Mittag-Leffler Family of Distributions

Implements the Mittag-Leffler function, distribution, random variate generation, and estimation. Based on the Laplace-Inversion algorithm by Garrappa, R. (2015) .

The MittagLeffleR R package

The first type Mittag-Leffler distribution is a heavy-tailed distribution, and occurs mainly as a waiting time distribution in problems with “fractional” time scales, e.g. times between earthquakes.

The second type Mittag-Leffler distribution is light-tailed, and “inverse” to the sum-stable distributions. It typically models the number of events in fractional systems and is used for time-changes of stochastic processes, e.g. anomalous diffusion processes.

Installation

Stable release on CRAN

You can install MittagLeffleR from CRAN via

Development version on Github

Install the devtools package first, then

Usage

See the reference manual at strakaps.github.io/MittagLeffleR/reference/index.html

Examples

Fitting a Mittag-Leffler distribution

Generate a dataset first:

Fit the distribution:

• the shape parameter (0 < \nu < 1),
• the scale parameter (\delta > 0),
• their 95% confidence intervals.

Calculate the probability density of an anomalous diffusion process

Standard Brownian motion with drift (1) has, at time (t), has a normal probability density (n(x|\mu = t, \sigma^2 = t)). A fractional diffusion at time (t) has the time-changed probability density

[p(x,t) = \int n(x| \mu = u, \sigma^2 = u)h(u,t) du]

where (h(u,t)) is a second type Mittag-Leffler probability density with scale (t^\alpha). (We assume (t=1).)

Vignettes

See the page strakaps.github.io/MittagLeffleR/articles/ for vignettes on

• Plots of the Mittag-Leffler distributions
• Details of Mittag-Leffler random variate generation
• Probabilities and Quantiles

News

MittagLeffleR 0.2.0 [2017-11-07]

• Added log-moment estimator by Dexter Cahoy

MittagLeffleR 0.1.0 [2017-06-16]

• First version on CRAN

Reference manual

install.packages("MittagLeffleR")

0.4.1 by Peter Straka, 5 months ago

https://strakaps.github.io/MittagLeffleR/

Report a bug at https://github.com/strakaps/MittagLeffleR/issues

Browse source code at https://github.com/cran/MittagLeffleR

Authors: Katharina Hees [aut] , Peter Straka [aut, cre] , Gurtek Gill [aut] , Roberto Garrappa [ctb]

Documentation:   PDF Manual