ADCV {dCovTS} | R Documentation |
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 |
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,
, between
and
. Formal
definition of
can be found in Zhou (2012) and Fokianos and
Pitsillou (2017).
The empirical auto-distance covariance function, ,
is the non-negative square root defined by
and , for
,
where
and
are Euclidean distances with elements
given by
with ,
,
,
.
is given analogously based on
, where
.
and
are independent if and only if
.
See Fokianos and Pitsillou (2017) for more information on theoretical
properties of
including consistency.
If unbiased = TRUE, ADCV
returns the unbiased estimator of
squared auto-distance covariance function,
, proposed by Szekely and Rizzo (2014).
In the context of time series data, this is given by
for , where
is the
element of the so-called U-centered matrix
, defined by
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 , we observe that
, 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
Examples
x <- rnorm(500)
ADCV(x, 18)
ADCV(BJsales, 25)