TestIndVars (version 0.1.0)

covMatAR: Generate a covariance matrix with Autoregressive (AR) structure.

Description

This function generates generates an Autoregressive (AR) covariance structure matrix of size \(p \times p\) based on the specified autoregressive coefficient (\(\rho\)) and variance (\(\sigma^2\)).

Usage

covMatAR(p, sigma2 = 1, rho)

Value

A \(p \times p\) numeric matrix representing the Autoregressive (AR) covariance structure.

Arguments

p

An integer specifying the number of dimensions of the covariance matrix.

sigma2

A numeric value specifying the variance parameter (default = 1).

rho

A numeric value specifying the autoregressive coefficient. If not provided, a random value between 0 and 1 will be generated.

The Autoregressive structure is defined as follows:

$$\Sigma = \Sigma_{AR} = \sigma^2 \begin{bmatrix} 1 & \rho & \rho^2 & \cdots & \rho^{\lvert p-1 \rvert} \\ \rho & 1 & \rho & \cdots & \rho^{\lvert p-2 \rvert} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \rho^{\lvert p-1 \rvert} & \rho^{\lvert p-2 \rvert} & \rho^{\lvert p-3 \rvert} \cdots & 1 \end{bmatrix}$$ where \(\Sigma \) is the covariance matrix, \(\sigma^2\) is the variance parameter, and \(\rho \) is the correlation parameter.

Examples

# generate a covariance matrix for \eqn{p = 5}, \eqn{\sigma^2 = 1}, and \eqn{\rho = 0.9}.
covMatAR(p = 5, rho = 0.9)

# generate a covariance matrix for \eqn{p = 5},  \eqn{\sigma^2 = 5}, and \eqn{\rho = 0.9}.
covMatAR(p = 5, sigma2 = 5, rho = 0.9)

# generate  covariance matrix for \eqn{p = 5},  and no value is considered for \eqn{\rho}
covMatAR(p = 5)