Gaussian mean estimation in simulated data sets

Last updated: 2018-12-20

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File	Version	Author	Date	Message
Rmd	86da808	Peter Carbonetto	2018-12-10	Added pointers to dsc/README in relevant R Markdown files.
Rmd	c589dbb	Peter Carbonetto	2018-12-06	wflow_publish(c(“index.Rmd”, “gaussian_signals.Rmd”,
html	ee71f27	Peter Carbonetto	2018-12-04	Made a few small adjustments to the text in the “gaussianmeanest” analysis.
Rmd	eb6cc34	Peter Carbonetto	2018-12-04	wflow_publish(“gaussmeanest.Rmd”)
html	05684ba	Peter Carbonetto	2018-12-04	Ran wflow_publish(“gaussmeanest.Rmd”) to populate the webpage.
Rmd	9a67b48	Peter Carbonetto	2018-12-02	Moved dsc results file.
Rmd	049dcbb	Peter Carbonetto	2018-11-08	Moved around some files and revised TOC in home page.

In this analysis, we assess the ability of different signal denoising methods to recover the true signal after being provided with Gaussian-distributed observations of the signal. We consider scenarios in which the data have homoskedastic errors (constant variance) and heteroskedastic errors (non-constant variance).

Since the simulation experiments are computationally intensive, they were implemented separately. (For instructions on re-running these simulation experiments, see the README in the “dsc” directory of the git repository). Here we create plots to summarize the results of these experiments.

Set up environment

Load the ggplot2 and cowplot packages, and the functions definining the mean and variances used to simulate the data.

library(ggplot2)
library(cowplot)
source("../code/signals.R")

Load results

Load the results of the simulation experiments.

load("../output/dscr.RData")

Simulated data with constant variances

This plot reproduces Fig. 2 of the manuscript, which compares the accuracy of the mean curves estimated from the data sets that were simulated using the “Spikes” mean function with constant variance.

First, extract the results used to generate this plot.

homo.data.smash <-
  res[res$.id    == "sp.3.v1" &
      res$method == "smash.s8",]
homo.data.smash.homo <-
  res[res$.id    == "sp.3.v1" &
      res$method == "smash.homo.s8",]
homo.data.tithresh <-
  res[res$.id == "sp.3.v1" &
      res$method == "tithresh.homo.s8",]
homo.data.ebayes <-
  res[res$.id    == "sp.3.v1" &
      res$method == "ebayesthresh",]
homo.data.smash.true <-
  res[res$.id == "sp.3.v1" &
  res$method  == "smash.true.s8",]
homo.data <-
  res[res$.id == "sp.3.v1" &
  (res$method == "smash.s8" |
   res$method == "ebayesthresh" |
   res$method == "tithresh.homo.s8"),]

Transform these results into a data frame suitable for ggplot2.

pdat <-
  rbind(data.frame(method      = "smash",
                   method.type = "est",
                   mise        = homo.data.smash$mise),
        data.frame(method      = "smash.homo",
                   method.type = "homo",
                   mise        = homo.data.smash.homo$mise),
        data.frame(method      = "tithresh",
                   method.type = "homo",
                   mise        = homo.data.tithresh$mise),
        data.frame(method      = "ebayesthresh",
                   method.type = "homo",
                   mise        = homo.data.ebayes$mise),
        data.frame(method      = "smash.true",
                   method.type = "true",
                   mise        = homo.data.smash.true$mise))
pdat <-
  transform(pdat,
            method = factor(method,
                            names(sort(tapply(pdat$mise,pdat$method,mean),
                                       decreasing = TRUE))))

Create the combined boxplot and violin plot using ggplot2.

p <- ggplot(pdat,aes(x = method,y = mise,fill = method.type)) +
     geom_violin(fill = "skyblue",color = "skyblue") +
     geom_boxplot(width = 0.15,outlier.shape = NA) +
     scale_y_continuous(breaks = seq(6,16,2)) +
     scale_fill_manual(values = c("darkorange","dodgerblue","gold"),
                       guide = FALSE) +
     coord_flip() +
     labs(x = "",y = "MISE") +
     theme(axis.line = element_blank(),
           axis.ticks.y = element_blank())
print(p)

Expand here to see past versions of plot-1-create-1.png:

Version	Author	Date
05684ba	Peter Carbonetto	2018-12-04

From this plot, we see that the three variations of SMASH all outperformed EbayesThresh and TI thresholding in this setting.

Next, we compare the same methods in simulated data sets with heteroskedastic errors.

Simulated data with heteroskedastic errors: “Spikes” mean signal and “Clipped Blocks” variance

In this scenario, the data sets were simulated using the “Spikes” mean function and the “Clipped Blocks” variance function. The next two plots reproduce part of Fig. 3 in the manuscript.

This plot shows the mean function as a block line, and the +/- 2 standard deviations as orange lines:

t         <- (1:1024)/1024
mu        <- spikes.fn(t,"mean")
sigma.ini <- sqrt(cblocks.fn(t,"var"))
sd.fn     <- sigma.ini/mean(sigma.ini) * sd(mu)/3
par(cex.axis = 1,cex.lab = 1.25)
plot(mu,type = "l", ylim = c(-0.05,1),xlab = "position",ylab = "",
     lwd = 1.75,xaxp = c(0,1024,4),yaxp = c(0,1,4))
lines(mu + 2*sd.fn,col = "darkorange",lty = 5,lwd = 1.75)
lines(mu - 2*sd.fn,col = "darkorange",lty = 5,lwd = 1.75)

Expand here to see past versions of spikes-signal-1.png:

Version	Author	Date
05684ba	Peter Carbonetto	2018-12-04

Extract the results from running the simulations.

