edrGraphicalTools (version 2.2)

edr: Main function for estimation of the EDR space

Description

It creates objects of class edr to estimate the effective dimension regression (EDR) space. Several helper functions that require an edr object can then be applied to the output from this function.

Usage

edr(Y, X, H, K, method, submethod="SIR-QZ", ...)

Arguments

Y

A numeric vector representing the dependent variable (a response vector).

X

A matrix representing the quantitative explanatory variables (bind by column).

H

The chosen number of slices.

K

The chosen dimension K.

method

This character string specifies the method of fitting. The option includes "SIR-I", "SIR-II", and "SAVE".

submethod

This character string specifies the method of fitting when the number of lines of \(X\) is greater than its number of columns. It should be either "SIR-QZ", "RSIR" or "SR-SIR".

...

Arguments to be passed to edrUnderdet when the number of lines of \(X\) is greater than its number of columns.

Value

edr returns an object of class edr, with attributes:

matEDR

A matrix corresponding of the eigenvectors of the interest matrix

eigvalEDR

The eigenvalues of the matrix of interest

K

The chosen dimension.

H

The chosen number of slices.

n

Sample size.

method

The dimension reduction method used.

X

The matrix of the quantitative explanatory variables (bind by column).

Y

The numeric vector of the dependent variable (a response vector).

Details

We are interested in the following semiparametric dimension reduction model proposed by Li (1991) $$y=f(b1'x,b2'x,...,bK'x,e)$$ where the univariate response variable \(y\) is associated with the p-dimensional regressor \(p\) only through the reduced \(K\)-dimensional variable \((b1'x,b2'x,...,bK'x)\) with \(K < p\). The error term \(e\) is independent of \(x\). The link function \(f\) and the \(b\)-vectors are unknown. We are interested in finding the linear subspace spanned by the \(K\) unknown \(b\)-vector, called the effective dimension reduction (EDR) space. We focus on the SIR, SIR-II and SAVE methods to estimate the EDR space. The slicing step of these methods depends on the number \(H\) of slices. We propose with the function criterionRkh a naive bootstrap estimation of the square trace correlation criterion to allow selection of an ``optimal'' number H of slices and simultaneously the corresponding suitable dimension \(K\) (number of the linear combination of \(x\)). After choosing an optimal couple \((H,K)\) for the best estimation method (the square trace correlation criterion closest to one), the EDR space could be estimate with this function. Each method consists in a spectral decomposition of a matrix of interest. The eigenvectors of this matrix associated of the K largest eigenvalues are EDR directions.

References

Liquet, B. and Saracco, J. (2012). A graphical tool for selecting the number of slices and the dimension of the model in SIR and SAVE approaches. Computational Statistics, 27(1), 103-125.

Li, K.C. (1991). Sliced inverse regression for dimension reduction, with discussions. Journal of the American Statistical Association 86, 316-342.

Cook, R. D. and Weisberg, S. (1991). Discussion of ``Sliced inverse regression''. Journal of the American Statistical Association, 86, 328-332.

See Also

criterionRkh, summary.edr, plot.edr

Examples

# NOT RUN {
set.seed(10)
n <- 500
beta1 <- c(1,1,rep(0,8))
beta2 <- c(0,0,1,1,rep(0,6))
X <- rmvnorm(n,sigma=diag(1,10))
eps <- rnorm(n)
Y <- (X%*%beta1)**2+(X%*%beta2)**2+eps

## Estimation of the trace square criterion 
## grid.H <- c(2,5,10,15,20,30)
## res2 <- criterionRkh(Y,X,H=grid.H,B=50,method="SIR-II")
## summary(res2)
## plot(res2)

## Estimation of the EDR direction for K=2 and H=2 and SIR-II method
edr2 <- edr(Y,X,H=2,K=2,method="SIR-II")
summary(edr2)
#plot(edr2)
# }