Estimate the autocorrelation function of the series
autocorrelation.RdObtain the empirical autocorrelation 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:
$$\hat{C}_{h}(u,v) = \frac{1}{N} \sum_{i=1}^{N-|h|}(X_{i}(u) -
\overline{X}_{N}(u))(X_{i+|h|}(v) - \overline{X}_{N}(v))$$
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. The
autocorrelation functions are defined over the range \((0,1)\) by
normalizing these functions using the factor \(\int\hat{C}_{0}(u,u)du\).
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.
Value
Return a list or data.frame with the lagged autocorrelation 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.
Examples
N <- 100
v <- seq(from = 0, to = 1, length.out = 10)
bbridge <- generate_brownian_bridge(N = N, v = v)
lagged_autocor <- autocorrelation(X = bbridge, lags = 0:1)