Exploring methods for large-scale multiple linear regression
Peter Carbonetto
Last updated: 2023-05-23
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Knit directory: pcarbo/analysis/
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made to the R Markdown (analysis/linreg_methods_demo.Rmd)
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click on the hyperlinks in the table below to view the files as they
were in that past version.
| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | f73afa4 | Peter Carbonetto | 2023-05-23 | workflowr::wflow_publish("linreg_methods_demo.Rmd") |
| Rmd | 3e6b925 | Peter Carbonetto | 2023-05-23 | Added trimmed lasso script and started incorporating the results into the linreg_methods_demo. |
| html | f1ba3dd | Peter Carbonetto | 2023-05-17 | Build site. |
| Rmd | 3083353 | Peter Carbonetto | 2023-05-17 | workflowr::wflow_publish("linreg_methods_demo.Rmd") |
| html | 2ea7aaa | Peter Carbonetto | 2023-05-13 | Fixed one of the scatterplots in the linreg_methods_demo. |
| Rmd | 8d721cf | Peter Carbonetto | 2023-05-13 | workflowr::wflow_publish("linreg_methods_demo.Rmd", verbose = TRUE) |
| html | d548844 | Peter Carbonetto | 2023-05-13 | Added L0Learn to linreg_methods_demo. |
| Rmd | eaecd02 | Peter Carbonetto | 2023-05-13 | workflowr::wflow_publish("linreg_methods_demo.Rmd", verbose = TRUE) |
| html | 6182121 | Peter Carbonetto | 2023-05-09 | Added emvs to linreg_methods_demo. |
| Rmd | 97c6939 | Peter Carbonetto | 2023-05-09 | wflow_publish("linreg_methods_demo.Rmd", verbose = TRUE) |
| html | 1f32078 | Peter Carbonetto | 2023-05-09 | First build of linreg_methods_demo. |
| Rmd | 0b8238e | Peter Carbonetto | 2023-05-09 | workflowr::wflow_publish("linreg_methods_demo.Rmd", verbose = TRUE) |
Add overview here.
Load the packages and set the seed so that we may reproduce the results more easily.
library(MASS)
library(L0Learn)
library(SSLASSO)
library(EMVS)
library(R.matlab)
# R.matlab v3.6.2 (2018-09-26) successfully loaded. See ?R.matlab for help.
#
# Attaching package: 'R.matlab'
# The following objects are masked from 'package:base':
#
# getOption, isOpen
library(ggplot2)
library(cowplot)
set.seed(1)
source("../R/linreg_methods_demo_functions.R")Simulate a data set with correlated variables in a similar way to Example 1 from Zou & Hastie 2005.
simulate_predictors_decaying_corr <- function (n, p, s = 0.5)
return(mvrnorm(n,rep(0,p),s^abs(outer(1:p,1:p,"-"))))
simulate_outcomes <- function (X, b, se)
return(drop(X %*% b - 1 + rnorm(nrow(X),sd = se)))
p <- 1000
se <- 3
b <- rep(0,p)
b[1:3] <- c(3,1.5,2)
Xtrain <- simulate_predictors_decaying_corr(200,p,0.5)
Xtest <- simulate_predictors_decaying_corr(500,p,0.5)
train <- list(X = Xtrain,y = simulate_outcomes(Xtrain,b,se))
test <- list(X = Xtest,y = simulate_outcomes(Xtest,b,se))
btrue <- bwriteMat("train.mat",
X = scale(train$X,center = TRUE,scale = FALSE),
y = with(train,y - mean(y)))L0Learn
First, let’s try the L0Learn from Hazimeh and Mazumder 2020. In the vignette, the authors suggest using the “L0L1” or “L0L2” penalties for better predictive performance. The package includes an interface to automatically select the penalty parameters \(\lambda, \gamma\) that minimize the test-set error. (L0Learn has two model fitting algorithms; the CD algorithm is faster whereas the CDPSI can sometimes produces better fits. For expediency we’ll use the CD algorithm.)
l0learn_cv <- with(train,L0Learn.cvfit(X,y,penalty = "L0L1",algorithm = "CD"))
i <- which.min(sapply(l0learn_cv$cvMeans,min))
j <- which.min(l0learn_cv$cvMeans[[i]])
gamma <- l0learn_cv$fit$gamma[i]
lambda <- l0learn_cv$fit$lambda[[i]][j]Compare the coefficient estimates against the ground truth:
b <- as.vector(coef(l0learn_cv,gamma = gamma,lambda = lambda))
b <- b[-1]
plot_coefs(btrue,b)
As expected, the L0 penalty shrinks most of the coefficients to zero.
