Nonparametric test for change-point detection particularly sensitive to changes in Spearman's rho in multivariate time series. The observations can be serially independent or dependent (strongly mixing). Approximate p-values for the test statistic are obtained by means of a multiplier approach or by estimating the asymptotic null distribution. Details can be found in first reference.
cpRho(x, method = c("mult", "asym.var"),
statistic = c("pairwise", "global"),
b = NULL, weights = c("parzen", "bartlett"),
N = 1000, init.seq = NULL, include.replicates = FALSE)An object of class
htest which is a list,
some of the components of which are
value of the test statistic.
corresponding approximate p-value.
the values of the nrow(x)-1 intermediate
change-point statistics; the test statistic is defined as
the maximum of those.
the value of parameter b.
a data matrix whose rows are multivariate continuous observations.
a string specifying the method for computing the
approximate p-value for the test statistic; can be either
"mult" (the multiplier approach 'tilde' in the first
reference) or "asym.var" (the approach based on the
estimation of the asymptotic null distribution of the test statistic
described in the first reference). The 'mult' approach appears to lead to
better behaved tests.
a string specifying the test statistic; can be either
"pairwise" (the statistic \(S_{n,3}\) in the first
reference) or "global" (the statistic \(S_{n,1}\)
in the first reference).
strictly positive integer specifying the value of the
bandwidth parameter determining the serial dependence when
generating dependent multiplier sequences using the 'moving average
approach'; see Section 5 of the second reference. The value 1
will create i.i.d. multiplier
sequences suitable for serially independent observations. If set to
NULL, b will be estimated from x using the
procedure described in the first reference.
a string specifying the kernel for creating the weights used in the generation of dependent multiplier sequences within the 'moving average approach'; see Section 5 of the second reference.
number of multiplier replications.
a sequence of independent standard normal variates of
length N * (nrow(x) + 2 * (b - 1)) used to generate dependent
multiplier sequences.
a logical specifying whether the
object of class htest returned by the function
(see below) will include the multiplier replicates, if generated.
When method == "mult", the approximate p-value is computed as
$$(0.5 +\sum_{i=1}^N\mathbf{1}_{\{S_i\ge S\}})/(N+1),$$
where \(S\) and \(S_i\) denote the test statistic and
a multiplier replication, respectively. This ensures that the
approximate p-value is a number strictly between 0 and 1, which is
sometimes necessary for further treatments.
When method == "asym.var", the approximate p-value is computed
from the estimated asymptotic null distribution, which involves the
Kolmogorov distribution. The latter is dealt with reusing code from
the ks.test() function; credit to RCore.
I. Kojadinovic, J-F. Quessy and T. Rohmer (2016), Testing the constancy of Spearman's rho in multivariate time series, Annals of the Institute of Statistical Mathematics 68:5, pages 929-954, https://arxiv.org/abs/1407.1624.
A. Bücher and I. Kojadinovic (2016), A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing, Bernoulli 22:2, pages 927-968, https://arxiv.org/abs/1306.3930.
cpTau() for a related test based on
Kendall's tau, cpDist() for a related test
based on the multivariate
empirical d.f., cpCopula() for a related test based on
the empirical copula.
if (FALSE) {
require(copula)
n <- 100
k <- 50 ## the true change-point
u <- rCopula(k,gumbelCopula(1.5))
v <- rCopula(n-k,gumbelCopula(3))
x <- rbind(u,v)
cp <- cpRho(x, b = 1)
cp
## Estimated change-point
which(cp$rho == max(cp$rho))}
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