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We deal with Tsallis entropy in MV-algebraic dynamical systems. We prove that Tsallis entropy is a submeasure and that it is invariant under isomorphisms. We also provide two examples which show that Tsallis entropy allows one to distinguish some non-isomorphic MV-dynamical systems. [ABSTRACT FROM AUTHOR]

This paper considers the only sectors to constitute the present universe as the dark matter and dark energy. The proposed new agegraphic dark energy model studies the gradual transition from matter to dark energy in sequential order, including the current dominance of the matter sector by dark energy. The model utilizes concepts of Tsallis entropy and the Karolyhazy relation with conformal time η as the time scale. Various evolutionary aspects of the universe are characterized by Tsallis parameter δ and the model constant c. Expressions for dynamic behavior of dark energy, as a differential equation, equation of state and deceleration parameters are obtained to describe the thermal history of universe evolution. Also, the stability of the model is analyzed through squared sound speed parameter. The analysis is made by considering an interaction as well as independence among the constituent sectors of the universe. [ABSTRACT FROM AUTHOR]

In generalized logit dynamics, optimal probabilistic actions of agents in iterative games with perturbed utilities are described using partial integro-differential equations. Any equilibrium of a (generalized) logit dynamic is usually expected to converge to a Nash equilibrium of the original iterative game as the perturbation approaches zero. It has also been pointed out that using different perturbations, and in particular different entropies, approximates different equilibria. We explain this phenomenon by focusing on the tail behavior of the logit functions and demonstrate mathematically and numerically that different equilibria are obtained for different q-exponential type logit functions for different values of q. We also demonstrate that the generalized logit dynamic admits a unique measure-valued solution with theoretical estimates of the difference between the solutions corresponding to different values of q. Finally, we apply generalized logit dynamics to a problem inspired by hydropower generation and fishery resource management, both of which are key to sustainable and comprehensive development, to analyze the differences between different dynamics. [ABSTRACT FROM AUTHOR]

Current inspection strives to perform a detailed elaboration of the Tsallis entropy‐based magnetocaloric effect (MCE) of GaAs quantum dot (QD) doped with Gaussian impurity and under applied Gaussian white noise. Noise takes additive and multiplicative pathways for its entry to the doped QD. MCE has been found to decrease following the enhancement of temperature. The enquiry manifests magnetic field‐induced confinement and consequent decline in the disorder of the system during the gradual variation of several physical parameters. However, harnessing the extent of above confinement involves extremely sensitive and delicate control over the resultant influence of the particular physical parameter undergoing change, its range of magnitude, application of noise and its mode of inclusion. Importantly, MCE has been found to maximize only during the change in the strength of the dopant potential under multiplicative noise. And, on most occasions, multiplicative noise profoundly raises the MCE over its value under the state devoid of noise and under additive noise. [ABSTRACT FROM AUTHOR]

Shannon differential entropy is extensively applied in the literature as a measure of dispersion or uncertainty. Nonetheless, there are other measurements, such as the cumulative residual Tsallis entropy (CRTE), that reveal interesting effects in several fields. Motivated by this, we study and compute Tsallis measures for the concomitants of the generalized order statistics (CGOS) from the iterated Farlie-Gumbel-Morgenstern (IFGM) bivariate family. Some newly introduced information measures are also being considered for CGOS within the framework of the IFGM family, including Tsallis entropy, CRTE, and an alternative measure of CRTE of order $ \eta $. Applications of these results are given for order statistics and record values with uniform, exponential, and power marginals distributions. In addition, the empirical cumulative Tsallis entropy is suggested as a method to calculate the new information measure. Finally, a real-world data set has been analyzed for illustrative purposes, and the performance is quite satisfactory.

Tsallis entropy has been widely used in image thresholding because of its non-extensive properties. The non-extensive parameter q contained in this entropy plays an important role in various adaptive algorithms and has been successfully applied in bi-level image thresholding. In this paper, the relationships between parameter q and pixels' long-range correlations have been further studied within multi-threshold image segmentation. It is found that the pixels' correlations are remarkable and stable for images generated by a known physical principle, such as infrared images, medical CT images, and color satellite remote sensing images. The corresponding non-extensive parameter q can be evaluated by using the self-adaptive Tsallis entropy algorithm. The results of this algorithm are compared with those of the Shannon entropy algorithm and the original Tsallis entropy algorithm in terms of quantitative image quality evaluation metrics PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity). Furthermore, we observed that for image series with the same background, the q values determined by the adaptive algorithm are consistently kept in a narrow range. Therefore, similar or identical scenes during imaging would produce similar strength of long-range correlations, which provides potential applications for unsupervised image processing. [ABSTRACT FROM AUTHOR]

