dataset. This cleaning process encompasses various pre-processing steps, including tokenization, lemmatization, and stop word removal, as depicted in Fig. 2. This pre-processing ensures that the data is appropriately structured and prepared for subsequent analytical procedures. After the cleaning process, the focus shifts towards the extraction of textual features from the refined dataset. The next step entails the retrieval of two distinct features from the text, namely context scores ((PCS⊗), (NCS⊗)) and emotion scores ((PES⊗), (NES⊗)), as delineated in Fig. 2. The concluding step of this phase involves the generation of a decision matrix of order 2 × 2. This matrix is the normal form representation of the game consisting of two players (context and emotion), two strategies (positive and negative), and payoffs of the players ((PCS⊗), (NCS⊗), (PES⊗) and (NES⊗)) for playing certain strategies. Detailed explanations of these payoff computations are explained in step 1 and step 2 below.Fig. 2 Flowchart depicting phase I of the framework.

Step 1: Evaluate the normalized context scores of reviews ((PCS⊗), (NCS⊗))

Context scores are objective numerical measures allocated to assess the contextual sentiment in a given text, effectively quantifying the magnitude of positive or negative sentiment conveyed within. These scores are calculated utilizing the Python-based libraries, leveraging the capabilities of SentiWordnet (SWN) for accurate computation. The resulting context score values reside within the closed interval [0, 1]. (PCS⊗) denotes the normalized positive context score and (NCS⊗) denotes the normalized negative context score. Algorithm 1 is employed to calculate these scores.Algorithm 1 Retrieve textual Context scores

Step 2: Evaluate the normalized emotion scores of reviews ((PES⊗), (NES⊗))

For this study, emotions are categorized into five distinct categories viz., happy (H), angry (A), sad (S), surprised (Sp), and fear. However, the emotion of fear is not taken into account. To quantify the emotions, the text2emotion library in Python is utilized. This library enables the computation of emotion scores, which provide numerical representations of the intensity or prevalence of the identified emotions in the text. After obtaining the emotion scores for the five emotions, these emotions are further categorized into two groups: Normalized positive emotion (PES⊗) and normalized negative emotion (NES⊗). The categorization of emotions into these two categories is performed following Algorithm 2. The resulting values of (PES⊗) and (NES⊗), representing the degree of positive and negative emotions, respectively, fall within the range of 0–1.Algorithm 2 Retrieve textual Emotion scores

Table 3 contains the information about game components i.e. two players (context and emotion) having two strategies (positive and negative) and their respective numeric values (PCS⊗ = c+, PES⊗ = c−, NCS⊗ = e+ and NES⊗ = e−). Integrating context and emotion features leads to a more holistic sentiment analysis, offering a comprehensive view by combining the specific details (context) with the intensity and nature of the sentiment (emotion). This integration resolves ambiguities where one feature alone might be misleading, ensuring accurate sentiment classification even in mixed or contradictory scenarios. For instance, a review mentioning "excellent battery life" (positive context) with "I am extremely happy with my purchase" (positive emotion) solidifies the positive sentiment, while a contextually positive statement with a negative emotional tone indicates a complex sentiment that requires deeper analysis. By considering both the rationale (context) and the emotional response (emotion), the proposed methodology ensures accurate sentiment classification, leveraging the strengths of both features to enhance the model's efficiency and reliability in sentiment tagging. This comprehensive approach improves the accuracy of predictions and the robustness of the analysis, making it essential for applications in natural language processing, customer feedback analysis, and automated review systems, where understanding the full spectrum of sentiments is vital.Table 3 Normalized contextual and emotional scores of a review.

Players with strategies	Given values	
Positive context PCS⊗	c+	
Negative context NCS⊗	c−	
Positive emotion PES⊗	e+	
Negative emotion NES⊗	e−	

Phase II: sentiment tagging

To apply the population game to the sentiment tagging task, we consider a review R as a game with two players: context and emotion. Each player has two strategies viz. positive (p1) and negative (p2). These strategies have values between 0 and 1, ensuring that their sum is always 1 (p1 + p2 = 1). We evaluated the numeric scores from Phase I are fed as input to Phase II. We follow Algorithm 3 to perform sentiment tagging of reviews using the population game model. In the algorithm, input is the payoff. The first step is the evaluation of the fitness of both players. The fitness of player 1 (context) is denoted as F1π1,i, and for player 2 (emotion), it is denoted as F2π2,j. These fitness measures capture the effectiveness of each strategy in the game. In the next step, the expected fitness of players 1 and 2 is E1π1,π2 and E2π1,π2. The third step involves determining the replicator dynamics, which describes the rate of change of strategies over time. For player 1, this is represented by pi˙=dpidt, and for player 2, it is represented by qj˙=dqjdt. As the time approaches infinity (t → ∞), the strategies of both players reach a point known as Nash equilibrium, where they no longer change their strategies. At this point, the rate of change of strategies for both players tends to zero i.e., dpidt=0anddqjdt=0. The maximum value between p1, p2, q1, and q2 indicates the sentiment tag assigned to review R. A detailed explanation of each step of Algorithm 3 is illustrated below from step 1 to step 6.Algorithm 3 Sentiment tagging of reviews using population game

Step 1: Define the Payoff matrix

The payoff matrix for the game is shown in Table 4, which is the normal form representation of the game. In this study the choices made by the Context player and the Emotion player are not fully correlated; their strategies impact each other’s results. Here, the payoffs of player 1 (Context) are represented by the rows, and player 2 (emotion) are represented by the columns. Here, aij represents the payoff received by player 1 (context) and player 2 (emotion). Similarly, bij represents the payoff for player 2 under the same condition. For the proposed approach, the values in the table are derived from the context and emotion scores, as shown in matrix A and matrix B.Table 4 Game theoretical normal form matrix depicting strategic payoffs.

	Emotion (player 2)	
Positive	Negative	
Context (player1)	Positive	(a11 = c+, b11 = e+)	(a12 = c+, b12 = e−)	
Negative	(a21 = c−, b21 = e+)	(a22 = c−, b22 = e−)	

Table 4 shows that the strategies of the players are non-correlated, i.e., each player's choice is independent of the other player's choice. This means that the decision made by the context (Player 1) does not influence the decision made by the emotion (Player 2), and vice versa. In other words, we can say the strategy proportion of player 1 (pi) is independent of (qj). The outcome is determined by the combination of strategies chosen independently by each player.

Step 2: Calculation of fitness of the players

The fitness of a player reflects its effectiveness in achieving desired outcomes within a given context. It measures how well the chosen strategy aligns with the goals of the situation. This evaluation considers the weighted combination of scores extracted from Phase I, encompassing factors like preferences, capabilities, and contextual constraints relevant to decision-making. Through this integration with chosen strategies, player fitness is assessed. This process determines how much each strategy contributes to achieving desirable outcomes based on the combined scores. Ultimately, these fitness values offer insights into the effectiveness of different strategies in meeting specific goals or requirements. The fitness of player 1 for the positive strategy is given by Eq. (11), and for the negative strategy is given by Eq. (12). The fitness of player 2 for the positive strategy is given by Eq. (13), and for negative is given by Eq. (14). Here, pi and qj are the proportions of individuals using strategy i and j respectively. The strategy proportion of player 1 is p1 and p2 and q1 and q2 is the strategy proportion of player 2.

