mADCV: Auto-Distance Covariance Matrix

In dCovTS: Distance Covariance and Correlation for Time Series Analysis

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	The lag order at which to calculate the mADCV. No default is given.
unbiased	A 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	A logical value. If output=FALSE, no output is given. Default value is TRUE.

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 (2018).

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).

Value

If lags is a single number then the function will return a matrix. If lags is a vector of many values the function will return an array. For either case, the matrix (matrices) will contain 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.

Author(s)

Maria Pitsillou, Michail Tsagris and Konstantinos Fokianos.

References

Edelmann, D, K. Fokianos. and M. Pitsillou. (2019). An Updated Literature Review of Distance Correlation and Its Applications to Time Series. International Statistical Review, 87, 237-262.

Fokianos K. and Pitsillou M. (2018). Testing independence for multivariate time series via the auto-distance correlation matrix. Biometrika, 105, 337-352.

Huo, X. and G. J. Szekely. (2016). Fast Computing for Distance Covariance. Technometrics, 58, 435-447.

Pitsillou M. and Fokianos K. (2016). dCovTS: Distance Covariance/Correlation for Time Series. R Journal, 8, 324-340.

See Also

ADCV, mADCF

Examples

x <- matrix( rnorm(200), ncol = 2 )
mADCV(x, lags = 1)
mADCV(x, lags = 15)

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

dCovTS documentation built on Sept. 29, 2023, 1:06 a.m.