Tsallis entropy is a flexible device to measure indeterminacy of uncertain sets. A formula is obtained to calculate Tsallis entropy of uncertain sets via inversion of membership functions. Also, by considering Tsallis entropy as a risk measure, we optimize portfolio selection problems via mean-entropy models. [ABSTRACT FROM AUTHOR]

Shannon differential entropy is extensively applied in the literature as a measure of dispersion or uncertainty. Nonetheless, there are other measurements, such as the cumulative residual Tsallis entropy (CRTE), that reveal interesting effects in several fields. Motivated by this, we study and compute Tsallis measures for the concomitants of the generalized order statistics (CGOS) from the iterated Farlie-Gumbel-Morgenstern (IFGM) bivariate family. Some newly introduced information measures are also being considered for CGOS within the framework of the IFGM family, including Tsallis entropy, CRTE, and an alternative measure of CRTE of order η. Applications of these results are given for order statistics and record values with uniform, exponential, and power marginals distributions. In addition, the empirical cumulative Tsallis entropy is suggested as a method to calculate the new information measure. Finally, a real-world data set has been analyzed for illustrative purposes, and the performance is quite satisfactory [ABSTRACT FROM AUTHOR]

The Petrinja earthquake sequence started on December 28, 2020, with a destructive ML 6.2 mainshock occurring in the area, preceded by a ML 5.08 foreshock, following a long period of relative seismic quiescence. Over the first six months of the Petrinja earthquake sequence, almost 14,000 events were recorded. In the present work, we separated seismic events based on their spatial concentration using a density-based clustering algorithm, DBSCAN. We identified four main clusters and analyzed their spatiotemporal properties using the notions of Non-Extensive Statistical Physics (NESP). This framework, which relies on Tsallis entropy (Sq), describes the scaling behavior of complex systems. In this frame, we investigated the inter-event time (T) and distance (D) distributions, providing the qT and qD entropic parameters, respectively. Additionally, we studied the frequency–magnitude distributions in terms of the fragment–asperity model, leading to the determination of the non-extensive parameter qM. The results of the analysis suggest that the statistical properties of the Petrinja earthquake sequence can be effectively reproduced utilizing NESP. Furthermore, the coseismic static Coulomb stress changes were estimated, indicating that the clusters' seismic events may have resulted from a complex fault system's (re)activation. In addition, the effective static stress drop was estimated for each spatial cluster. Lastly, the temporal patterns of the earthquake evolution are discussed using the superstatistics approach, indicating that the temporal progression of the Petrinja earthquake clusters is governed by a very low number of degrees of freedom, highlighting the spatiotemporal organization of each cluster. [ABSTRACT FROM AUTHOR]

A universal method is proposed for performing calculations within the framework of generalized statistics generated by the parametric Tsallis, Rényi, and Sharma–Mittal entropies. The essence of the approach lies in the use of an auxiliary gamma distribution whose parameters correspond to a particular variant of the statistics. Equations are derived that allow the generalised partition function and the mean energy to be expressed in terms of canonical quantities. The effectiveness of the proposed method is demonstrated using the example of Rényi statistics. The Maxwell–Rényi distribution is obtained and its properties are calculated, based on which assumptions about the possible nature of the generalised parameter are formulated. [ABSTRACT FROM AUTHOR]

Core distributions of Maximum Entropy Theory of Ecology (METE) are the Spatial Structure Function (SSF) and the Ecosystem Structure Function (ESF). SSF is a by-species prediction of the clustering of individuals over space. ESF is a kind of container function that describes the probability space of how abundances are assigned to species and how metabolic energy is partitioned over individuals in a community. In this study, these core functions of METE are generalized by deriving the corresponding functions in the Tsallis q-entropy. Derivation used the method of Lagrange multipliers. The generalized SSF and ESF are expressed in terms of the q-exponential function. Numerical examples are provided to illustrate the generalized SSF. [ABSTRACT FROM AUTHOR]

AbstractThe 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. [ABSTRACT FROM AUTHOR]

