get_Sigs: Compute the covariance matrix $\Sigma_{\mathrm{ALL}}^*$ for observations within a moving window

Description

Calculates the covariance matrix $\Sigma_{\mathrm{ALL}}^*$ for all observations within a moving window of length $w$, given the random walk innovation covariance $\Sigma_{\boldsymbol{\eta}}$, the VAR(1) innovation covariance $\Sigma_{\boldsymbol{\nu}}$, the autoregressive coefficient matrix $\Phi$, and the initial-state covariance matrix $\Gamma_{\boldsymbol{\epsilon}}(0)$ (denoted here by `Sig_e1`). The resulting matrix accounts for both the random walk component and the temporal dependence introduced by the VAR(1) structure.

Usage

get_Sigs(w, p, Sig_eta, Sig_nu, Phi, Sig_e1)

Value

A list with the following components:

`Sig_AR` — Covariance contribution from the VAR(1) component ($\Sigma_{\mathrm{AR}}$).
`Sig_RW` — Covariance contribution from the random walk component ($\Sigma_{\mathrm{RW}}$).
`Sig_all` — Combined covariance matrix ($\Sigma_{\mathrm{ALL}}^* = \Sigma_{\mathrm{AR}} + \Sigma_{\mathrm{RW}}$).
`Sig_all_inv` — Inverse of the combined covariance matrix $(\Sigma_{\mathrm{ALL}}^*)^{-1}$.

Arguments

w: Integer; window size.
p: Integer; data dimension.
Sig_eta: Numeric $p \times p$ matrix representing the covariance of the random walk innovation $\Sigma_{\boldsymbol{\eta}}$.
Sig_nu: Numeric $p \times p$ matrix representing the covariance of the VAR(1) innovation $\Sigma_{\boldsymbol{\nu}}$.
Phi: Numeric $p \times p$ autoregressive coefficient matrix $\Phi$.
Sig_e1: Numeric $p \times p$ matrix representing the covariance of the initial state $\Gamma_{\boldsymbol{\epsilon}}(0)$.

Details

The function decomposes the overall covariance matrix $\Sigma_{\mathrm{ALL}}^*$ into two additive components corresponding to the random walk contribution $\Sigma_{\mathrm{RW}}$ and the autoregressive contribution $\Sigma_{\mathrm{AR}}$, so that $$\Sigma_{\mathrm{ALL}}^* = \Sigma_{\mathrm{RW}} + \Sigma_{\mathrm{AR}}.$$ When $p = 1$, the construction reduces to the scalar random walk and AR(1) case, for which closed-form covariance expressions are available. For higher-dimensional settings, block-matrix structures are constructed using functions from the blockmatrix package to capture both cross-sectional and temporal dependence. The returned inverse matrix $(\Sigma_{\mathrm{ALL}}^*)^{-1}$ is used in the main change point detection algorithm to adjust for the effects of random walk drift and vector autoregressive correlation.

Examples

Run this code

set.seed(1)
p <- 3
w <- 20
Sig_eta <- diag(0.01, p)
Sig_nu  <- random_Signu(p, 0)
Phi <- random_Phi(p, p)
Sig_e1 <- get_Sig_e1_approx(Sig_nu, Phi)
res <- get_Sigs(w, p, Sig_eta, Sig_nu, Phi, Sig_e1)

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