Comparing two different ways of estimating the mixture weights in mr-ash
Peter Carbonetto
February 20, 2020
Last updated: 2020-02-24
Checks: 7 0
Knit directory: mr-ash/analysis/
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| File | Version | Author | Date | Message |
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| Rmd | d02866e | Peter Carbonetto | 2020-02-24 | Fixed bugs in mr_ash_with_mixsqp and updated em_vs_mixsqp with Youngseok’s test case. |
| html | d02866e | Peter Carbonetto | 2020-02-24 | Fixed bugs in mr_ash_with_mixsqp and updated em_vs_mixsqp with Youngseok’s test case. |
| html | 23ad0c2 | Peter Carbonetto | 2020-02-20 | Made a few adjustments to the plots in the em_vs_mixsqp analysis. |
| Rmd | 1e5af9f | Peter Carbonetto | 2020-02-20 | wflow_publish(“em_vs_mixsqp.Rmd”) |
| html | 4cbeb36 | Peter Carbonetto | 2020-02-20 | Built first draft of the em_vs_mixsqp analysis. |
| html | 38f8a38 | Peter Carbonetto | 2020-02-20 | Build site. |
| Rmd | 825688b | Peter Carbonetto | 2020-02-20 | wflow_publish(“em_vs_mixsqp.Rmd”) |
| Rmd | 582e465 | Peter Carbonetto | 2020-02-20 | Created first draft of mr_ash_demo workflowr page. |
Here we compare two different ways of implementing the mixture weight updates in mr-ash: EM and mix-SQP.
Script parameters
These are the data simulation settings.
n <- 100
p <- 400
sd <- c(0, 1, 2)
w <- c(0.9, 0.05, 0.05)
s <- 0.1This specifies the variances for the mixture-of-normals prior on the regression coefficients.
s0 <- 10^seq(-4,0,length.out = 12)Load functions
These are the packages used in this analysis.
library(ggplot2)
library(cowplot)
library(MASS)
library(mixsqp)This R code provides a very simple implementation of the mr-ash algorithm.
source("../code/misc.R")
source("../code/mr_ash.R")
source("../code/mr_ash_with_mixsqp.R")Simulate data
The predictors are drawn from the multivariate normal with zero mean and covariance matrix S, in which all diagonal entries are 1, and all off-diagonal entries are s. Setting s = 0.5 reproduces the simulation of the predictors used in Example 3 of Zou & Hastie (2005).
set.seed(2)
S <- matrix(s,p,p)
diag(S) <- 1
X <- mvrnorm(n,rep(0,p),S)
k <- sample(length(w),p,replace = TRUE,prob = w)
beta <- sd[k] * rnorm(p)
y <- drop(X %*% beta + rnorm(n))Fit model
These are the initial estimates of residual variance (s), mixture weights (w0), and posterior mean estimates of the regression coefficients (b).
k <- length(s0)
se <- 1
w0 <- rep(1/k,k)
b <- rep(0,p)Fit the model by running 200 EM updates for the mixture weights.
fit1 <- mr_ash(X,y,se,s0,w0,b,maxiter = 200,verbose = FALSE)Fit the model a second time using the mix-SQP updates for the mixture weights.
fit2 <- mr_ash_with_mixsqp(X,y,se,s0,w0,b,numiter = 10)
# iter elbo max|b-b'| max|w0-w0'| EM mix alpha
# 1 -3.786110149612e+02 5.782e-05 4.59556e-01 40 14 5.0e-01
# 2 -3.742189639930e+02 4.261e-05 2.12984e-01 13 17 5.0e-01
# 3 -3.739975329896e+02 7.374e-05 1.77119e-01 12 15 1.0e+00
# 4 -3.738117653767e+02 3.802e-05 1.15550e-02 14 16 5.0e-01
# 5 -3.737628735795e+02 9.657e-05 3.12331e-03 10 15 5.0e-01
# 6 -3.737604578864e+02 8.854e-05 1.31017e-03 8 15 5.0e-01
# 7 -3.737599689561e+02 3.380e-05 4.95211e-04 8 15 5.0e-01
# 8 -3.737599026351e+02 8.254e-05 1.92471e-04 6 15 5.0e-01
# 9 -3.737598918976e+02 5.168e-05 7.11962e-05 5 15 5.0e-01
# 10 -3.737598904765e+02 6.541e-05 1.65833e-05 3 15 2.5e-01Review model fit
Compare the posterior mean estimates from the two fits.
ggplot(data.frame(em = fit1$b,mixsqp = fit2$b),
aes(x = em,y = mixsqp)) +
geom_point(color = "darkblue") +
geom_abline(intercept = 0,slope = 1,col = "magenta",lty = "dotted") +
theme_cowplot()
Compare the posterior mean estimates against the values used to simulate the data.
ggplot(data.frame(true = beta,mixsqp = fit2$b),
aes(x = true,y = mixsqp)) +
geom_point(color = "darkblue") +
geom_abline(intercept = 0,slope = 1,col = "magenta",lty = "dotted") +
theme_cowplot()
Plot the improvement in the solution over time.
elbo.best <- max(c(fit1$elbo,fit2$elbo))
pdat <- rbind(data.frame(update = "em",
iter = 1:length(fit1$elbo),
elbo = fit1$elbo),
data.frame(update = "mixsqp",
iter = cumsum(fit2$numem),
elbo = fit2$elbo))
pdat$elbo <- elbo.best - pdat$elbo + 1e-4
ggplot(pdat,aes(x = iter,y = elbo,color = update)) +
geom_line() +
geom_point() +
scale_y_log10() +
scale_color_manual(values = c("royalblue","darkorange")) +
labs(y = "distance to \"best\" elbo") +
theme_cowplot()
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.3
#
# 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] mixsqp_0.3-17 MASS_7.3-51.4 cowplot_1.0.0 ggplot2_3.2.1
#
# loaded via a namespace (and not attached):
# [1] Rcpp_1.0.3 compiler_3.6.2 pillar_1.4.3 later_1.0.0
# [5] git2r_0.26.1 workflowr_1.6.0 tools_3.6.2 digest_0.6.23
# [9] lattice_0.20-38 evaluate_0.14 lifecycle_0.1.0 tibble_2.1.3
# [13] gtable_0.3.0 pkgconfig_2.0.3 rlang_0.4.2 Matrix_1.2-18
# [17] yaml_2.2.0 xfun_0.11 withr_2.1.2 stringr_1.4.0
# [21] dplyr_0.8.3 knitr_1.26 fs_1.3.1 rprojroot_1.3-2
# [25] grid_3.6.2 tidyselect_0.2.5 glue_1.3.1 R6_2.4.1
# [29] rmarkdown_2.0 irlba_2.3.3 farver_2.0.1 purrr_0.3.3
# [33] magrittr_1.5 whisker_0.4 backports_1.1.5 scales_1.1.0
# [37] promises_1.1.0 htmltools_0.4.0 assertthat_0.2.1 colorspace_1.4-1
# [41] httpuv_1.5.2 labeling_0.3 stringi_1.4.3 lazyeval_0.2.2
# [45] munsell_0.5.0 crayon_1.3.4