F2dfa {DCCA}

Detrended Variance

Description

Calculates the detrended variance based on a given time series.

Usage

F2dfa(y, m = 3, nu = 0, overlap = TRUE)

Arguments

y

vector corresponding to the time series data.

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 variance 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
y = rnorm(100)
F2.dfa = F2dfa(y, m = 3, nu = 0, overlap = TRUE)
F2.dfa

vF2.dfa = F2dfa(y, m = 3:5, nu = 0, overlap = TRUE)
vF2.dfa


###################################################
# AR(1) example showing how the DFA varies with phi

phi = (1:8)/10
n = 300
z = matrix(nrow = n, ncol = length(phi))
for(i in 1:length(phi)){
  z[,i] = arima.sim(model = list(ar = phi[i]), n)
}

ms = 3:50
F2.dfa = matrix(ncol = length(phi), nrow = length(ms))

for(j in 1:length(phi)){
  F2.dfa[,j] = F2dfa(z[,j], m = ms  , nu = 0, overlap = TRUE)
}

cr = rainbow(length(phi))
plot(ms, F2.dfa[,1], type = "o", xlab = "m", col = cr[1],
    ylim = c(0,max(F2.dfa)), ylab = "F2.dfa")
for(j in 2:length(phi)){
  points(ms, F2.dfa[,j], type = "o", col = cr[j])
}
legend("topleft", lty = 1, legend = phi, col = cr, bty = "n", title = expression(phi), pch=1)


##############################################################################
# An MA(2) example showcasing why overlapping windows are usually advantageous
n = 300
ms = 3:50
theta = c(0.4,0.5)

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

EF2.dfa = EF2dfa(m = ms, nu = 0, G = G)

z = arima.sim(model = list(ma = theta), n)

ms = 3:50
OF2.dfa = F2dfa(z, m = ms, nu = 0, overlap = TRUE)
NOF2.dfa = F2dfa(z, m = ms, nu = 0, overlap = FALSE)

plot(ms, OF2.dfa, type = "o", xlab = "m", col = "blue",
    ylim = c(0,max(OF2.dfa,NOF2.dfa,EF2.dfa)), ylab = "F2.dfa")
points(ms, NOF2.dfa, type = "o", col = "darkgreen")
points(ms, EF2.dfa, type = "o", col = "red")
legend("bottomright", legend = c("overlapping","non-overlapping","expected"),
            col = c("blue", "darkgreen","red"), lty= 1, bty = "n", pch=1)