ADCV {dCovTS}

Auto-distance Covariance Function

Description

Computes the auto-distance covariance function of a univariate time series. It also computes the unbiased estimator of squared auto-distance covariance.

Usage

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

Arguments

x

A numeric vector or univariate time series.

MaxLag

The maximum lag order at which to calculate the ADCV. Default is 15.

unbiased

A logical value. If unbiased = TRUE, the unbiased estimator of squared auto-distance covariance is returned. Default value is FALSE.

Details

Szekely et al. (2007) proposed distance covariance function between two random vectors. Zhou (2012) extended this measure of dependence to a time series framework by calling it auto-distance covariance function.

ADCV computes the sample auto-distance covariance function, V_X(\cdot), between \{X_t\} and \{X_{t+j}\}. Formal definition of V_X(\cdot) can be found in Zhou (2012) and Fokianos and Pitsillou (2017).

The empirical auto-distance covariance function, \hat{V}_X(\cdot), is the non-negative square root defined by

\hat{V}_X^2(j) = \frac{1}{(n-j)^2}\sum_{r,l=1+j}^{n}{A_{rl}B_{rl}}, \quad 0 \leq j \leq (n-1)

and \hat{V}_X^2(j) = \hat{V}_X^2(-j), for -(n-1) \leq j < 0, where A=A_{rl} and B=B_{rl} are Euclidean distances with elements given by

A_{rl} = a_{rl} - \bar{a}_{r.} - \bar{a}_{.l} + \bar{a}_{..}

with a_{rl}=|X_r-X_l|, \bar{a}_{r.}=\Bigl(\sum_{l=1+j}^{n}{a_{rl}}\Bigr)/(n-j), \bar{a}_{.l}=\Bigl(\sum_{r=1+j}^{n}{a_{rl}}\Bigr)/(n-j) , \bar{a}_{..}=\Bigl(\sum_{r,l=1+j}^{n}{a_{rl}}\Bigr)/(n-j)^2. B_{rl} is given analogously based on b_{rl}=|Y_r-Y_l|, where Y_t=X_{t+j}. X_t and X_{t+j} are independent if and only if V_X^2(j)=0. See Fokianos and Pitsillou (2017) for more information on theoretical properties of V_X^2(\cdot) including consistency.

If unbiased = TRUE, ADCV returns the unbiased estimator of squared auto-distance covariance function, \tilde{V}_X^2(j), proposed by Szekely and Rizzo (2014). In the context of time series data, this is given by

\tilde{V}_X^2(j) = \frac{1}{(n-j)(n-j-3)}\sum_{r\neq l}{\tilde{A}_{rl}\tilde{B}_{rl}},

for n > 3, where \tilde{A}_{rl} is the (r,l) element of the so-called U-centered matrix \tilde{A}, defined by

\tilde{A}_{rl} = \frac{1}{n-j-2}\sum_{t=1+j}^{n}{a_{rt}}- \frac{1}{n-j-2}\sum_{s=1+j}^{n}{a_{sl}+\frac{1}{(n-j-1)(n-j-2)}\sum_{t,s=1+j}^{n}{a_{ts}}}, \quad i \neq j,

with zero diagonal.

mADCV gives the auto-distance covariance function of a multivariate time series.

Value

A vector whose length is determined by MaxLag and contains the biased estimator of ADCV or the unbiased estimator of squared ADCV.

Note

Based on the definition of \hat{V}_X(\cdot), we observe that \hat{V}^2_X(j)=\hat{V}^2_X(-j), and thus results based on negative lags are omitted.

Author(s)

Maria Pitsillou, Michail Tsagris and Konstantinos Fokianos.

References

Dominic, E, K. Fokianos and M. Pitsillou Maria (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., 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

ADCF, mADCV

Examples

x <- rnorm(500)
ADCV(x, 18)

ADCV(BJsales, 25)