rhoE {DCCA}R Documentation

The limit value of the detrended cross-covariance

Description

Calculates the theoretical counterpart of the cross-correlation coefficient. This is expression (11) in Prass and Pumi (2019). For trend-stationary processes under mild assumptions, this is equivalent to the limit of the detrended cross correlation coefficient calculated with window of size m+1m+1 as mm tends to infinity (see theorem 3.2 in Prass and Pumi, 2019).

Usage

rhoE(m = 3, nu = 0, G1, G2, G12, K = NULL)

Arguments

m

an integer or integer valued vector indicating the size (or sizes) of the window for the polinomial fit. min(m)min(m) must be greater or equal than nunu or else it will return an error.

nu

a non-negative integer denoting the degree of the polinomial fit applied on the integrated series.

G1, G2

the autocovariance matrices for the original time series. Both are max(m)+1max(m)+1 by max(m)+1max(m)+1 matrices.

G12

the cross-covariance matrix for the original time series. The dimension of G12G12 must be max(m)+1)max(m)+1) by max(m)+1)max(m)+1).

K

optional: the matrix KK. See the details.

Details

The optional argument KK is an m+1m+1 by m+1m+1 matrix defined by K=JQJK = J'QJ, where JJ is a m+1m+1 by m+1m+1 lower triangular matrix with all non-zero entries equal to one and QQ is a m+1m+1 by m+1m+1 given by Q=IPQ = I - P where PP is the projection matrix into the subspace generated by degree nu+1nu+1 polynomials and II is the m+1m+1 by m+1m+1 identity matrix. KK is equivalent to expression (18) in Prass and Pumi (2019). If this matrix is provided and mm is an integer, then nunu are ignored.

Value

A list containing the following elements, calculated considering windows of size m+1m+1, for each mm supplied:

EF2dfa1, EF2dfa2

the expected values of the detrended variances.

EFdcca

the expected value of the detrended cross-covariance.

rhoE

the vector with the theoretical counterpart of the cross-correlation coefficient.

Author(s)

Taiane Schaedler Prass

References

Prass, T.S. and Pumi, G. (2019). On the behavior of the DFA and DCCA in trend-stationary processes <arXiv:1910.10589>.

See Also

Km which creates the matrix KK, Jn which creates the matrix JJ, Qm which creates QQ and Pm which creates PP.

Examples

m = 3
K = Km(m = m, nu = 0)
G1 = G2 =  diag(m+1)
G12 = matrix(0,ncol = m+1, nrow = m+1)
rhoE(G1 = G1, G2 = G2, G12 = G12, K = K)
# same as
rhoE(m = 3, nu = 0, G1 = G1, G2 = G2, G12 = G12)

[Package DCCA version 0.1.1 Index]