dCovTS (version 1.1)

mADCF: 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

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

unbiased

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

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 correlation functions or the bias-corrected estimators of squared pairwise auto-distance correlation functions at lag, \(j\), determined by the argument lags.

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 (2016).

If unbiased = TRUE, mADCF returns a matrix that contains the bias-corrected estimators of squared pairwise auto-distance correlation functions, namely $$ \tilde{R}^{(2)}(j) = [\tilde{R}^2_{rm}(j)]_{r,m=1}^d, \quad j=0, \pm 1, \pm 2, \dots. $$ \(\tilde{R}_{rm}^2(\cdot)\) are based on the unbiased estimator of pairwise auto-distance covariance, \(\tilde{V}_{rm}^2(\cdot)\). The definition of \(\tilde{V}_{rm}^2(\cdot)\) can be found in mADCV.

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

ADCF,mADCV

Examples

# NOT RUN {
x <- MASS::mvrnorm(100,rep(0,2),diag(2))
mADCF(x,2)

mADCF(x,-2)

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