Entropy Measures Research Articles

A novel BWM-entropy-COPRAS group decision framework with spherical fuzzy information for digital supply chain partner selection

The transformation and upgrading of traditional supply chain models through digital technology receive widespread attention from the fields of circular economy, manufacturing, and sustainable development. Enterprises need to choose a digital supply chain partner (DSCP) during the process of digital transformation in uncertain and sustainable environments. Thus, the research constructs an innovative decision methodology for selecting the optimal DSCP to achieve digital transformation. The proposed methodology is propounded based upon the entropy measure, generalized Dombi operators, integrated weight-determination model, and complex proportional assessment (COPRAS) method under spherical fuzzy circumstances. Specifically, a novel entropy measure is proposed for measuring the fuzziness of spherical fuzzy (SF) sets, while generalized Dombi operators are presented for fusing SF information. The related worthwhile properties of these operators are discussed. Further, an integrated criteria weight-determination model is presented by incorporating objective weights obtained from the SF entropy-based method and subjective weights from the SF best worst method. Afterward, an improvement of the COPRAS method is proposed based on the presented generalized Dombi operators with SF information. Lastly, the practicability and validity of the proposed methodology are verified by an empirical study that selects an appropriate DSCP for a new energy vehicle enterprise to finish the goal of digital transformation. The sensitivity and comparative analysis are carried out to illustrate the stability, reliability, and superiority of the propounded methodology from multiple perspectives. The results and conclusions indicate that the propounded method affords a synthetic and systematic uncertain decision-making framework for identifying the optimal DSCP with incomplete weight information.

A link of extropy to entropy for continuous random variables via the generalized ϕ–entropy

The concepts of entropy and divergence, along with their past, residual, and interval variants are revisited in a reliability theory context and generalized families of them that are based on ϕ-functions are discussed. Special emphasis is given in the parametric family of entropies and divergences of Cressie and Read. For non-negative and absolutely continuous random variables, the dual to Shannon entropy measure of uncertainty, the extropy, is considered and its link to a specific member of the ϕ-entropies family is shown. A number of examples demonstrate the implementation of the generalized entropies and divergences, exhibiting their utility.

On the higher-order smallest ring-star network of Chialvo neurons under diffusive couplings.

Network dynamical systems with higher-order interactions are a current trending topic, pervasive in many applied fields. However, our focus in this work is neurodynamics. We numerically study the dynamics of the smallest higher-order network of neurons arranged in a ring-star topology. The dynamics of each node in this network is governed by the Chialvo neuron map, and they interact via linear diffusive couplings. This model is perceived to imitate the nonlinear dynamical properties exhibited by a realistic nervous system where the neurons transfer information through multi-body interactions. We deploy the higher-order coupling strength as the primary bifurcation parameter. We start by analyzing our model using standard tools from dynamical systems theory: fixed point analysis, Jacobian matrix, and bifurcation patterns. We observe the coexistence of disparate chaotic attractors. We also observe an interesting route to chaos from a fixed point via period-doubling and the appearance of cyclic quasiperiodic closed invariant curves. Furthermore, we numerically observe the existence of codimension-1 bifurcation points: saddle-node, period-doubling, and Neimark-Sacker. We also qualitatively study the typical phase portraits of the system, and numerically quantify chaos and complexity using the 0-1 test and sample entropy measure, respectively. Finally, we study the synchronization behavior among the neurons using the cross correlation coefficient and the Kuramoto order parameter. We conjecture that unfolding these patterns and behaviors of the network model will help us identify different states of the nervous system, further aiding us in dealing with various neural diseases and nervous disorders.

Transcendental properties of entropy-constrained sets II

In this work, we address the question of the impossibility of certain single-letter formulas by exploiting the semi-algebraic nature of various entropy-constrained sets. The focus lies on studying the properties of the level sets of relative entropy, mutual information, and Rényi entropies. We analyze the transcendental structure of the set of states in which one of the aforementioned entropy quantities is fixed. Our results rule out (semi)algebraic single-shot characterizations of these entropy measures with bounded ancilla for both the classical and quantum cases.

