ICtest (version 0.3-5)

SIRladle: Ladle Estimate for SIR

Description

In the supervised dimension reduction context with response y and explaining variables x, this functions provides the ladle estimate for the dimension of the central subspace for SIR.

Usage

SIRladle(X, y, h = 10, n.boot = 200, 
         ncomp = ifelse(ncol(X) > 10, floor(ncol(X)/log(ncol(X))), ncol(X) - 1), ...)

Arguments

X

numeric data matrix.

y

numeric response vector.

h

number of slices in SIR.

n.boot

number of bootstrapping samples to be used.

ncomp

The number of components among which the ladle estimator is to be searched. The default here follows the recommendation of Luo and Li 2016.

…

arguments passed on to quantile.

Value

A list of class ladle containing:

method

the string SIR.

k

the estimated value of k.

fn

vector giving the measures of variation of the eigenvectors using the bootstrapped eigenvectors for the different number of components.

phin

normalized eigenvalues of the M matrix in the SIR case.

gn

the main criterion for the ladle estimate - the sum of fn and phin. k is the value where gn takes its minimum

lambda

the eigenvalues of the M matrix in the SIR case.

W

the transformation matrix to supervised components.

S

data matrix with the centered supervised components.

MU

the location of the data which was substracted before calculating the supervised components.

data.name

the name of the data for which the ladle estimate was computed.

Details

The idea here is that the eigenvalues of the SIR-M matrix are of the form \(\lambda_1 \geq ... \geq \lambda_k > 0 = ... = 0\) and the eigenvectors of the non-zero eigenvalue span the central subspace.

The ladle estimate for k for this purpose combines the values of the scaled eigenvalues and the variation of the eigenvectors based on bootstrapping. The idea there is that for distinct eigenvales the variation of the eigenvectors is small and for equal eigenvalues the corresponding eigenvectors have large variation.

This measure is then computed assuming k=0,..., ncomp and the ladle estimate for k is the value where the measure takes its minimum.

References

Luo, W. and Li, B. (2016), Combining Eigenvalues and Variation of Eigenvectors for Order Determination, Biometrika, 103. 875--887. <doi:10.1093/biomet/asw051>

See Also

ladleplot

Examples

# NOT RUN {
n <- 1000
X <- cbind(rnorm(n), rnorm(n), rnorm(n), rnorm(n), rnorm(n))
eps <- rnorm(n, sd=0.02)
y <- 4*X[,1] + 2*X[,2] + eps


test <- SIRladle(X, y)
test
summary(test)
plot(test)
pairs(cbind(y, components(test)))
ladleplot(test)
ladleplot(test, crit = "fn")
ladleplot(test, crit = "lambda")
ladleplot(test, crit = "phin")
# }