autocov: Autocovariance matrix

In SChangeBlock: Spatial Structural Change Detection by an Analysis of Variability Between Blocks of Observations

Autocovariance matrix

Description

Estimates the autocovariance matrix for a given data matrix X. Via the parameter direction, it is possible to estimate only row- or columnwise autocovariance matrices, which is useful if the autocovariance function is separable.

Usage

autocov(X, b, M = as.integer(c(1, 1)), direction = 0L, type = 0L)

Arguments

X	numeric matrix,
b	numeric vector containing two integer values: the bandwidths for the row- resp. column-wise estimation. (Up to which lag should the autocovariances be estimated?) If direction > 0: only one integer must be supplied. Bandwidths must be smaller than the dimensions of X.
M	numeric vector containing two integer values, only needed for type = 1, see Details.
direction	0: all directions, 1: only row-wise autocovariances, 2: only column-wise autocovariances.
type	0: ordinary autocovariance estimation, 1: difference-based autocovariance estimation. See Details.

Details

In this function, the autocovariance matrix of X is interpreted as the autocovariance matrix of x = as.vector(X), i.e. where X is ordered into a vector column-wise. If type = 0, the autocovariance to lags h_1, h_2 is estimated using the regular estimator

\hat{\gamma}_{\text{reg}}(h_1, h_2) = \frac{1}{(n-h_1)(m-h_2)}\sum_{i = 1}^{n-h_1} \sum_{j = 1}^{m-h_2} (Y_{i, j} - \bar{Y})(Y_{i+h_1, j+h_2} - \bar{Y}).

If type = 1, the autocovariance to lags h_1, h_2 is estimated by a difference-based version, inspired by the estimator of Tecuapetla-Gómez and Munk (2017) for time series:

\hat{\gamma}_{\text{diff}}(h_1, h_2) = \hat{\sigma}_{\text{diff}}^2 - \frac{1}{2(n - h_1)(m - h_2)}\sum_{i = 1}^{n-h_1} \sum_{j = 1}^{m-h_2} (Y_{i, j} - Y_{i + h_1, j + h_2})^2

with

\hat{\sigma}_{\text{diff}}^2 = \frac{1}{4} \left(\frac{1}{n(m - M_2)}\sum_{i = 1}^n \sum_{j = 1}^{m - M_2} (Y_{i, j} - Y_{i, j+M_2})^2 + \frac{1}{(n - M_1)m}\sum_{i = 1}^{n-M_1} \sum_{j = 1}^{m} (Y_{i, j} - Y_{i+M_1, j})^2 \right),

where M_1 = M[1], M_2 = M[2].

Value

A numeric matrix of size N \times N. If direction = 0 then N = prod(dim(X)). If direction = 1 then N = ncol(X), if direction = 2 then N = nrow(X).

References

Tecuapetla‐Gómez, I., & Munk, A. (2017). Autocovariance estimation in regression with a discontinuous signal and m‐dependent errors: A difference‐based approach. Scandinavian Journal of Statistics, 44(2), 346-368.

Examples

X <- genField(c(20, 20))
autocov(X, c(4, 4))[1:10, 1:100]

# if separable:
Sigma1 <- autocov(X, 4, direction = 1)
Sigma2 <- autocov(X, 4, direction = 2)
kronecker(Sigma1, Sigma2)[1:10, 1:100]

SChangeBlock documentation built on May 26, 2026, 5:06 p.m.