the q gamma and q q polygamma functions of tsallis statistics abstract an axiomatic definition is given for the q gamma function q x q r q 0 x r of tsallis non extensive statistical physics the continuous analogue of the q factorial of suyari h suyari physica a 368 1 2006 63 and the q analogue of the gamma function x of euler and gauss a working definition in closed form based on the hurwitz and riemann zeta functions including their analytic continuations is shown to satisfy this definition several relations involving the q gamma and other functions are obtained the q q polygamma functions q q m x m n defined by successive derivatives of ln q q x where ln q a 1 q 1 a 1 q 1 a 0 is the q logarithmic function are also reported the new functions are used to calculate the inferred probabilities and multipliers for tsallis systems with finite numbers of particles n

κ deformed fourier transform abstract we present a new formulation of fourier transform in the picture of the algebra derived in the framework of the generalized statistical mechanics the fourier transform is obtained from a fourier series recently introduced by scarfone 2013 the kernel of this transform that reduces to the usual exponential phase in the 0 limit is composed by a deformed phase and a damping factor that gives a wavelet like behaviour we show that the fourier transform is isomorph to the standard fourier transform through a changing of time and frequency variables nevertheless the new formalism is useful to study according to fourier analysis those functions defined in the realm of the algebra as a relevant application we discuss the central limit theorem for the sum of n iterate statistically independent random variables

nonextensive triangle equality and other properties of tsallis relative entropy minimization kullback leibler relative entropy has unique properties in cases involving distributions resulting from relative entropy minimization tsallis relative entropy is a one parameter generalization of kullback leibier relative entropy in the nonextensive thermostatistics in this paper we present the properties of tsallis relative entropy minimization and present some differences with the classical case in the representation of such a minimum relative entropy distribution we highlight the use of the q product an operator that has been recently introduced to derive the mathematical structure behind the tsallis statistics one of our main results is the generalization of triangle equality of relative entropy minimization to the nonextensive case

when shannon and khinchin meet shore and johnson equivalence of information theory and statistical inference axiomatics we propose a unified framework for both shannon khinchin and shore johnson axiomatic systems we do it by rephrasing shannon khinchine axioms in terms of generalized arithmetics of kolmogorov and nagumo we prove that the two axiomatic schemes yield identical classes of entropic functionals the uffink class of entropies this allows to re establish the entropic parallelism between information theory and statistical inference that has seemed to be broken by the use of non shannonian entropies

probability distribution of schwammle and tsallis two parameter entropies and the lambert w function abstract we investigate a two parameter entropy introduced by schwammle and tsallis and obtain its probability distribution in the canonical ensemble the probability distribution is given in terms of the lambert w function which has been used in many branches of physics especially in fractal structures also extensivity of s q q is discussed and a relationship is found to exist between the probabilities of a composite system and its subsystems so that the two parameter entropy s q q is extensive

binary indices of time series complexity measures and entropy plane abstract complexity of time series is an important feature of dynamical systems such as financial systems in order to bring out the complete non linear behavior of financial time series non linear tools of complexity measurement like entropy measures are indispensable sample entropy sampen and distribution entropy disten are popular methods of assessing the complexity in various fields however both sample entropy and distribution entropy show some limitations in detecting the complexity of stock markets therefore we use two entropies as binary indices to analyze the complexity of time series and structure the entropy plane in order to further improve the accuracy of the research results we take the embedding dimension m as the variable expand the entropy point into the entropy curve for further research furthermore considering the shortcomings of sample entropy and distribution entropy in practical application we generalize them by i replacing sample entropy and distribution entropy with multi scale sample entropy and multi scale distribution entropy ii generalizing shannon entropy as tsallis entropy also scale factor and entropic index q are taken respectively as variables to get the entropy curves by using the artificial data we confirm the rationality of using entropy points and entropy curves to study the complexity of time series finally we apply this method to measure the complexity of real world financial time series the results show that the entropy curves plotted by the financial time series obtained from different areas have significant differences

linear regression analysis for fuzzy input and output data using the extension principle abstract the method for obtaining the fuzzy least squares estimators with the help of the extension principle in fuzzy sets theory is proposed the membership functions of fuzzy least squares estimators will be constructed according to the usual least squares estimators in order to obtain the membership value of any given value taken from the fuzzy least squares estimator optimization problems have to be solved we also provide the methodology for evaluating the predicted fuzzy output from the given fuzzy input data