A NEW INVERSE TWO-PARAMETER LINDLEY DISTRIBUTION AND ITS APPLICATION

This paper proposes an inverse two-parameter Lindley distribution by utilizing the one and two parameter Lindley distributions. The key properties of the novel distribution like, its survival function, shape characteristics of the density, entropy measure, hazard rate function, stochastic ordering and stress-strength reliability were examined. Two data sets were employed in the empirical studies. Method of maximum likelihood was used to estimate the parameters. The goodness- of -fit was accessed using the HQIC, BIC, CAIC, and AIC. The proposed distribution was compared with the inverse Lindley and the inverse Akash distributions in order to access its superiority over the two distributions. Ultimately, the new inverse two parameter Lindley distribution was found to be superior by providing a better fit.

Renewable energy resource management using an integrated robust decision making model under entropy and similarity measures of fuzzy hypersoft set

The demand for renewable energy has significantly increased over the last decade with increased attention to the preservation of the environment and sustainable, optimal resource management. As traditional sources of energy production are depleting at an alarming rate and causing long-lasting environmental damage, it is essential to explore green and cost-effective methodologies for meeting energy demand. With each country having different geographical, political, social, and natural factors, the problem arises of which renewable energy should be utilized for optimal resource management. This multi-criteria decision making (MCDM) challenge is tackled by applying a dynamic fuzzy hypersoft set-based Method for the evaluation of currently deployed Renewable Energy systems and providing a decision support system for the installation of new ones based on the factors mentioned above for Turkey. As the installation of new renewable energy projects and the evaluation of old ones is significantly influenced by human judgment, it leaves great room for uncertainty primarily because of the psychological factors of the expert. The novel concept of Fuzzy Hypersoft Sets (FHSs) and their Entropy (EN) and TOPSIS-based operations are first discussed with reference to the problem at hand. The presented structure is superior to the ones in the literature by allowing access to data parameters as sub-parametric values while utilizing the versatility of Fuzzy structures to deal with uncertainty. The technique has great potential to serve as a potential decision support system in any setting. For now, hypothetical expert ratings are used to illustrate the working of the dynamic structure along with a sensitivity analysis to investigate the primary criterion weights in sorting. The evaluation of currently deployed renewable energy systems using our methodology revealed an average improvement in system performance compared to traditional methods. Furthermore, the decision support system for the installation of new projects based on geographical, political, social, and natural factors exhibited a potential increase in overall system efficiency. These numeric outcomes highlight the effectiveness and practical applicability of our approach in optimizing resource management and fostering sustainable energy practices.

Response to Discretising and validating Keyfitz' entropy for any demographic classification

Abstract In recent years, demographers have parsed variation in survivorship into two distinct components: the shape and the pace of ageing. The pace of ageing is defined by measures of longevity such as mean life expectancy or maximum longevity, whereas measures of the shape of ageing attempt to classify different shapes of the survivorship curve. We recently published a paper pointing out that a commonly used discretization of a shape measure, Keyfitz' entropy, does not correctly classify survivorship curves into negatively and positively senescing curves (de Vries et al., 2023). In that paper, we also suggested an alternative, accurate discrete‐time version of Keyfitz' entropy. de Vries et al. (2023) ended with two open questions, both of which have been answered by Giaimo (2024) now: (1) Can a discrete‐time entropy measure of survivorship be generalized beyond age‐based population models? Giaimo (2024) introduce a new formula that achieves this. (2) Will a discretization of Keyfitz' derivation of his measure as the elasticity of lifespan to a uniform change in mortality lead to the same formula as a discretization of Keyfitz' result? Giaimo (2024) answers: no. Here, we briefly discuss the implications of the results obtained by Giaimo (2024), and the implementation of his new formula into the Rpackage Rage, which is widely used for comparative demographic studies. We showcase the strength of the new method by reanalysing a comparison of Keyfitz' entropy to another shape measure. We find that the comparison is significantly altered by using Giaimo's new Keyfitz' formula. This example strengthens Giaimo's (2004) words of warning in approaching discretizations with attentiveness.

Longitudinal detection of gait alterations associated with hypertension-induced cerebral microhemorrhages in mice: predictive role of stride length and stride time asymmetry and increased gait entropy

