Adaptive slicing for functional slice inverse regression

Abstract

In the paper, we propose a functional dimension reduction method for functional predictors and a scalar response. In the past study, the most popular functional dimension reduction method is the functional sliced inverse regression (FSIR) and people usually use a fixed slicing scheme to implement the estimation of FSIR. However, in practical, there are two main questions for the fixed slicing scheme: how many slices should be chosen and how to divide all samples into different slices. To solve these problems, we first expand the functional predictor and functional regression parameters on the functional principal component basis or a given basis such as B-spline basis. Then the functional regression parameters will be estimated by using the adaptive slicing for FSIR approach. Simulation results and real data analysis are presented to show the merit of the new proposed method.

Appendix

Proof of Theorem 3.1

According to the theorem, that is to say, as $n \rightarrow \infty$,

$$\begin{aligned} P \left\{ \max \limits _{H{'} \in \mathcal {H}_{+} \cup \mathcal {H}_{-}} BIC(H{'},\tilde{B})<BIC(H_0,\tilde{B})\right\} \rightarrow 1, \end{aligned}$$

where $H_0=\{H_{01},H_{02},\ldots ,H_{0G}\}$ is the optimal scheme.

At first, we consider an over-slicing scheme $H{'} \in \mathcal {H}_{+}$. For the sake of convenience, we assume that $H{'}=\{H_{11},H_{12},H_{02},\ldots ,H_{0G}\}$, where $H_{11}$ and $H_{12}$ are two sub-slices formed from $H_{01}$. That is to say, the slice $H_{01}$ becomes two slices $H_{11}$ and $H_{12}$ in this case. We have

$$\begin{aligned} BIC(H,B)= trace(B^\intercal \hat{\Upsilon } B) -\frac{log(n)}{n}\times K\times S, \end{aligned}$$

$$\begin{aligned} \hat{\Upsilon }={\Phi }\hat{\Gamma }^{-1}\sum _{s=1}^S f_s(\hat{\lambda }^{(g:i)}-\bar{x})\otimes (\hat{\lambda }^{(g:i)}-\bar{x})^\intercal {\Phi },\\ \end{aligned}$$

so

$$\begin{aligned} BIC(H{'},\tilde{B})&=f_{H_{11}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{11}}-\bar{x})\otimes (\bar{x}_{H_{11}}-\bar{x})^\intercal {\Phi } \right] \tilde{B} \right\} \\&\ \ \ +f_{H_{12}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{12}}-\bar{x})\otimes (\bar{x}_{H_{12}}-\bar{x})^\intercal {\Phi } \right] \tilde{B} \right\} \\&\ \ \ + \cdots \\&\ \ \ +f_{H_{0G}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{0G}}-\bar{x})\otimes (\bar{x}_{H_{0G}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ -\frac{log(n)}{n}\times K \times (G+1), \end{aligned}$$

$$\begin{aligned} BIC(H_0,\tilde{B})&=f_{H_{01}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{01}}-\bar{x})\otimes (\bar{x}_{H_{01}}-\bar{x})^\intercal {\Phi } \right] \tilde{B} \right\} \\&\ \ \ + \cdots \\&\ \ \ +f_{H_{0G}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{0G}}-\bar{x})\otimes (\bar{x}_{H_{0G}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ -\frac{log(n)}{n}\times K \times G. \end{aligned}$$

Therefore,

$$\begin{aligned} BIC(H{'},\tilde{B})-BIC(H_0,\tilde{B})&=f_{H_{11}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{11}}-\bar{x})\otimes (\bar{x}_{H_{11}}-\bar{x})^\intercal {\Phi } \right] \tilde{B} \right\} \\&\ \ \ +f_{H_{12}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{12}}-\bar{x})\otimes (\bar{x}_{H_{12}}-\bar{x})^\intercal {\Phi } \right] \tilde{B} \right\} \\&\ \ \ -f_{H_{01}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{01}}-\bar{x})\otimes (\bar{x}_{H_{01}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ -\frac{log(n)}{n}\times K, \end{aligned}$$