For player 1 (Context)11 F1π1,1=a11∗q1+a12∗q2=c+∗q1+c+∗q2

12 F1π1,2=a21∗q1+a22∗q2=c-∗q1+c-∗q2

For player 2 (Emotion)13 F2π2,1=b11∗p1+b21∗p2=e+∗p1+e+∗p2

14 F2π2,2=b12∗p1+b22∗p2=e-∗p1+e-∗p2

Step 3: Estimation of Expected Fitness

In this step, we evaluate the average expected fitness of players in the population by considering the proportions of different strategies. The expected fitness for player 1 (E1(π, π)) is given by Eq. (15), and for player 2 (E2(π, π)) is given by Eq. (16).15 E1π1,π2=p1∗F1π1,1+p2∗F1π1,2

16 E2π1,π2=q1∗F2π2,1+q2∗F2π2,2

Step 4: Evaluate Replicator dynamics

In this step, we use replicator dynamics to figure out the proportion of change in strategies over time. This is represented by Eqs. (17) and (18) for player 1. Equations (19) and (20) represent the replicator dynamics for player 2.

For player 1 (Context)17 p˙1=dp1dt=p1∗F1π1,1-E1π1,π2forpositivestrategy

18 p˙2=dp2dt=p2∗F1π1,2-E2π1,π2fornegativestrategy

For player 2 (Emotion)19 q˙1=dq1dt=q1∗F2π2,1-E2π1,π2forpositivestrategy

20 q˙2=dq2dt=q2∗F2π2,2-E2π1,π2fornegativestrategy

Step 5: Steady state

After examining the rate of change of strategies over time, we analyze the population game's dynamics by observing the long-term behavior of replicator dynamics. This involves identifying a solution where strategies no longer change over time. This condition implies that as time progresses, a point comes when the rate at which strategies change slows down and eventually stops, i.e. dpidt = 0 for player 1 and dqjdt = 0 for player 2. This point is the time when players cannot enhance their performances and are hence restricted to the same strategy. This stable state is called Nash equilibrium (NE). Equations (21)–(24) shows the conditions for a steady state for both the players.

For player 1 (Context)21 dp1dt=0⇒p1∗F1π1,1-E1π1,π2=0

22 dp2dt=0⇒p2∗F1π1,2-E1π1,π2=0

For player 2 (Emotion)23 dq1dt=0⇒q1∗F2π2,1-E2π1,π2=0

24 dq2dt=0⇒q2∗F2π2,2-E2π1,π2=0

Step 6: Determine Sentiment Tag

In the population game model, the sentiment tag of reviews is determined by analyzing the proportion and stability of Nash equilibria. It is important to note that while Nash equilibria represent stable states, such that no player can improve their payoff by unilaterally changing their strategy, the dynamics of reaching such equilibria in replicator systems are highly dependent on the initial conditions. This dependency arises due to the nature of the differential equations governing the system, which can lead to convergence to different equilibria based on the starting point. The system's trajectory and the eventual equilibrium state are influenced by the initial values.

During our experimentation with various initial conditions, we observed that the results fluctuated and showed bias. To ensure no initial bias towards any strategy, we fix the initial conditions at p1(0) = 0.5, p2(0) = 0.5, q1(0) = 0.5, q2(0) = 0.5. This consistent starting point mitigates the influence of initial condition biases on the resulting sentiment tags. By fixing these neutral initial conditions, we achieve a more stable and repeatable process for determining the sentiment tag based on the dynamics of the population game model. By analyzing the most frequently occurring sentiment tags across player interactions and maintaining consistent initial conditions, we can reliably classify the sentiment of the text. We solve Eqs. (21)–(24) by using an analytic differential equation with the given initial conditions and evaluating the values of p1, p2, q1 and q2. The ultimate tag to the text is given by Eq. (25).25 Maxp1,p2andMax(q1,q2)

In sentiment classification, context, and emotion tags are assigned based on the analysis of the text. Each sentiment tag's strategy proportion is denoted by p1, p2, q1, and q2 for both players. To determine the most appropriate tag, we consider the maximum probabilities from both context and emotion analyses. Specifically, by taking the maximum of p1 and p2, we select the tag with the highest confidence based on the context. Similarly, by taking the maximum of q1 and q2, we select the tag with the highest confidence based on the emotion. When both the maximum confidence values from context (max (p1, p2)) and emotion (max (q1, q2)) converge to the same tag, it indicates that both analyses agree on the sentiment. This convergence is crucial as it provides higher reliability and confidence in the classification. The agreement of two independent assessments—context and emotion—reinforces the conclusion, making the sentiment classification more robust and dependable. In some cases, when context and emotion analyses disagree, it might be due to several factors such as sarcasm, mixed emotions, or complex contexts that are difficult to interpret accurately. Some of the cases are addressed in Section “Recommendations for the future work” as research limitations.

Numerical illustration

In this subsection, we took three illustrative examples of the above process being implemented on three reviews one with a different star rating.

Illustrative example 1

Let us consider an example review with a 4-star rating.

Rating: ★★★★☆: “The Headphones are fantastic! The sound quality is crystal clear, and they are incredibly comfortable to wear for long periods. I'm thoroughly impressed with the battery life and the sleek design.”	

Initially, the context scores and emotion scores of the written review comment are computed as explained in Phase I using Algorithms 1 and 2, respectively. Table 5 contains the normalized context and emotion scores of the review.Table 5 Context and emotion scores for an illustrative review.

Players with strategies	Scores	
Positive context PCS⊗	c+  = 0.75	
Negative context NCS⊗	c− = 0	
Positive emotion PES⊗	e+  = 0.9	
Negative emotion NES⊗	e− = 0.1	

Step 1: Create Payoff Matrices

Table 6 is the normal form representation of the game, consisting of two players with two strategies along with their payoffs. With the help of Table 6, we generate the two decision matrices—A and B as shown in Eqs. (26) and (27). Now we use these payoff matrices as input in Phase II for playing the population game model and sentiment tagging.Table 6 Normal form representation of strategic game interactions.

	Emotion (player 2)	
Positive	Negative	
Context (player1)	Positive	(0.75, 0.9)	(0.75, 0.1)	
Negative	(0, 0.9)	(0, 0.1)	

26 A=c+c+c-c-=0.750.7500

27 B=e+e-e+e-=0.90.10.90.1

Step 2: Calculate the Fitness of players. The fitness of player 1 (F1(π, i)) and player 2 (F2(π, j)) is expressed mathematically by Eqs. (28)–(31).

For player 1 (Context)28 F1(π,1)=a11∗q1+a12∗q2=0.75∗q1+0.75∗q2

29 F1(π,2)=a21∗q1+a22∗q2=0∗q1+0∗q2

For Player 2 (Emotion)30 F2(π,1)=b11∗p1+b21∗p2=0.9∗p1+0.9∗p2=0.9

31 F2(π,2)=b12∗p1+b22∗p2=0.1∗p1+0.1∗p2=0.1

Step 3: Calculate Expected Fitness. The expected fitness of the players in the game, is given by Eqs. (32) and (33).

For player 1 (Context)32 E1π,π=p1∗F1π,1+p2∗F1π,2=p1∗0.75∗q1+0.75∗q2=p1∗0.75∗q1+0.75∗q2

For Player 2 (emotion)33 E2π,π=q1∗F2π,1+q2∗F2π,2=q1∗0.9+q2∗0.1=0.9q1+0.1q2

Step 4: Replicator Dynamics Equation: We evaluate the replicator dynamics for player 1 and player 2 using Eqs. (34)–(37).

For player 1 (Context),34 p1˙=dp1dt=p1∗F1π,1-E1π,π=p1∗0.75-0.75p1=0.75p1∗(1-p1)

35 p2˙=dp2dt=p2∗F1π,2-E1π,π=p2∗0-0.75p1=-0.75p1p2

For player 2 (Emotion),36 q1˙=dq1dt=q1∗F2π,1-E2π,π=q1∗0.9-(0.9q1+0.1q2)

37 q2˙=dq2dt=q2∗F2π,2-E2π,π=q2∗0.1-(0.9q1+0.1q2

Using the constraints, p1 + p2 = 1 and q1 + q2 = 1 we get the Eqs. (38)–(41).

