Computes the sample auto-distance covariance matrices of a multivariate time series.
mADCV(x, lags, unbiased = FALSE, output = TRUE)Multivariate time series.
The lag order at which to calculate the mADCV. No default is given.
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.
A logical value. If output=FALSE, no output is given. Default value is TRUE.
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.
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).
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.
# NOT RUN {
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)
# }
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