DCCA (version 0.1.1)

covFdcca: Autocovariance function of the detrended cross-covariance

Description

Calculates the autocovariance of the detrended cross-covariance.

Usage

covFdcca(m = 3, nu = 0, h = 0, overlap = TRUE, G1, G2, G12, Cumulants = NULL)

Arguments

m

an integer or integer valued vector indicating the size 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.

h

an integer or integer valued vector indicating the lags for which the autocovariance function is to be calculated. Negative values are not allowed.

overlap

logical: if true (the default), overlapping boxes are used for calculations. Otherwise, non-overlapping boxes are applied.

G1, G2

the autocovariance matrices for the original time series. The dimension of \(G1\) and \(G2\) must be compatible with the highest values in vectors \(m\) and \(h\). More specifically, the dimension of \(G1\) and \(G2\) is \((max(m)+max(h)+1)\) by \((max(m)+max(h)+1)\) if overlap = TRUE and \(dim(G1) = dim(G2) = (max(m)+max(h))(max(h)+1)\) by \((max(m)+max(h))(max(h)+1)\) otherwise.

G12

the cross-covariance matrix for the original time series. The dimension of \(G12\) must be compatible with the highest values in vectors \(m\) and \(h\). If overlap = TRUE, \(dim(G12) = [(max(m)+1)*(max(h)+1) - max(m)*max(h)]\) by \([(max(m)+1)*(max(h)+1) - max(m)*max(h)]\) and \(dim(G12) = [(max(m)+1)*(max(h)+1)]\) by \([max(m)+1)*(max(h)+1)]\), otherwise

Cumulants

The matrix of cumulants. If not provided, it is assumed that the cumulants are all zero.

Value

A matrix of dimension \(lenght(h)\) by \(length(m)\) with the autocovariance of lag \(h\) (rows), for each value of \(m\) (columns) provided. This matrix is obtained from expressions (24) for \(h = 0\) and (25) for \(h > 0\) in Prass and Pumi (2019).

References

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

Examples

# NOT RUN {
ms = seq(3,100,1)
hs = seq(0,50,1)
overlap = TRUE
nu = 0
m_max = (max(ms)+1)*(max(hs)+1) - max(ms)*max(hs)*as.integer(overlap)

theta = c(c(1,(20:1)/10), rep(0, m_max - 20))
Gamma1 = diag(m_max+1)
Gamma2 = matrix(0, ncol = m_max+1, nrow = m_max+1)
Gamma12 = matrix(0, ncol = m_max+1, nrow = m_max+1)
for(t in 1:(m_max+1)){
    for(h in 0:(m_max+1-t)){
        Gamma2[t,t+h] = sum(theta[1:(length(theta)-h)]*theta[(1+h):length(theta)])
        Gamma2[t+h,t] = Gamma2[t,t+h]
        Gamma12[t,t+h] = theta[h+1]
    }
}

covdcca = covFdcca(m = ms, nu = 0, h = hs,
                   G1 = Gamma1, G2 = Gamma2, G12 = Gamma12)
                   
# }