For player 1 (Context),38 dp1dt=0.75p1∗1-p1

39 dp2dt=-0.75p1p2

For Player 2 (emotion),40 dq1dt=0.8q1∗(1-q1)

41 dq2dt=-0.8q1q2

Using the analytic differential equation method with initial condition p1(0) = 0.5, p2(0) = 0.5, q1(0) = 0.5, q2(0) = 0.5, we get the values of p1(t), p2(t), q1(t), q2(t) as given below.p1t=e0.75t1+e0.75tp2t=11+e0.75tq1t=e0.8t1+e0.8tq2t=11+e0.8t

Step 5: Determine Steady-State Value: For a further long time (t→∞) and at a steady state, the rate of change of strategies becomes zero with time. Given that both p1 and q1 initially start at 0.5, we observe the dynamic at (t→∞) for p1 and q1 increases over time towards 1, i.e., p1 = 1, p2 = 0 and q1 = 1, q2 = 0.maxp1,p2=max1,0=1positivetag

maxq1,q2=max1,0=1positivetag

This ensures that the equilibrium solutions are consistent with the initial conditions and the dynamics defined by the replicator equations, providing a stable solution for the probabilities p1, p2, q1, and q2. The sentiment tag is determined by the maximum proportion of the strategies. Since both p1 and q1 represent positive strategies with higher values, the sentiment tag of the review is deduced as positive.

The plot of the replicator dynamics of the illustrative example is generated to see the sentiment dynamics. Figure 3 represents the plot generated from the replicator dynamics model. It illustrates the evolution of strategy proportions for both context and emotion over time. The x-axis represents the time span of the observation, while the y-axis shows the probability values of the strategies, ranging from 0 to 1.Fig. 3 Dynamic visualization of replicator dynamics.

The blue line represents that p1 (Context Positive), starting at an initial probability of 0.5, rapidly increased to 1. This indicates the increasing proportion of the population adopting the positive context strategy, stabilizing at 1, which suggests the dominance of the positive context strategy. Conversely, the red line for p2 (Context Negative) starts at 0.5 and quickly decreases to 0, indicating the extinction of the negative context strategy. Similarly, the green line represents q1 (Emotion Positive), also starting at 0.5 and rapidly increasing to 1, showing the dominance of the positive emotion strategy. The yellow line for q2 (Emotion Negative) starts at 0.5 and decreases to 0, indicating the extinction of the negative emotion strategy.

The rapid change observed, with positive strategies p1 and q1 quickly increasing and negative strategies p2 and q2 decreasing, demonstrates the system's tendency to Favor positive interactions. The system reaches a steady state where p1 and q1 are near 1, and p2 and q2 are near 0, indicating the overwhelming adoption of positive strategies. The dominance of p1 and q1 over p2 and q2 leads to the sentiment classification being "positive." This dynamic behavior aligns with the provided payoff matrices, where positive interactions are more beneficial for both context and emotion players. The rapid convergence to positive strategies suggests that the model strongly favors positive sentiment in the given setup.

Illustrative example 2

Let us consider an example review with a 2-star rating.

Rating: ★★☆☆☆: “I recently purchased this product hoping it would meet my needs, but unfortunately, it didn't live up to my expectations. On the positive side, the quality is decent, and the delivery was on time, which I appreciate. However, I was quite disappointed with the overall performance. The product did not perform as advertised, and I found several issues that impacted its usability. The experience left me feeling frustrated and dissatisfied.”	

Initially, the context and emotion scores for the review comment are computed using Algorithms 1 and 2, as described in Phase I. These scores are detailed in Table 7. In Phase II, we then apply the proposed methodology to these extracted scores to determine the sentiment tag. The evolution of the two strategies, based on different features, over time, is illustrated in Fig. 4. Ultimately, we observe the convergence of these two strategies, which will determine the final sentiment tag of the text.Table 7 Context and emotion scores for an illustrative review.

Players with strategies	Scores	
Positive context PCS⊗	c+ = 0.8	
Negative context NCS⊗	c- = 0.2	
Positive emotion PES⊗	e+ = 0	
Negative emotion NES⊗	e- = 0.3	

Fig. 4 Dynamic visualization of replicator dynamics.

In the analysis conducted using Algorithm 3, the replicator dynamics plot (Fig. 4) demonstrates that as time progresses, the scores for context-negative and emotion-negative strategies converge towards 1, while the scores for context-positive and emotion-positive strategies approach 0. This trend indicates that the review sentiment classification is negative. This result shows that, despite any conflicting scores in the initial stages, the system stabilizes over time, consistently converging to an accurate negative classification for the review. Thus, the dynamics effectively resolve the sentiment classification to reflect the correct sentiment as negative, regardless of the initial discrepancies.

Illustrative example 3

Let us consider an example review with a 3-star rating.

Rating: ★★★☆☆: “Royal Inn is a decent place to stay, but it has its ups and downs. The lobby is really nice and the staff are super friendly, which is a big plus. However, there are some issues that need attention. The cleanliness of the rooms could definitely be better, and the noise from outside was pretty annoying. If they can fix these problems, Royal Inn could be a much more enjoyable place to stay.”	

In accordance with the methodology described in Phase I, numeric scores are extracted, as presented in Table 8. Subsequently, in Phase II, these scores are utilized to determine the sentiment tag, with the results illustrated in Fig. 5.Table 8 Context and emotion scores for an illustrative review.

Players with strategies	Scores	
Positive context PCS⊗	c+ = 0.4	
Negative context NCS⊗	c- = 0.6	
Positive emotion PES⊗	e+ = 0.3	
Negative emotion NES⊗	e- = 0.5	

Fig. 5 Dynamic visualization of replicator dynamics.

Table 8 illustrates that, initially, the positive scores for both context and emotion features are dominated by the negative scores. After implementing the proposed methodology, we observe the following dynamics: At t = 0, the game begins with initial strategies, and over time, the positive strategy for both context and emotion features converges to 1 as illustrated in Fig. 5. Initially, the negative strategy for context increases but subsequently declines and converges to 0. Similarly, the negative strategy for emotion also converges to 0. This indicates that, ultimately, the positive strategy prevails over the negative strategy.

The population game model effectively captures these dynamics, demonstrating its robustness across different scenarios. The proposed methodology has proven effective in resolving conflicting scores and accurately tagging sentiment, as evidenced by the consistency and accuracy of the results across three different scenarios.

Human and animal ethics

No humans or animals were harmed in any way.

Result and evaluation

This section presents an overview of the datasets utilized for evaluating the performance of our population game model in both English and Hindi languages. We assessed our model's performance using several metrics, including accuracy, F1 score, and precision score. These metrics were compared against those achieved by state-of-the-art techniques to establish a benchmark for the effectiveness of our model. To ensure the validity of our comparisons, we adhered to the data annotation guidelines detailed in Section “Annotation guidelines for model output verification”. These guidelines ensured consistency in annotation across different methodologies. We evaluated the overall performance of our model by calculating macro and micro versions of accuracy, precision, F1 score, and recall. These metrics, presented in Sections “Comparison of the proposed model with mathematical optimization techniques on the English dataset” to “Comparison of macro and micro evaluation”, offer a comprehensive assessment of our model’s performance, reflecting both aggregated and individual performance metrics. Furthermore, we conducted a statistical analysis, as outlined in Section “Statistical validation of the proposed model”, to assess the significance of our proposed model. This analysis provided insights into the extent to which our model outperformed existing techniques, thereby highlighting its relative effectiveness.

Datasets

The proposed algorithm was applied to six distinct datasets, covering both English and Hindi reviews. We used three datasets in English from diverse domains and three datasets in the Hindi language from varied domains. Each dataset contains written reviews paired with their respective ratings. Detailed data statistics are illustrated in Table 9.Table 9 Summary statistics of datasets employed in experimental evaluation.

