In rickygill01/MittagLeffler: The Mittag-Leffler Distribution

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MittagLeffleR

The MittagLeffleR R package calculates probabilities, quantiles and random variables from both types Mittag-Leffler distributions.

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, in a sense, inverse to the family of sum-stable distributions. It is used for time-changes of stochastic processes, to create "time-fractional" evolutions, e.g. anomalous diffusion processes.

Installation

You can install MittagLeffleR from GitHub with:

# install.packages("devtools")
devtools::install_github("strakaps/MittagLeffler")

Examples

Fitting a Mittag-Leffler distribution to a random dataset

Fit a Mittag-Leffler distribution (first type) to a collection of Mittag-Leffler random variables:

library(MittagLeffleR)
y = rml(n = 100, tail = 0.9, scale = 2)
mlml <- function(X) {
  log_l <- function(theta) {
    #transform parameters so can do optimization unconstrained
    theta[1] <- 1/(1+exp(-theta[1]))
    theta[2] <- exp(theta[2])
    -sum(log(dml(X,theta[1],theta[2])))
  }
  ml_theta <- stats::optim(c(0.5,0.5), fn=log_l)$par
  #transform back
  ml_a <- 1/(1 + exp(-ml_theta[1]))
  ml_d <- exp(ml_theta[2])
  return(list("tail" = ml_a, "scale" = ml_d))
}
mlml(y)

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 $t=1$.)

tail <- 0.65
dx <- 0.01
x <- seq(-2,5,dx)
umax <- qml(p = 0.99, tail = tail, scale = 1, second.type = TRUE)
u <- seq(0.01,umax,dx)
h <- dml(x = u, tail = tail, scale = 1, second.type = TRUE)
N <- outer(x,u,function(x,u){dnorm(x = x, mean = u, sd = sqrt(u))})
p <- N %*% h * dx
plot(x,p, type='l', main = "Fractional diffusion with drift at t=1")

Vignettes

Vignettes are written in R Markdown.

• MLdist.Rmd: shows plots with the basic properties of the two distributions.
• probsNquantiles.Rmd: explains how probability densities, quantiles and random variables are calculated using the Mittag-Leffler function and the package stabledist
• parametrisation.Rmd: explains how the parameters for the stable distribution were chosen.

rickygill01/MittagLeffler documentation built on May 24, 2019, 6:17 a.m.