The mean squared error (MSE) summarizes the accuracy of the predictions in the test set examples:
y <- as.vector(predict(l0learn_cv,newx = test$X,gamma = gamma,lambda = lambda))
plot_responses(test$y,y)
Let’s compare this to L0Learn with the simpler L0 penalty (which is a special case of the L0L1 penalty in which \(\gamma = 0\)):
l0learn_cv <- with(train,L0Learn.cvfit(X,y,penalty = "L0",algorithm = "CD"))
i <- which.min(l0learn_cv$cvMeans[[1]])
lambda <- l0learn_cv$fit$lambda[[1]][i]
y <- as.vector(predict(l0learn_cv,newx = test$X,lambda = lambda))
qplot(test$y,y) +
geom_abline(intercept = 0,slope = 1,color = "deepskyblue",
linetype = "dashed") +
ggtitle(sprintf("mse = %0.3f",mse(test$y,y))) +
theme_cowplot(font_size = 12) +
theme(plot.title = element_text(face = "plain",size = 12))
| Version | Author | Date |
|---|---|---|
| 2ea7aaa | Peter Carbonetto | 2023-05-13 |
Indeed, in this one example at least, L0Learn with the L0L1 penalty has better prediction performance than the L0 penalty.
Trimmed Lasso
ADD BRIEF DESCRIPTION OF TRIMMED LASSO HERE.
Use trimmed_lasso.m to run the method. Then load the
results:
b <- drop(readMat("trimmed_lasso_b.mat")$b)Compute the MLE of the intercept:
b0 <- with(train,mean(y - X %*% b))Compare the coef estimates vs. the ground truth:
plot_coefs(btrue,b)
Compare predicted Y vs. true Y:
y <- drop(b0 + test$X %*% b)
plot_responses(test$y,y)
Notes: (1) Works particularly well in this example because the true is indeed very sparse, and we happened to choose the “correct” k (in general this will be hard to do). (2) Cross-validation will be need to be used to get the right k.
SSLASSO
Next let’s try both the “adaptive” and “separable” variants of the Spike-and-Slab LASSO:
sslasso_adapt <-
with(train,SSLASSO(X,y,penalty = "adaptive",variance = "unknown"))
sslasso_sep <-
with(train,SSLASSO(X,y,penalty = "separable",variance = "unknown"))Let’s take a look at the SSLASSO with separable penalty first. As far as I can tell, the SSLASSO does not provide an automated way to select the spike and slab penalties, so I’ll choose one by hand:
b <- sslasso_sep$beta[,16]
qplot(btrue,b) +
geom_abline(intercept = 0,slope = 1,color = "deepskyblue",
linetype = "dashed") +
theme_cowplot(font_size = 12) 
| Version | Author | Date |
|---|---|---|
| 1f32078 | Peter Carbonetto | 2023-05-09 |
Compare these estimates to the estimates obtained by the adaptive SSLASSO:
b <- sslasso_adapt$beta[,16]
qplot(btrue,b) +
geom_abline(intercept = 0,slope = 1,color = "deepskyblue",
linetype = "dashed") +
theme_cowplot(font_size = 12) 
| Version | Author | Date |
|---|---|---|
| 1f32078 | Peter Carbonetto | 2023-05-09 |
Another annoying thing about the SSLASSO package is that it does not provide a “predict” method. So we will compute the predictions by hand after extracting the :
b0 <- sslasso_sep$intercept[16]
b <- sslasso_sep$beta[,16]
y <- drop(b0 + test$X %*% b)
p1 <- qplot(test$y,y) +
geom_abline(intercept = 0,slope = 1,color = "deepskyblue",
linetype = "dashed") +
ggtitle(sprintf("separable (mse = %0.3f)",mse(test$y,y))) +
theme_cowplot(font_size = 12)
b0 <- sslasso_adapt$intercept[16]
b <- sslasso_adapt$beta[,16]
y <- drop(b0 + test$X %*% b)
p2 <- qplot(test$y,y) +
geom_abline(intercept = 0,slope = 1,color = "deepskyblue",
linetype = "dashed") +
ggtitle(sprintf("adaptive (mse = %0.3f)",mse(test$y,y))) +
theme_cowplot(font_size = 12)
plot_grid(p1,p2)
EMVS
The EMVS package seems to be better documented (unfortunately, it was removed from CRAN). It also has two variants with different priors, the “independent” prior (which is recommended by the authors) and the “conjugate” prior. Let’s compare their performance in this simulated example.
