Functional time series

A functional time series is a time series whose observation at each time is a whole function, often modeled as a random element of the Hilbert space $L^2[0,1]$. For example, $X_j(s)$ might be the temperature profile within day $j$, with $s$ marking time of day. Dependence across days is different from dependence between nearby positions within one curve.

For a simple model, let $X_j(s)=A_j+B_{j}s$, where $(A_j,B_j)$ is a stationary bivariate time series. Its mean curve is $\mu (s)=\mathbb{E}{A}_j+s\mathbb{E}{B}_j$. The covariance between curves at lag $h$ is a kernel $C_h(s,t)=\mathrm{Cov}(X_0(s),X_h(t))$, retaining both within-curve coordinates.

Estimating $\mu$ by $n^{-1}{\sum }_{j=1}^n{X}_j$ resembles scalar averaging, but its uncertainty depends on serial covariance operators rather than only the variance of one curve. Treating every densely measured point as an independent observation can severely overstate precision.^[ref]