History index model

Definition

In Functional data analysis, functional data are considered as realizations of a Stochastic process $X(t),t\in {\mathcal {I}}$ that is an $L^{2}$ process on a bounded and closed interval ${\mathcal {I}}$.

Let the current functional response process $Y(t)$ at time $t$ depends on the recent history of the predictor process $X$ in a sliding window of length $\Delta$, $\mathrm {E} \{Y(t)|X(t)\}=\beta _{0}+\beta _{1}(t)\int _{0}^{\Delta }\gamma (u)X(t-u)du,$ for $t\in [\Delta ,T]$ with a suitable $T>0$. Then, a ''history index function'' is $\gamma (\cdot )$ defining the history index factor at $\beta _{1}(\cdot )$ by quantifying the influence of the recent history of the predictor values on the response. In most cases, $\gamma (\cdot )$ is assumed to be smooth. For identifiability, $\gamma (\cdot )$ is normalized by requiring that $\int _{0}^{\Delta }\gamma ^{2}(u)du=1$ and that $\gamma (0)>0$, which is no real restriction as $\{-\beta _{1}(t)\}\{-\gamma (u)\}=\beta _{1}(t)\gamma (u)$.