on generalisations of the log normal distribution by means of a new product definition in the kapteyn process abstract we discuss the modification of the kapteyn multiplicative process using the q product of borges e p borges a possible deformed algebra and calculus inspired in nonextensive thermostatistics physica a 340 2004 95 depending on the value of the index q a generalisation of the log normal distribution is yielded namely the distribution increases the tail for small when q 1 or large when q 1 values of the variable upon analysis the usual log normal distribution is retrieved when q 1 which corresponds to the traditional kapteyn multiplicative process the main statistical features of this distribution as well as related random number generators and tables of quantiles of the kolmogorov smirnov distance are presented finally we illustrate the validity of this scenario by describing a set of variables of biological and financial origin

deformed exponentials and logarithms in generalized thermostatistics abstract criteria are given that deformed logarithmic and exponential functions should satisfy with a pair of such functions one can associate another function called the deduced logarithmic function it is shown that generalized thermostatistics can be formulated in terms of deformed exponential functions together with the associated deduced logarithmic functions

group entropies correlation laws and zeta functions the notion of group entropy is proposed it enables the unification and generaliztion of many different definitions of entropy known in the literature such as those of boltzmann gibbs tsallis abe and kaniadakis other entropic functionals are introduced related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic the associated thermostatistics are discussed the mathematical structure underlying our construction is that of formal group theory which provides the general structure of the correlations among particles and dictates the associated entropic functionals as an example of application the role of group entropies in information theory is illustrated and generalizations of the kullback leibler divergence are proposed a new connection between statistical mechanics and zeta functions is established in particular tsallis entropy is related to the classical riemann zeta function

a new one parameter deformation of the exponential function abstract recently in kaniadakis physica a 296 2001 405 a new one parameter deformation for the exponential function exp x 1 2 x 2 x 1 exp 0 x exp x which presents a power law asymptotic behaviour has been proposed the statistical distribution f z 1 exp e has been obtained both as stable stationary state of a proper nonlinear kinetics and as the state which maximizes a new entropic form in the present contribution starting from the algebra and after introducing the analysis we obtain the exponential exp x as the eigenstate of the derivative and study its main mathematical properties

generalized fisher information matrix in nonextensive systems with spatial correlation by using the q gaussian distribution derived by the maximum entropy method for spatially correlated n unit nonextensive systems we have calculated the generalized fisher information matrix of g ensuremath theta n ensuremath theta m for ensuremath theta 1 ensuremath theta 2 ensuremath theta 3 ensuremath mu q ensuremath sigma q 2 s where ensuremath mu q ensuremath sigma q 2 and s denote the mean variance and degree of spatial correlation respectively for a given entropic index q it has been shown from the cram er rao theorem that 1 an accuracy of an unbiased estimate of ensuremath mu q is improved degraded by a negative positive correlation s 2 that of ensuremath sigma q 2 is worsen with increasing s and 3 that of s is much improved for s ensuremath simeq ensuremath 1 n ensuremath 1 or s ensuremath simeq 1 0 though it is worst at s n ensuremath 2 2 n ensuremath 1 our calculation provides a clear insight to the long standing controversy whether the spatial correlation is beneficial or detrimental to decoding in neuronal ensembles we discuss also a calculation of the q gaussian distribution applying the superstatistics to the langevin model subjected to spatially correlated inputs

law of error in tsallis statistics in order to theoretically explain the ubiquitous existence of power law behavior such as chaos and fractals in nature tsallis entropy has been successfully applied to the generalization of the traditional boltzmann gibbs statistics the fundamental information measure of which is shannon entropy tsallis entropy s sub q is a one parameter generalization of shannon entropy s sub 1 in the sense that lim sub q spl rarr 1 s sub q s sub 1 the generalized statistics using tsallis entropy are referred to as tsallis statistics in order to present the law of error in tsallis statistics as a generalization of gauss law of error and prove it mathematically we apply the new multiplication operation determined by q logarithm and q exponential the fundamental functions in tsallis statistics to the definition of the likelihood function in gauss law of error the present maximum likelihood principle mlp leads us to determine the so called q gaussian distribution which coincides with one of the tsallis distributions derived from the maximum entropy principle for tsallis entropy under the second moment constraint

