ldr (version 1.3.3)

ldr: Likelihood-based Dimension Reduction

Description

Main function of the package. It creates objects of one of classes core, lad, or pfc to estimate a sufficient dimension reduction subspace using covariance reducing models (CORE), likelihood acquired directions (LAD), or principal fitted components (PFC).

Usage

ldr(X, y = NULL, fy = NULL, Sigmas = NULL, ns = NULL, numdir = NULL, nslices = NULL, model = c("core", "lad", "pfc"), numdir.test = FALSE, ...)

Arguments

X

Design matrix with n rows of observations and p columns of predictors. The predictors are assumed to have a continuous distribution.

y

The response vector of length n. It can be continuous or categorical.

fy

Basis function to be obtained using bf or defined by the user. It is a function of y alone and has independent column vectors. It is used exclusively with pfc. See bf for detail.

Sigmas

A list object of sample covariance matrices corresponding to the different populations. It is used exclusively with core.

ns

A vector of number of observations of the samples corresponding to the different populations.

numdir

The number of directions to be used in estimating the reduction subspace. When calling pfc, the dimension numdir must be less than or equal to the minimum of p and r, where r is the number of columns of fy. When calling lad and y is continuous, numdir is the number of slices to use.

nslices

Number of slices for a continuous response. It is used exclusively with lad.

model

One of the following: "pfc", "lad", "core".

numdir.test

Boolean. If FALSE, the chosen model fits with the provided numdir. If TRUE, the model is fit for all dimensions less or equal to numdir.

...

Additional arguments for specific models and/or Grassmannoptim.

Value

core, lad, or pfc . The output depends on the model used. See pfc, lad, and core for further detail.

Details

Likelihood-based methods to sufficient dimension reduction are model-based inverse regression approaches using the conditional distribution of the $p$-vector of predictors $X$ given the response $Y=y$. Three methods are implemented in this package: covariance reduction (CORE), principal fitted components (PFC), and likelihood acquired directions (LAD). All three assume that $X|(Y=y) \sim N(\mu_y, \Delta_y)$.

For CORE, given a set of $h$ covariance matrices, the goal is to find a sufficient reduction that accounts for the heterogeneity among the population covariance matrices. See the documentation of "core" for details.

For PFC, $\mu_y=\mu + \Gamma \beta f_y$, with various structures of $\Delta$. The simplest is the isotropic ("iso") with $\Delta=\delta^2 I_p$. The anisotropic ("aniso") PFC model assumes that $\Delta=\mathrm{diag}(\delta_1^2, ..., \delta_p^2)$, where the conditional predictors are independent and on different measurement scales. The unstructured ("unstr") PFC model allows a general structure for $\Delta$. Extended structures are considered. See the help file of pfc for more detail.

LAD assumes that the response $Y$ is discrete. A continuous response is sliced into finite categories to meet this condition. It estimates the central subspace $\mathcal{S}_{Y|X}$ by modeling both $\mu_y$ and $\Delta_y$. See lad for more detail.

References

Adragni, KP and Cook, RD (2009): Sufficient dimension reduction and prediction in regression. Phil. Trans. R. Soc. A 367, 4385-4405.

Cook, RD (2007): Fisher Lecture - Dimension Reduction in Regression (with discussion). Statistical Science, 22, 1--26.

Cook, R. D. and Forzani, L. (2008a). Covariance reducing models: An alternative to spectral modelling of covariance matrices. Biometrika 95, 799-812.

Cook, R. D. and Forzani, L. (2008b). Principal fitted components for dimension reduction in regression. Statistical Science 23, 485--501.

Cook, R. D. and Forzani, L. (2009). Likelihood-based sufficient dimension reduction. Journal of the American Statistical Association, Vol. 104, 485, pp 197--208.

See Also

pfc, lad, core

Examples

data(bigmac)
fit1 <- ldr(X=bigmac[,-1], y=bigmac[,1], fy=bf(y=bigmac[,1], case="pdisc",
        degree=0, nslices=5), numdir=3, structure="unstr", model="pfc")
summary(fit1)
plot(fit1)

fit2 <- ldr(X=bigmac[,-1], y=bigmac[,1], fy=bf(y=bigmac[,1], case="poly",
        degree=2), numdir=2, structure="aniso", model="pfc")
summary(fit2)
plot(fit2)

fit3 <- ldr(X=as.matrix(bigmac[,-1]), y=bigmac[,1], model="lad", nslices=5)
summary(fit3)
plot(fit3)