Processing math: 47%
  • Introduction
    • Design setting
    • Signal setting
    • PVE
  • Methods
    • Optimal Ridge
    • Elastic Net
    • Packages / Libraries
  • Results
  • Source code
  • System Configuration

Last updated: 2019-10-22

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Introduction

The experiment is based on the following simulation setting.

Design setting

We sample the standard i.i.d. Gaussian measurement XijN(0,1) anda construct XRp with n=500 and p{50,100,200,500,1000,2000}.

Signal setting

We sample the i.i.d. normal coefficients βjN(0,σ2β) for j=1,,p, or βN(0,σ2βIp).

This signal will be called normal.

PVE

We fix PVE = 0.5. The relative performance does not very much dependent on the PVE value.

Methods

In what follows, we briefly describe the comparison methods.

Optimal Ridge

Let us recall that we sample the i.i.d. normal coefficients βjN(0,σ2β) for j=1,,p, or βN(0,σ2βIp).

We expect that in this simulation setting, the ridge regression with the optimal tuning parameter λ will perform the best.

p(β|y,X,σ2)p(y|X,β,σ2)p(β)exp(12σ2

This implies that p(\beta|y,X,\sigma^2) is again a multivariate normal distribution and thus the posterior mode is equal to the posterior mean. Thus the optimal \lambda is \sigma^2 / (n\sigma_\beta^2).

Elastic Net

The glmnet R package provides an elastic net implementation. It seeks to minimize the following objective function.

\frac{1}{2n} \| y - X\beta \|^2 + \lambda \left(\alpha \|\beta\|_1 + 0.5 (1 - \alpha) \|\beta\|_2^2 \right)

\lambda and \alpha are tuning parameters. For a fixed \alpha in \{ 0.1 * (a-1) : a = 1,\cdots, 11, we run cv.glmnet with the default setting to tune \lambda by cross-validation. Then we select a best tuple of \alpha and \lambda that minimizes the cross-validation error.

Packages / Libraries

A list of packages we have loaded is collapsed. Please click “code” to see the list.

library(Matrix); library(ggplot2); library(cowplot); library(susieR); library(BGLR);
library(glmnet); library(mr.ash); library(ncvreg); library(L0Learn); library(varbvs);
standardize = FALSE
source('code/method_wrapper.R')
source('code/sim_wrapper.R')

Results

The result is summarized below.

res_df       = readRDS("results/ridge_pve0.5.RDS")
p_list       = c(50,100,200,500,1000,2000)
method_list  = c("Mr.ASH","VarBVS","BayesB","Blasso","SuSiE","E-NET","Lasso","Ridge","SCAD","MCP","L0Learn","Ridge.opt")
col          = gg_color_hue(13)[1:11]
sdat = data.frame()
for (i in 1:6) {
  sdat       = rbind(sdat, data.frame(pred = colMeans(matrix(res_df[[i]]$pred,20,12)),
                                      time = colMeans(matrix(res_df[[i]]$time,20,12)),
                                      p = p_list[i],
                                      fit = method_list))
}
sdat$fit    = factor(sdat$fit, levels =  c("Mr.ASH","E-NET","Lasso","Ridge",
                                           "SCAD","MCP","L0Learn",
                                           "VarBVS","BayesB","Blasso","SuSiE",
                                           "Ridge.opt"))
sdat1       = sdat[sdat$fit %in% c("Mr.ASH","E-NET","Lasso","Ridge","Ridge.opt"),]
sdat2       = sdat[sdat$fit %in% c("Mr.ASH","E-NET","SCAD","MCP","L0Learn","Ridge.opt"),]
sdat3       = sdat[sdat$fit %in% c("Mr.ASH","VarBVS","BayesB","Blasso","SuSiE","Ridge.opt"),]