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

continuous growth models in terms of generalized logarithm and exponential functions abstract consider the one parameter generalizations of the logarithmic and exponential functions which are obtained from the integration of non symmetrical hyperboles these generalizations coincide to the one obtained in the context of non extensive thermostatistics we show that these functions are suitable to describe and unify the great majority of continuous growth models which we briefly review physical interpretation to the generalization function parameter is given for the richards model which has an underlying microscopic model to justify it

a q generalization of laplace transforms the laplace transform is generalized by using the q exponential function eqx 1 1 q x 1 1 q that emerges from tsallis non extensive statistical mechanics and some of its properties are obtained the usual transform is recovered as a limiting case q1 the use of the q laplace transform is illustrated by establishing a relation between the classical canonical q partition function and the density of states

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

asymptotically scale invariant occupancy of phase space makes the entropy sq extensive phase space can be constructed for n equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated if they are locally correlated we expect the boltzmann gibbs entropy s bg k i p i ln p i to be extensive i e s bg n n for n in particular if they are independent s bg is strictly additive i e s bg n ns bg 1 n however if the subsystems are globally correlated we expect for a vast class of systems the entropy s q k 1 i p q i q 1 with s 1 s bg for some special value of q 1 to be the one which is extensive i e s q n n for n another concept which is relevant is strict or asymptotic scale freedom or scale invariance defined as the situation for which all marginal probabilities of the n system coincide or asymptotically approach for n the joint probabilities of the n 1 system if each subsystem is a binary one scale freedom is guaranteed by what we hereafter refer to as the leibnitz rule i e the sum of two successive joint probabilities of the n system coincides or asymptotically approaches the corresponding joint probability of the n 1 system the kinds of interplay of these various concepts are illustrated in several examples one of them justifies the title of this paper we conjecture that these mechanisms are deeply related to the very frequent emergence in natural and artificial complex systems of scale free structures and to their connections with nonextensive statistical mechanics summarizing we have shown that for asymptotically scale invariant systems it is s q with q 1 and not s bg the entropy which matches standard clausius like prescriptions of classical thermodynamics

distributivity and deformation of the reals from tsallis entropy we propose a one parameter family rq of deformations of the reals which is motivated by the generalized additivity of the tsallis entropy we introduce a generalized multiplication which is distributive with respect to the generalized addition of the tsallis entropy these operations establish a one parameter family of field isomorphisms q between r and rq through which an absolute value on rq is introduced this turns out to be a quasisymmetric map whose metric and measure theoretical implications are pointed out

a possible deformed algebra and calculus inspired in nonextensive thermostatistics we present a deformed algebra related to the q exponential and the q logarithm functions that emerge from nonextensive statistical mechanics we also develop a q derivative and consistently a q integral for which the q exponential is an eigenfunction the q derivative and the q integral have a dual nature that is also presented

use of the geometric mean as a statistic for the scale of the coupled gaussian distributions abstract the geometric mean is shown to be an appropriate statistic for the scale of a heavy tailed coupled gaussian distribution or equivalently the student s t distribution the coupled gaussian is a member of a family of distributions parameterized by the nonlinear statistical coupling which is the reciprocal of the degree of freedom and is proportional to fluctuations in the inverse scale of the gaussian existing estimators of the scale of the coupled gaussian have relied on estimates of the full distribution and they suffer from problems related to outliers in heavy tailed distributions in this paper the scale of a coupled gaussian is proven to be equal to the product of the generalized mean and the square root of the coupling from our numerical computations of the scales of coupled gaussians using the generalized mean of random samples it is indicated that only samples from a cauchy distribution with coupling parameter one form an unbiased estimate with diminishing variance for large samples nevertheless we also prove that the scale is a function of the geometric mean the coupling term and a harmonic number numerical experiments show that this estimator is unbiased with diminishing variance for large samples for a broad range of coupling values

