mbrdr {mbrdr}

Main function for model-based response dimension reduction regression

Description

This is the main function in the mbrdr package. It creates objects of class mbrdr to estimate the response mean subspace and perform tests concerning its dimension. Several helper functions that require a mbrdr object can then be applied to the output from this function.

Usage

mbrdr (formula, data, subset, na.action = na.fail, weights, ...)

mbrdr.compute (y, x, weights, method = "upfrr", ...)
 

Arguments

formula

a two-sided formula like cbind(y1,y2,y3,y4)~x1+x2+x3, where the left-side variables are a matrix of the response variables, and the right-hand side variables represent the predictors. The left-hand side of the formula must be a matrix, since the package reduces the dimension of the responses variables.

data

an optional data frame containing the variables in the model. By default the variables are taken from the environment from which ‘mbrdr’ is called.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

weights

an optional vector of weights to be used where appropriate. In the context of dimension reduction methods, weights are used to obtain elliptical symmetry, not constant variance.

na.action

a function which indicates what should happen when the data contain ‘NA’s. The default is ‘na.fail,’ which will stop calculations. The option 'na.omit' is also permitted, but it may not work correctly when weights are used.

x

The design matrix. This will be computed from the formula by dr and then passed to dr.compute, or you can create it yourself.

y

The response vector or matrix

method

This character string specifies the method of fitting. The default is "upfrr". The options include "yc", "prr", "pfrr". Each method may have its own additional arguments, or its own defaults; see the details below for more information.

...

For mbrdr, all additional arguments passed to mbrdr.compute. For mbrdr.compute, additional arguments may be required for particular dimension reduction method. For example, numdir is the maximum number of directions to compute, with default equal to 4. Other methods may have other defaults.

Details

The general regression problem mainly focuses on studying E(y|x), the conditional mean of a response y given a set of predictors x, where y is r-dimensional response variables with r geq 2 and

This function provides methods for estimating the response dimension subspace of a general regression problem. That is, we want to find a r \times d matrix B of minimal rank d such that

E(y|x)=E(P(B)y|x)

, where P(B) is an orthogonal projections onto the column space of B. Both the dimension d and the subspace P(B) are unknown. These methods make few assumptions.

For the methods "yc", "prr", "pfrr" and "upfrr", B is estimated and returned. And, only for "pfrr" and "upfrr", chi-squared test results for estimating d is provided.

Weights can be used, essentially to specify the relative frequency of each case in the data.

The option fx.choice is required to fit "pfrr" and "upfrr" and has the following four values.

fx.choice=1: This is default and returns the original predictor matrice X, centered at zero as fx.

fx.choice=2: This returns the original predictor matrice X, centered at zero and its squared values.

fx.choice=3: This returns the original predictor matrice X, centered at zero and its exponentiated values.

fx.choice=4: This clusters X with K-means algoritm with the number of clusters equal to the value in nclust. Then, the cluster results are expanded to \code{nclust}-1 dummy variables, like factor used in lm function. Finally, it returns nclust-1 categorical basis. The option of nclust works only with fx.choice=4.

Value

mbrdr returns an object that inherits from mbrdr (the name of the type is the value of the method argument), with attributes:

y

The response matrix

x

The design matrix

weights

The weights used, normalized to add to n.

cases

Number of cases used.

call

The initial call to mbrdr.

evectors

The eigenvectors from kernel matrices to estimate B computed from each response dimension reduction methods. It is the estimate of B.

evalues

The eigenvalues corresponding to the eigenvectors.

stats

This is the dimension test statistics for pfrr and "upfrr". It is the cumulatative sum of the eigenvalues for "yc" and "prr"

fx

This returns the user-selection of fx for "pfrr" and "upfrr".

numdir

The maximum number of directions to be found. The output value of numdir may be smaller than the input value.

method

the dimension reduction method used.

Author(s)

Jae Keun Yoo, <[e-mail]>.

References

Yoo, JK. (2018). Response dimension reduction: model-based approach. Statistics : A Journal of Theoretical and Applied Statistic, 52, 409-425. "prr" and "pfrr"

Yoo, JK. (2019). Unstructured principal fitted response reduction in multivariate regression. Journal of the Korean Statistical Society, 48, 561-567. "upfrr"

Yoo, JK. and Cook, R. D. (2008), Response dimension reduction for the conditional mean in multivariate regression. Statistics and Probability Letters, 47, 381-389. "yc".

Examples

data(mps)
# default fitting method is "upfrr"
s0 <- mbrdr(cbind(A4, B4, A6, B6)~AFDC+Attend+B+Enrol+HS+Minority+Mobility+Poverty+PTR, data=mps)
summary(s0)

# Refit, using different choice of fx.
summary(s1 <- update(s0, fx.choice=2))

# Refit again, using pfrr with fx.choice=2
summary(s2<-update(s1, method="pfrr", fx.choice=1))

# Refit, using prr, which does not require the choice of fx.
summary(s3<- update(s1,method="prr"))

# fit using Yoo-Cook method:
summary(s4 <- update(s1,method="yc"))