Cerebral microhemorrhages (CMHs) are of paramount importance as they not only signify underlying vascular pathology but also have profound implications for cognitive function and neurological health, serving as a critical indicator for the early detection and management of vascular cognitive impairment (VCI). This study aimed to investigate the effects of hypertension-induced CMHs on gait dynamics in a mouse model, focusing on the utility of advanced gait metrics as sensitive indicators of subclinical neurological alterations associated with CMHs. To induce CMHs, we employed a hypertensive mouse model, using a combination of Angiotensin II and L-NAME to elevate blood pressure, further supplemented with phenylephrine to mimic transient blood pressure fluctuations. Gait dynamics were analyzed using the CatWalk system, with emphasis on symmetry indices for Stride Length (SL), Stride Time (ST), and paw print area, as well as measures of gait entropy and regularity. The study spanned a 30-day experimental period, capturing day-to-day variations in gait parameters to assess the impact of CMHs. Temporary surges in gait asymmetry, detected as deviations from median gait metrics, suggested the occurrence of subclinical neurological signs associated with approximately 50% of all histologically verified CMHs. Our findings also demonstrated that increases in gait entropy correlated with periods of increased gait asymmetry, providing insights into the complexity of gait dynamics in response to CMHs. Significant correlations were found between SL and ST symmetry indices and between these indices and the paw print area symmetry index post-hypertension induction, indicating the interdependence of spatial and temporal aspects of gait affected by CMHs. Collectively, advanced gait metrics revealed sensitive, dynamic alterations in gait regulation associated with CMHs, resembling the temporal characteristics of transient ischemic attacks (TIAs). This underscores their potential as non-invasive indicators of subclinical neurological impacts. This study supports the use of detailed gait analysis as a valuable tool for detecting subtle neurological changes, with implications for the early diagnosis and monitoring of cerebral small vessel disease (CSVD) in clinical settings.

Investigating Cortical Complexity in Mixed Dementia through Nonlinear Dynamic Analyses: A Resting-State EEG Study.

Objective: Dementia is a broad term referring to a decline in problem-solving abilities, language skills, memory, and other cognitive functions to a degree that it significantly disrupts everyday activities. The underlying cause of dementia is the impairment or loss of nerve cells and their connections within the brain. The particular symptoms experienced are contingent upon specific regions of the brain affected by this damage. In this research, we aimed to investigate the nonlinear dynamics of the mixed demented brain compared to healthy subjects using electroencephalogram (EEG) analysis. Method : For this purpose, EEG was recorded from 66 patients with mixed dementia and 65 healthy subjects during rest. After signal preprocessing, sample entropy and Katz fractal dimension analyses were applied to the preprocessed EEG data. Analysis of variance with repeated measures was utilized to compare the nonlinear dynamics of brain activity between dementia and healthy states and partial correlation analysis was employed to explore the relationship between EEG complexity measures and cognitive and neuropsychiatric symptoms of patients. Results: Based on repeated measures ANOVA, there was a significant main effect between groups for both Katz fractal dimension (F = 4.10, P = 0.01) and sample entropy (F = 4.81, P = 0.009) measures. Post hoc comparisons revealed that EEG complexity was significantly reduced in dementia mainly in the occipitoparietal and temporal areas (P < 0.05). MMSE scores were positively correlated with EEG complexity measures, while NPI scores were negatively correlated with EEG complexity measures, mainly in the occipitoparietal and temporal areas (P < 0.05). Moreover, using a KNN classifier, all significant complexity measures yielded the best classification performance with an accuracy of 98.05%, sensitivity of 97.03% and specificity of 99.16% in detecting dementia. Conclusion: This study demonstrated a unique dynamic system within the brain impacted by dementia that results in more predictable patterns of cortical activity mainly in the occipitoparietal and temporal areas. These abnormal patterns were associated with patients' cognitive capacity and neuropsychiatric symptoms.

Pattern formation by the drying of saline droplets on pillars

The application of saline solutions for surface coating is pertinent across multiple biomedical fields, various technological sectors, and industries, including agriculture. The drying of salt solution droplets is key to understanding and controlling morphological structures on surfaces. In this paper, we report the study of pattern formation from the evaporation of saline drops (NaCl, CsCl, and KCl) placed on pillars. Our findings indicate that regardless of saline concentration and type, the drying process can be categorized into three modes: stable, metastable, and unstable. In both the stable and metastable modes, the droplet fixes to the pillar during the drying process and ensuing pattern formation; however, the crucial distinction lies in the metastable mode, where the droplet additionally undergoes mass loss through fluid drainage from the walls of the pillars. Conversely, in the unstable drying mode, the droplet undergoes rapid collapse, leading to a substantial loss of mass. The distinct drying modes dictate the resulting patterns on the pillars. We employ measurements of configurational entropy and fractal dimension as quantitative metrics to assess the complexity, reproducibility, and similarity of patterns. The texture analysis reveals that, at low concentrations of both CsCl and KCl, significant differences emerge among the patterns produced by the different drying modes, especially highlighting differences in patterns of the stable drying mode. Finally, we analyze fertilizer deposition patterns to prove that all three drying modes can occur in complex fluids.

