The MittagLeffleR R package

- calculates probability densities, probabilities and quantiles, based on a

Laplace-Inversion algorithm by Roberto Garrappa. - simulates random variables from both types Mittag-Leffler distributions
- fits a Mittag-Leffler distribution to data, using the log-moments estimator (for the first type of distribution) by Dexter Cahoy.

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.

You can install MittagLeffleR from CRAN via

```
install.packages("MittagLeffleR")
library(MittagLeffleR)
```

Install the `devtools`

package first, then

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

See reference manual.

Generate a dataset first:

```
library(MittagLeffleR)
y = rml(n = 10000, tail = 0.9, scale = 2)
```

Fit the distribution:

```
logMomentEstimator(y, 0.95)
#> tail scale tailLo tailHi scaleLo scaleHi
#> 0.9030906 2.0045782 0.9027436 0.9034377 2.0026354 2.0065210
```

Read off

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

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 \(t=1\).)

```
library(ggplot2)
library(tidyr)
tail <- 0.65
dx <- 0.01
x <- seq(-2,5,dx)
t <- 3^(-1:2)
# cut off time so that only 1 % of probability is lost
umax <- qml(p = 0.99, tail = tail, scale = max(t), second.type = TRUE)
u <- seq(0.01,umax,dx)
H <- outer(u,t, function(u,t) {dml(x = u, tail = tail, scale = t^tail)})
N <- outer(x,u,function(x,u){dnorm(x = x, mean = u, sd = sqrt(u))})
p <- N %*% H * dx
df <- data.frame(p)
names(df) <- sapply(t, function(t){paste0("T=",round(t,2))})
df['x'] <- x
df %>%
gather(key = "time", value = "density", -x) %>%
ggplot(mapping = aes(x=x, y=density, col=time)) +
geom_line() +
labs(ggtitle("Subdiffusion with drift"))
```

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