tensorBSS (version 0.3.9)

mModeAutoCovariance: The m-Mode Autocovariance Matrix

Description

Estimates the m-mode autocovariance matrix from an array of array-valued observations with the specified lag.

Usage

mModeAutoCovariance(x, m, lag, center = TRUE, normalize = TRUE)

Value

The m-mode autocovariance matrix of x with respect to lag having the size \(p_m \times p_m\).

Arguments

x

Array of order higher than two with the last dimension corresponding to the sampling units.

m

The mode with respect to which the autocovariance matrix is to be computed.

lag

The lag with respect to which the autocovariance matrix is to be computed.

center

Logical, indicating whether the observations should be centered prior to computing the autocovariance matrix. Default is TRUE.

normalize

Logical, indicating whether the resulting matrix is divided by p_1 ... p_{m-1} p_{m+1} ... p_r or not. Default is TRUE.

Author

Joni Virta

Details

The m-mode autocovariance matrix provides a higher order analogy for the ordinary autocovariance matrix of a random vector and is computed for a random tensor \(X_t\) of size \(p_1 \times p_2 \times \ldots \times p_r\) as \(Cov_{m \tau}(X_t) = E(X_t^{(m)} X_{t+\tau}^{(m)T})/(p_1 \ldots p_{m-1} p_{m+1} \ldots p_r)\), where \(X_t^{(m)}\) is the centered \(m\)-flattening of \(X_t\) and \(\tau\) is the desired lag. The algorithm computes the estimate of this based on the sample x.

References

Virta, J. and Nordhausen, K., (2017), Blind source separation of tensor-valued time series, Signal Processing, 141, 204-216, tools:::Rd_expr_doi("10.1016/j.sigpro.2017.06.008")

See Also

mModeCovariance

Examples

n <- 1000
S <- t(cbind(as.vector(arima.sim(n = n, list(ar = 0.9))),
             as.vector(arima.sim(n = n, list(ar = -0.9))),
             as.vector(arima.sim(n = n, list(ma = c(0.5, -0.5)))),
             as.vector(arima.sim(n = n, list(ar = c(-0.5, -0.3)))),
             as.vector(arima.sim(n = n, list(ar = c(0.5, -0.3, 0.1, -0.1), ma=c(0.7, -0.3)))),
             as.vector(arima.sim(n = n, list(ar = c(-0.7, 0.1), ma = c(0.9, 0.3, 0.1, -0.1))))))
dim(S) <- c(3, 2, n)

mModeAutoCovariance(S, m = 1, lag = 1)
mModeAutoCovariance(S, m = 1, lag = 4)