Mexico is a well-known seismically active country, which is primarily affected by several tectonic plate interactions along the southern Pacific coastline and by active structures in the Gulf of California. In this paper, we investigate this seismicity using the classical Gutenberg–Richter (GR) law and a non-extensive statistical approach based on Tsallis entropy. The analysis is performed using data from the corrected Mexican seismic catalog provided by the National Seismic Service, spanning the period from January 2000 to October 2023, and unlike previous work, it includes six different regions along the entire west coastline of Mexico. The Gutenberg–Richter law fitting to the earthquake sub-catalogs for all six regions studied indicates magnitudes of completeness between 3.30 and 3.76, implying that the majority of seismic movements occur for magnitudes less than 4. The cumulative distribution of earthquakes as derived from the Tsallis entropy was fitted to the corrected catalog data to estimate the q-entropic index for all six regions, which for values greater than one is a measure of the non-extensivity (i.e., non-equilibrium) of the system. All regions display values of the entropic index in the range 1.52 ≲ q ≲ 1.61 , slightly lower than previously estimated ( 1.54 ≲ q ≲ 1.70 ) using catalog data from 1988 to 2010. The reason for this difference is related to the use of modern recording devices, which are sensitive to the detection of a larger number of low-magnitude events compared to older instrumentation. [ABSTRACT FROM AUTHOR]

Due to its capacity to unveil the dynamic characteristics of time series data, entropy has attracted growing interest. However, traditional entropy feature extraction methods, such as permutation entropy, fall short in concurrently considering both the absolute amplitude information of signals and the temporal correlation between sample points. Consequently, this limitation leads to inadequate differentiation among different time series and susceptibility to noise interference. In order to augment the discriminative power and noise robustness of entropy features in time series analysis, this paper introduces a novel method called Tsallis entropy-based complexity-improved permutation entropy casualty plane (TC-IPE-CP). TC-IPE-CP adopts a novel symbolization approach that preserves both absolute amplitude information and inter-point correlations within sequences, thereby enhancing feature separability and noise resilience. Additionally, by incorporating Tsallis entropy and weighting the probability distribution with parameter q, it integrates with statistical complexity to establish a feature plane of complexity and entropy, further enriching signal features. Through the integration of multiscale algorithms, a multiscale Tsallis-improved permutation entropy algorithm is also developed. The simulation results indicate that TC-IPE-CP requires a small amount of data, exhibits strong noise resistance, and possesses high separability for signals. When applied to the analysis of heart rate signals, fault diagnosis, and underwater acoustic signal recognition, experimental findings demonstrate that TC-IPE-CP can accurately differentiate between electrocardiographic signals of elderly and young subjects, achieve precise bearing fault diagnosis, and identify four types of underwater targets. Particularly in underwater acoustic signal recognition experiments, TC-IPE-CP achieves a recognition rate of 96.67%, surpassing the well-known multi-scale dispersion entropy and multi-scale permutation entropy by 7.34% and 19.17%, respectively. This suggests that TC-IPE-CP is highly suitable for the analysis of complex time series. [ABSTRACT FROM AUTHOR]

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 ( S H p ) and a new approach based on Tsallis Spectral Permutation Entropy ( S q s ). Notably, our observations reveal the segregation of reaction terms in this S H p × S q s 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. [ABSTRACT FROM AUTHOR]

A methodology to study statistical properties of anomalous transport in fusion plasma is investigated. Three time traces generated by the full-f gyrokinetic code GKNET are analyzed for this purpose. The time traces consist of heat flux as a function of the radial position, which is studied in a novel manner using statistical methods. The simulation data exhibit transport processes with both medium and long correlation length along the radius. A typical example of a phenomenon with long correlation length is avalanches. In order to investigate the evolution of the turbulent state, two basic configurations are studied, one flux-driven and one gradient-driven with decaying turbulence. The information length concept in tandem with Boltzmann–Gibbs and Tsallis entropy is used in the investigation. It is found that the dynamical states in both flux-driven and gradient-driven cases are surprisingly similar, but the Tsallis entropy reveals differences between them. This indicates that the types of probability distribution function are nevertheless quite different since the higher moments are significantly different. [ABSTRACT FROM AUTHOR]

In this article, we focus on the inflation dynamics of the early Universe using an inverse power law potential scalar field (V(ϕ) = V0ϕ-n) within the framework of Tsallis entropy. First, we derive the modified Friedmann equations from the non-additive Tsallis entropy by applying the first law of thermodynamics to the apparent horizon of the Friedmann-Robertson-Walker (FRW) Universe. We assume that the inflationary era of the Universe consists of two phases; the slow roll inflation phase and the kinetic inflation phase. We obtained the scalar spectral index ηs and tensor-to-scalar ratio r and compared our results with the latest Planck data for these phases. By choosing the appropriate values for the Tsallis parameters, which bounded by α < 2, and the inverse power-term of the potential η, we determined that the inflation era of the Universe in Tsallis entropy can only occur in the second phase (kinetic inflation), while the slow-roll inflation phase is incompatible with the Planck data. [ABSTRACT FROM AUTHOR]

