criterionRkh: Estimation of the Rkh criterion by bootstrap method
In edrGraphicalTools: Provides Tools for Dimension Reduction Methods

Description Usage Arguments Details Value Author(s) References See Also Examples

This is the main function in the edrGraphicalTools package. This function estimates the square trace correlation criterion Rkh by bootstrap in order to simultaneously choose the number H of slices and the dimension K of the EDR space reduction. It creates objects of class criterionRkh. Several helper functions that require a criterionRkh object can then be applied to the output from this function.

1	criterionRkh(Y, X, H, K, indices, B = 50, method)

`Y`	A numeric vector representing the dependent variable (a response vector).
`X`	A matrix representing the quantitative explanatory variables (bind by column).
`H`	An integer vector representing the different number of slices to be investigated. By default H takes integer values in `2,3,...,round(n/4)`; `n` is the sample size.
`K`	An integer vector representing the different dimension K to be investigated. By default K takes integer values in `1,2,...,min(p,25)`; `p` is the number of explanatory variables.
`indices`	An optional integer vector specifying the indices of the bootstrap samples. By default, non-paramatric resampling is performed. If used, an integer vector of length `B`*`n` will be specified; (`n` is the length of `Y`).
`B`	The number of bootstrap replicates. By default `B` equals 50.
`method`	This character string specifies the method of fitting. The options include "SIR-I", "SIR-II", and "SAVE".

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 (named SIR-I), 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 a naive bootstrap estimation of the square trace correlation criterion to allow selection of an “optimal” number of slices and to simultaneously select the corresponding suitable dimension K (number of the linear combinations of x).

criterionRkh returns an object of class criterionRkh (the name of the type is the value of the method argument), with attributes:

`Rkhbootmean`	A matrix corresponding of the estimation by bootstrap of the square trace criterion Rkh (h in rows and k in columns).
`Rkhboot`	A list including the result of the estimation of the square trace criterion for each bootsrap replicate.
`method`	the dimension reduction method used.
`n`	Number of subject.
`H`	A vector representing the different numbers H of slices investigated.
`K`	A vector representing the different dimensions K investigated.
`indices`	An vector of integers representing the indices of the bootstrap sample used.

Benoît Liquet, <benoit.liquet@isped.u-bordeaux2.fr> and Jérôme Saracco <jerome.saracco@math.u-bordeaux1.fr>

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.

edr, summary.criterionRkh, plot.criterionRkh

## simulated example 1 
set.seed(10)
n <- 500
beta <- c(1,rep(0,9))
X <- rmvnorm(n,sigma=diag(10))
eps <- rnorm(n)
Y <- (X%*%beta)**3+eps*((X%*%beta)**2)
## Choice a grid of values for H
grid.H <- c(2,5,10,15,20,30)
res1 <- criterionRkh(Y,X,H=grid.H,B=50,method="SIR-I")
res1
#plot(res1,choice.H=c(2,5),choice.K=c(1,2))
## Estimation for SIR-II method with the same bootstrap replicate than for SIR-I
res2 <- criterionRkh(Y,X,H=grid.H,indices=res1$indices,B=50,method="SIR-II")
res2