A Novel Three-Parameter Nadarajah Haghighi Model: Entropy Measures, Inference, and Applications

The fitting and modeling of skewed, complex, symmetric, and asymmetric datasets is an exciting research topic in many fields of applied sciences: notably, lifetime, medical, and financial sciences. This paper introduces a heavy-tailed Nadarajah Haghighi model by compounding the heavy-tailed family and Nadarajah Haghighi distribution. The model obtained has three parameters that account for the scale and shape of the distribution. The proposed distribution’s fundamental characteristics, such as the probability density, cumulative distribution, hazard rate, and survival functions, are provided, several key statistical properties are established, and several entropy information measures are proposed. Estimation of model parameters is performed via a maximum likelihood estimator procedure. Further, different simulation experiments are conducted to demonstrate the proposed estimator’s performance using measures like the average estimate, the average bias, and the associated mean square error. Finally, we apply our proposed model to analyze three different real datasets. In our illustration, we compare the practicality of the recommended model with several well-known competing models.

CEO duality and corporate social responsibility: A moderation effect of founder CEO

With a panel of 6613 firm-year observations of NSE-listed firms during 2015–2020, we construct a corporate social responsibility (CSR) engagement score based on entropy measures to analyse the impact of chief executive officer (CEO) duality on CSR practices in India. This study shows that CEO duality negatively influences the CSR practices of the firm. We also examine the moderation impact of the founder status of CEO on the relationship between CEO duality and CSR practices and find a positive moderation effect. We have also checked the validity of CEO duality and CSR relationship under high and low external and internal monitoring. The results are in line with agency theory and the Uday Kotak committee and suggest the non-duality structure for strong corporate governance. We also extend the study by analysing the relationship between CEO duality and CSR practices across different business life cycles. The result shows that the duality structure of mature and declining firms negatively influences the CSR practices of the firm compared to introduction, growth, and shakeout firms. This result suggests that CEO duality of mature and declining firms utilize their power to maximise available resources for their personal benefits at the cost of other stakeholders. Further, the findings are robust to applying propensity score matching, two stage least squares, and three stage least squares.

A novel multi-criteria group decision making algorithm for enhancing supply chain efficiency under high uncertainty during crisis based on q-rung orthopair fuzzy information

In recent years, q-rung orthopair fuzzy sets (q-ROFSs) have been becoming gradually more advantageous in handling information with high uncertainty. Several multi-criteria group decision-making (MCGDM) algorithms under q-ROFS information have been proposed in literature and used to solve various real-life problems. However, several shortcomings of some existing MCGDM algorithms under certain circumstances also emerged which limit their applicability. To overcome these challenges and deal with information under high uncertainty more accurately and reasonably, we propose a novel MCGDM algorithm under q-ROF information, i.e., (q-ROF-MCGDM), that retains the advantages of currently available methods, but extend its applicability by introducing a new q-rung orthopair fuzzy weighted averaging aggregation operator (q-ROFWAAO) along with a new entropy measure. The proposed q-ROF-MCGDM algorithm involves the steps: firstly, the input requirement in terms of alternatives, criteria and expert evaluations are required. Secondly, aggregating expert opinions using weights and normalizing decision matrices, and finally, calculating entropy weights and aggregating overall evaluations for final prioritization. Further, new operational laws have been developed and several necessary properties of the proposed q-ROFWAAO are also proved. Moreover, a sensitivity and comparative analysis has been carried out for validity and effectiveness of the proposed q-ROF-MCGDM algorithm. The proposed q-ROF-MCGDM algorithm has been implemented to enhance the efficiency of the United Arab Emirates (UAE) food industry under high uncertainty during the recent crisis. Finally, an evaluation of identifying and prioritizing the most severe food SC disruptions and appropriate mitigation strategies under crisis is provided to demonstrate applicability of the proposed q-ROF-MCGDM algorithm, and the obtained real case results confirm its usefulness. Findings of the study offer valuable insights to both food industry researchers and managers in developing effective recovery strategies, mitigating risks, and improving overall efficiency to ensure the survival of food businesses during the crisis.

Characterizing Complex Spatiotemporal Patterns from Entropy Measures.

