ADCV(x, MaxLag)ADCV. Default is 15.MaxLag, and contains the estimated ADCV.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 (2016).The empirical auto-distance covariance function, $\hat{V}_X(\cdot)$, is the non-negative square root defined by
$$\hat{V}_X^2(j) = \frac{1}{(n-|j|)^2}\sum_{r,l=1+|j|}^{n}{A_{rl}B_{rl}}, ~~~~~ j=0, \pm 1, \pm 2, \dots.$$
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 (2016) for more information on theoretical properties
of $V_X^2(\cdot)$ including consistency.
mADCV gives the auto-distance covariance function of a multivariate time series.ADCF, mADCVx <- rnorm(500)
ADCV(x,18)
ADCV(BJsales,25)Run the code above in your browser using DataLab