Data set	Language	No. of reviews	
TripAdvisor reviews	English	2643	
Electronic reviews	English	2603	
IMBD movies review	English	2478	
Hotel reviews	Hindi	2977	
Electronic reviews	Hindi	1789	
Movie reviews	Hindi	1101	

Annotation guidelines for model output verification

To evaluate the performance of the algorithm, the outcomes were compared with the sentiments annotated in the dataset. Manual annotation by domain experts was employed to classify the reviews based on sentiment. This manual annotation process involved the formation of an expert team with a strong understanding of sentiment analysis and the specific domain. Sentiment labels, positive and negative, were determined for the reviews. Annotation guidelines were created to compare the results of the population game algorithm with the rating scores specified in Eq. (42). Each written review was associated with a normalized rating score between 0 and 1, and based on this range, the reviews were labelled with positive or negative sentiment. Conflicts were resolved through expert discussions and collaboration within the team. Through this scientific approach, the manual annotation process by experts ensured reliable and accurate sentiment labelling for the reviews.42 S=Ifratingscore∈0,0.5→NegativeSentimentTagIfratingscore∈0.5,1→PositiveSentimentTag

The purpose of this manual annotation was to evaluate and assess the accuracy of the population game model by comparing its results with the sentiments annotated in the dataset. This allowed us to measure the proposed model's performance, understand its strengths and weaknesses, and determine how well it aligned with the human-annotated sentiments. The comparison between the population game model's outputs and the annotated dataset served as a means of validating and verifying the model's accuracy by evaluating various metrics for performance measurement of the proposed population game model.

Why evaluating the neutral tag in reviews considered less critical?

While evaluating neutral sentiment can provide a more comprehensive understanding of sentiment distribution within textual data, in some contexts, it might be less critical or impactful for decision-making or analysis purposes compared to strongly polarized sentiments. Therefore, the importance of evaluating the neutral tag in reviews may vary based on the specific objectives, applications, and priorities of the sentiment analysis task at hand. Some of the main reasons we only consider the positive and negative tags are listed below:Focus on polarized sentiments: Many sentiment analysis tasks prioritize distinguishing between positive and negative sentiments, as they often carry more significant implications for decision-making processes. Neutral sentiments, while valuable, might not influence decision-making or opinions to the same extent.

Application specificity: In certain applications, particularly those where decision-making relies heavily on polarized sentiments (e.g., customer feedback analysis, product reviews), assessing only positive or negative sentiments might hold more relevance than the neutral category.

Complexity and interpretation: Identifying neutral sentiment can be inherently more complex than identifying clear positive or negative sentiments. The ambiguity or subjectivity in what constitutes neutral sentiment makes its evaluation more challenging.

Resource allocation: Resources, including time and computational power, might be better allocated towards distinguishing and analyzing polarized sentiments, which often have a more direct impact on decision-making.

Evaluation of the population game model’s performance

Evaluating performance metrics for frameworks is crucial as it provides a systematic and quantitative approach to assess the performance, reliability, and effectiveness of the framework. These metrics offer objective measures to gauge how well the framework achieves its intended goals, whether it's classification accuracy, anomaly detection, or any other task. By evaluating these metrics, researchers and practitioners can pinpoint the strengths, weaknesses, and areas for improvement within the framework. Additionally, it enables comparison with alternative frameworks or approaches, facilitating the selection of the most suitable tool for a given problem domain. The use of various metrics such as accuracy, F1-score, recall, precision, and MCC score is fundamental in assessing models as they provide diverse insights into performance. Accuracy reflects overall correctness by measuring the proportion of correctly classified instances. Precision underscores the accuracy of positive predictions, crucial when minimizing false positives is paramount. Recall focuses on capturing all relevant instances, especially vital in scenarios where missing positives incur significant costs. F1-score, harmonizing precision and recall, delivers a balanced assessment, particularly valuable in dealing with imbalanced datasets. Together, these metrics offer a comprehensive understanding of a model's efficiency, facilitating model selection, parameter tuning, and optimization, ultimately elevating the reliability and effectiveness of machine learning systems. These metrics are evaluated using Eqs. (43), (44) which employs TP, TN, FP, and FN as essential indicators for assessing classification models. True positives (TP) refer to cases that are correctly identified as positive, whereas true negatives (TN) are instances that are accurately classified as negative. False positives (FP) arise when the model erroneously classifies an instance as positive when it should be negative, while false negatives (FN) occur when the model mistakenly classifies an instance as negative when it should be positive. TP and TN denote accurate forecasts, but FP and FN signify prediction inaccuracies. These measures are crucial for evaluating the model's accuracy, precision, recall, and other performance indicators, offering vital insights into its capacity to effectively categorize cases. In the appendix subsection, we assessed the performance of the population game model across several datasets illustrated in Tables 9, 10, and 11.43 Accuracy=TP+TNTP+TN+FP+FN

44 Precision=TPTP+FP

Table 10 Two-Proportion Z-test statistics across multiple datasets.

Parameters	Sample 1	Sample 2	
Sample size (n1)	1000	500	
Sample proportion (p1)	0.91	0.896	
Favorable cases (X1)	910	448	

Table 11 Context and Emotion Scores for an Illustrative Review.

Players with strategies	Scores	
Positive context PCS⊗	c+ = 0.5	
Negative context NCS⊗	c- = 0.5	
Positive emotion PES⊗	e+ = 0.5	
Negative emotion NES⊗	e- = 0.5	

Comparison of the proposed model with mathematical optimization techniques on the English dataset

In this section, we conduct a comparative analysis between the proposed model and several established sentiment analysis approaches, including SA-MpMcDM22, COPGT-ST28, NEGVOT29, SOTM5, GRA-ST24, BGM-ST23, and TOGT-ST3. Among these, six models are rooted in Multiple Criteria Decision Making (MCDM) techniques, while BGM-ST is based on the Bayesian game model. The performance of these models is depicted through the evaluation metrics of accuracy (refer Fig. 6), precision (refer Fig. 7), and F1-score (refer Fig. 8). The numeric scores of Eqs. (26)–(28) metrics are in Table 1 of the appendix section. SA-MpMcDM, an MCDM-based model, achieved accuracies of 0.71 on the IMDB movie reviews dataset, and 0.68 and 0.67 on the Electronic and Trip-Advisor datasets, respectively. However, the dependency of SA-MpMcDM on the selection and weighting criteria may lead to varying performance across different datasets, making it less robust in diverse settings. COPGT-ST, integrating the COPRAS MCDM technique with a non-cooperative game model, attained accuracies of approximately 0.79 on IMDB movies, 0.78 on electronic reviews, and 0.76 on TripAdvisor reviews. While commendable, the performance of COPGT-ST may suffer from scalability issues and the need for fine-tuning parameters to achieve optimal results. NEGVOT, utilizing the VIKOR MCDM technique, scored around 0.79 on IMDB movies, 0.8 on electronic reviews, and 0.79 on TripAdvisor reviews datasets. Despite its competitive accuracy, NEGVOT's reliance on specific optimization strategies may limit its generalizability and adaptability to different domains.SOTM, employing the Simple Additive Weighting (SAW) technique with a non-cooperative game model, achieved the highest accuracy of 0.85 on the electronic dataset. However, SOTM's performance may be sensitive to the choice of weighting scheme and may struggle with capturing complex relationships in the data. GRA-ST, employing the Grey Relational Analysis (GRA) MCDM technique, achieved the highest accuracy of 0.81 on the IMDB movie dataset. While effective, GRA-ST may face challenges in handling high-dimensional data and may not scale well to larger datasets. Similarly, BGM-ST, a Bayesian game model-based sentiment tagger incorporating context and rating score, reached the highest accuracy of around 0.81 on the TripAdvisor dataset. However, BGM-ST's reliance on additional contextual information such as ratings alongside text may limit its applicability in scenarios where such information is unavailable or irrelevant. TOGT-ST, an integrated MCDM and game theory-based model assigning sentiment tags to text, achieved an accuracy of 0.82 on the TripAdvisor review dataset. Nonetheless, TOGT-ST's performance may be affected by the complexity of the decision-making process and the need for domain-specific adjustments. In contrast, the proposed model consistently outperforms existing approaches across all evaluations. The comparatively lower accuracies of MCDM-based models stem from their dependency on the number and weighting of criteria used. Additionally, BGM-ST's dependency on the presence of ratings alongside text restricts its domain of implementation. In contrast, the proposed model overcomes these limitations. It does not rely on weights and maintains consistent sentiment tags over time, making it a robust framework for sentiment analysis. Its decentralized decision-making process harnesses collective intelligence, mitigating individual biases and offering superior performance in diverse data settings. With its scalable and adaptable nature, the proposed model emerges as a reliable solution for sentiment analysis tasks, achieving the highest accuracy of 0.85 on the trip-advisor reviews dataset out of three in the English language.Fig. 6 Comparison of the proposed model with optimization techniques-based sentiment taggers in terms of accuracy.