variational approach and deformed derivatives recently we have demonstrated that there exists a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely the tsallis framework and the kaniadakis scenario with a local form of fractional derivative operators for fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure here in this paper we present an extension of the traditional calculus of variations for systems containing deformed derivatives embedded into the lagrangian and the lagrangian densities for classical and field systems the results extend the classical euler lagrange equations and the hamiltonian formalism the resulting dynamical equations seem to be compatible with those found in the literature specially with mass dependent and with nonlinear equations for systems in classical and quantum mechanics examples are presented to illustrate applications of the formulation also the conserved noether current is worked out

properties of the maximum q likelihood estimator for independent random variables abstract in the last two decades the nonextensive statistics proposed by tsallis have been extensively discussed in terms of the tsallis entropy which is a generalization of the boltzmann gibbs shannon entropy within the nonextensive framework many kinds of generalizations have been made suyari h suyari ieee trans on inf theory 51 2005 753 has proposed the generalized likelihood called the q likelihood in which a traditional product operator is replaced by the q product we study properties of the maximum q likelihood estimator mqle which is a generalization of the conventional maximum likelihood estimator with the use of the q likelihood we discuss mqle from the viewpoint of the minimization of the divergences in the nonextensive statistics it has been shown that optimum parameters determined by the mqle are nearly in agreement with those yielding the minimum distance of the divergence proposed by rajagopal a k rajagopal in s abe y okamoto eds nonextensive statistical mechanics and its applications springer 2000 p 99 we here show that the consistency of mqle is achieved along with the non negativity of rajagopal s divergence the asymptotic gaussianity and robustness of the mqle for independent random variables are discussed both by analytical methods and simulations

a note on the definition of deformed exponential and logarithm functions the recent generalizations of the boltzmann gibbs statistics mathematically rely on the deformed logarithmic and exponential functions defined through some deformation parameters in the present work we investigate whether a deformed logarithmic exponential map is a bijection from r r set of positive real numbers all real numbers to r r as their undeformed counterparts we show that their inverse map exists only in some subsets of the aforementioned co domains furthermore we present conditions which a generalized deformed function has to satisfy so that the most important properties of the ordinary functions are preserved the fulfillment of these conditions permits us to determine the validity interval of the deformation parameters we finally apply our analysis to tsallis q deformed functions and discuss the interval of concavity of the renyi entropy

tsallis renyi and nonextensive gaussian entropy derived from the respective multinomial coefficients abstract we explore the generalization of the ordinary multinomial coefficient based on the deformed q multiplication and q division aim of this study is to construct the appropriate multinomial coefficients from which one can obtain the tsallis renyi and nonextensive gaussian entropy respectively we show that for all three above entropies there are two possible ways to define the generalized multinomial coefficient its consequence is discussed

a note on the relevance of the q exponential function in the context of intertemporal choices abstract this paper shows that the q exponential function well known in the deformed algebra inspired in the tsallis s nonextensive thermodynamics may be used to model discount functions in intertemporal choices which present the phenomenon known as increasing patience moreover we show that this insight may also be used to provide a measure of the degree of dynamic inconsistency presented in such decisions

on q gaussians and exchangeability the q gaussian distributions introduced by tsallis are discussed from the point of view of variance mixtures of normals and exchangeability for each there is a q gaussian distribution that maximizes the tsallis entropy under suitable constraints this paper shows that q gaussian random variables can be represented as variance mixtures of normals when q 1 these variance mixtures of normals are the attractors in central limit theorems for sequences of exchangeable random variables thereby providing a possible model that has been extensively studied in probability theory the formulation provided has the additional advantage of yielding for each q a process which is naturally the q analog of the brownian motion explicit mixing distributions for q gaussians should facilitate applications to areas such as option pricing the model might provide insight into the study of superstatistics

non linear kinetics underlying generalized statistics abstract the purpose of the present effort is threefold firstly it is shown that there exists a principle that we call kinetical interaction principle kip underlying the non linear kinetics in particle systems independently on the picture kramers boltzmann used to describe their time evolution secondly the kip imposes the form of the generalized entropy associated to the system and permits to obtain the particle statistical distribution both as stationary solution of the non linear evolution equation and as the state which maximizes the generalized entropy thirdly the kip allows on one hand to treat all the classical or quantum statistical distributions already known in the literature in a unifying scheme and on the other hand suggests how we can introduce naturally new distributions finally as a working example of the approach to the non linear kinetics here presented a new non extensive statistics is constructed and studied starting from a one parameter deformation of the exponential function holding the relation f x f x 1

