Testing for the number of interesting components in SIR'

In ICtest: Estimating and Testing the Number of Interesting Components in Linear Dimension Reduction

Sliced inverse regression

Denote the $p$-variate predictors $x_i$, $i=1,...,n$ with corresponding responses $y_i$. The predictors are assumed to follow the model $$ x_i = A s_i + b, $$ where $A$ is a non-singular $p \times p$ matrix, $b$ a $p$-vector and the random vector $s$ can be decomposed into $(s_1^T,s_2^T)^T$ with $E(s)=0$ and $Cov(s)=I_p$ where $s_1$ has dimension $k$ and $s_2$ dimension $p-k$. It is then assumed that $s_1$ is the signal part, the interesting components, which are relevant to model $y$, whereas $s_2$ is the noise part.

The working model assumption is then that $$ (y,s_1^T)^T \ is \ independent \ of \ s_2. $$ Defining $S_1=E((x-b)(x-b)^T)$ and $S_2=E(E(x-b|y)E(x-b|y)^T)$ the sliced inverse regression (SIR) can be interpreted as finding the transformation matrix $W$ such that $$ W S_1 W^T = I_p \ and \ W S_2 W^T = D, $$ where $D$ is a diagonal matrix with diagonal elements $d_1 \geq d_2 \geq ... \geq d_k > d_{k+1} = ... = d_p = 0$.

In practice $S_2$ is estimated by approximating $E(x-b|y)$ by dividing $y$ into $h$ slices where in this package $y$ is divided into $h$ intervals containing an equal number of observations.

The practical problem is to decide then the value $k$.

Asymptotic test a specific value of $k$

For an asymptotic test using the test statistic $$ T= n\sum_{i=k+1}^p d_i, $$ the limiting distribution under the null is then $T \sim \chi^2_{(p-k)(h-k-1)}$. Therefore for the hypothetical value $k$ and the number of slices $h$ is required that $k \geq h-1$.

Bootstrapping test a specific value of $k$

Bootstrap tests can be constructed as follows

1. Compute $\hat s_i = \hat W(x_i - \bar x_i)$, and decompose into $\hat s_{i,1}$ and $\hat s_{i,2}$.
2. Sample a bootstrap sample $(y_i^, \hat s_{i,1}^)$ of size $n$ from $(y_i, \hat s_{i,1})$.
3. Sample a bootstrap sample $\hat s_{i,2}^*$ of size $n$ from $\hat s_{i,2}$.
4. Compute $x_i^ = \hat W^{-1} ((\hat s_{i,1}^)^T,(\hat s_{i,2}^*)^T)^T$.
5. Recompute the test statistic $T^$ based on the bootstrap data $x_i^$, $i=,1,...,n$.
6. Repeat the steps above $m$ times to obtain bootstrap test statistics $T^_1,...,T^_m$.

The p-value is then $$ \frac{#(T_i^* \geq T)+1}{m+1}. $$

Example

Some simulated data with true $k=2$:

set.seed(1234)
n <- 200
p <- 10
X <- matrix(rnorm(p*n), ncol = p)
eps <- rnorm(n, sd=0.25)
y <- X[, 1]/ (0.5+(X[, 2]+1.5)^2)
pairs(cbind(y,X))

First performing the asymptotic test

library(ICtest)
SIRasympk2 <- SIRasymp(X,y,2)
screeplot(SIRasympk2)
SIRasympk2

Then the bootstrap test

SIRbootk2 <- SIRboot(X,y,2)
SIRbootk2

Looking at the first two components and their relationship with the response

pairs(cbind(y, components(SIRbootk2, which="k")))

ICtest documentation built on May 18, 2022, 9:05 a.m.