Dimension reduction in functional regression with categorical predictor

Abstract

In the present paper, we consider dimension reduction methods for functional regression with a scalar response and the predictors including a random curve and a categorical random variable. To deal with the categorical random variable, we propose three potential dimension reduction methods: partial functional sliced inverse regression, marginal functional sliced inverse regression and conditional functional sliced inverse regression. Furthermore, we investigate the relationships among the three methods. In addition, a new modified BIC criterion for determining the dimension of the effective dimension reduction space is developed. Real and simulation data examples are then presented to show the effectiveness of the proposed methods.

Appendix

Lemma 4

For generic random variables or random curves $V_1,V_2,V_3, V_4$, the following equivalences hold:

We define the independent for random variable $V_1$ and random curve $V_2\in L^2(T)$ by for any $t\in T$. A discussion of this result can be found in Chiaromonte et al. (2002). The proof of Lemmas 1, 2 and 3 are similar to Propositions 1–3 of Chiaromonte et al. (2002).

1.1 Proof of Lemma 1

From Lemma 4 we have that, for a generic subspace S,

Thus, under the assumption that ,

and therefore $S_{y|x}^{(W)}\supseteq S_{y|x}$. Again using Lemma 4,

Thus, under the assumption that ,

and therefore $S_{y|x}^{(W)}\supseteq S_{y|x}$.

1.2 Proof of Lemma 2

From the Lemma 4, we know that

As in the proof of Lemma 1, noting that is obviously guaranteed by . Under such an assumption

and therefore $S_{y|x}\supseteq S_{y|x}^{(W)}$.

1.3 Proof of Lemma 3

It is immediate to see that, for a generic subspace S,

(8)

Because $S_{y|x}^{(W)}$ satisfies the left-hand side of (8), then it also satisfies for any $w=1, \ldots , C.$ And this implies $S_{y|x}^{(W)}\supseteq S_{y_w|x_w},\ w=1,\ldots ,C$, and therefore

$$\begin{aligned} S_{y|x}^{(W)}\supseteq \oplus _{w=1}^CS_{y_w|x_w}. \end{aligned}$$

Since $\oplus _{w=1}^CS_{y_w|x_w}\supseteq S_{y_w|x_w}, \ w=1,\ldots ,C$, the sum space satisfies the right-hand side of (8). Hence

which implies

$$\begin{aligned} \oplus _{w=1}^CS_{y_w|x_w} \supseteq S_{y|x}^{(W)}. \end{aligned}$$

Then, we have proven the Lemma 3.

1.4 Proof of Theorem 1

Let K be the true value of the dimension of the EDR space. By the law of large numbers, we can get

$$\begin{aligned} \hat{\lambda }_i-\lambda _i=O_p(1/\sqrt{n}). \end{aligned}$$

Therefore, If $K>k,$

$$\begin{aligned} G(K)-G(k)= & {} \frac{n\sum _{i=k+1}^K(\log (1+\hat{\lambda }_i) -\hat{\lambda }_i)}{2\sum _{i=1}^p(\log (1+\hat{\lambda }_i)-\hat{\lambda }_i)}+\frac{c_n}{2p}[k(k+1)-K(K+1)]\nonumber \\ \end{aligned}$$

(9)

$$\begin{aligned}&\quad \rightarrow \frac{c_n}{2p}[k(k+1)-K(K+1)]>0. \end{aligned}$$

(10)

If $K<k$,

$$\begin{aligned} G(K)-G(k)= & {} \frac{n\sum _{i=K+1}^k(\log (1+\hat{\lambda }_i) -\hat{\lambda }_i)}{2\sum _{i=1}^p(\log (1+\hat{\lambda }_i)-\hat{\lambda }_i)} +\frac{c_n}{2p}[k(k+1)-K(K+1)] \end{aligned}$$

(11)

$$\begin{aligned}&\rightarrow \frac{c_n}{2p}[k(k+1)-K(K+1)]>0. \end{aligned}$$

(12)

It follows from (10) and (12) that $\hat{K}\rightarrow K$ in probability.