Result1_Ridge
Last updated: 2019-10-31
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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 \(X_{ij} \sim N(0,1)\) anda construct \(X \in \mathbb{R}^p\) with \(n = 500\) and \(p \in \{50,100,200,500,1000,2000\}\).
Signal setting
We sample the i.i.d. normal coefficients \(\beta_j \sim N(0,\sigma_\beta^2)\) for \(j = 1,\cdots,p\), or \(\beta \sim N(0,\sigma_\beta^2 I_p)\).
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 \(\beta_j \sim N(0,\sigma_\beta^2)\) for \(j = 1,\cdots,p\), or \(\beta \sim N(0,\sigma_\beta^2 I_p)\).
We expect that in this simulation setting, the ridge regression with the optimal tuning parameter \(\lambda\) will perform the best.
\[ p(\beta|y,X,\sigma^2) \propto p(y|X,\beta,\sigma^2) p(\beta) \propto \exp\left( - \frac{1}{2\sigma^2} \|y - X\beta\|_2^2 - \frac{1}{2\sigma_\beta^2} \|\beta\|_2^2 \right) \]
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.alpha); 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
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
Source code
The source code will be popped up when you click code on the right side.
setwd("..")
library(Matrix); library(ggplot2); library(cowplot); library(susieR); library(BGLR);
library(glmnet); library(mr.ash.alpha); 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_list = c("enet","lasso","ridge")
method_list2 = c("mr.ash", method_list,"enet2","lasso2","ridge2","ridge.opt")
method_num = length(method_list2)
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$t
if (method_list[j] == "lasso") {
lasso.path.order = mr.ash.alpha:::path.order(fit$fit$glmnet.fit)
lasso.beta = as.vector(coef(fit$fit))[-1]
lasso.time = c(fit$t, fit$t2)
}
if (method_list[j] == "lasso") {
pred[i,method_num - 2] = fit$rsse2 / data$sigma / sqrt(n)
} else if (method_list[j] == "enet") {
pred[i,method_num - 3] = fit$rsse2 / data$sigma / sqrt(n)
} else if (method_list[j] == "ridge") {
pred[i,method_num - 1] = fit$rsse2 / data$sigma / sqrt(n)
}
}
fit = fit.mr.ash(data$X, data$y, data$X.test, data$y.test, seed = i,
sa2 = (2^((0:19) / 20) - 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
print(c(pred[i,]))
}
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 Mojave 10.14
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_2.5-16 L0Learn_1.2.0 ncvreg_3.11-1
[4] mr.ash.alpha_0.1-2 glmnet_2.0-18 foreach_1.4.7
[7] BGLR_1.0.8 susieR_0.8.0 cowplot_1.0.0
[10] ggplot2_3.2.1 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_1.4.2 git2r_0.26.1
[7] workflowr_1.4.0 iterators_1.0.12 tools_3.5.3
[10] digest_0.6.21 evaluate_0.14 tibble_2.1.3
[13] gtable_0.3.0 lattice_0.20-38 pkgconfig_2.0.3
[16] rlang_0.4.0 yaml_2.2.0 xfun_0.9
[19] withr_2.1.2 stringr_1.4.0 dplyr_0.8.3
[22] knitr_1.25 fs_1.3.1 rprojroot_1.3-2
[25] grid_3.5.3 tidyselect_0.2.5 glue_1.3.1
[28] R6_2.4.0 rmarkdown_1.15 latticeExtra_0.6-28
[31] reshape2_1.4.3 purrr_0.3.2 magrittr_1.5
[34] whisker_0.4 codetools_0.2-16 backports_1.1.4
[37] scales_1.0.0 htmltools_0.3.6 assertthat_0.2.1
[40] colorspace_1.4-1 nor1mix_1.3-0 stringi_1.4.3
[43] lazyeval_0.2.2 munsell_0.5.0 truncnorm_1.0-8
[46] crayon_1.3.4