Measures of uncertainty are a topic of considerable and growing interest. Recently, the introduction of extropy as a measure of uncertainty, dual to Shannon entropy, has opened up interest in new aspects of the subject. Since there are many versions of entropy, a unified formulation has been introduced to work with all of them in an easy way. Here we consider the possibility of defining a unified formulation for extropy by introducing a measure depending on two parameters. For particular choices of parameters, this measure provides the well-known formulations of extropy. Moreover, the unified formulation of extropy is also analyzed in the context of the Dempster–Shafer theory of evidence, and an application to classification problems is given. [ABSTRACT FROM AUTHOR]

Because of its flexibility and multiple meanings, the concept of information entropy in its continuous or discrete form has proven to be very relevant in numerous scientific branches. For example, it is used as a measure of disorder in thermodynamics, as a measure of uncertainty in statistical mechanics as well as in classical and quantum information science, as a measure of diversity in ecological structures and as a criterion for the classification of races and species in population dynamics. Orthogonal polynomials are a useful tool in solving and interpreting differential equations. Lately, this subject has been intensively studied in many areas. For example, in statistics, by using orthogonal polynomials to fit the desired model to the data, we are able to eliminate collinearity and to seek the same information as simple polynomials. In this paper, we consider the Tsallis, Kaniadakis and Varma entropies of Chebyshev polynomials of the first kind and obtain asymptotic expansions. In the particular case of quadratic entropies, there are given concrete computations. [ABSTRACT FROM AUTHOR]

Abstract Human emotion recognition remains a challenging and prominent issue, situated at the convergence of diverse fields, such as brain–computer interfaces, neuroscience, and psychology. This study utilizes an EEG data set for investigating human emotion, presenting novel findings and a refined approach for EEG-based emotion detection. Tsallis entropy features, computed for q values of 2, 3, and 4, are extracted from signal bands, including theta-θ (4–7 Hz), alpha-α (8–15 Hz), beta-β (16–31 Hz), gamma-γ (32–55 Hz), and the overall frequency range (0–75 Hz). These Tsallis entropy features are employed to train and test a KNN classifier, aiming for accurate identification of two emotional states: positive and negative. In this study, the best average accuracy of 79% and an F-score of 0.81 were achieved in the gamma frequency range for the Tsallis parameter q = 3. In addition, the highest accuracy and F-score of 84% and 0.87 were observed. Notably, superior performance was noted in the anterior and left hemispheres compared to the posterior and right hemispheres in the context of emotion studies. The findings show that the proposed method exhibits enhanced performance, making it a highly competitive alternative to existing techniques. Furthermore, we identify and discuss the shortcomings of the proposed approach, offering valuable insights into potential avenues for improvements.

This study introduces the Inverse Burr-X Burr-XII (IBXBXII) distribution as a novel approach for handling asymmetric-bimodal claims and revenues. It explores the distribution's statistical properties and evaluates its performance in three contexts. The analysis includes assessing entropy, highlighting the distribution's significance in various fields, and comparing it to rival distributions using practical examples. The IBXBXII model is then applied to analyze risk indicators in actuarial data, focusing on bimodal insurance claims and income. Simulation analysis shows its preference for right-skewed data, making it suitable for mathematical modeling and actuarial risk assessments. The study emphasizes the IBXBXII model's versatility and effectiveness, suggesting it as a flexible framework for actuarial data analysis, particularly in cases of large samples and right-skewed data.

With the continuous increase and complexity of network traffic, traditional network traffic recognition technology is facing numerous difficulties, especially in dealing with outlier data and improving recognition accuracy. Therefore, an improved expectation maximization algorithm based on the constraint matrix Z and Tsallis entropy is proposed. The core goal of this algorithm is to accelerate the convergence of classification and improve accuracy. Furthermore, to enhance the classification accuracy, the spatial expectation maximization algorithm is introduced, which innovatively converts the sample mean and covariance matrix into $L_{1}$ -median and modified rank covariance matrix. According to the experimental data, the recall rate of the original expectation maximization algorithm is only 74%. However, the recall rate of the spatial expectation maximization algorithm in the Attack service has significantly increased to 85%. In other tests, such as Www and Peer-to-peer services, the recall rate has also significantly improved, increasing from 96% and 95.3% to 97.7% and 96.1%, respectively. These experimental results highlight the superior robustness of the spatial expectation maximization algorithm in handling outlier data. It further proves the outstanding performance in improving the accuracy of network traffic recognition. This research has brought significant innovation and potential practical value to the network traffic identification.

