# msir.bic: BIC-type criterion for dimensionality In msir: Model-Based Sliced Inverse Regression

## Description

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

## Usage

 ```1 2 3``` ```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

## 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:

• `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 = 4` Penalty(p,d,n) = 1/2 d \log(n)

## 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.

## Author(s)

Luca Scrucca [email protected]

## 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.

`msir`
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```# 1-dimensional symmetric response curve n <- 200 p <- 5 b <- as.matrix(c(1,-1,rep(0,p-2))) x <- matrix(rnorm(n*p), nrow = n, ncol = 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) ```