# fitHeavyTail

Robust estimation methods for the mean vector, scatter matrix, and
covariance matrix (if it exists) from data (possibly containing NAs)
under multivariate heavy-tailed distributions such as angular Gaussian
(via Tyler’s method), Cauchy, and Student’s t distributions.
Additionally, a factor model structure can be specified for the
covariance matrix. The latest revision also includes the multivariate
skewed t distribution.

## Installation

The package can be installed from CRAN or GitHub:

```
# install stable version from CRAN
install.packages("fitHeavyTail")
# install development version from GitHub
devtools::install_github("convexfi/fitHeavyTail")
```

To get help:

```
library(fitHeavyTail)
help(package = "fitHeavyTail")
?fit_mvt
```

To cite `fitHeavyTail`

in publications:

## Quick Start

To illustrate the simple usage of the package `fitHeavyTail`

,
let’s start by generating some multivariate data under a Student’s \(t\) distribution with significant heavy
tails (degrees of freedom \(\nu=4\)):

```
library(mvtnorm) # package for multivariate t distribution
N <- 10 # number of variables
T <- 80 # number of observations
nu <- 4 # degrees of freedom for heavy tails
set.seed(42)
mu <- rep(0, N)
U <- t(rmvnorm(n = round(0.3*N), sigma = 0.1*diag(N)))
Sigma_cov <- U %*% t(U) + diag(N) # covariance matrix with factor model structure
Sigma_scatter <- (nu-2)/nu * Sigma_cov
X <- rmvt(n = T, delta = mu, sigma = Sigma_scatter, df = nu) # generate data
```

We can first estimate the mean vector and covariance matrix via the
traditional sample estimates (i.e., sample mean and sample covariance
matrix):

```
mu_sm <- colMeans(X)
Sigma_scm <- cov(X)
```

Then we can compute the robust estimates via the package `fitHeavyTail`

:

```
library(fitHeavyTail)
fitted <- fit_mvt(X)
```

We can now compute the estimation errors and see the significant
improvement:

```
sum((mu_sm - mu)^2)
#> [1] 0.2857323
sum((fitted$mu - mu)^2)
#> [1] 0.1487845
sum((Sigma_scm - Sigma_cov)^2)
#> [1] 5.861138
sum((fitted$cov - Sigma_cov)^2)
#> [1] 4.663539
```

To get a visual idea of the robustness, we can plot the shapes of the
covariance matrices (true and estimated ones) on two dimensions. Observe
how the heavy-tailed estimation follows the true one more closely than
the sample covariance matrix:

## Documentation

For more detailed information, please check the vignette.

## Links

Package: CRAN and GitHub.

README file: GitHub-readme.

Vignette: CRAN-vignette
and GitHub-vignette.