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This paper introduces a novel application of Tsallis entropy for complexity analysis in electroencephalography (EEG) data. Tsallis entropy, a generalization of Shannon entropy, is employed to uncover hidden structures and distinguish varying complexity levels in EEG signals. By leveraging this framework on publicly available EEG datasets, the study demonstrates that Tsallis entropy is highly effective in categorizing brain activity patterns across different levels of complexity. The results highlight the method's potential for clinical and experimental neurodata analysis.
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Alzheimer's disease (AD) is a neurodegenerative disorder that dramatically affects cognitive abilities and represents the most common cause of dementia. Currently, pharmacological interventions represent the main treatment to deal with the symptoms of AD; however, alternative approaches are readily sought. Transcranial pulse stimulation (TPS) is an emerging non-invasive neuromodulation technique that uses short, repetitive shockwaves with the potential to provide a wide range of vascular, metabolic, and neurotrophic changes and that has recently been shown to improve cognitive abilities in AD. This exploratory study aims to gain insight into the neurophysiological effect of one session of TPS in AD as reflected in electroencephalographic measures, e.g., spectral power, coherence, Tsallis entropy (TE), and cross-frequency coupling (cfc). We document changes in power (frontal and occipital), coherence (frontal, occipital and temporal), and TE (temporal and frontal) as well as changes in cfc (parietal-frontal, parietal-temporal, frontal-temporal). Our results emphasize the role of electroencephalographic measures as prospective markers for the neurophysiological effect of TPS.
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Precipitation patterns are critical for understanding the hydrological and climatological dynamics of any region. Sicily, the largest island in the Mediterranean sea, with its diverse topography and climatic conditions, serves as an ideal case study for analyzing precipitation data, to gain insights into regional water resources, agricultural productivity, and climate change impacts. This paper employs advanced statistical physics methods, particularly Tsallis q-statistics, to analyze sub-hourly precipitation data from 2002 to 2023, provided by the Sicilian Agrometeorological Informative System (SIAS). We investigate several critical variables related to rainfall events, including duration, depth, maximum record, and inter-event time. The study spans two decades (2002-2012 and 2013-2023), analyzing the distributions of relevant variables. Additionally, we examine the simple returns of these variables to identify significant temporal changes, fitting these returns with q-Gaussian distributions. Our findings reveal the scale-invariant nature of precipitation events, the presence of long-range interactions, and memory effects, characteristic of complex environmental processes.
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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.
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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.
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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.
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We present a phenomenological framework based on the MIT bag model to estimate the pressure experienced by quarks and gluons inside nucleons. This is accomplished by implementing non-extensive Tsallis statistics for the two-component system. In this model of hadrons, the strong interaction generates correlations effectively described by the q-Tsallis parameter. The resulting hadron pressure exhibits general agreement with recent calculations derived from Lattice QCD. Additionally, we compared this pressure with data extracted from deep virtual Compton scattering experiments and gravitational form factor analyses. The extended bag model provides an alternative interpretation of bag pressure in terms of the q-Tsallis parameter. Consequently, the MIT bag model can be expressed without requiring the inclusion of the bag pressure parameter.
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Currently, there is no widely accepted consensus regarding a consistent thermodynamic framework within the special relativity paradigm. However, by postulating that the inverse temperature 4-vector, denoted as ß, is future-directed and time-like, intriguing insights emerge. Specifically, it is demonstrated that the q-dependent Tsallis distribution can be conceptualized as a de Sitterian deformation of the relativistic Maxwell-Jüttner distribution. In this context, the curvature of the de Sitter space-time is characterized by Λ/3, where Λ represents the cosmological constant within the ΛCDM standard model for cosmology. For a simple gas composed of particles with proper mass m, and within the framework of quantum statistical de Sitterian considerations, the Tsallis parameter q exhibits a dependence on the cosmological constant given by q=1+âcΛ/n, where âc=â/mc is the Compton length of the particle and n is a positive numerical factor, the determination of which awaits observational confirmation. This formulation establishes a novel connection between the Tsallis distribution, quantum statistics, and the cosmological constant, shedding light on the intricate interplay between relativistic thermodynamics and fundamental cosmological parameters.
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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-ß (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.
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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.
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Recent innovations have focused on the creation of new families that extend well-known distributions while providing a huge amount of practical flexibility for data modeling. Weighted distributions offer an effective approach for addressing model building and data interpretation problems. The main objective of this work is to provide a novel family based on a weighted generator called the length-biased truncated Lomax-generated (LBTLo-G) family. Discussions are held about the characteristics of the LBTLo-G family, including expressions for the probability density function, moments, and incomplete moments. In addition, different measures of uncertainty are determined. We provide four new sub-distributions and investigated their functionalities. Subsequently, a statistical analysis is given. The LBTLo-G family's parameter estimation is carried out using the maximum likelihood technique on the basis of full and censored samples. Simulation research is conducted to determine the parameters of the LBTLo Weibull (LBTLoW) distribution. Four genuine data sets are considered to illustrate the fitting behavior of the LBTLoW distribution. In each case, the application outcomes demonstrate that the LBTLoW distribution can, in fact, fit the data more accurately than other rival distributions.
