In statistical analysis, 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, is disadvantageous when it is assumed that past and present values of the predictor process influence current response. In contrast to these approaches, the history index model includes the effect of recent past values of the predictor through the history index function. Specifically, the influence of past predictor values is modeled by a smooth history index functions, while the effects on the response are described by smooth varying coefficient functions.
Definition In Functional data analysis, functional data are considered as realizations of a Stochastic process X ( t ) , t ∈ I {\displaystyle X(t),t\in {\mathcal {I}}} that is an L 2 {\displaystyle L^{2}} process on a bounded and closed interval I {\displaystyle {\mathcal {I}}} . Let the current functional response process Y ( t ) {\displaystyle Y(t)} at time t {\displaystyle t} depends on the recent history of the predictor process X {\displaystyle X} in a sliding window of length Δ {\displaystyle \Delta } . Then the history index model is defined as
E { Y ( t ) | X ( t ) } = β 0 + β 1 ( t ) ∫ 0 Δ γ ( u ) X ( t − u ) d u , {\displaystyle \mathrm {E} \{Y(t)|X(t)\}=\beta _{0}+\beta _{1}(t)\int _{0}^{\Delta }\gamma (u)X(t-u)du,} (1) for t ∈ [ Δ , T ] {\displaystyle t\in [\Delta ,T]} with a suitable T > 0 {\displaystyle T>0} . Then, a ''history index function'' is γ ( ⋅ ) {\displaystyle \gamma (\cdot )} defining the history index factor at β 1 ( ⋅ ) {\displaystyle \beta _{1}(\cdot )} by quantifying the influence of the recent history of the predictor values on the response. In most cases, γ ( ⋅ ) {\displaystyle \gamma (\cdot )} is assumed to be smooth. For identifiability, γ ( ⋅ ) {\displaystyle \gamma (\cdot )} is normalized by requiring that ∫ 0 Δ γ 2 ( u ) d u = 1 {\displaystyle \int _{0}^{\Delta }\gamma ^{2}(u)du=1} and that γ ( 0 ) > 0 {\displaystyle \gamma (0)>0} , which is no real restriction as { − β 1 ( t ) } { − γ ( u ) } = β 1 ( t ) γ ( u ) {\displaystyle \{-\beta _{1}(t)\}\{-\gamma (u)\}=\beta _{1}(t)\gamma (u)} .
Estimation of the history index model
Estimation of the history index function At each fixed time point t {\displaystyle t} , the model in (1) reduces to a functional linear model between the scalar response Y ( t ) {\displaystyle Y(t)} and the functional predictor X ( t ) , t − Δ ≤ s ≤ t . {\displaystyle X(t),t-\Delta \leq s\leq t.} Also, X C ( s ) = X ( s ) − E { X ( s ) } {\displaystyle X^{C}(s)=X(s)-\mathrm {E} \{X(s)\}} is a centered functional covariate and Y C ( s ) = Y ( s ) − E { Y ( s ) } {\displaystyle Y^{C}(s)=Y(s)-\mathrm {E} \{Y(s)\}} is a centered response process. Writing the model as
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