Fractional-order formulation of power-law and exponential distributions

Alexopoulos, Aris; Weinberg, Graham V.

doi:10.1016/j.physleta.2014.07.007

Cited by 13 publications

(13 citation statements)

References 5 publications

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“…For a comprehensive review of the many versions that have been derived, see [31] and the references therein. Research into fractional order mathematics has been prevalent in recent times in many fields of science, mathematics and engineering [32][33][34][35][36][37][38][39][40][41]. In this paper, the interest is in the fractional derivative of functions only and the Riemann-Liouville formulation for the fractional derivative will be considered:…”

Section: Fractional Divergence Of Exponential and Pareto Densitiesmentioning

confidence: 99%

Fractional Divergence of Probability Densities

Alexopoulos¹

2017

Fractal Fract

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Abstract:The divergence or relative entropy between probability densities is examined. Solutions that minimise the divergence between two distributions are usually "trivial" or unique. By using a fractional-order formulation for the divergence with respect to the parameters, the distance between probability densities can be minimised so that multiple non-trivial solutions can be obtained. As a result, the fractional divergence approach reduces the divergence to zero even when this is not possible via the conventional method. This allows replacement of a more complicated probability density with one that has a simpler mathematical form for more general cases.

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Section: Fractional Divergence Of Exponential and Pareto Densitiesmentioning

confidence: 99%

Fractional Divergence of Probability Densities

Alexopoulos¹

2017

Fractal Fract

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“…The development and application of fractional calculus in the analysis of data has been introduced in [2]. Using fractional order derivatives or integrals of a function allows unprecedented asymptotic variation of the function which can be utilised in the analysis of data such as clutter.…”

Section: Preliminariesmentioning

confidence: 99%

“…The problem with the fractional integro-differential (1) and (2) is that in most cases the differentiation and integration of functions cannot be performed analytically. Various ways around this problem have been proposed, but there is a simple and straightforward way to deal with this issue.…”

Section: Iet Radar Sonar and Navigationmentioning

confidence: 99%

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Fractional order Pareto distributions with application to X‐band maritime radar clutter

Alexopoulos

Weinberg

2015

IET Radar, Sonar & Navigation

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The theory of fractional order calculus is applied to construct variations of the fundamental Pareto clutter model density, used in X-band maritime surveillance radar. Such data are characterised, at high resolution, by spiky clutter returns whose density function possesses heavy tails. By an application of fractional calculus to the Pareto distribution function, together with optimisation applied to determine the fractional derivative order, it is possible to obtain a new model for radar clutter. This is shown to fit real radar clutter returns better than a standard Pareto model. Additionally, it accounts for receiver thermal noise without the complexity of an additional noise distribution.

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“…Fractional mathematics has the potential to change the way statistical analysis of problems is made. The reader is referred to [16][17][18] for details and the references therein. It is worth noting that the word 'fractional' is a historical misnomer.…”

Section: Introductionmentioning

confidence: 99%

One-Parameter Weibull-Type Distribution, Its Relative Entropy with Respect to Weibull and a Fractional Two-Parameter Exponential Distribution

Alexopoulos¹

2019

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A new one-parameter distribution is presented with similar mathematical characteristics to the two parameter conventional Weibull. It has an estimator that only depends on the sample mean. The relative entropy with respect to the Weibull distribution is derived in order to examine the level of similarity between them. The performance of the new distribution is compared to the Weibull and in some cases the Gamma distribution using real data. In addition, the Exponential distribution is modified to include an extra parameter via a simple transformation using fractional mathematics. It will be shown that the modified version also exhibits Weibull characteristics for particular values of the second parameter.

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Fractional-order formulation of power-law and exponential distributions

Cited by 13 publications

References 5 publications

Fractional Divergence of Probability Densities

Fractional Divergence of Probability Densities

Fractional order Pareto distributions with application to X‐band maritime radar clutter

One-Parameter Weibull-Type Distribution, Its Relative Entropy with Respect to Weibull and a Fractional Two-Parameter Exponential Distribution

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