nu_POP_estimator {fitHeavyTail}

Estimate the degrees of freedom of a heavy-tailed t distribution based on the POP estimator

Description

This function estimates the degrees of freedom of a heavy-tailed t distribution based on the POP estimator from paper [Pascal-Ollila-Palomar, EUSIPCO2021, Alg. 1]. Traditional nonparametric methods or likelihood methods provide erratic estimations of the degrees of freedom unless the number of observations is very large. The POP estimator provides a stable estimator based on random matrix theory. A number of different versions are provided, but the default POP method will most likely be the desired choice.

Usage

nu_POP_estimator(
  Xc = NULL,
  N = NULL,
  T = NULL,
  Sigma = NULL,
  nu = NULL,
  r2 = NULL,
  method = c("POP", "POP-approx-1", "POP-approx-2", "POP-approx-3", "POP-approx-4",
    "POP-exact", "POP-sigma-corrected", "POP-sigma-corrected-true"),
  alpha = 1
)

Arguments

Xc

Centered data matrix (with zero mean) containing the multivariate time series (each column is one time series).

N

Number of variables (columns of data matrix) in the multivariate time series.

T

Number of observations (rows of data matrix) in the multivariate time series.

Sigma

Current estimate of the scatter matrix.

nu

Current estimate of the degrees of freedom of the t distribution.

r2

Vector containing the values of diag( Xc %*% inv(scatter) %*% t(Xc) ).

method

String indicating the version of the POP estimator (default is just "POP" and should work well in all cases). Other versions include: "POP-approx-1", "POP-approx-2", "POP-approx-3", "POP-approx-4", "POP-exact", "POP-sigma-corrected", "POP-sigma-corrected-true".

alpha

Value for the acceleration technique (cf. fit_mvt()).

Value

Estimated value of the degrees of freedom nu of a heavy-tailed t distribution.

Author(s)

Frédéric Pascal, Esa Ollila, and Daniel P. Palomar

References

Frédéric Pascal, Esa Ollila, and Daniel P. Palomar, "Improved estimation of the degree of freedom parameter of multivariate t-distribution," in Proc. European Signal Processing Conference (EUSIPCO), Dublin, Ireland, Aug. 23-27, 2021. <https://doi.org/10.23919/EUSIPCO54536.2021.9616162>

Examples

library(mvtnorm)       # to generate heavy-tailed data
library(fitHeavyTail)

# parameters
N <- 5
T <- 100
nu_true <- 4           # degrees of freedom
mu_true <- rep(0, N)   # mean vector
Sigma_true <- diag(N)  # scatter matrix

# generate data
X <- rmvt(n = T, sigma = Sigma_true, delta = mu_true, df = nu_true)  # generate Student's t data
mu <- colMeans(X)
Xc <- X - matrix(mu, T, N, byrow = TRUE)    # center data

# usage #1
nu_POP_estimator(Xc = Xc, nu = 10, Sigma = Sigma_true)

# usage #2
r2 <- rowSums(Xc * (Xc %*% solve(Sigma_true)))
nu_POP_estimator(r2 = r2, nu = 10, N = N)

# usage #3
nu_POP_estimator(r2 = r2, nu = 10, N = N, method = "POP-approx-1")