Last updated: 2019-10-22
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The experiment is based on the following simulation setting.
We sample the standard i.i.d. Gaussian measurement Xij∼N(0,1) anda construct X∈Rp with n=500 and p∈{50,100,200,500,1000,2000}.
We sample the i.i.d. normal coefficients βj∼N(0,σ2β) for j=1,⋯,p, or β∼N(0,σ2βIp).
This signal will be called normal
.
We fix PVE = 0.5. The relative performance does not very much dependent on the PVE value.
In what follows, we briefly describe the comparison methods.
Let us recall that we sample the i.i.d. normal coefficients βj∼N(0,σ2β) for j=1,⋯,p, or β∼N(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).
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.
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')
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
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))
}
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