ADCF {dCovTS}

Auto-Distance Correlation Function

Description

Computes the auto-distance correlation function of a univariate time series. It also computes the bias-corrected estimator of (squared) auto-distance correlation.

Usage

ADCF(x, MaxLag = 15, unbiased = FALSE)

Arguments

x

A numeric vector or univariate time series.

MaxLag

Maximum lag order at which to calculate the ADCF. Default is 15.

unbiased

Logical value. If unbiased = TRUE, the bias-corrected estimator of squared auto-distance correlation is returned. Default value is FALSE.

Details

Distance covariance and correlation firstly introduced by Szekely et al. (2007) are new measures of dependence between two random vectors. Zhou (2012) extended this measure to univariate time series framework.

For a univariate time series, ADCF computes the auto-distance correlation function, R_X(j), between \{X_t\} and \{X_{t+j}\}, whereas ADCV computes the auto-distance covariance function between them, denoted by V_X(j). Formal definition of R_X(\cdot) and V_X(\cdot) can be found in Zhou (2012) and Fokianos and Pitsillou (2017). The empirical auto-distance correlation function, \hat{R}_X(j), is computed as the positive square root of

\hat{R}_X^2(j)=\frac{\hat{V}_X^2(j)}{\hat{V}_X^2(0)}, \quad j=0, \pm 1, \pm 2, \dots

for \hat{V}_X^2(0) \neq 0 and zero otherwise, where \hat{V}_X(\cdot) is a function of the double centered Euclidean distance matrices of the sample X_t and its lagged sample X_{t+j} (see ADCV for more details). Theoretical properties of this measure can be found in Fokianos and Pitsillou (2017).

If unbiased = TRUE, ADCF computes the bias-corrected estimator of the squared auto-distance correlation, \tilde{R}_X^2(j), based on the unbiased estimator of auto-distance covariance function, \tilde{V}_X^2(j). The definition of \tilde{V}_X^2(j) relies on the U-centered matrices proposed by Szekely and Rizzo (2014) (see ADCV for a brief description).

mADCF computes the auto-distance correlation function of a multivariate time series.

Value

Returns a vector, whose length is determined by MaxLag, and contains the biased estimator of ADCF or the bias-corrected estimator of squared ADCF.

Note

Based on the definition of ADCF, one can observe that R^2_X(j)=R^2_X(-j) \forall j, and so results based on negative lags are omitted.

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 M. Pitsillou (2017). Consistent testing for pairwise dependence in time series. Technometrics, 159(2), 262-3270.

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.

Szekely, G. J. and M. L. Rizzo (2014). Partial distance correlation with methods for dissimilarities. The Annals of Statistics, 42, 2382-2412.

Szekely, G. J. and M. L. Rizzo and N. K. Bakirov (2007). Measuring and testing dependence by correlation of distances. The Annals of Statistics, 35, 2769-2794.

Zhou, Z. (2012). Measuring nonlinear dependence in time series, a distance correlation approach. Journal of Time Series Analysis 33, 438-457.

See Also

ADCFplot, ADCV, mADCF

Examples

x <- rnorm(100)
ADCF(x)

ADCF(ldeaths, 18)

ADCF(mdeaths, unbiased = TRUE)