bootLOPR {HDCI}

Bootstrap Lasso OLS Partial Ridge

Description

Combines residual (or paired) bootstrap Lasso+Partial Ridge and residual (or paired) bootstrap Lasso+OLS, and produces confidence intervals for regression coefficients.

Usage

bootLOPR(x, y, lambda2, B = 500, type.boot = "residual", thres = 0.5, alpha = 0.05, 
         OLS = TRUE, cv.method = "cv", nfolds = 10, foldid, cv.OLS = TRUE, tau = 0, 
         parallel = FALSE, standardize = TRUE, intercept = TRUE, parallel.boot = 
         FALSE, ncores.boot = 1, ...)

Arguments

x

Input matrix as in glmnet, of dimension nobs x nvars; each row is an observation vector.

y

Response variable.

lambda2

Tuning parameter in the Partial Ridge. If missing, lambda2 will be set to 1/nobs, where nobs is the number of observations.

B

Number of replications in the bootstrap – default is 500.

type.boot

Bootstrap method which can take one of the following two values: "residual" or "paired". The default is residual.

thres

A threshold parameter. For the variables/predictors with selection probability (obtained by bootstrap) larger than thres, this function uses bootstrap Lasso+OLS (if OLS=TRUE) or bootstrap Lasso (if OLS=FALSE) to produce confidence intervals; while, for the variables/predictors with selection probability (obtained by bootstrap) smaller than thres, this function uses bootstrap Lasso+Partial Ridge to produce confidence intervals.

alpha

Significance level – default is 0.05.

OLS

If TRUE, this function uses Lasso+OLS estimator to compute the residuals for residual bootstrap Lasso+Partial Ridge; otherwise, it uses Lasso estimator to compute the residuals for residual bootstrap Lasso+Partial Ridge. The default value is TRUE. This argument can be ignored for paired bootstrap Lasso+Partial Ridge.

cv.method

The method used to select lambda in the Lasso – can be cv, cv1se, and escv; the default is cv.

nfolds, foldid, cv.OLS, tau, parallel

Arguments that can be passed to escv.glmnet.

standardize

Logical flag for x variable standardization, prior to fitting the model. Default is standardize=TRUE.

intercept

Should intercept be fitted (default is TRUE) or set to zero (FALSE).

parallel.boot

If TRUE, use parallel foreach to run the bootstrap replication. Must register parallel before hand, such as doParallel or others. See the example below.

ncores.boot

Number of cores used in the bootstrap replication.

...

Other arguments that can be passed to glmnet.

Details

The function combines the performance of bootstrap Lasso+Partial Ridge and bootstrap Lasso+OLS (if OLS=TRUE). For "large" regression coefficient in the sense that their selection probability (obtained by bootstrap) is larger than a threshold value (thres), it uses bootstrap Lasso+OLS to produce confidence intervals which can decrease the interval length ; while, for "small" regression coefficients meaning that their selection probability (obtained by bootstrap) is smaller than a threshold value (thres), it uses bootstrap Lasso+Partial Ridge to produce confidence intervals which can guarantee coverage. Note that there are two arguments related to parallel, "parallel" and "parallel.boot": "parallel" is used for parallel foreach in the escv.glmnet; while, "paralle.boot" is used for the parallel foreach in the bootstrap replication precodure.

Value

A list consisting of the following elements is returned.

lambda.opt

The optimal value of lambda selected by cv/cv1se/escv.

Beta

Lasso+OLS (if OLS=TRUE) or Lasso (if OLS=FALSE) estimate of the regression coefficients.

Beta.LPR

Lasso+Partial Ridge estimate of the regression coefficients.

interval

A 2 by p matrix containing the bootstrap Lasso+OLS (if OLS=TRUE) or bootstrap Lasso (if OLS=FALSE) confidence intervals – the first row is the lower bounds of the confidence intervals for each of the coefficients and the second row is the upper bounds of the confidence intervals.

interval.LPR

A 2 by p matrix containing the bootstrap Lasso+Partial Ridge confidence intervals – the first row is the lower bounds of the confidence intervals for each of the coefficients and the second row is the upper bounds of the confidence intervals.

interval.LOPR

A 2 by p matrix containing the combining confidence intervals of bootstrap Lasso+Partial Ridge and bootstrap Lasso+OLS (or bootstrap Lasso if OLS=FALSE) – the first row is the lower bounds of the confidence intervals for each of the coefficients and the second row is the upper bounds of the confidence intervals.

Examples

library("glmnet")
library("mvtnorm") 

## generate the data
set.seed(2015)
n <- 200      # number of obs
p <- 500
s <- 10
beta <- rep(0, p)
beta[1:s] <- runif(s, 1/3, 1)
x <- rmvnorm(n = n, mean = rep(0, p), method = "svd")
signal <- sqrt(mean((x %*% beta)^2))
sigma <- as.numeric(signal / sqrt(10))  # SNR=10
y <- x %*% beta + rnorm(n)

## residual bootstrap Lasso OLS + Partial Ridge
set.seed(0)
obj <- bootLOPR(x = x, y = y, B = 10)
# confidence interval
obj$interval
sum((obj$interval[1,]<=beta) & (obj$interval[2,]>=beta))

## using parallel in the bootstrap replication
#library("doParallel")
#registerDoParallel(2)
#set.seed(0)
#system.time(obj <- bootLOPR(x = x, y = y))
#system.time(obj <- bootLOPR(x = x, y = y, parallel.boot = TRUE, ncores.boot = 2))