Fig. 7 Comparison of the proposed model with optimization techniques-based sentiment taggers in terms of precision.

Fig. 8 Comparison of the proposed model with optimization techniques-based sentiment taggers in terms of F1-measure.

Comparison of the proposed model with existing approaches in English Text

In this subsection, we conduct a comparison between the proposed model and several existing approaches in terms of accuracy. These approaches include W2VLDA, LSVM classifier, SDA model, B-MLCNN, LDA, Apriori model, and DOC-ABSADeepL model. We applied these models to three datasets from different domains and evaluated their accuracy for comparative analysis, as illustrated in Fig. 9. Numeric scores of Eqs. (28)–(30) metrics are tabulated in Table 2 of the appendix section.Fig. 9 Comparison of the performance of the proposed model with existing approaches in terms of accuracy.

Atandoh et al.21 introduced the BERT-MultiLayer Convolutional Neural Network (B-MLCNN) as an integrated deep learning paradigm for sentiment analysis. Despite achieving a commendable accuracy of approximately 0.81 for IMDB movie reviews, B-MLCNN's reliance on treating entire textual reviews as a single document limits its ability to capture nuanced sentiments within longer texts. Furthermore, while B-MLCNN outperforms in certain domains, its performance diminishes in others, highlighting its lack of domain adaptability. Additionally, the proposed model exhibited better performance as compared to Buon Appetito30. Apriori on IMBD and Trip-Advisor its accuracy is 0.79, and on the electronic dataset, it is recorded as 0.8. The Latent Dirichlet Allocation (LDA)31 based model achieved respectable accuracies, including 0.78 on IMDB movie reviews and 0.85 and 0.83 on electronic and Trip-Advisor reviews, respectively. However, LDA's performance may suffer from its inherent reliance on topic modeling, which may not fully capture sentiment nuances. In contrast, the proposed model offers a more holistic approach to sentiment analysis, overcoming the limitations of topic-based models by directly analyzing sentiment expressions, thereby ensuring more accurate sentiment classification across various datasets. Daniel and Meena’s12 LSVM classifier approach achieved accuracies of around 0.71 and 0.68 on IMDB and Amazon electronic review datasets, respectively. However, LSVM's performance may be limited by its dependency on specific feature representations, which may not fully capture the complexity of natural language sentiment. In contrast, the proposed model does not rely on predefined feature representations, allowing it to adapt more effectively to different datasets and capture subtle sentiment nuances more accurately. Similarly, W2VLDA by Garcia-Pablos et al.7 achieved accuracies of 0.8 for the electronic dataset, 0.79 for IMDB, and 0.81 for the TripAdvisor dataset. While W2VLDA combined word embeddings with LDA for sentiment analysis, it may still face challenges in accurately capturing sentiment nuances, especially in complex language expressions. In contrast, the proposed model leverages population dynamics to capture the collective sentiment of a group, thereby providing a more robust and accurate sentiment analysis across various linguistic contexts.

Selective Domain Adaptation (SDA)15 focused on selectively transferring knowledge from source to target domains at the feature level and achieved its highest accuracy of 0.81 for the TripAdvisor dataset. However, SDA's performance may be hindered by its reliance on specific domain features, which may not generalize well to other domains. In contrast, the proposed model does not rely on domain-specific features, allowing it to perform consistently well across diverse datasets without the need for domain-specific adaptations. DOC-ABSADeepL22 methodology incorporated expert evaluations based on natural language reviews and numerical ratings. However, the proposed model demonstrates superior precision, recall, and f1-measure when compared to DOC-ABSADeepL. DOC-ABSADeepL's reliance on expert evaluations may limit its scalability and adaptability to different datasets and linguistic contexts. In contrast, the proposed model's data-driven approach ensures robust and accurate sentiment analysis across diverse datasets and linguistic expressions achieving an accuracy of 0.86.

Comparison of the proposed model with mathematical optimization techniques on Hindi text

In this section, we compare the performance of the proposed model on the Hindi dataset against several mathematical optimization models, including TOGT-ST3, BGM-ST23, GRA-ST24, SOTM5, NEGVOT29, and COPDT-ST28. Numeric scores of Eqs. (28)–(30) metrics are tabulated in Table 3 of the appendix section. To evaluate the performance of the proposed model on Hindi reviews, we tweaked the proposed approach a bit. To fetch the context scores of the Hindi review comments, instead of SWN, we employed Hindi SentiWordnet (HSWN). HSWN is a collection of Hindi words and their associated positive and negative sentiment values. Once, we had the context scores of Hindi reviews, the rest of the approach remained the same.

COPDT-ST, formulated by integrating the COPRAS MCDM technique with a non-cooperative game model, achieved an accuracy of approximately 0.74 for electronic reviews and around 0.76 for movie reviews. While achieving commendable accuracy, COPDT-ST may suffer from computational complexity and scalability issues due to its integration of multiple decision-making techniques. NEGVOT demonstrates the highest accuracy of approximately 0.7 for movie reviews. However, it may struggle with generalizability and robustness across different domains due to its reliance on specific optimization strategies tailored for movie review datasets. SOTM, an amalgamation of the SAW technique and other non-cooperative game models, achieved accuracies of 0.66 for electronics, 0.69 for hotels, and 0.689 for movies. Despite its versatility, SOTM may lack adaptability to dynamic data environments and could be sensitive to parameter settings. GRA-ST, utilizing the GRA, MCDM technique, achieves its peak accuracy at 0.76. While GRA-ST offers a robust methodology for decision-making, it may face challenges in handling complex decision landscapes and may be less effective in scenarios with high-dimensional data. Similarly, BGM-ST, grounded on the Bayesian game model, achieves an accuracy of around 0.66 on electronic review datasets. Bayesian game models, while theoretically sound, may require strong prior knowledge and assumptions about the underlying data distribution, limiting their applicability in real-world scenarios. TOGT-ST, an integration of the TOPSIS MCDM technique and a non-cooperative game model achieves the highest accuracy in hotel domain reviews at around 0.77. However, its performance in other domains may vary, indicating potential domain-specific biases or limitations in generalizability. Evaluation metric measures for all these models consistently reveal the superior performance of the proposed model when compared to other models, as shown in Figs. 10, 11, and 12. The proposed model not only demonstrates competitive accuracy but also offers advantages in terms of scalability, adaptability, and robustness across diverse datasets and archived accuracy of 0.88 on movie reviews dataset. By leveraging collective intelligence and decentralized decision-making, it mitigates individual biases and outperforms traditional optimization models in real-world applications.Fig. 10 Comparison of accuracy of the population game model with optimization techniques across various domains.