comment on renyi entropy yields artificial biases not in the data and incorrect updating due to the finite size data in their recent paper phys rev e 99 032134 2019 2470 004510 1103 physreve 99 032134 oikonomou and bagci have argued that renyi entropy is ill suited for inference purposes because it is not consistent with the shore johnson axioms of statistical estimation theory in this comment we seek to clarify the latter statement by showing that there are several issues in oikonomou s and bagci s reasonings which lead to erroneous conclusions when all these issues are properly accounted for no violation of shore johnson axioms is found

the deformed exponential functions of two variables in the context of various statistical mechanics in the recent development in various disciplines of physics it is noted the need for including the deformed versions of the exponential functions in last two decades the tsallis and kaniadakis versions have found a lot of applications in this paper we consider the deformations which have two purposes first we introduce them like beginning of a more general mathematical approach where the tsallis and kaniadakis exponential functions are the special cases then we wish to pay attention to the mathematical community that they have a lot of interesting properties from mathematical point of view and possibilities in applications really we will show the differential and difference properties of our deformations which are important for the formation and explanation of continuous and discrete models of numerous phenomena

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

on a q generalization of circular and hyperbolic functions a generalization of the circular and hyperbolic functions is proposed based on the tsallis statistics and on a consistent generalization of the euler formula some properties of the presently proposed q trigonometry are then investigated the generalized functions are exact solutions of a nonlinear oscillator original circular and hyperbolic relations are recovered as the limiting case

scaled bregman divergences in a tsallis scenario abstract there exist two different versions of the kullback leibler divergence k ld in tsallis statistics namely the usual generalized k ld and the generalized bregman k ld problems have been encountered in trying to reconcile them a condition for consistency between these two generalized k ld forms is derived by recourse to the additive duality of tsallis statistics it is also shown that the usual generalized k ld subjected to this additive duality known as the dual generalized k ld is a scaled bregman divergence this leads to an interesting conclusion the dual generalized mutual information is a scaled bregman information the utility and implications of these results are discussed

probability distribution and entropy as a measure of uncertainty the relationship between three probability distributions and their maximizable entropy forms is discussed without postulating entropy property for this purpose the entropy i is defined as a measure of uncertainty of the probability distribution of a random variable x by a variational relationship a definition underlying the maximization of entropy for corresponding distribution

generalized statistics variational perturbation approximation using q deformed calculus abstract a principled framework to generalize variational perturbation approximations vpas formulated within the ambit of the nonadditive statistics of tsallis statistics is introduced this is accomplished by operating on the terms constituting the perturbation expansion of the generalized free energy gfe with a variational procedure formulated using q deformed calculus a candidate q deformed generalized vpa gvpa is derived with the aid of the hellmann feynman theorem the generalized bogoliubov inequality for the approximate gfe are derived for the case of canonical probability densities that maximize the tsallis entropy numerical examples demonstrating the application of the q deformed gvpa are presented the qualitative distinctions between the q deformed gvpa model vis a vis prior gvpa models are highlighted

on the average uncertainty for systems with nonlinear coupling abstract the increased uncertainty and complexity of nonlinear systems have motivated investigators to consider generalized approaches to defining an entropy function new insights are achieved by defining the average uncertainty in the probability domain as a transformation of entropy functions the shannon entropy when transformed to the probability domain is the weighted geometric mean of the probabilities for the exponential and gaussian distributions we show that the weighted geometric mean of the distribution is equal to the density of the distribution at the location plus the scale i e at the width of the distribution the average uncertainty is generalized via the weighted generalized mean in which the moment is a function of the nonlinear source both the renyi and tsallis entropies transform to this definition of the generalized average uncertainty in the probability domain for the generalized pareto and student s t distributions which are the maximum entropy distributions for these generalized entropies the appropriate weighted generalized mean also equals the density of the distribution at the location plus scale a coupled entropy function is proposed which is equal to the normalized tsallis entropy divided by one plus the coupling