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Title: Identifying essence codings and effects in functional linear models Authors:  Shao-Wei Cheng - National Tsing Hua University (Taiwan) [presenting]
Abstract: The functional linear model $Y(t)=\beta_0(t)+\bm{X}\bm{\beta}(t)+\varepsilon(t)$ is considered. We assume that the vector of coefficient functions $\bm{\beta}(t)$ is a linear combination of some unknown essence codings $\bm{\Phi}(t)$. That is, there exists a matrix of parameters $\bm{\Gamma}$ such that $\bm{\beta}(t)=\bm{\Gamma}\bm{\Phi}(t)$. The parameters in $\bm{\Gamma}$ are called essence effects. We are interested in identifying meaningful and important essence codings and effects. Because the equation $\bm{\beta}(t)=\bm{\Gamma}\bm{\Phi}(t)$ has infinitely many solutions of $\bm{\Gamma}$ and $\bm{\Phi}(t)$, we propose a criterion for this equation to be uniquely defined, and therefore $\bm{\Gamma}$ and $\bm{\Phi}(t)$ become identifiable. This criterion treats essence codings as projection directions. By projecting $Y(t)$ onto every essence codings, the functional linear model is transformed into a uni-variate linear model. We obtain a set of orthogonal estimators $\hat{\bm{\Phi}}(t)$ of essence codings by sequentially maximizing the amount of variation in the response of this uni-variate linear model explained by the model matrix $\bm{X}$. For the analysis about the essence effects in $\bm{\Gamma}$, including estimation and testing, we suggest performing them by conditioning on $\hat{\bm{\Phi}}(t)$. Finally, the methods developed are applied on some functional data of wafer thickness to estimate essence codings and identify important essence effects.