In addition to their importance in statistical thermodynamics, probabilistic entropy measurements are crucial for understanding and analyzing complex systems, with diverse applications in time series and one-dimensional profiles. However, extending these methods to two- and three-dimensional data still requires further development. In this study, we present a new method for classifying spatiotemporal processes based on entropy measurements. To test and validate the method, we selected five classes of similar processes related to the evolution of random patterns: (i) white noise; (ii) red noise; (iii) weak turbulence from reaction to diffusion; (iv) hydrodynamic fully developed turbulence; and (v) plasma turbulence from MHD. Considering seven possible ways to measure entropy from a matrix, we present the method as a parameter space composed of the two best separating measures of the five selected classes. The results highlight better combined performance of Shannon permutation entropy (SHp) and a new approach based on Tsallis Spectral Permutation Entropy (Sqs). Notably, our observations reveal the segregation of reaction terms in this SHp×Sqs space, a result that identifies specific sectors for each class of dynamic process, and it can be used to train machine learning models for the automatic classification of complex spatiotemporal patterns.

Assessing resiliency in scale-free supply chain networks: a stress testing approach based on entropy measurements and value-at-risk analysis

Supply chain (SC) resiliency and risk management have garnered increasing attention recently. While several studies have explored the use of scale-free network models to design and optimise SC networks, there remains a lack of a generalised stress-testing method that can be applied to various types and sizes of SCs. To address this, we propose a novel approach to assess the resiliency of scale-free SC networks based on entropy measurements. By considering various sources of disruptions, including demand fluctuations, capacity constraints, and transportation disruptions, we create a stress test that measures the network's resiliency. Our method categorises the SC into plants, ports, and customers and analyses the impact of disruptions on individual nodes within similar categories. We utilise real-world SC logistics data and map it into a scale-free network representation. By calculating the global vulnerability of the entire SC network and the localised impact of node vulnerability, we gain insights into how disruptions propagate and affect interconnected nodes. We use value-at-risk (VaR) and conditional-VaR (CVaR) to identify high-risk nodes, highlighting nodes at risk of extreme performance fluctuations or failures. The analytical results have led to developing six propositions, which serve as references for practitioners and help generalise the findings.

Dendropsophus minutus repertoire complexity and its relationship with environmental traits

ABSTRACT Anurans have several characteristics in their acoustic communication that distinguish them. These characteristics are influenced by body-related, environmental and social variables. However, the complexity of anuran repertoire and the effect of such variables on it are still understudied. Dendropsophus minutus is a Neotropical treefrog that has a wide repertoire composed of three notes, grouped in at least 71 unique combinations. To quantify complexity, we used the entropy measurement, since it measures the randomness of a sequence. Therefore, we could test if repertoire complexity in D. minutus is related with body condition, social context, and environmental variables. We recorded vocalisations of 80 males in three sites and found 44 unique note combinations not described before, increasing the species’ repertoire to 115 combinations. We observed that the species has near maximum entropy for note frequency, but a low entropy for note sequencies, showing a clear a bias for some note combinations. We found a positive relationship between repertoire complexity and air temperature. Thus, based on our results, the complexity of this species’ repertoire is an additional trait influenced by climate, and should be considered in future studies.

Divergent functional connectivity changes associated with white matter hyperintensities

Age-related white matter hyperintensities are a common feature and are known to be negatively associated with structural integrity, functional connectivity, and cognitive performance. However, this has yet to be fully understood mechanistically. We analyzed multiple MRI modalities acquired in 465 non-demented individuals from the Swedish BioFINDER study including 334 cognitively normal and 131 participants with mild cognitive impairment. White matter hyperintensities were automatically quantified using fluid-attenuated inversion recovery MRI and parameters from diffusion tensor imaging were estimated in major white matter fibre tracts. We calculated fMRI resting state-derived functional connectivity within and between predefined cortical regions structurally linked by the white matter tracts. How change in functional connectivity is affected by white matter lesions and related to cognition (in the form of executive function and processing speed) was explored. We examined the functional changes using a measure of sample entropy. As expected hyperintensities were associated with disrupted structural white matter integrity and were linked to reduced functional interregional lobar connectivity, which was related to decreased processing speed and executive function. Simultaneously, hyperintensities were also associated with increased intraregional functional connectivity, but only within the frontal lobe. This phenomenon was also associated with reduced cognitive performance. The increased connectivity was linked to increased entropy (reduced predictability and increased complexity) of the involved voxels’ blood oxygenation level-dependent signal. Our findings expand our previous understanding of the impact of white matter hyperintensities on cognition by indicating novel mechanisms that may be important beyond this particular type of brain lesions.

