Estimate the autocovariance function of the series

Obtain the empirical autocovariance function for the given lags of a functional time series, X. Given a functional time series, the sample autocovariance functions \(\hat{C}_{h}(u,v)\) are given by: $$\hat{C}_{h}(u,v) = \frac{1}{N} \sum_{i=1}^{N-|h|}(Y_{i}(u) - \overline{X}_{N}(u))(Y_{i+|h|}(v) - \overline{X}_{N}(v))$$ where \( \overline{X}_{N}(u) = \frac{1}{N} \sum_{i = 1}^{N} X_{i}(t)\) denotes the sample mean function and \(h\) is the lag parameter.

Usage

autocovariance(X, lags = 0:1, center = TRUE)

Arguments

X

A dfts object or data which can be automatically converted to that format. See dfts().

lags

Numeric(s) for the lags to estimate the lagged operator.

center

Boolean if the data should be centered. Default is true.

Value

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

See also

autocorrelation(), var()

Examples

v <- seq(0,1,length.out=20)
lagged_autocov <- autocovariance(
  X = generate_brownian_bridge(100,v=v),
  lags = 1)