p1 = ggplot(sdat1) + geom_line(aes(x = p, y = pred, color = fit)) +
  geom_point(aes(x = p, y = pred, color = fit, shape = fit), size = 2.5) +
  theme_cowplot(font_size = 14) +
  scale_x_continuous(trans = "log10", breaks = p_list) +
  labs(y = "predictior error (rmse / sigma)", x = "number of coefficients (p)") +
  theme(axis.line = element_blank(),
        plot.title = element_text(hjust = 0.5)) +
  scale_color_manual(values = c(col[c(1,2,3,4)],"gray50")) +
  scale_shape_manual(values = c(19,17,24,25,15)) +
  scale_y_continuous(trans = "log10", limits = c(1.04,1.46), breaks = c(1.1,1.2,1.3,1.4))
p2 = ggplot(sdat2) + geom_line(aes(x = p, y = pred, color = fit)) +
  geom_point(aes(x = p, y = pred, color = fit, shape = fit), size = 2.5) +
  theme_cowplot(font_size = 14) +
  scale_x_continuous(trans = "log10", breaks = p_list) +
  labs(y = "", x = "number of coefficients (p)") +
  theme(axis.line = element_blank(),
        plot.title = element_text(hjust = 0.5)) +
  scale_color_manual(values = c(col[c(1,2,5,6,7)],"gray50")) +
  scale_shape_manual(values = c(19,17,9,3,11,15)) +
  scale_y_continuous(trans = "log10", limits = c(1.04,1.46), breaks = c(1.1,1.2,1.3,1.4))
p3 = ggplot(sdat3) + geom_line(aes(x = p, y = pred, color = fit)) +
  geom_point(aes(x = p, y = pred, color = fit, shape = fit), size = 2.5) +
  theme_cowplot(font_size = 14) +
  scale_x_continuous(trans = "log10", breaks = p_list) +
  labs(y = "", x = "number of coefficients (p)") +
  theme(axis.line = element_blank(),
        plot.title = element_text(hjust = 0.5)) +
  scale_color_manual(values = c(col[c(1,8,9,10,11)],"gray50")) +
  scale_shape_manual(values = c(19,4,5,7,8,15)) +
  scale_y_continuous(trans = "log10", limits = c(1.04,1.46), breaks = c(1.1,1.2,1.3,1.4))
fig_main  = plot_grid(p1,p2,p3, nrow = 1, rel_widths = c(0.35,0.35,0.3))
title     = ggdraw() + draw_label("Prediction Error (log-scale)", fontface = 'bold', size = 20) 
subtitle  = ggdraw() + draw_label("Scenario: IndepGauss + Normal, n = 500, p = 50,100,200,500,1000,2000, pve = 0.5", fontface  = 'bold', size = 18) 
fig       = plot_grid(title,subtitle,fig_main, ncol = 1, rel_heights = c(0.1,0.06,0.95))
fig

Version Author Date
2443c8b Youngseok Kim 2019-10-22
bd36a79 Youngseok 2019-10-13
p4  = my.box2(sdat[sdat$fit != "Ridge.opt",], "fit", "time",
             gg_color_hue(13)[1:11]) +
  theme(axis.line    = element_blank(),
        axis.text.x  = element_text(angle = 45,hjust = 1),
        legend.position = "none") +
  scale_y_continuous(trans = "log10")
title     = ggdraw() + draw_label("Computation time (log-scale)", fontface = 'bold', size = 20) 
subtitle  = ggdraw() + draw_label("Scenario: IndepGauss + Normal, n = 500, p = 50,100,200,500,1000,2000, pve = 0.5", fontface  = 'bold', size = 18) +
  labs(y = "computation time (sec)")
p0        = ggplot() + geom_blank() + theme_cowplot() + theme(axis.line = element_blank())
fig_main  = plot_grid(p0,p4,p0, nrow = 1, rel_widths = c(0.3,0.6,0.3))
fig       = plot_grid(title,subtitle,fig_main, ncol = 1, rel_heights = c(0.1,0.06,0.95))
fig

Version Author Date
2443c8b Youngseok Kim 2019-10-22
bd36a79 Youngseok 2019-10-13

Source code

The source code will be popped up when you click code on the right side.

