# fitHeavyTail

Robust estimation methods for the mean vector and covariance matrix from data (possibly containing NAs) under multivariate heavy-tailed distributions such as angular Gaussian, Cauchy, and Student’s t. Additionally, a factor model structure can be specified for the covariance matrix.

## 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("dppalomar/fitHeavyTail")``````

To get help:

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

To cite `fitHeavyTail` in publications:

``citation("fitHeavyTail")``

## 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:

``````library(mvtnorm)  # package for multivariate t distribution
N <- 10   # number of variables
T <- 80   # number of observations
nu <- 4   # degrees of freedom for tail heavyness

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 big improvement:

``````sum((mu_sm     - mu)^2)
#>  0.2857323
sum((fitted\$mu - mu)^2)
#>  0.1404855

sum((Sigma_scm  - Sigma_cov)^2)
#>  5.861138
sum((fitted\$cov - Sigma_cov)^2)
#>  4.107825``````

To get a visual idea of the robustness, we can plot the shapes of the covariance matrices (true and estimated ones) projected 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.

Package: CRAN and GitHub.