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+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. |
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 |
G12 |
the cross-covariance matrix for the original time series. The dimension of |
K |
optional: the matrix |
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)