Fig. 11 Comparison of Precision the population game model with optimization techniques across various domains.

Fig. 12 Comparison of F1-score the population game model with optimization techniques across various domains.

Comparison of macro and micro evaluation

In this section, we conduct a comparative analysis between the macro and micro-level scores of the Bayesian game model and the proposed model. The examination reveals the superior performance of the proposed model in terms of accuracy, precision, and recall metrics. With a macro F1-score of 0.88 for the proposed model and micro precision is 0.89 across the four datasets, the former demonstrates enhanced reliability. Figure 13 presents an overview of the performance across six datasets, reinforcing the superiority of the proposed model.Fig. 13 Performance of population game model over Bayesian game model.

Statistical validation of the proposed model

Two distinct samples were extracted from datasets containing hotel reviews and movie reviews, respectively. In the first sample, a total of 1000 reviews (n1) were gathered and subsequently analyzed. Of these reviews, 910 were accurately classified, yielding a proportion of 0.91 (p1). Likewise, the sample size (n2) in the second group was 252, and out of a total of 500 reviews, 410 were accurately categorized, resulting in a sample proportion of 0.896 (p2). A Z-test was conducted to analyze the proportions of two populations (p1 and p2). The data statistics are presented in Table 10.

To perform the z-test, we define a null hypothesis (H0) and an alternate hypothesis (Ha) as given below.

Ho

p1 = p2 i.e., the accuracy of sample 1 is equal to the accuracy of sample 2

Ha

p1 ≠ p2 i.e., the accuracy of sample 1 is not equal to the accuracy of sample 2

Two population proportions were analyzed using a two-tailed test and a z-test as shown in Fig. 14. The z-statistic is computed utilizing Eq. (45).45 z=p1-p2√P1-P1n1+1n2=0.91-0.8960.9053∗1-0.905311000+1500=0.873

Fig. 14 An illustration of the hypothesis’s critical zone.

The critical region of the assumed hypothesis is depicted in graphical form in Fig. 14. Based on the statistical analysis results, the null hypothesis (Ho) was not rejected, indicating a lack of sufficient evidence to support the claim that the population proportions p1 and p2 are different at the 0.05 significance level. Hence, it can be concluded that there is insufficient statistical support to assert a significant disparity between the two proportions. This suggests that the percentage of correctly classified reviews remains consistent across datasets with different sample sizes. As a result, it can be inferred that the proposed model consistently produces reliable outcomes.

Discussion

In this section, we discuss the proposed model’s error rate, significance, practical applications, limitations, and areas for improvement.

Error analysis

The error rate in model evaluation is a fundamental metric used to gauge the performance of machine learning models. It represents the proportion of incorrect predictions made by a model on a given dataset. This metric holds significant importance as it quantifies the model's ability to generalize to unseen data and its overall predictive accuracy. Figure 15 illustrates a comparison of error rates between the proposed model and existing models. Notably, the LSVM classifier and SA-MpMcDM model exhibit the highest error rates, approximately 0.29, while the proposed model demonstrates the lowest, at 0.16. The remaining models fall within the error rate spectrum of 0.29–0.16. This analysis underscores the effectiveness of the proposed model in minimizing prediction errors compared to its counterparts.Fig. 15 Error rate comparison across different models.

Significance of the proposed model

This study proposes a methodology for conducting sentiment analysis by utilizing a mathematical optimization framework. The proposed model's unsupervised nature is a significant advantage. No extensive dataset annotation is necessary. The proposed model utilizes data that are unlabelled and unclassified. The sole prerequisite is the availability of SWN or analogous lexicon resources that can be utilized for the computation of context scores of reviews. The model is efficient in both space and time. The study paves the way for the implementation of mathematical optimizations in diverse NLP applications.

Real-time insights: utilizing dynamic models for effective sentiment analysis

The proposed framework applies to real-time textual data as it relies on two key inputs: context and emotion scores. These scores can be generated using Algorithms 1 and 2, making the framework adaptable to various textual data domains. The four scores extracted in Phase I serve as inputs for Phase II, allowing implementation across different domains. Potential real-time domains for implementing this framework include:(i) Market Intelligence and Brand Management:

The model analyzes textual data such as social media posts, comments, reviews, and feedback to understand consumer sentiments towards brands, products, or services. It identifies strategic interactions among consumers, modeling how sentiments evolve and spread within the population. By considering the dynamics of sentiment propagation as strategic choices made by individuals, the model provides more accurate predictions of market trends and brand perceptions. It also identifies key influencers whose sentiments significantly impact overall trends and perceptions, enabling targeted marketing strategies.

(ii) Social Media Monitoring and Trend Analysis:

For social media data, the model captures the strategic interactions among users through their textual engagements. It analyzes how sentiments are expressed, shared, and influenced within online communities, identifying influential individuals or groups driving trends. By modeling sentiment spread as strategic interactions, the model can anticipate emerging trends, crises, or opportunities in real time. It provides insights into sentiment dynamics and helps organizations respond effectively to changing public opinion on social media platforms.

(iii) Political Analysis and Public Opinion Research:

In the political domain, the model examines textual data such as social media discussions, news articles, and polling data to understand voter sentiments and behavior. It models political sentiments as strategic choices made by individuals within the population, considering factors like candidate preferences and policy perceptions. By analyzing these interactions, the model predicts electoral outcomes more accurately and helps policymakers gauge public sentiment toward policies and initiatives. It enables evidence-based decision-making and responsive governance by understanding the strategic dynamics of sentiment within the population.

(iv) Customer Feedback Analysis and Service Improvement:

For customer feedback data, the model analyzes textual inputs from various channels like surveys, reviews, and social media comments. It identifies patterns, trends, and outliers in customer sentiments, considering factors like product experiences and service interactions. By modeling customer sentiments as strategic choices, the model provides actionable insights for service improvement strategies. It helps businesses understand the underlying dynamics of customer sentiment within the population, enabling targeted interventions and personalized approaches to address customer needs effectively.

Overall, the population game model excels in sentiment analysis by capturing the strategic interactions among individuals based on textual data, providing deeper insights into market trends, public opinion, and customer preferences across diverse domains.

Research limitations

In this subsection, we examine the error detection in one case where the proposed methodology fails.

Case 1: Ambiguous Sentiment Tags: A significant limitation of the proposed model is when p1 = p2 = q1 = q2 = 0.5. This situation indicates an equal probability for both positive and negative sentiments, leading to ambiguity illustrated in example 2.Rating: ★★★☆☆: "The presentation was fine. It wasn't particularly exciting, but it was informative."	

In Phase I, we initially extract the numeric scores as demonstrated in Table 11. Subsequently, these numeric scores are input into Phase II, where sentiment tags are assigned to the text. The final output is illustrated in Fig. 16. Output illustrated by the replicator dynamics plot shown in Fig. 16, the results indicate that the positive context score converges to 0, while the negative context score converges to 1. Meanwhile, both the positive and negative emotion scores stabilize at an initial condition value of 0.5. This pattern reveals significant conflicting results among the four parameters. The dynamics exhibit a clear shift where context strategies reach a definitive state, but the emotion strategies remain unresolved at their initial levels, highlighting inconsistencies in sentiment classification. In this case, both the context and emotion analyses result from different strategies. This indicates an ambiguous sentiment where neither positive nor negative sentiment is dominant. This is also a unique scenario where there is no clear dominant sentiment, and both are equally probable. In this case, we get the absurd and incorrect answers. This might require additional rules or more sophisticated handling to determine the final tag or to recognize it as an inherently ambiguous sentiment.Fig. 16 Replicator dynamic plot of the review.

