Fdcca {DCCA}

Detrended Cross-covariance

Description

Calculates the detrended cross-covariance between two time series y1 and y2.

Usage

Fdcca(y1, y2, m = 3, nu = 0, overlap = TRUE)

Arguments

y1, y2

vectors corresponding to the time series data. If length(y1) and length(y2) differ, the longer time series is coerced to match the lenght of the shorter.

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.

overlap

logical: if true (the default), uses overlapping windows. Otherwise, non-overlapping boxes are applied.

Value

A vector of size length(m) containing the detrended cross-covariance considering windows of size m+1, for each m supplied.

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>.

Examples

# Simple usage
y1 = rnorm(100)
y2 = rnorm(100)
F.dcca = Fdcca(y1, y2, m = 3, nu = 0, overlap = TRUE)
F.dcca

# A simple example where y1 and y2 are independent.

ms = 3:50
F.dcca1 = Fdcca(y1, y2, m = ms, nu = 0, overlap = TRUE)
F.dcca2 = Fdcca(y1, y2, m = ms, nu = 0, overlap = FALSE)

plot(ms, F.dcca1, type = "o", xlab = "m", col = "blue",
     ylim = c(min(F.dcca1,F.dcca2),max(F.dcca1,F.dcca2)),
     ylab = expression(F[DCCA]))
points(ms, F.dcca2, type = "o", col = "red")
legend("bottomright", legend = c("overlapping","non-overlapping"),
       col = c("blue", "red"), lty= 1, bty = "n", pch=1)


# A more elaborated example where y1 and y2 display cross-correlation for non-null lags.
# This example also showcases why overlapping windows are usually advantageous.
# The data generating process is the following:
# y1 is i.i.d. Gaussian while y2 is an MA(2) generated from y1.

n = 500
ms = 3:50
theta = c(0.4,0.5)

# Calculating the expected value of the DCCA in this scenario
m_max = max(ms)
vtheta = c(1,theta, rep(0, m_max - length(theta)))
G12 = 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)){
    G12[t,t+h] = vtheta[h+1]
  }
}

EF.dcca = EFdcca(m = ms, nu = 0, G = G12)

# generating the series and calculating the DCCA
burn.in = 100
eps = rnorm(burn.in)

y1 = rnorm(n)
y2 = arima.sim(model = list(ma = theta), n, n.start = burn.in, innov = y1, start.innov = eps)

ms = 3:50
OF.dcca = Fdcca(y1, y2, m = ms, nu = 0, overlap = TRUE)
NOF.dcca = Fdcca(y1, y2, m = ms, nu = 0, overlap = FALSE)

plot(ms, OF.dcca, type = "o", xlab = "m", col = "blue",
     ylim = c(min(NOF.dcca,OF.dcca,EF.dcca),max(NOF.dcca,OF.dcca,EF.dcca)),
     ylab = expression(F[DCCA]))
points(ms, NOF.dcca, type = "o", col = "darkgreen")
points(ms, EF.dcca, type = "o", col = "red")
legend("bottomright", legend = c("overlapping","non-overlapping","expected"),
       col = c("blue", "darkgreen","red"), lty= 1, bty = "n", pch=1)