We deal with Tsallis entropy in MV-algebraic dynamical systems. We prove that Tsallis entropy is a submeasure and that it is invariant under isomorphisms. We also provide two examples which show that Tsallis entropy allows one to distinguish some non-isomorphic MV-dynamical systems.

A comprehensive understanding of velocity distribution is the fundamental information hydraulic engineers need to estimate an alluvial channel's discharge and stage characteristics. This study examines the Tsallis entropy approach for estimating streamwise velocity patterns in open channels. Entropy, which quantifies system uncertainty, has been applied in hydraulic research to account for variables such as shear strength, silt content, and flow velocities. However, its applicability to non-uniform channel sections remains unexplored. In the current work, the velocities estimation under various experimental conditions was calculated using the Tsallis entropy approach, wherein the random variable employed for constructing the velocity estimations was the time-averaged normalised velocity. This study considered two experimental conditions: (1) channels with attached spurs under seepage and non-seepage conditions and (2) channels with a 31-degree bank slope with and without an upstream pit. The velocity pattern obtained closely corresponds to the experimental data, exhibiting significant accuracy. However, it should be noted that the accuracy of the velocity pattern is slightly diminished in the region near the spur field when the y / D value is below 0.3. The difference can be caused by factors such as the area's non-uniform cross-section, sediment interaction along the bed, and secondary currents, which ultimately affect the velocity profile. [ABSTRACT FROM AUTHOR]

The inverse Rayleigh distribution finds widespread applications within life testing and reliability research. Particularly, it proves invaluable in scenarios involving multiple censored data points. In this context, the Renyi, Havrda, Charvat, and Tsallis entropies of the inverse Rayleigh distribution are efficiently calculated. The maximum likelihood approach is used to get the estimators, as well as the approximate confidence interval. The mean squared errors, approximate confidence interval, and their related average length are computed. To illuminate the behavior of estimates across varying sample sizes, a comprehensive simulation study is conducted. The outcomes of the simulation study consistently reveal a downward trend in mean squared errors and average lengths as the sample size increases. Additionally, an interesting finding emerges as the censoring level diminishes. The entropy estimators progressively converge towards their true values. For practical demonstration, the effectiveness of the approach is showcased through the analysis of two real-world datasets. These applications underscore the real-world relevance of the methodology, further validating its utility in addressing complex scenarios involving censored data and inverse Rayleigh distributions. [ABSTRACT FROM AUTHOR]

This study introduces the Inverse Burr-X Burr-XII (IBXBXII) distribution as a novel approach for handling asymmetric-bimodal claims and revenues. It explores the distribution's statistical properties and evaluates its performance in three contexts. The analysis includes assessing entropy, highlighting the distribution's significance in various fields, and comparing it to rival distributions using practical examples. The IBXBXII model is then applied to analyze risk indicators in actuarial data, focusing on bimodal insurance claims and income. Simulation analysis shows its preference for right-skewed data, making it suitable for mathematical modeling and actuarial risk assessments. The study emphasizes the IBXBXII model's versatility and effectiveness, suggesting it as a flexible framework for actuarial data analysis, particularly in cases of large samples and right-skewed data. [ABSTRACT FROM AUTHOR]

In quantum information theory, one needs to consider systems with incomplete information. To estimate a quantum system as an information resource, one uses various characteristics of non-classical correlations. Currently, much attention is paid to coherence quantifiers averaged over a set of specially selected states. In particular, mutually unbiased bases, symmetric informationally complete measurements, and some of their generalizations are of importance in this regard. The aim of the present study is to derive uncertainty relations for coherence quantifiers based on divergences of the Tsallis type. The obtained inequalities concern quantifiers averaged over a set of mutually unbiased bases and a set of states that form an equiangular tight frame. [ABSTRACT FROM AUTHOR]

