HDBRR {HDBRR}

High Dimensional Bayesian Ridge Regression without MCMC.

Description

Ridge regression provide biased estimators of the regression parameters with lower variance. The HDBRR ("High Dimensional Bayesian Ridge Regression") function fits Bayesian Ridge regression without MCMC, this one uses the SVD or QR decomposition for the posterior computation.

Usage

HDBRR(y, X, n0 = 5, p0 = 5, s20 = NULL, d20 = NULL, h = 0.5,
    intercept = TRUE, vpapp = TRUE,npts = NULL,c = NULL,
    corpred = NULL, method = c("svd","qr"),bigmat = TRUE, ncores = 2, svdx = NULL)

## S3 method for class 'HDBRR'
summary(object, all.coef = FALSE, crit = log(4), ...)

## S3 method for class 'HDBRR'
plot(x, crit = log(4), var_select = FALSE, post = FALSE, ...)

## S3 method for class 'HDBRR'
predict(object,  ...)

## S3 method for class 'summary.HDBRR'
print(x, ...)

## S3 method for class 'HDBRR'
print(x, ...)

## S3 method for class 'HDBRR'
coef(object, all = FALSE, ...)

Arguments

y

The data vector (numeric, n) NAs allowed.

X

Design Matrix of dimension n x p.

n0, p0

n0/2 and p0/2 are the shape parameter of the Gamma Inverse prior assigned to the residual variance and the shape parameter of the Gamma Inverse prior assigned to the Beta's variance respectively. The default value for n0/2 and p0/2 parameter is 5.

s20, d20

(n0s20)/2 and (p0d20)/2 are the scale parameter of the Gamma Inverse prior assigned to the residual variance and the scale parameter of the Gamma Inverse prior assigned to the Beta's variance respectively. The default value for the s20 and d20 is NULL. If the scale is not specified a value is calculated with h and quantiles.

h

(numeric, 0<h<1) shrinkage factor. Only used if the hyper-parameters are not specified; If h -> 0 then we have greater shrinkage, this is, \beta -> 0. If h -> 1 then we have less shrinkage.

intercept

Logic value. The default value for the intercept is TRUE.

vpapp

Logic value. Compute an approximation of the predictive variance. The default value for the vpapp is TRUE.

npts

Number (integer) of points used to evaluate the u's density for the numeric aprroach. The default value for the npts parameter is 200.

c

ratio of Gaussian densities (Spike/Slab) in the prior mixture density of each Beta for variable selection.

corpred

The method for the compute of the correlation, there are two methods, Empirical Bayes ("eb") and Bayesian ("b") method. The default value for the parameter corpred is NULL. If the values is NULL then the corr and edf values will be NULL.

method

Options for the posterior computation. There are two methods available: "qr" decomposition of X*t(X) and "svd" decomposition of matrix X. The default value for the method is SVD decomposition.

bigmat

Use of the bigstatsr package. The default value for bigmat is TRUE.

ncores

Number of the cores for computation. The default value for the ncores is 2, you can detect your number of cores with detectCores() and use it (iOS and Linux).

object

A HDBRR object, typically generated by a call to HDBRR.

all.coef

Logical. Should results be returned for all ridge regression penalty parameters (all.coef = TRUE), or only those whose log(bayes factor)>crit.

crit

Numerical. The lower bound of the log Bayes factor in favour to include a variable in the model. The default value for crit is log(4).

...

Additional arguments to be passed to or from other methods.

x

A HDBRR object, typically generated by a call to HDBRR (for the print.HDBRR and plot.HDBRR functions) or an object of class summary.HDBRR (for the print.summary.HDBRR function).

var_select

Logical. If is TRUE a plot with variable selection is returned. The default value is FALSE.

post

Logical. If is TRUE a plot with marginal posterior of u is returned. The default value is FALSE.

all

Logical. All coefficients are returned. If is FALSE, then, if p > 250 only 250 coefficients are returned. The default value es FALSE.

svdx

It is possible to add the svd. The default value es NULL.

Details

Ridge regression is a useful tool to deal with colinerity in the homocesastic linear regression model providing biased estimators of the regression parameters with lower variance than the least square estimators. The model

y = X\beta + \epsilon

where \epsilon vector is assumed Normal with mean vector 0 and covariance matrix \sigma^2 I_n. For further details see the vignettes in the package.

Value

List containing the following components:

betahat

Vector (numeric, p) with the betas estimates.

yhat

Vector (numeric, n) with the y's estimates.

sdyhat

Vector (numeric, n) with the standard deviation of the predicts values.

sdpred

Vector (numeric, n) with the standard deviation of predict variances.

varb

Vector (numeric, p) with the beta's variance.

sigsqhat

Value (numeric) of the residual variance estimate.

sigbsqhat

Value (numeric) of the Beta's variance estimate.

u

Vector (numeric, npts) with the u's values.

postu

Vector (numeric, npts) with the values of the u posterior.

uhat

Value (numeric) of u estimated.

umode

Value (numeric) of the posterior mode of u.

whichNa

Value (integer) of NAs in the y vector.

phat

Vector (numeric, p), selection probability of x_i.

delta

Used in the variable selection.

edf

Value (numeric) of the effective degrees of freedom for regression.

corr

Vector (numeric, n) of the correlation between y_i estimates and y_i.

svdx

The svd decomposition.

Author(s)

Sergio Perez-Elizalde, Blanca E. Monroy-Castillo, Paulino Perez-Rodriguez, Jose Crossa.

Examples

## Not run: 

data("phenowheat")
mod <- lmer(pheno$HD~pheno$env+(1|pheno$Line))
y <- unlist(ranef(mod))
n <- length(y)
X <- scale(X, scale=F)
fitall <- HDBRR(y,X/sqrt(ncol(X)),intercept = FALSE, corpred = "eb", c = 100)
fitall
sumarry(fitall, crit = 0)
plot(fitall, crit = 0)
predict(fitall)


## End(Not run)