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This work focuses on exploring the properties of past Tsallis entropy as it applies to order statistics. The relationship between the past Tsallis entropy of an ordered variable in the context of any continuous probability law and the past Tsallis entropy of the ordered variable resulting from a uniform continuous probability law is worked out. For order statistics, this method offers important insights into the characteristics and behavior of the dynamic Tsallis entropy, which is associated with past events. In addition, we investigate how to find a bound for the new dynamic information measure related to the lifetime unit under various conditions and whether it is monotonic with respect to the time when the device is idle. By exploring these properties and also investigating the monotonic behavior of the new dynamic information measure, we contribute to a broader understanding of order statistics and related entropy quantities.
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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.
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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.
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The imminent threat of Mongolian montane forests facing extinction due to climate change emphasizes the pressing need to study these ecosystems for sustainable development. Leveraging multispectral remote sensing data from Landsat 8 OLI TIRS (2013-2021), we apply Tsallis non-extensive thermodynamics to assess spatiotemporal fluctuations in the absorbed solar energy budget (exergy, bound energy, internal energy increment) and organizational parameters (entropy, information increment, q-index) within the mountain taiga-meadow landscape. Using the principal component method, we discern three functional subsystems: evapotranspiration, heat dissipation, and a structural-informational component linked to bioproductivity. The interplay among these subsystems delineates distinct landscape cover states. By categorizing ecosystems (pixels) based on these processes, discrete states and transitional areas (boundaries and potential disturbances) emerge. Examining the temporal dynamics of ecosystems (pixels) within this three-dimensional coordinate space facilitates predictions of future landscape states. Our findings indicate that northern Mongolian montane forests utilize a smaller proportion of received energy for productivity compared to alpine meadows, which results in their heightened vulnerability to climate change. This approach deepens our understanding of ecosystem functioning and landscape dynamics, serving as a basis for evaluating their resilience amid ongoing climate challenges.
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By employing Tsallis' extensive but non-additive δ-entropy, we formulate the first two laws of thermodynamics for gravitating systems. By invoking Carathéodory's principle, we pay particular attention to the integrating factor for the heat one-form. We show that the latter factorizes into the product of thermal and entropic parts, where the entropic part cannot be reduced to a constant, as is the case in conventional thermodynamics, due to the non-additive nature of Sδ. The ensuing two laws of thermodynamics imply a Tsallis cosmology, which is then applied to a radiation-dominated universe to address the Big Bang nucleosynthesis and the relic abundance of cold dark matter particles. It is demonstrated that the Tsallis cosmology with the scaling exponent δâ¼1.499 (or equivalently, the anomalous dimension Δâ¼0.0013) consistently describes both the abundance of cold dark matter particles and the formation of primordial light elements, such as deuterium 2H and helium 4He. Salient issues, including the zeroth law of thermodynamics for the δ-entropy and the lithium 7Li problem, are also briefly discussed.
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The healthy function of the vestibular system (VS) is of vital importance for individuals to carry out their daily activities independently and safely. This study carries out Tsallis entropy (TE)-based analysis on insole force sensor data in order to extract features to differentiate between healthy and VS-diseased individuals. Using a specifically developed algorithm, we detrend the acquired data to examine the fluctuation around the trend curve in order to consider the individual's walking habit and thus increase the accuracy in diagnosis. It is observed that the TE value increases for diseased people as an indicator of the problem of maintaining balance. As one of the main contributions of this study, in contrast to studies in the literature that focus on gait dynamics requiring extensive walking time, we directly process the instantaneous pressure values, enabling a significant reduction in the data acquisition period. The extracted feature set is then inputted into fundamental classification algorithms, with support vector machine (SVM) demonstrating the highest performance, achieving an average accuracy of 95%. This study constitutes a significant step in a larger project aiming to identify the specific VS disease together with its stage. The performance achieved in this study provides a strong motivation to further explore this topic.
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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.
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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 Lat-22-18 and Lon-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 MW8.1 earthquake. (2) Tsallis entropy (mutability) sharply decreases (increases) a few weeks before the MW8.1 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.
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Skin Cancer (SC) is among the most hazardous due to its high mortality rate. Therefore, early detection of this disease would be very helpful in the treatment process. Multilevel Thresholding (MLT) is widely used for extracting regions of interest from medical images. Therefore, this paper utilizes the recent Coronavirus Disease Optimization Algorithm (COVIDOA) to address the MLT issue of SC images utilizing the hybridization of Otsu, Kapur, and Tsallis as fitness functions. Various SC images are utilized to validate the performance of the proposed algorithm. The proposed algorithm is compared to the following five meta-heuristic algorithms: Arithmetic Optimization Algorithm (AOA), Sine Cosine Algorithm (SCA), Reptile Search Algorithm (RSA), Flower Pollination Algorithm (FPA), Seagull Optimization Algorithm (SOA), and Artificial Gorilla Troops Optimizer (GTO) to prove its superiority. The performance of all algorithms is evaluated using a variety of measures, such as Mean Square Error (MSE), Peak Signal-To-Noise Ratio (PSNR), Feature Similarity Index Metric (FSIM), and Normalized Correlation Coefficient (NCC). The results of the experiments prove that the proposed algorithm surpasses several competing algorithms in terms of MSE, PSNR, FSIM, and NCC segmentation metrics and successfully solves the segmentation issue.