obtain_autocovariance {fdaACF}

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

# 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)


# 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)