Recommendations for the future work

The future of this research holds promising advancements driven by technological innovations and evolving needs in various industries. Some potential directions for future work in sentiment analysis include:Fine-grained sentiment analysis: Current sentiment analysis often categorizes text into positive, negative, or neutral sentiments. Future work aims to enhance this by detecting and analyzing more nuanced emotions, opinions, and intentions within the text, such as sarcasm, irony, humor, or subtle sentiments.

Multimodal sentiment analysis: We may aim to integrate multiple data modalities like text, images, videos, and audio for sentiment analysis. This involves understanding sentiment expressed not just through text but also through visual and auditory cues, enhancing the accuracy and depth of sentiment interpretation.

Contextual understanding: Improving algorithms to better understand the context in which sentiment is expressed. Considering the broader context of conversation, cultural nuances, user-specific contexts, and historical data can help in more accurate sentiment analysis.

Aspect-based sentiment analysis: The target will be to focus on identifying sentiments towards specific aspects or entities within the text. For instance, in a product review, understanding sentiments towards different features of the product rather than general sentiment about the whole product.

Overall, the future of sentiment analysis is likely to focus on more precise, context-aware, and adaptable models that can understand human sentiments more accurately across various mediums and contexts while addressing ethical considerations and user-specific needs.

Conclusion

In this study, we employed a population game model specifically designed for sentence-level sentiment analysis. We conducted a rigorous evaluation across diverse datasets in both English and Hindi, covering various domains such as movie reviews, hotel reviews, electronic device reviews, and IMDB movie ratings. Our findings demonstrate the model's language and domain independence. Utilizing a range of evaluation metrics, we achieved a maximum macro accuracy of 89%. We thoroughly examined error cases, and limiting scenarios, and discussed real-life implementation considerations, alongside offering recommendations for future program enhancements. Furthermore, our framework incorporates statistical cross-validation techniques, including Z-tests, to assess its performance robustly. The results reveal the promising performance of our model and affirm its competency in sentiment classification tasks. In our forthcoming research, we aim to delve deeper into sentiment analysis nuances, such as aspect-based sentiment analysis, and address challenges such as negation handling, irony detection, and multimodal framework generation. Our future endeavors will focus on utilizing advanced mathematical and statistical frameworks by integrating additional textual features and employing fuzzy-based approaches to effectively address various challenges of sentiment classification within the text.

Supplementary Information

Supplementary Information.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-70766-z.

Acknowledgements

Vaibhav sharma.

Author contributions

All authors contributed equally.

Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

Code availability

The code generated during the current study is available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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References

