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 $\{-\beta _{1}(t)\}\{-\gamma (u)\}=\beta _{1}(t)\gamma (u)$.

Motivation

The standard framework of varying coefficient models (also known as concurrent regression models), where the current value of a response process is modeled in dependence on the current value of a predictor process,

Estimation of the history index model

Estimation of the history index function

At each fixed time point $t$, the model in (\ref{him}) reduces to a functional linear model between the scalar response $Y(t)$ and the functional predictor $X(s),$ $t-\Delta\leq s\leq t.$ Also, $X^{C}(s)=X(s)-\mathrm{E}\{X(s)\}$ is a centered functional covariate and $Y^{C}(s)=Y(s)-\mathrm{E}\{Y(s)\}$ is a centered response process.