It is easy to show that

$$\begin{aligned} f_{H_{01}}(\bar{x}_{H_{01}}-\bar{x})\otimes (\bar{x}_{H_{01}}-\bar{x})^\intercal&= f_{H_{11}}(\bar{x}_{H_{11}}-\bar{x})\otimes (\bar{x}_{H_{11}}-\bar{x})^\intercal \\&\ \ \ +f_{H_{12}}(\bar{x}_{H_{12}}-\bar{x})\otimes (\bar{x}_{H_{12}}-\bar{x})^\intercal \\&\ \ \ -\frac{f_{H_{11}}f_{H_{12}}}{f_{H_{01}}}(\bar{x}_{H_{11}}-\bar{x}_{H_{12}})\otimes (\bar{x}_{H_{11}}-\bar{x}_{H_{12}})^\intercal . \end{aligned}$$

Therefore,

$$\begin{aligned}&BIC(H{'},\tilde{B})-BIC(H_0,\tilde{B})\\&\ \ \ =\frac{f_{H_{11}}f_{H_{12}}}{f_{H_{01}}} trace\left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{11}}-\bar{x}_{H_{12}})\otimes (\bar{x}_{H_{11}}-\bar{x}_{H_{12}})^\intercal {\Phi } \right] \tilde{B}\right\} -\frac{log(n)}{n}\times K. \end{aligned}$$

Since $\bar{x}_{H_{1i}}=\bar{x}_{H_{01}}+O_p(n^{-\frac{1}{2}}),i=1,2$, we can obtain

$$\begin{aligned} BIC(H{'},\tilde{B})-BIC(H_0,\tilde{B})&=O_p(\frac{1}{n})-\frac{log(n)}{n}\times K. \end{aligned}$$

In general, this result will hold for any $H{'} \in \mathcal {H}_{+}$, i.e, as $n \rightarrow \infty$,

$$\begin{aligned} P \left\{ \max \limits _{H{'} \in \mathcal {H}_{+}} BIC(H{'},\tilde{B})<BIC(H_0,\tilde{B})\right\} \rightarrow 1. \end{aligned}$$

Then consider the case that $H{'}$ is under-slicing scheme, that is to say $H{'} \in \mathcal {H}_{-}$. For simplicity, we can assume that $H{'}=\{H_{00},H_{03},\ldots ,H_{0G}\}$, where $H_{00}$ is a new slice constructed by merging two slices $H_{01}$ and $H_{02}$. We have

$$\begin{aligned} BIC(H{'},\tilde{B})&=f_{H_{00}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{00}}-\bar{x})\otimes (\bar{x}_{H_{00}}-\bar{x})^\intercal {\Phi } \right] \tilde{B} \right\} \\&\ \ \ +f_{H_{03}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{03}}-\bar{x})\otimes (\bar{x}_{H_{03}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ + \cdots \\&\ \ \ +f_{H_{0G}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{0G}}-\bar{x})\otimes (\bar{x}_{H_{0G}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ -\frac{log(n)}{n}\times K \times (G-1), \end{aligned}$$

$$\begin{aligned} BIC(H_0,\tilde{B})&=f_{H_{01}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{01}}-\bar{x})\otimes (\bar{x}_{H_{01}}-\bar{x})^\intercal {\Phi } \right] \tilde{B} \right\} \\&\ \ \ +f_{H_{02}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{02}}-\bar{x})\otimes (\bar{x}_{H_{02}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ + \cdots \\&\ \ \ +f_{H_{0G}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1} (\bar{x}_{H_{0G}}-\bar{x})\otimes (\bar{x}_{H_{0G}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ -\frac{log(n)}{n}\times K \times G. \end{aligned}$$

