- Introduction
- Popular shrinkage operators
- MR.ASH shrinkage operator may resemble the popular shrinkage operators
Figures_for_the_paper
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Introduction
This .Rmd documentation is to reproduce figures in the paper/manuscript.
library(ggplot2); library(cowplot)Popular shrinkage operators
- Soft thresholding
Ssoft,λ(b)=sign(b)max
- Hard thresholding
S_{{\rm hard}, \lambda}(b) = {\rm sign}(b) (\max \{|b|, \lambda\} - \lambda) = \left\{\begin{array}{ll} 0, & |b| \leq \lambda \\ b, & b > \lambda \\ b, & b < -\lambda \end{array} \right.
- SCAD (Smoothly Clipped Absolute Deviation)
S_{{\rm scad}, \lambda, \gamma}(b) = \left\{\begin{array}{ll} S_{{\rm soft}, \lambda}(b), & |b| \leq 2 \lambda \\ \frac{S_{{\rm soft}, \gamma\lambda/(\gamma - 1)}(b)}{1 - (1 / (\gamma - 1))}, & 2\lambda < |b| \leq \gamma \lambda \\ b, & \gamma \lambda < |b| \end{array}\right.
- MCP (Minimax Concave Penalty)
S_{{\rm mcp}, \lambda, \gamma}(b) = \left\{\begin{array}{ll} \frac{S_{{\rm soft}, \lambda}(b)}{1 - (1 / (\gamma - 1))}, & |b| \leq \gamma \lambda \\ b, & \gamma \lambda < |b| \end{array}\right.
scad = function(b, lambda = 1, gamma = 3) {
b1 = pmax(abs(b) - lambda, 0) * sign(b) * (abs(b) <= 2 * lambda)
b2 = ((gamma-1) * b - sign(b) * gamma * lambda) / (gamma-2) * (abs(b) <= gamma * lambda) * (abs(b) > 2 * lambda)
b3 = b * (abs(b) > gamma * lambda)
return (b1+b2+b3)
}
mcp = function(b, lambda = 1, gamma = 2) {
b1 = pmax(abs(b) - lambda, 0) * sign(b) * (b <= gamma * lambda) * gamma / (gamma-1)
b2 = b * (abs(b) > gamma * lambda)
return (b1+b2)
}
softt = function(b, lambda = 1) {
return (pmax(abs(b) - lambda, 0) * sign(b))
}
hardt = function(b, lambda = 1) {
return (b * (abs(b) >= lambda))
}MR.ASH shrinkage operator may resemble the popular shrinkage operators
postmean = function(b, pi, sa2 = 0:100, sigma2 = 1){
phi = outer(b^2, 1 / 2 / (1 + 1 / sa2) / sigma2);
phi = exp(phi - apply(phi, 1, max))
phi = t(pi * t(phi) / sqrt(1 + sa2));
phi = phi / rowSums(phi);
out = c(colSums(t(phi) / (1 + 1 / sa2))) * b
return (out)
}
sa2 = 0:100; b = seq(0,10,0.001)
pi = exp(-sa2)
pi = pi / sum(pi)
b_softt = postmean(b, pi)
pi = double(101)
pi[1] = 0.8; pi[101] = 0.2
b_mcp = postmean(b, pi)
pi = double(101)
pi[1] = 0.6; pi[101] = 0.4
b_hardt = postmean(b, pi, sa2 = sa2^5)
pi = double(101)
pi[1] = 0.5; pi[2:51] = 0.2/50; pi[101] = 0.3
b_scad = postmean(b, pi, sa2 = sa2)
df = data.frame(b = c(b,b,b,b), sb = c(b_softt, b_hardt, b_scad, b_mcp),
sb2 = c(softt(b, lambda = 1.43),
hardt(b, lambda = 5),
scad(b, lambda = 1, gamma = 4),
mcp(b, lambda = 2)),
operator = rep(c("soft","hard","scad","mcp"), each = length(b)))
df$operator = factor(df$operator, levels = c("soft","hard","scad","mcp"))
p1 = ggplot(df) + geom_line(aes(x = b, y = sb, color = operator)) +
theme_cowplot(font_size = 14) + theme(axis.line = element_blank()) +
labs(y = "shrinakge of b (S(b))", title = "MR.ASH shrinkage operator")
p2 = ggplot(df) + geom_line(aes(x = b, y = sb2, color = operator)) +
theme_cowplot(font_size = 14) + theme(axis.line = element_blank()) +
labs(title = "123") +