library(Matrix); library(ggplot2); library(cowplot); library(susieR); library(BGLR);
library(glmnet); library(mr.ash); library(ncvreg); library(L0Learn); library(varbvs);
standardize = FALSE
source('code/method_wrapper.R')
source('code/sim_wrapper.R')
tdat1        = list()
n            = 500
p_range      = c(50,100,200,500,1000,2000)
method_list  = c("varbvs","bayesb","blasso","susie","enet","lasso","ridge","scad","mcp","l0learn")
method_list2 = c("mr.ash", method_list, "ridge.opt")
method_num   = length(method_list) + 2
iter_num     = 20
pred         = matrix(0, iter_num, method_num);
time         = matrix(0, iter_num, method_num);
colnames(pred) <- colnames(time) <- method_list2

for (iter in 1:6) {
p               = p_range[iter]
for (i in 1:20) {
  data          = simulate_data(n, p, s = p, seed = i, signal = "normal", pve = 0.5)
 
  for (j in 1:length(method_list)) {
    fit.method    = get(paste("fit.",method_list[j],sep = ""))
    fit           = fit.method(data$X, data$y, data$X.test, data$y.test, seed = i)
    pred[i,j+1]   = fit$rsse / data$sigma / sqrt(n)
    time[i,j+1]   = fit$t
    
    if (method_list[j] == "lasso") {
      lasso.path.order = mr.ash:::path.order(fit$fit$glmnet.fit)
      lasso.beta       = as.vector(coef(fit$fit))[-1]
      lasso.time       = c(fit$t, fit$t2)
    }
  }
 
  fit         = fit.mr.ash(data$X, data$y, data$X.test, data$y.test, seed = i,
                           sa2 = (2^((0:19) / 5 / 2^iter) - 1)^2)
  pred[i,1]   = fit$rsse / data$sigma / sqrt(n)
  time[i,1]   = fit$t
  
  fit         = fit.ridge.opt(data$X, data$y, data$X.test, data$y.test, data$sigma, seed = i)
  pred[i,method_num] = fit$rsse / data$sigma / sqrt(n)
  time[i,method_num] = -Inf
  cat(i," ")
}
cat("\n")
print(c(colMeans(pred)))
tdat1[[iter]] = data.frame(pred = c(pred), time = c(time),
                           fit = rep(method_list2, each = 20))
}

System Configuration

Click the below Session Info.


sessionInfo()
R version 3.5.3 (2019-03-11)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.3

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/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] varbvs_2.6-5  L0Learn_1.1.0 ncvreg_3.11-1 mr.ash_0.1-2  glmnet_2.0-16
 [6] foreach_1.4.4 BGLR_1.0.8    susieR_0.7.1  cowplot_0.9.4 ggplot2_3.2.1
[11] Matrix_1.2-17

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.2          RColorBrewer_1.1-2  plyr_1.8.4         
 [4] compiler_3.5.3      pillar_1.3.1        git2r_0.25.2       
 [7] workflowr_1.3.0     iterators_1.0.10    tools_3.5.3        
[10] digest_0.6.18       evaluate_0.13       tibble_2.1.1       
[13] gtable_0.3.0        lattice_0.20-38     pkgconfig_2.0.2    
[16] rlang_0.4.0         yaml_2.2.0          xfun_0.6           
[19] withr_2.1.2         stringr_1.4.0       dplyr_0.8.3        
[22] knitr_1.22          fs_1.3.0            rprojroot_1.3-2    
[25] grid_3.5.3          tidyselect_0.2.5    glue_1.3.1         
[28] R6_2.4.0            rmarkdown_1.12      latticeExtra_0.6-28
[31] reshape2_1.4.3      purrr_0.3.2         magrittr_1.5       
[34] whisker_0.3-2       codetools_0.2-16    backports_1.1.4    
[37] scales_1.0.0        htmltools_0.4.0     assertthat_0.2.1   
[40] colorspace_1.4-1    labeling_0.3        nor1mix_1.2-3      
[43] stringi_1.4.3       lazyeval_0.2.2      munsell_0.5.0      
[46] truncnorm_1.0-8     crayon_1.3.4