DCCA (version 0.1.1)

rhoE: 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 \(m+1\) 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.

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.

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.

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

# NOT RUN {
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)
# }