Computes the auto-distance correlation function of a univariate time series. It also computes the bias-corrected estimator of (squared) auto-distance correlation.
ADCF(x, MaxLag = 15, unbiased = FALSE)
numeric vector or univariate time series.
maximum lag order at which to calculate the ADCF
. Default is 15.
logical value. If unbiased = TRUE, the bias-corrected estimator of squared auto-distance correlation is returned. Default value is FALSE.
Returns a vector, whose length is determined by MaxLag
, and contains the biased estimator of ADCF
or the bias-corrected
estimator of squared ADCF
.
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 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 (2016). 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 (2016).
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.
Fokianos K. and M. Pitsillou (2016). Consistent testing for pairwise dependence in time series. Technometrics, http://dx.doi.org/10.1080/00401706.2016.1156024.
Szekely, G. J. and M. L. Rizzo (2014). Partial distance correlation with methods for dissimilarities. The Annals of Statistics \(\textbf{42}\), 2382-2412, dx.doi.org/10.1214/14-AOS1255.
Szekely, G. J. and M. L. Rizzo and N. K. Bakirov (2007). Measuring and testing dependence by correlation of distances. The Annals of Statistics \(\textbf{35}\), 2769-2794, http://dx.doi.org/10.1214/009053607000000505.
Zhou, Z. (2012). Measuring nonlinear dependence in time series, a distance correlation approach. Journal of Time Series Analysis \(\textbf{33}\), 438-457, http://dx.doi.org/10.1111/j.1467-9892.2011.00780.x.
# NOT RUN {
x <- rnorm(1000)
# }
# NOT RUN {
ADCF(x)
# }
# NOT RUN {
ADCF(ldeaths,18)
ADCF(mdeaths,unbiased=TRUE)
# }
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