Computes the auto-distance correlation matrix of a multivariate time series.
mADCF(x, lags, unbiased = FALSE, output = 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 correlation functions or the
bias-corrected estimators of squared pairwise auto-distance correlation
functions at lag, \(j\), determined by the argument lags.
Multivariate time series.
The lag order at which to calculate the mADCF. No default is given.
This can be a single number or a vector of numbers with different lag orders.
A logical value. If unbiased = TRUE, the individual elements of auto-distance correlation matrix correspond to the bias-corrected estimators of squared auto-distance correlation functions. Default value is FALSE.
A logical value. If output=FALSE, no output is given. Default value is TRUE.
Maria Pitsillou, Michail Tsagris and Konstantinos Fokianos.
If \(\textbf{X}_t=(X_{t;1}, \dots, X_{t;d})'\) is a multivariate time series
of dimension \(d\), then mADCF computes the sample auto-distance
correlation matrix, \(\hat{R}(\cdot)\), of \(\textbf{X}_t\). It is defined by
$$ \hat{R}(j) = [\hat{R}_{rm}(j)]_{r,m=1}^d, \quad j=0, \pm 1, \pm 2, \dots,
$$
where \(\hat{R}_{rm}(j)\) is the biased estimator of the so-called pairwise auto-distance correlation function between \(X_{t;r}\) and \(X_{t+j;m}\) given by the positive square root of $$ \hat{R}_{rm}^2(j) = \frac{\hat{V}_{rm}^2(j)}{\hat{V}_{rr}(0)\hat{V}_{mm}(0)} $$ for \(\hat{V}_{rr}(0)\hat{V}_{mm}(0) \neq 0\) and zero otherwise.
\(\hat{V}_{rm}(j)\) is the \((r,m)\) element of the corresponding
mADCV matrix at lag \(j\).
Formal definition and more details can be found in Fokianos and Pitsillou (2017).
If unbiased = TRUE, mADCF returns a matrix that contains the
bias-corrected estimators of squared pairwise
auto-distance correlation functions.
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.
ADCF, mADCV
x <- matrix( rnorm(200), ncol = 2 )
mADCF(x, 2)
mADCF(x, -2)
mADCF(x, lags = 4, unbiased = TRUE)
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