emvs_conj = with(train,
EMVS(y,X,v0 = seq(0.1,2,length.out = 20),v1 = 10,independent = FALSE))
emvs_ind = with(train,
EMVS(y,X,v0 = exp(seq(-18,-1,length.out = 20)),v1 = 1,independent = TRUE))Add text here.
i <- which.max(emvs_conj$log_g_function)
b <- with(emvs_conj,betas[i,] * prob_inclusion[i,])
qplot(btrue,b) +
geom_abline(intercept = 0,slope = 1,color = "deepskyblue",
linetype = "dashed") +
theme_cowplot(font_size = 12)
| Version | Author | Date |
|---|---|---|
| 6182121 | Peter Carbonetto | 2023-05-09 |
Add text here.
b <- with(emvs_ind,betas[14,] * prob_inclusion[14,])
qplot(btrue,b) +
geom_abline(intercept = 0,slope = 1,color = "deepskyblue",
linetype = "dashed") +
theme_cowplot(font_size = 12)
| Version | Author | Date |
|---|---|---|
| 6182121 | Peter Carbonetto | 2023-05-09 |
Add text here.
b0 <- emvs_conj$intersects[i]
b <- emvs_conj$betas[i,]
y <- drop(b0 + test$X %*% b)
p1 <- qplot(test$y,y) +
geom_abline(intercept = 0,slope = 1,color = "deepskyblue",
linetype = "dashed") +
ggtitle(sprintf("conjugate (mse = %0.3f)",mse(test$y,y))) +
theme_cowplot(font_size = 12)
b0 <- emvs_ind$intersects[14]
b <- emvs_ind$beta[14,]
y <- drop(b0 + test$X %*% b)
p2 <- qplot(test$y,y) +
geom_abline(intercept = 0,slope = 1,color = "deepskyblue",
linetype = "dashed") +
ggtitle(sprintf("independent (mse = %0.3f)",mse(test$y,y))) +
theme_cowplot(font_size = 12)
plot_grid(p1,p2)
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.7
#
# Matrix products: default
# BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
#
# locale:
# [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#
# attached base packages:
# [1] stats graphics grDevices utils datasets methods base
#
# other attached packages:
# [1] cowplot_1.1.1 ggplot2_3.3.6 R.matlab_3.6.2 EMVS_1.2.1 SSLASSO_1.2-2
# [6] L0Learn_2.1.0 MASS_7.3-51.4
#
# loaded via a namespace (and not attached):
# [1] tidyselect_1.1.1 xfun_0.29 bslib_0.3.1 purrr_0.3.4
# [5] reshape2_1.4.3 lattice_0.20-38 colorspace_1.4-1 vctrs_0.3.8
# [9] generics_0.0.2 htmltools_0.5.4 yaml_2.2.0 utf8_1.1.4
# [13] rlang_1.0.6 R.oo_1.24.0 jquerylib_0.1.4 later_1.0.0
# [17] pillar_1.6.2 withr_2.5.0 R.utils_2.11.0 glue_1.4.2
# [21] DBI_1.1.0 lifecycle_1.0.3 plyr_1.8.5 stringr_1.4.0
# [25] munsell_0.5.0 gtable_0.3.0 workflowr_1.7.0.3 R.methodsS3_1.8.1
# [29] evaluate_0.14 labeling_0.3 knitr_1.37 fastmap_1.1.0
# [33] httpuv_1.5.2 fansi_0.4.0 highr_0.8 Rcpp_1.0.8
# [37] promises_1.1.0 scales_1.1.0 jsonlite_1.7.2 farver_2.0.1
# [41] fs_1.5.2 digest_0.6.23 stringi_1.4.3 dplyr_1.0.7
# [45] rprojroot_2.0.3 grid_3.6.2 cli_3.5.0 tools_3.6.2
# [49] magrittr_2.0.1 sass_0.4.0 tibble_3.1.3 crayon_1.4.1
# [53] whisker_0.4 pkgconfig_2.0.3 ellipsis_0.3.2 Matrix_1.3-4
# [57] assertthat_0.2.1 rmarkdown_2.11 R6_2.4.1 git2r_0.29.0
# [61] compiler_3.6.2