dCovTS (version 1.1)

mADCV: Auto-Distance Covariance Matrix

Description

Computes the sample auto-distance covariance matrices of a multivariate time series.

Usage

mADCV(x, lags, unbiased = FALSE, output = TRUE)

Arguments

x

multivariate time series.

lags

lag order at which to calculate the mADCV. No default is given.

unbiased

logical value. If unbiased = TRUE, the individual elements of auto-distance covariance matrix correspond to the unbiased estimators of squared auto-distance covariance functions. Default value is FALSE.

output

logical value. If output=FALSE, no output is given. Default value is TRUE.

Value

Returns a matrix containing either the biased estimators of the pairwise auto-distance covariance functions or the unbiased estimators of squared pairwise auto-distance covariance functions at lag, \(j\), determined by the argument lags.

Details

Suppose that \(\textbf{X}_t=(X_{t;1}, \dots, X_{t;d})'\) is a multivariate time series of dimension \(d\). Then, mADCV computes the \(d \times d\) sample auto-distance covariance matrix, \(\hat{V}(\cdot)\), of \(\textbf{X}_t\) given by $$ \hat{V}(j) = [\hat{V}_{rm}(j)]_{r,m=1}^d , \quad j=0, \pm 1, \pm 2, \dots, $$ where \(\hat{V}_{rm}(j)\) denotes the biased estimator of the pairwise auto-distance covariance function between \(X_{t;r}\) and \(X_{t+j;m}\). The definition of \(\hat{V}_{rm}(j)\) is given analogously as in the univariate case (see ADCV). Formal definitions and theoretical properties of auto-distance covariance matrix can be found in Fokianos and Pitsillou (2016).

If unbiased = TRUE, mADCV computes the matrix, \(\tilde{V}^{(2)}(j)\), whose elements correspond to the unbiased estimators of squared pairwise auto-distance covariance functions, namely $$ \tilde{V}^{(2)}(j) = [\tilde{V}^2_{rm}(j)]_{r,m=1}^d , \quad j=0, \pm 1, \pm 2, \dots. $$ The definition of \(\tilde{V}_{rm}^2(\cdot)\) is defined analogously as explained in the univariate case (see ADCV).

References

Fokianos K. and M. Pitsillou (2016). Testing pairwise independence for multivariate time series by the auto-distance correlation matrix. Submitted for publication.

See Also

ADCV,mADCF

Examples

# NOT RUN {
x <- MASS::mvrnorm(100,rep(0,2),diag(2))
mADCV(x,lags=1)
mADCV(x,lags=15)

y <- as.ts(swiss)
mADCV(y,15)
mADCV(y,15,unbiased=TRUE)
# }