1. Peng Y Xiao T Yuan H Cooperative gating network based on a single BERT encoder for aspect term sentiment analysis Appl. Intell. 2022 52 5867 5879 10.1007/s10489-021-02724-5
Peng, Y., Xiao, T. & Yuan, H. Cooperative gating network based on a single BERT encoder for aspect term sentiment analysis. Appl. Intell. 52, 5867–5879 (2022).10.1007/s10489-021-02724-5
2. Perikos I Kardakis S Hatzilygeroudis I Sentiment analysis using novel and interpretable architectures of Hidden Markov Models Knowl. Based Syst. 2021 229 107332 10.1016/j.knosys.2021.107332
Perikos, I., Kardakis, S. & Hatzilygeroudis, I. Sentiment analysis using novel and interpretable architectures of Hidden Markov Models. Knowl. Based Syst. 229, 107332 (2021).10.1016/j.knosys.2021.107332
3. Punetha N Jain G Game theory and MCDM-based unsupervised sentiment analysis of restaurant reviews Appl. Intell. 2023 53 20152 20173 10.1007/s10489-023-04471-1
Punetha, N. & Jain, G. Game theory and MCDM-based unsupervised sentiment analysis of restaurant reviews. Appl. Intell. 53, 20152–20173 (2023).10.1007/s10489-023-04471-1
4. Vashishtha S Susan S Highlighting keyphrases using senti-scoring and fuzzy entropy for unsupervised sentiment analysis Expert Syst. Appl. 2021 169 114323 10.1016/j.eswa.2020.114323
Vashishtha, S. & Susan, S. Highlighting keyphrases using senti-scoring and fuzzy entropy for unsupervised sentiment analysis. Expert Syst. Appl. 169, 114323 (2021).10.1016/j.eswa.2020.114323
5. Punetha N Jain G Aspect and orientation-based sentiment analysis of customer feedback using mathematical optimization models Knowl. Inf. Syst. 2023 65 2731 2760 10.1007/s10115-023-01848-z
Punetha, N. & Jain, G. Aspect and orientation-based sentiment analysis of customer feedback using mathematical optimization models. Knowl. Inf. Syst. 65, 2731–2760 (2023).10.1007/s10115-023-01848-z
6. Jha V Savitha R Shenoy PD Venugopal KR Sangaiah AK A novel sentiment aware dictionary for multi-domain sentiment classification Comput. Electr. Eng. 2018 69 585 597 10.1016/j.compeleceng.2017.10.015
Jha, V., Savitha, R., Shenoy, P. D., Venugopal, K. R. & Sangaiah, A. K. A novel sentiment aware dictionary for multi-domain sentiment classification. Comput. Electr. Eng. 69, 585–597 (2018).10.1016/j.compeleceng.2017.10.015
7. García-Pablos A Cuadros M Rigau G W2VLDA: Almost unsupervised system for aspect based sentiment analysis Expert Syst. Appl. 2017 91 127 137 10.1016/j.eswa.2017.08.049
García-Pablos, A., Cuadros, M. & Rigau, G. W2VLDA: Almost unsupervised system for aspect based sentiment analysis. Expert Syst. Appl. 91, 127–137 (2017).10.1016/j.eswa.2017.08.049
8. Yiran, Y. & Srivastava, S. Aspect-based Sentiment Analysis on mobile phone reviews with LDA. In ACM International Conference Proceeding Series 101–105. 10.1145/3340997.3341012 (2019).
9. Song M Park H Shin Shik K Attention-based long short-term memory network using sentiment lexicon embedding for aspect-level sentiment analysis in Korean Inf. Process. Manag. 2019 56 637 653 10.1016/j.ipm.2018.12.005
Song, M., Park, H., Shin, & Shik, K. Attention-based long short-term memory network using sentiment lexicon embedding for aspect-level sentiment analysis in Korean. Inf. Process. Manag. 56, 637–653 (2019).10.1016/j.ipm.2018.12.005
10. Mee A Homapour E Chiclana F Engel O Sentiment analysis using TF–IDF weighting of UK MPs’ tweets on Brexit Knowl. Based Syst. 2021 228 107238 10.1016/j.knosys.2021.107238
Mee, A., Homapour, E., Chiclana, F. & Engel, O. Sentiment analysis using TF–IDF weighting of UK MPs’ tweets on Brexit. Knowl. Based Syst. 228, 107238 (2021).10.1016/j.knosys.2021.107238
11. Fiok K Karwowski W Gutierrez E Wilamowski M Analysis of sentiment in tweets addressed to a single domain-specific Twitter account: Comparison of model performance and explainability of predictions Expert Syst. Appl. 2021 186 115771 10.1016/j.eswa.2021.115771
Fiok, K., Karwowski, W., Gutierrez, E. & Wilamowski, M. Analysis of sentiment in tweets addressed to a single domain-specific Twitter account: Comparison of model performance and explainability of predictions. Expert Syst. Appl. 186, 115771 (2021).10.1016/j.eswa.2021.115771
12. Daniel DAJ Meena MJ A novel sentiment analysis for amazon data with TSA based feature selection Scalable Comput. Pract. Exp. 2021 22 53 66 10.12694/scpe.v22i1.1839
Daniel, D. A. J. & Meena, M. J. A novel sentiment analysis for amazon data with TSA based feature selection. Scalable Comput. Pract. Exp. 22, 53–66 (2021).10.12694/scpe.v22i1.1839
13. Basiri ME Nemati S Abdar M Cambria E Acharya UR ABCDM: An attention-based bidirectional CNN-RNN deep model for sentiment analysis Future Gener. Comput. Syst. 2021 115 279 294 10.1016/j.future.2020.08.005
Basiri, M. E., Nemati, S., Abdar, M., Cambria, E. & Acharya, U. R. ABCDM: An attention-based bidirectional CNN-RNN deep model for sentiment analysis. Future Gener. Comput. Syst. 115, 279–294 (2021).10.1016/j.future.2020.08.005
14. Nguyen DN Phan TT Do P Embedding knowledge on ontology into the corpus by topic to improve the performance of deep learning methods in sentiment analysis Sci. Rep. 2021 11 1 23541 10.1038/s41598-021-03011-6 34876635
Nguyen, D. N., Phan, T. T. & Do, P. Embedding knowledge on ontology into the corpus by topic to improve the performance of deep learning methods in sentiment analysis. Sci. Rep. 11(1), 23541 (2021).34876635 10.1038/s41598-021-03011-6
15. Dai Y Liu J Zhang J Fu H Xu Z Unsupervised sentiment analysis by transferring multi-source knowledge Cogn. Comput. 2021 13 1185 1197 10.1007/s12559-020-09792-8
Dai, Y., Liu, J., Zhang, J., Fu, H. & Xu, Z. Unsupervised sentiment analysis by transferring multi-source knowledge. Cogn. Comput. 13, 1185–1197 (2021).10.1007/s12559-020-09792-8
16. Chiha R Ayed MB Pereira CD A complete framework for aspect-level and sentence-level sentiment analysis Appl. Intell. 2022 10.1007/S10489-022-03279-9/TABLES/11
Chiha, R., Ayed, M. B. & Pereira, C. D. A complete framework for aspect-level and sentence-level sentiment analysis. Appl. Intell.10.1007/S10489-022-03279-9/TABLES/11 (2022).10.1007/S10489-022-03279-9/TABLES/11
17. Hu J Zhang X Yang Y Liu Y Chen X New doctors ranking system based on VIKOR method Int. Trans. Oper. Res. 2020 27 1236 1261 10.1111/itor.12569
Hu, J., Zhang, X., Yang, Y., Liu, Y. & Chen, X. New doctors ranking system based on VIKOR method. Int. Trans. Oper. Res. 27, 1236–1261 (2020).10.1111/itor.12569
18. Liu N Shen B ReMemNN: A novel memory neural network for powerful interaction in aspect-based sentiment analysis Neurocomputing 2020 395 66 77 10.1016/j.neucom.2020.02.018
Liu, N. & Shen, B. ReMemNN: A novel memory neural network for powerful interaction in aspect-based sentiment analysis. Neurocomputing 395, 66–77 (2020).10.1016/j.neucom.2020.02.018
19. Jin Z Zhao X Liu Y Heterogeneous graph network embedding for sentiment analysis on social media Cogn. Comput. 2021 13 81 95 10.1007/s12559-020-09793-7
Jin, Z., Zhao, X. & Liu, Y. Heterogeneous graph network embedding for sentiment analysis on social media. Cogn. Comput. 13, 81–95 (2021).10.1007/s12559-020-09793-7
20. Zhu F Zhong PA Xu B Wu YN Zhang Y A multi-criteria decision-making model dealing with correlation among criteria for reservoir flood control operation J. Hydroinf. 2016 18 531 543 10.2166/hydro.2015.055
Zhu, F., Zhong, P. A., Xu, B., Wu, Y. N. & Zhang, Y. A multi-criteria decision-making model dealing with correlation among criteria for reservoir flood control operation. J. Hydroinf. 18, 531–543 (2016).10.2166/hydro.2015.055
21. Atandoh P Zhang F Adu-Gyamfi D Atandoh PH Nuhoho RE Integrated deep learning paradigm for document-based sentiment analysis J. King Saud Univ. Comput. Inf. Sci. 2023 35 101578
Atandoh, P., Zhang, F., Adu-Gyamfi, D., Atandoh, P. H. & Nuhoho, R. E. Integrated deep learning paradigm for document-based sentiment analysis. J. King Saud Univ. Comput. Inf. Sci. 35, 101578 (2023).
22. Zuheros C Martínez-Cámara E Herrera-Viedma E Herrera F Sentiment Analysis based Multi-Person Multi-criteria Decision Making methodology using natural language processing and deep learning for smarter decision aid. Case study of restaurant choice using TripAdvisor reviews Inf. Fusion 2021 68 22 36 10.1016/j.inffus.2020.10.019
Zuheros, C., Martínez-Cámara, E., Herrera-Viedma, E. & Herrera, F. Sentiment Analysis based Multi-Person Multi-criteria Decision Making methodology using natural language processing and deep learning for smarter decision aid. Case study of restaurant choice using TripAdvisor reviews. Inf. Fusion 68, 22–36 (2021).10.1016/j.inffus.2020.10.019
23. Punetha N Jain G Bayesian game model based unsupervised sentiment analysis of product reviews Expert Syst. Appl. 2023 214 119128 10.1016/j.eswa.2022.119128
Punetha, N. & Jain, G. Bayesian game model based unsupervised sentiment analysis of product reviews. Expert Syst. Appl. 214, 119128 (2023).10.1016/j.eswa.2022.119128
24. Punetha, N. & Jain, G. Sentiment analysis of stock prices and news headlines using the MCDM framework. In AIST 2022—4th International Conference on Artificial Intelligence and Speech Technology 1–4. 10.1109/AIST55798.2022.10065221 (2022).
25. Punetha N Jain G Integrated Shannon entropy and COPRAS optimal model-based recommendation framework Evol. Intell. 2023 10.1007/s12065-023-00886-4
Punetha, N. & Jain, G. Integrated Shannon entropy and COPRAS optimal model-based recommendation framework. Evol. Intell.10.1007/s12065-023-00886-4 (2023).10.1007/s12065-023-00886-4
26. Vincent TL Brown JS Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics 2005 Cambridge University Press
Vincent, T. L. & Brown, J. S. Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics (Cambridge University Press, 2005).
27. Grant AP Game Theory in Action: An Introduction to Classical and Evolutionary Models 2017 Princeton University Press 131 133
Grant, A. P. Game Theory in Action: An Introduction to Classical and Evolutionary Models 131–133 (Princeton University Press, 2017).
28. Punetha N Jain G Unsupervised sentiment analysis of Hindi reviews using MCDM and game model optimization techniques Sādhanā 2023 48 1 13 10.1007/s12046-023-02255-4
Punetha, N. & Jain, G. Unsupervised sentiment analysis of Hindi reviews using MCDM and game model optimization techniques. Sādhanā 48, 1–13 (2023).10.1007/s12046-023-02255-4
29. Punetha N Jain G Optimizing sentiment analysis: A cognitive approach with negation handling via mathematical modelling Cogn. Comput. 2023 10.1007/s12559-023-10227-3
Punetha, N. & Jain, G. Optimizing sentiment analysis: A cognitive approach with negation handling via mathematical modelling. Cogn. Comput.10.1007/s12559-023-10227-3 (2023).10.1007/s12559-023-10227-3
30. Trevisiol, M., Chiarandini, L. & Baeza-Yates, R. Buon appetito: Recommending personalized menus. In HT 2014—Proceedings of the 25th ACM Conference on Hypertext and Social Media 327–329. 10.1145/2631775.2631784 (2014).
31. Huang F Multi-granular document-level sentiment topic analysis for online reviews Appl. Intell. 2022 52 7723 7733 10.1007/s10489-021-02817-1
Huang, F. et al. Multi-granular document-level sentiment topic analysis for online reviews. Appl. Intell. 52, 7723–7733 (2022).10.1007/s10489-021-02817-1
