fdaACF (version 1.0.0)

obtain_autocovariance: Estimate the autocovariance function of the series

Description

Obtain the empirical autocovariance function for lags \(= 0,...,\)nlags of the functional time series. Given \(Y_{1},...,Y_{T}\) a functional time series, the sample autocovariance functions \(\hat{C}_{h}(u,v)\) are given by: $$\hat{C}_{h}(u,v) = \frac{1}{T} \sum_{i=1}^{T-h}(Y_{i}(u) - \overline{Y}_{T}(u))(Y_{i+h}(v) - \overline{Y}_{T}(v))$$ where \( \overline{Y}_{T}(u) = \frac{1}{T} \sum_{i = 1}^{T} Y_{i}(t)\) denotes the sample mean function.

Usage

obtain_autocovariance(Y, nlags)

Arguments

Y

Matrix containing the discretized values of the functional time series. The dimension of the matrix is \((n x m)\), where \(n\) is the number of curves and \(m\) is the number of points observed in each curve.

nlags

Number of lagged covariance operators of the functional time series that will be used to estimate the autocorrelation function.

Value

Return a list with the lagged autocovariance functions estimated from the data. Each function is given by a \((m x m)\) matrix, where \(m\) is the number of points observed in each curve.

Examples

# NOT RUN {
# Example 1

N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 1
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
image(x = v, y = v, z = lagged_autocov$Lag0)

# }
# NOT RUN {
# Example 2

N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 10
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
image(x = v, y = v, z = lagged_autocov$Lag0)
image(x = v, y = v, z = lagged_autocov$Lag10)

# Example 3

require(fields)
N <- 500
v <- seq(from = 0, to = 1, length.out = 50)
sig <- 2
bbridge <- simulate_iid_brownian_bridge(N, v, sig)
nlags <- 4
lagged_autocov <- obtain_autocovariance(Y = bbridge,
                                        nlags = nlags)
z_lims <- range(lagged_autocov$Lag0)
colors <- heat.colors(12)
opar <- par(no.readonly = TRUE)
par(mfrow = c(1,5))
par(oma=c( 0,0,0,6)) 
for(k in 0:nlags){
   image(x=v,
         y=v,
         z = lagged_autocov[[paste0("Lag",k)]],
         main = paste("Lag",k),
         col = colors,
         xlab = "u",
         ylab = "v")
}
par(oma=c( 0,0,0,2.5)) # reset margin to be much smaller.
image.plot( legend.only=TRUE, legend.width = 2,zlim=z_lims, col = colors)
par(opar)
# }