In this article, we study the helium atom confined in a spherical impenetrable cavity by using informational measures. We use the Ritz variational method to obtain the energies and wave functions of the confined helium atom as a function of the cavity radius r0$$ {r}_0 $$. As trial wave functions we use one uncorrelated function and five explicitly correlated basis sets in Hylleraas coordinates with different degrees of electronic correlation. We computed the Shannon entropy, Fisher information, Kullback–Leibler entropy, Tsallis entropy, disequilibrium and Fisher–Shannon complexity, as a function of r0$$ {r}_0 $$. We found that these entropic measures are sensitive to electronic correlation and can be used to measure it. As expected these entropic measures are less sensitive to electron correlation in the strong confinement regime (r0<1$$ {r}_0<1 $$ a.u.). [ABSTRACT FROM AUTHOR]

With the help of Tsallis residual entropy, we introduce Tsallis quantile entropy order between two random variables. We give necessary and sufficient conditions, study closure and reversed closure properties under parallel and series operations and show that this order is preserved in the proportional hazard rate model, proportional reversed hazard rate model, proportional odds model and record values model. [ABSTRACT FROM AUTHOR]

In medical domain, diseases in critical internal organs are generally inspected using invasive/non-invasive imaging techniques. Magnetic resonance imaging (MRI) is one of the commonly considered imaging approaches to confirm the abnormality in various internal organs. After recording the MRI, an appropriate image processing exercise is to be implemented to investigate and infer the severity of the disease and its location. This paper proposes a semi-automated tool to investigate the medical MRI captured with contrast improved T1 modality (T1C). This technique considers the integration of Bat algorithm (BA) and Tsallis based thresholding along with region growing (RG) segmentation. Proposed approach is tested on RGB/gray scale images of brain and breast MRI recorded along with a contrast agent. After mining the infected region, its texture features are extracted with Haralick function to assess the surface details of abnormal section. Performance of RG is confirmed against other segmentation methods, such as level set (LS), principal component analysis (PCA) and watershed. The clinical significance of the proposed technique is also validated using the brain images of BRATS recorded using T1C modality. The experiment outcome confirms that, the implemented procedure provides better values of Jaccard (87.41%), Dice (90.36%), sensitivity (98.27%), specificity (97.72%), accuracy (97.53%) and precision (95.85%) for the considered BRATS brain MRI. [ABSTRACT FROM AUTHOR]

The random forest algorithm is one of the most popular and commonly used algorithms for classification and regression tasks. It combines the output of multiple decision trees to form a single result. Random forest algorithms demonstrate the highest accuracy on tabular data compared to other algorithms in various applications. However, random forests and, more precisely, decision trees, are usually built with the application of classic Shannon entropy. In this article, we consider the potential of deformed entropies, which are successfully used in the field of complex systems, to increase the prediction accuracy of random forest algorithms. We develop and introduce the information gains based on Renyi, Tsallis, and Sharma-Mittal entropies for classification and regression random forests. We test the proposed algorithm modifications on six benchmark datasets: three for classification and three for regression problems. For classification problems, the application of Renyi entropy allows us to improve the random forest prediction accuracy by 19≥96% in dependence on the dataset, Tsallis entropy improves the accuracy by 20≥98%, and Sharma-Mittal entropy improves accuracy by 22≥111% compared to the classical algorithm. For regression problems, the application of deformed entropies improves the prediction by 2≥23% in terms of R2 in dependence on the dataset. [ABSTRACT FROM AUTHOR]

Velocity distribution plays a fundamental role in understanding the hydrodynamics of open-channel flow. Among a multitude of approaches, the entropy-based approach holds great promise in achieving a reasonable characterisation of the velocity distribution. In entropy-based methods, the distribution depends on a key parameter, known as the entropy parameter (a function of the time-averaged mean velocity and maximum velocity), that relates to channel characteristics, such as channel roughness and channel bed slopes. The entropy parameter was regarded as constant for lack of experimental evidence, which would otherwise demonstrate if it had any correlation with channel properties. A series of experiments were conducted to collect velocity data in the laboratory flume for seven different values of the channel bed slope. The experimental data analysis revealed dissimilar fluctuations in entropy parameter values with varying bed slopes, with the lowest coefficient of variation in Renyi's (∼0.5%) and the highest in Shannon's case (∼10%). Performance evaluation of the predicted results substantiated good accuracy for all three entropies with the best results of Renyi entropy and lent strong support for using a constant (overall average) value of the entropy parameter for a specific channel cross-section rather than separate values for each channel bed slope. HIGHLIGHTS Verification of the influence of the channel bed slope on entropy parameters.; Velocity observations for mild, horizontal, and adverse channel bed slopes.; Shannon, Tsallis, and Renyi entropy-based velocity distributions.; Statistical and experimental evidence supporting the constant nature of all the entropy parameters.; Modified equation to estimate mean and maximum velocity ratio in terms of the Renyi entropy parameter.;