Therefore,

$$\begin{aligned} BIC(H{'},\tilde{B})-BIC(H_0,\tilde{B})&=f_{H_{00}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi }\hat{\Gamma }^{-1} (\bar{x}_{H_{00}}-\bar{x})\otimes (\bar{x}_{H_{00}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ -f_{H_{01}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi }\hat{\Gamma }^{-1} (\bar{x}_{H_{01}}-\bar{x})\otimes (\bar{x}_{H_{01}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ -f_{H_{02}} trace \left\{ \tilde{B}^\intercal \left[ {\Phi }\hat{\Gamma }^{-1} (\bar{x}_{H_{02}}-\bar{x})\otimes (\bar{x}_{H_{02}}-\bar{x})^\intercal {\Phi }\right] \tilde{B} \right\} \\&\ \ \ +\frac{log(n)}{n}\times K. \end{aligned}$$

Meanwhile, it is easy to show that

$$\begin{aligned} f_{H_{00}}(\bar{x}_{H_{00}}-\bar{x})\otimes (\bar{x}_{H_{00}}-\bar{x})^\intercal&= f_{H_{01}}(\bar{x}_{H_{11}}-\bar{x})\otimes (\bar{x}_{H_{01}}-\bar{x})^\intercal \\&\ \ \ +f_{H_{02}}(\bar{x}_{H_{02}}-\bar{x})\otimes (\bar{x}_{H_{02}}-\bar{x})^\intercal \\&\ \ \ -\frac{f_{H_{01}}f_{H_{02}}}{f_{H_{01}}}(\bar{x}_{H_{01}}-\bar{x}_{H_{02}})\otimes (\bar{x}_{H_{01}}-\bar{x}_{H_{02}})^\intercal . \end{aligned}$$

Therefore,

$$\begin{aligned}&BIC(H{'},\tilde{B})-BIC(H_0,\tilde{B})\\&\ \ \ =-\frac{f_{H_{01}}f_{H_{02}}}{f_{H_{01}}} \left\{ \tilde{B}^\intercal \left[ {\Phi } \hat{\Gamma }^{-1}(\bar{x}_{H_{01}}-\bar{x}_{H_{02}})\otimes (\bar{x}_{H_{01}}-\bar{x}_{H_{02}})^\intercal {\Phi }\right] \tilde{B} \right\} +\frac{log(n)}{n}\times K. \end{aligned}$$

Since $\tilde{B} \tilde{B}^\intercal =B_0 B_0 ^\intercal +O_p(n^{-\frac{1}{2}})$, we can obtain

$$\begin{aligned}&BIC(H{'},\tilde{B})-BIC(H_0,\tilde{B})\\&\ \ \ =-\frac{f_{H_{01}}f_{H_{02}}}{f_{B_{01}}}\left\{ B_0^\intercal \left[ {\Phi }\hat{\Gamma }^{-1}(\bar{x}_{H_{01}}-\bar{x}_{H_{02}})\otimes (\bar{x}_{H_{01}}-\bar{x}_{H_{02}})^\intercal {\Phi } \right] B_0 \right\} +O_p(n^{-\frac{1}{2}}). \end{aligned}$$

That is to say, there exists a constant $c<0$ so that $BIC(H{'},\tilde{B})-BIC(H_0,\tilde{B})<c$, with probability tending to 1 as $n \rightarrow \infty$. In general, this result will hold for any $H{'} \in \mathcal {H}_{-}$, i.e, as $n \rightarrow \infty$,

$$\begin{aligned} P \left\{ \max \limits _{H{'} \in \mathcal {H}_{-}} BIC(H{'},\tilde{B})<BIC(H_0,\tilde{B})\right\} \rightarrow 1. \end{aligned}$$

Combining these two cases, we can obtain

$$\begin{aligned} P \left\{ \max \limits _{H{'} \in \mathcal {H}_{+} \cup \mathcal {H}_{-} } BIC(H{'},\tilde{B})<BIC(H_0,\tilde{B})\right\} \rightarrow 1, \end{aligned}$$

the proof is complete. $\square$