msir (version 1.1)

msir.bic: BIC-type criterion for dimensionality

Description

BIC-type criterion for selecting the dimensionality of a dimension reduction subspace.

Usage

msir.bic(object, type = 1, plot = FALSE)
bicDimRed(M, x, nslices, type = 1, tol = sqrt(.Machine$double.eps))

Arguments

object

a 'msir' object

plot

if TRUE a plot of the criterion is shown.

M

the kernel matrix. See details below.

x

the predictors data matrix. See details below.

type

See details below.

nslices

the number of slices. See details below.

tol

a tolerance value

Value

• Returns a list with components:
• evalues
  eigenvalues
• l
  log-likelihood
• crit
  BIC-type criterion
• d
  selected dimensionality
• The msir.bic also assign the above information to the corresponding 'msir' object.

Details

This BIC-type criterion for the determination of the structural dimension selects $d$ as the maximizer of $$G(d) = l(d) - Penalty(p,d,n)$$ where $l(d)$ is the log-likelihood for dimensions up to $d$, $p$ is the number of predictors, and $n$ is the sample size. The term $Penalty(p,d,n)$ is the type of penalty to be used: ll{ type = 1 $Penalty(p,d,n) = -(p-d) \log(n)$ type = 2 $Penalty(p,d,n) = 0.5 C d (2p-d+1)$ where $C = (0.5 \log(n) + 0.1 n^(1/3))/2 nslices/n$ type = 3 $Penalty(p,d,n) = 0.5 C d (2p-d+1)$ where $C = \log(n) nslices/n$ type = 3 $Penalty(p,d,n) = 1/2 d \log(n)$ }

References

Zhu, Miao and Peng (2006) "Sliced Inverse Regression for CDR Space Estimation", JASA. Zhu, Zhu (2007) "On kernel method for SAVE", Journal of Multivariate Analysis.

See Also

msir

Examples

# 1-dimensional symmetric response curve
n = 200; p = 5
b = as.matrix(c(1,-1,rep(0,p-2)))
x = mvrnorm(n, rep(0,p), diag(p))
y = (0.5 * x%*%b)^2 + 0.1*rnorm(n)
MSIR = msir(x, y)
msir.bic(MSIR, plot = TRUE)
summary(MSIR)
msir.bic(MSIR, type = 3, plot = TRUE)
summary(MSIR)