This article introduces the concept of residual and past Tsallis extropy as a continuous information measure within the context of continuous distribution. Moreover, the characteristics and their relationships with other models are evaluated. Several stochastic comparisons are provided, along with outcomes concerning order statistics. Additionally, the models acquired include instances such as uniform and power function distributions. The measure incorporates its monotonic traits, and the outcomes defining its characteristics are presented. On the other hand, a different portrayal of the Tsallis extropy is introduced, expressed in relation to the hazard rate function. The Tsallis extropy of the lifetime for both mixed and coherent systems is explored. In the case of mixed systems, components' lifetimes are considered independent and identically distributed. Additionally, constraints on the Tsallis extropy of these systems are established, along with a clarification of their practical applicability. Non-parametric estimation using an alternative form of Tsallis function extropy for simulated and real data is performed.

Tsallis entropy has been widely used in image thresholding because of its non-extensive properties. The non-extensive parameter q contained in this entropy plays an important role in various adaptive algorithms and has been successfully applied in bi-level image thresholding. In this paper, the relationships between parameter q and pixels’ long-range correlations have been further studied within multi-threshold image segmentation. It is found that the pixels’ correlations are remarkable and stable for images generated by a known physical principle, such as infrared images, medical CT images, and color satellite remote sensing images. The corresponding non-extensive parameter q can be evaluated by using the self-adaptive Tsallis entropy algorithm. The results of this algorithm are compared with those of the Shannon entropy algorithm and the original Tsallis entropy algorithm in terms of quantitative image quality evaluation metrics PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity). Furthermore, we observed that for image series with the same background, the q values determined by the adaptive algorithm are consistently kept in a narrow range. Therefore, similar or identical scenes during imaging would produce similar strength of long-range correlations, which provides potential applications for unsupervised image processing.

In this paper, the Tsallis and Renyi extropy is presented as a continuous measure of information under the continuous distribution. Furthermore, the features and their connection to other information measures are introduced. Some stochastic comparisons and results on the order statistics and upper records are given. Moreover, some theorems about the maximum Tsallis and Renyi extropy are discussed. On the other hand, numerical results of the non-parametric estimation of Tsallis extropy are calculated for simulated and real data with application to time series model and its forecasting.

The dynamic cumulative residual (DCR) entropy is a helpful randomness metric that may be used in survival analysis. A challenging issue in statistics and machine learning is the estimation of entropy measures. This article uses progressive censored type II (PCT-II) samples to estimate the DCR Tsallis entropy (DCRTE) for the moment exponential distribution. The non-Bayesian and Bayesian approaches are the recommended estimating strategies. We obtain the DCRTE Bayesian estimator using the gamma and uniform priors via symmetric and asymmetric (linear exponential and general entropy) loss functions (LoFs). The Metropolis–Hastings algorithm is employed to generate Markov chain Monte Carlo samples from the posterior distribution. The accuracy of different estimates for various sample sizes, is implemented via Monte Carlo simulations. Generally, we note based on the simulation study that, in the majority of cases, the DCRTE Bayesian estimates under general entropy followed by linear exponential LoFs are preferable to the others. The accuracy measure of DCRTE Bayesian estimates using a gamma prior has smaller values than the others using a uniform prior. As sample sizes grow, the Bayesian estimates of the DCRTE are closer to the true value. Finally, analysis of the leukemia data confirmed the proposed estimators.

The random forest algorithm is one of the most popular and commonly used algorithms for classification and regression tasks. It combines the output of multiple decision trees to form a single result. Random forest algorithms demonstrate the highest accuracy on tabular data compared to other algorithms in various applications. However, random forests and, more precisely, decision trees, are usually built with the application of classic Shannon entropy. In this article, we consider the potential of deformed entropies, which are successfully used in the field of complex systems, to increase the prediction accuracy of random forest algorithms. We develop and introduce the information gains based on Renyi, Tsallis, and Sharma-Mittal entropies for classification and regression random forests. We test the proposed algorithm modifications on six benchmark datasets: three for classification and three for regression problems. For classification problems, the application of Renyi entropy allows us to improve the random forest prediction accuracy by 19–96% in dependence on the dataset, Tsallis entropy improves the accuracy by 20–98%, and Sharma-Mittal entropy improves accuracy by 22–111% compared to the classical algorithm. For regression problems, the application of deformed entropies improves the prediction by 2–23% in terms of R2 in dependence on the dataset.

