rhoE: The limit value of the detrended cross-covariance

In DCCA: Detrended Fluctuation and Detrended Cross-Correlation Analysis

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+1 as m 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) must be greater or equal than nu 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)+1 by max(m)+1 matrices.
G12	the cross-covariance matrix for the original time series. The dimension of G12 must be max(m)+1) by max(m)+1).
K	optional: the matrix K. See the details.

Details

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

Value

A list containing the following elements, calculated considering windows of size m+1, for each m 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 K, Jn which creates the matrix J, Qm which creates Q and Pm which creates P.

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)

DCCA documentation built on Jan. 1, 2020, 5:06 p.m.