The General Theory of Scientific Variability for Technological Evolution

The proposed general theory of scientific variability for technological evolution explains one of the drivers of technological change for economic progress in human society. Variability is the predisposition of the elements in systems to assume different values over time and space. In biology, the variability is basic to explaining differences and development in organisms. In economics of technical change, the effects of variability within research fields on evolutionary dynamics of related technologies are unknown. In a broad analogy with the principles of biology, suggested theoretical framework here can clarify a basic driver of technological evolution: the variability within research fields can explain the dynamics of scientific development and technological evolution. The study sees whether statistical evidence supports the hypothesis that the rate of growth of scientific and technological fields can be explained by the level of variability within scientific fields. The validation is based on emerging research fields in quantum technologies: quantum imaging, quantum meteorology, quantum sensing, and quantum optics. Statistical evidence seems in general to support the hypothesis stated that the rate of growth can be explained by the level of scientific variability within research fields, measured with the relative entropy (indicating the dispersion of scientific topics in a research field underlying a specific technology). Nonparametric correlation with Spearman’s rho shows a positive coefficient of 0.80 between entropy measures and rates of growth between scientific and technological fields. The linear model of the relation between rate of growth and scientific variability reveals a coefficient of regression equal to 1.63 (R2 = 0.60). The findings here suggest a general law that variability within research fields positively drives scientific development and technological evolution. In particular, a higher variability within research fields can support a high rate of growth in scientific development and technological evolution. The proposed general theory of scientific variability is especially relevant in turbulent environments of technology-based competition to clarify a basic determinant of technological development to design strategies of technological forecasting and management of promising innovations.

Is EEG Entropy a Useful Measure for Alzheimer's Disease?

The number of individuals diagnosed with Alzheimer's disease (AD) has increased, and it is estimated to continue rising in the coming years. The diagnosis of this disease is challenging due to variations in onset and course, its diverse clinical manifestations, and the indications for measuring deposit biomarkers. Hence, there is a need to develop more precise and less invasive diagnostic tools. Multiple studies have considered using electroencephalography (EEG) entropy measures as an indicator of the onset and course of AD. Entropy is deemed suitable as a potential indicator based on the discovery that variations in its complexity can be associated with specific pathologies such as AD. Following PRISMA guidelines, a literature search was conducted in 4 scientific databases, and 40 articles were analyzed after discarding and filtering. There is a diversity in entropy measures; however, Sample Entropy (SampEn) and Multiscale Entropy (MSE) are the most widely used (21/40). In general, it is found that when comparing patients with controls, patients exhibit lower entropy (20/40) in various areas. Findings of correlation with the level of cognitive decline are less consistent, and with neuropsychiatric symptoms (2/40) or treatment response less explored (2/40), although most studies show lower entropy with greater severity. Machine learning-based studies show good discrimination capacity. There is significant difficulty in comparing multiple studies due to their heterogeneity; however, changes in Multiscale Entropy (MSE) scales or a decrease in entropy levels are considered useful for determining the presence of AD and measuring its severity.

Enhancing Financial Time Series Prediction with Quantum-Enhanced Synthetic Data Generation: A Case Study on the S&amp;P 500 Using a Quantum Wasserstein Generative Adversarial Network Approach with a Gradient Penalty

This study introduces a novel Quantum Wasserstein Generative Adversarial Network approach with a Gradient Penalty (QWGAN-GP) model that leverages a quantum generator alongside a classical discriminator to synthetically generate time series data. This approach aims to accurately replicate the statistical properties of the S&amp;P 500 index. The synthetic data generated by this model were compared to the original series using various metrics, including Wasserstein distance, Dynamic Time Warping (DTW) distance, and entropy measures, among others. The outcomes demonstrate the model’s robustness, with the generated data exhibiting a high degree of fidelity to the statistical characteristics of the original data. Additionally, this study explores the applicability of the synthetic time series in enhancing prediction models. An LSTM (Long-Short Term Memory)-based model was developed to evaluate the impact of incorporating synthetic data on forecasting accuracy, particularly focusing on general trends and extreme market events. The findings reveal that models trained on a mix of synthetic and real data significantly outperform those trained solely on historical data, improving predictive performance.