Related papers: Fractional calculus approach to the statistical ch…
Fractional calculus has been used to describe physical systems with complexity. Here, we show that a fractional calculus approach can restore or include complexity in any physical systems that can be described by partial differential…
This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored, and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact…
Probability density estimation is a core problem of statistics and signal processing. Moment methods are an important means of density estimation, but they are generally strongly dependent on the choice of feasible functions, which severely…
We introduce a discrete-time fractional calculus of variations on the time scales $\mathbb{Z}$ and $(h\mathbb{Z})_a$. First and second order necessary optimality conditions are established. Some numerical examples illustrating the use of…
We introduce a novel method for obtaining a wide variety of moments of any random variable with a well-defined moment-generating function (MGF). We derive new expressions for fractional moments and fractional absolute moments, both central…
Many physical, biological, and engineered systems exhibit memory effects that challenge Markovian models. Fractional calculus provides nonlocal operators to capture hereditary dynamics. This survey connects modeling, analysis, and…
Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties by using integrations and…
This article provides an overview on the statistical modeling of complex data as increasingly encountered in modern data analysis. It is argued that such data can often be described as elements of a metric space that satisfies certain…
In this work, an inverse problem in the fractional diffusion equation with random source is considered. Statistical moments are used of the realizations of single point observation $u(x_0,t,\omega).$ We build the representation of the…
Functional data analysis offers a diverse toolkit of statistical methods tailored for analyzing samples of real-valued random functions. Recently, samples of time-varying random objects, such as time-varying networks, have been increasingly…
We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then…
Integer-order differential operators were originally used to describe local and isotropic effects, in both space and time. However, in fields like biology, the modelling of complex phenomena with spatial heterogeneity necessitates more…
Concentration inequalities, a major tool in probability theory, quantify how much a random variable deviates from a certain quantity. This paper proposes a systematic convex optimization approach to studying and generating concentration…
Getting tools that allow simple representations and comparisons of a set of categorical trajectories is of major interest for statisticians. Without loosing any information, we associate to each state a binary random indicator function,…
In this review, we present some fundamental classical and quantum phenomena in view of time fractional formalism. Time fractional formalism is a very useful tool in describing systems with memory and delay. We hope that this study can…
Viscoelasticity and related phenomena are of great importance in the study of mechanical properties of material especially, biological materials. Certain materials show some complex effects in mechanical tests, which cannot be described by…
Data series generated by complex systems exhibit fluctuations on many time scales and/or broad distributions of the values. In both equilibrium and non-equilibrium situations, the natural fluctuations are often found to follow a scaling…
First, we present a concise glossary of formulas for composition of standard, cumulant, factorial, and factorial cumulant moments in superposition (compound) models, where final particles are created via independent emission from a…
Frequentist inference typically is described in terms of hypothetical repeated sampling but there are advantages to an interpretation that uses a single random sample. Contemporary examples are given that indicate probabilities for random…
In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…