labs(y = "shrinakge of b (S(b))", title = "Well-known shrinkage/thresholding operator")
fig = plot_grid(p1,p2, nrow = 1)
title = ggdraw() + draw_label("MR.ASH Shrinkage operator may resemble the well-known shrinkage operators", fontface = 'bold', size = 20)
fig = plot_grid(title, fig, nrow = 2, rel_heights = c(0.06,0.95))
fig
R.hardt = function(b, lambda = 1) {
out = b^2/2
out[b > lambda] = lambda^2/2
exp(-out)
}
R.softt = function(b, lambda = 1) {
out = lambda^2/2 + (b - lambda) * lambda
out[b <= lambda] = b[b <= lambda]^2/2
exp(-out)
}
R.scad = function(b, lambda = 1, gamma = 3) {
out = lambda^2/2 + (b - lambda) * lambda
out[b <= lambda] = b[b <= lambda]^2/2
out[b > 2 * lambda] = lambda^2 * 3/2 + (gamma - 2) * lambda^2/2 *
(1 - (gamma - b[b > 2 * lambda] / lambda)^2 / (gamma - 2)^2)
out[b > gamma * lambda] = lambda^2 * 3/2 + (gamma - 2) * lambda^2/2
exp(-out)
}
R.mcp = function(b, lambda = 1, gamma = 2) {
out = lambda^2/2 + (gamma - 1) * lambda^2 / 2 *
(1 - (gamma - b / lambda)^2 / (gamma - 1)^2)
out[b > gamma * lambda] = lambda^2 / 2 + (gamma - 1) * lambda^2 / 2
out[b <= lambda] = b[b <= lambda]^2/2
exp(-out)
}
marginal_shape = function(b, lambda = 1, gamma = 2, operator = "ridge") {
if (operator == "ridge") {
return (exp(-lambda * b^2/2))
} else if (operator == "soft") {
out = b
out[b >= 0] = R.softt(b[b >= 0], lambda = lambda)
out[b < 0] = R.softt(-b[b < 0], lambda = lambda)
return (out)
} else if (operator == "hard") {
out = b
out[b >= 0] = R.hardt(b[b >= 0], lambda = lambda)
out[b < 0] = R.hardt(-b[b < 0], lambda = lambda)
return (out)
} else if (operator == "scad") {
out = b
out[b >= 0] = R.scad(b[b >= 0], lambda = lambda, gamma = gamma)
out[b < 0] = R.scad(-b[b < 0], lambda = lambda, gamma = gamma)
return (out)
} else if (operator == "mcp") {
out = b
out[b >= 0] = R.mcp(b[b >= 0], lambda = lambda, gamma = gamma)
out[b < 0] = R.mcp(-b[b < 0], lambda = lambda, gamma = gamma)
return (out)
}
}
b = seq(-5,5,0.001)
df = data.frame(b = c(b,b,b,b,b), sb = c(marginal_shape(b, operator = "ridge"),
marginal_shape(b, operator = "soft"),
marginal_shape(b, operator = "hard"),
marginal_shape(b, operator = "scad"),
marginal_shape(b, operator = "mcp")),
operator = rep(c("normal","soft","hard","scad","mcp"), each = length(b)))
df$operator = factor(df$operator, levels = c("normal","soft","hard","scad","mcp"))
ggplot(df) + geom_line(aes(x = b, y = sb, color = operator)) +
theme_cowplot(font_size = 14) + theme(axis.line = element_blank())
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] cowplot_0.9.4 ggplot2_3.2.1
loaded via a namespace (and not attached):
[1] Rcpp_1.0.2 knitr_1.22 whisker_0.3-2 magrittr_1.5
[5] workflowr_1.3.0 tidyselect_0.2.5 munsell_0.5.0 colorspace_1.4-1
[9] R6_2.4.0 rlang_0.4.0 dplyr_0.8.3 stringr_1.4.0
[13] tools_3.5.3 grid_3.5.3 gtable_0.3.0 xfun_0.6
[17] withr_2.1.2 git2r_0.25.2 htmltools_0.4.0 assertthat_0.2.1
[21] yaml_2.2.0 lazyeval_0.2.2 rprojroot_1.3-2 digest_0.6.18
[25] tibble_2.1.1 crayon_1.3.4 purrr_0.3.2 fs_1.3.0
[29] glue_1.3.1 evaluate_0.13 rmarkdown_1.12 labeling_0.3
[33] stringi_1.4.3 compiler_3.5.3 pillar_1.3.1 scales_1.0.0
[37] backports_1.1.4 pkgconfig_2.0.2