mADCF {dCovTS}

Auto-Distance Correlation Matrix

Description

Computes the auto-distance correlation matrix of a multivariate time series.

Usage

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

Arguments

x

Multivariate time series.

lags

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.

unbiased

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.

output

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

Details

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.

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 correlation functions or the bias-corrected estimators of squared pairwise auto-distance correlation 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

ADCF, mADCV

Examples

x <- matrix( rnorm(200), ncol = 2 )

mADCF(x, 2)

mADCF(x, -2)

mADCF(x, lags = 4, unbiased = TRUE)