Nonparametric test for change-point detection particularly sensitive to changes in the copula of multivariate continuous observations. 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. Details can be found in the first reference.
cpCopula(x, method = c("seq", "nonseq"), b = NULL,
weights = c("parzen", "bartlett"), m = 5,
L.method=c("max","median","mean","min"),
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
Cramér-von Mises 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 simulation method for
generating multiplier replicates of the test statistic;
can be either "seq" (the 'check' approach
in the first reference) or "nonseq" (the 'hat' approach
in the first reference). The 'check' approach appears to lead to
better behaved tests in the case of samples of moderate size. The
'hat' approach is substantially faster.
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
function bOptEmpProc(); see the procedure described in
Section 5 of the second 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.
a strictly positive integer specifying the number of points of the
uniform grid on \((0,1)^d\) (where \(d\) is
ncol(x)) involved in the estimation of the bandwidth
parameter; see Section 5 of the third reference. The number of
points of the grid is given by m^ncol(x) so that m needs to be
decreased as \(d\) increases.
a string specifying how the parameter \(L\) involved in the estimation of the bandwidth parameter is computed; 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.
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.
A. Bücher, I. Kojadinovic, T. Rohmer and J. Segers (2014), Detecting changes in cross-sectional dependence in multivariate time series, Journal of Multivariate Analysis 132, pages 111-128, https://arxiv.org/abs/1206.2557.
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.
cpRho() for a related test based on
Spearman's rho, cpTau() for a related test based on
Kendall's tau, cpDist() for a related test based
on the multivariate empirical d.f., bOptEmpProc() for the
function used to estimate b from x if b = NULL.
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 <- cpCopula(x, b = 1)
cp
## Estimated change-point
which(cp$cvm == max(cp$cvm))}
Run the code above in your browser using DataLab