We investigate the cumulative Tsallis entropy, an information measure recently introduced as a cumulative version of the classical Tsallis differential entropy, which is itself a generalization of the Boltzmann-Gibbs statistics. This functional is here considered as a perturbation of the expected mean residual life via some power weight function. This point of view leads to the introduction of the dual cumulative Tsallis entropy and of two families of coherent risk measures generalizing those built on mean residual life. We characterize the finiteness of the cumulative Tsallis entropy in terms of L p -spaces and show how they determine the underlying distribution. The range of the functional is exactly described under various constraints, with optimal bounds improving on all those previously available in the literature. Whereas the maximization of the Tsallis differential entropy gives rise to the classical q -Gaussian distribution which is a generalization of the Gaussian having a finite range or heavy tails, the maximization of the cumulative Tsallis entropy leads to an analogous perturbation of the Logistic distribution. [ABSTRACT FROM AUTHOR]

In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate. [ABSTRACT FROM AUTHOR]

Significant seismic activity has been witnessed in the area of Ridgecrest (Southern California) over the past 40 years, with the largest being the Mw 5.8 event on 20 September 1995. In July 2019, a strong earthquake of Mw 7.1, preceded by a Mw 6.4 foreshock, impacted Ridgecrest. The mainshock triggered thousands of aftershocks that were thoroughly documented along the activated faults. In this study, we analyzed the spatiotemporal variations of the frequency–magnitude distribution in the area of Ridgecrest using the fragment–asperity model derived within the framework of non-extensive statistical physics (NESP), which is well-suited for investigating complex dynamic systems with scale-invariant properties, multi-fractality, and long-range interactions. Analysis was performed for the entire duration, as well as within various time windows during 1981–2022, in order to estimate the qM parameter and to investigate how these variations are related to the dynamic evolution of seismic activity. In addition, we analyzed the spatiotemporal qM value distributions along the activated fault zone during 1981–2019 and during each month after the occurrence of the Mw 7.1 Ridgecrest earthquake. The results indicate a significant increase in the qM parameter when large-magnitude earthquakes occur, suggesting the system's transition in an out-of-equilibrium phase and its preparation for seismic energy release. [ABSTRACT FROM AUTHOR]

We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian motions, and compare the entropies of one-dimensional distributions of these processes. [ABSTRACT FROM AUTHOR]

Seismic data have improved in quality and quantity over the past few decades, enabling better statistical analysis. Statistical physics has proposed new ways to deal with these data to focus the attention on specific matters. The present paper combines these two progressions to find indicators that can help in the definition of areas where seismic risk is developing. Our data comes from the IPOC catalog for 2007 to 2014. It covers the intense seismic activity near Iquique in Northern Chile during March/April 2014. Centered in these hypocenters we concentrate on the rectangle L a t − 22 − 18 and L o n − 68 − 72 and deepness between 5 and 70 km, where the major earthquakes originate. The analysis was performed using two complementary techniques: Tsallis entropy and mutability (dynamical entropy). Two possible forecasting indicators emerge: (1) Tsallis entropy (mutability) increases (decreases) broadly about two years before the main M W 8.1 earthquake. (2) Tsallis entropy (mutability) sharply decreases (increases) a few weeks before the M W 8.1 earthquake. The first one is about energy accumulation, and the second one is because of energy relaxation in the parallelepiped of interest. We discuss the implications of these behaviors and project them for possible future studies. [ABSTRACT FROM AUTHOR]

In non-equilibrium plasmas, the temperature cannot be uniquely determined unless the energy-distribution function is approximated as a Maxwell–Boltzmann distribution. To overcome this problem, we applied Tsallis statistics to determine the temperature with respect to the excited-state populations in non-equilibrium state hydrogen plasma, which enables the description of its entropy that obeys q-exponential population distribution in the non-equilibrium state. However, it is quite difficult to apply the q-exponential distribution because it is a self-consistent function that cannot be solved analytically. In this study, a self-consistent iterative scheme was adopted to calculate q-exponential distribution using the similar algorithm of the Hartree–Fock method. Results show that the excited-state population distribution based on Tsallis statistics well captures the non-equilibrium characteristics in the high-energy region, which is far from the equilibrium-Boltzmann distribution. The temperature was calculated using the partial derivative of entropy with respect to the mean energy based on Tsallis statistics and using the coefficient of q-exponential distribution. An analytical expression was derived and compared with Boltzmann statistics, and the distribution was discussed from the viewpoint of statistical physics. [ABSTRACT FROM AUTHOR]