DCCA (version 0.1.1)

covF2dfa: Autocovariance function of the detrended variance

Description

Calculates the autocovariance of the detrended variance.

Usage

covF2dfa(m = 3, nu = 0, h = 0, overlap = TRUE, G, 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.

overlap

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

G

the autocovariance matrix for the original time series. The dimension of \(G\) must be \((max(m)+max(h)+1)\) by \((max(m)+max(h)+1)\) if overlap = TRUE and \((max(m)+max(h))(max(h)+1)\) by \((max(m)+max(h))(max(h)+1)\) otherwise.

Cumulants

The matrix containing the joint cumulants for lags. Dimension must be \((max(m)+1)*nrow(G)\). If not provided, it is assumed that the cumulants are all zero.

Value

A matrix with the autocovariance of lag \(h\), for each value of \(m\) provided. This matrix is obtained from expressions (21) for \(h = 0\) and (22) 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]
    }
}

covdfa1 = covF2dfa(m = ms, nu = 0, h = hs,
                   overlap = TRUE, G = Gamma1, Cumulants = NULL)

covdfa2 = covF2dfa(m = ms, nu = 0, h = hs,
                  overlap = TRUE, G = Gamma2, Cumulants = NULL)

cr = rainbow(100)
plot(ms, covdfa1[,1], type = "l", ylim = c(0,20),
    xlab = "m", ylab = expression(gamma[DFA](h)), col = cr[1])
for(i in 2:ncol(covdfa1)){
  points(ms, covdfa1[,i],  type = "l", col = cr[i])
}

lattice::wireframe(covdfa1, drape = TRUE,
    col.regions = rev(rainbow(150))[50:150],
    zlab = expression(gamma[DFA]), xlab = "m", ylab = "h")
# }