hetero.data.smash <-
  res[res$.id == "sp.3.v5" & res$method == "smash.s8",]
hetero.data.smash.homo <-
  res[res$.id == "sp.3.v5" & res$method == "smash.homo.s8",]
hetero.data.tithresh.homo <-
  res[res$.id == "sp.3.v5" & res$method == "tithresh.homo.s8",]
hetero.data.tithresh.rmad <-
  res[res$.id == "sp.3.v5" & res$method == "tithresh.rmad.s8",]
hetero.data.tithresh.smash <-
  res[res$.id == "sp.3.v5" & res$method == "tithresh.smash.s8",]
hetero.data.tithresh.true <-
  res[res$.id == "sp.3.v5" & res$method == "tithresh.true.s8",]
hetero.data.ebayes <-
  res[res$.id == "sp.3.v5" & res$method == "ebayesthresh",]
hetero.data.smash.true <-
  res[res$.id == "sp.3.v5" & res$method == "smash.true.s8",]

Transform these results into a data frame suitable for ggplot2.

pdat <-
  rbind(data.frame(method      = "smash",
                   method.type = "est",
                   mise        = hetero.data.smash$mise),
        data.frame(method      = "smash.homo",
                   method.type = "homo",
                   mise        = hetero.data.smash.homo$mise),
        data.frame(method      = "tithresh.rmad",
                   method.type = "tithresh",
                   mise        = hetero.data.tithresh.rmad$mise),
        data.frame(method      = "tithresh.smash",
                   method.type = "tithresh",
                   mise        = hetero.data.tithresh.smash$mise),
        data.frame(method      = "tithresh.true",
                   method.type = "tithresh",
                   mise        = hetero.data.tithresh.true$mise),
        data.frame(method      = "ebayesthresh",
                   method.type = "homo",
                   mise        = hetero.data.ebayes$mise),
        data.frame(method      = "smash.true",
                   method.type = "true",
                   mise        = hetero.data.smash.true$mise))
pdat <-
  transform(pdat,
            method = factor(method,
                            names(sort(tapply(pdat$mise,pdat$method,mean),
                                       decreasing = TRUE))))

Create the combined boxplot and violin plot using ggplot2.

p <- ggplot(pdat,aes(x = method,y = mise,fill = method.type)) +
     geom_violin(fill = "skyblue",color = "skyblue") +
     geom_boxplot(width = 0.15,outlier.shape = NA) +
     scale_fill_manual(values=c("darkorange","dodgerblue","limegreen","gold"),
                       guide = FALSE) +
     coord_flip() +
     scale_y_continuous(breaks = seq(10,70,10)) +
     labs(x = "",y = "MISE") +
     theme(axis.line = element_blank(),
           axis.ticks.y = element_blank())
print(p)

Expand here to see past versions of plot-2-create-1.png:

Version	Author	Date
05684ba	Peter Carbonetto	2018-12-04

In this scenario, we see that SMASH, when allowing for heteroskedastic errors, outperforms EbayesThresh and all variants of TI thresholding (including TI thresholding when provided with the true variance). Further, SMASH performs almost as well when estimating the variance compared to when provided with the true variance.

Simulated data with heteroskedastic errors: “Corner” mean signal and “Doppler” variance

In this next scenario, the data sets were simulated using the “Corner” mean function and the “Doppler” variance function. These plots were also used in Fig. 3 of the manuscript.

This plot shows the mean function as a block line, and the +/- 2 standard deviations as orange lines:

mu        <- cor.fn(t,"mean") 
sigma.ini <- sqrt(doppler.fn(t,"var"))
sd.fn     <- sigma.ini/mean(sigma.ini) * sd(mu)/3
plot(mu,type = "l", ylim = c(-0.05,1),xlab = "position",ylab = "",
     lwd = 1.75,xaxp = c(0,1024,4),yaxp = c(0,1,4))
lines(mu + 2*sd.fn,col = "darkorange",lty = 5,lwd = 1.75)
lines(mu - 2*sd.fn,col = "darkorange",lty = 5,lwd = 1.75)

Expand here to see past versions of corner-signal-1.png:

Version	Author	Date
05684ba	Peter Carbonetto	2018-12-04

Extract the results from running these simulations.

hetero.data.smash.2 <-
  res[res$.id == "cor.3.v3" & res$method == "smash.s8",]
hetero.data.smash.homo.2 <-
  res[res$.id == "cor.3.v3" & res$method == "smash.homo.s8",]
hetero.data.tithresh.homo.2 <-
  res[res$.id == "cor.3.v3" & res$method == "tithresh.homo.s8",]
hetero.data.tithresh.rmad.2 <-
  res[res$.id == "cor.3.v3" & res$method == "tithresh.rmad.s8",]
hetero.data.tithresh.smash.2 <-
  res[res$.id == "cor.3.v3" & res$method == "tithresh.smash.s8",]
hetero.data.tithresh.true.2 <-
  res[res$.id == "cor.3.v3" & res$method == "tithresh.true.s8",]
hetero.data.ebayes.2 <-
  res[res$.id == "cor.3.v3" & res$method == "ebayesthresh",]
hetero.data.smash.true.2 <-
  res[res$.id == "cor.3.v3" & res$method == "smash.true.s8",]

Transform these results into a data frame suitable for ggplot2.

pdat <-
  rbind(data.frame(method      = "smash",
                   method.type = "est",
                   mise        = hetero.data.smash.2$mise),
        data.frame(method      = "smash.homo",
                   method.type = "homo",
                   mise        = hetero.data.smash.homo.2$mise),
        data.frame(method      = "tithresh.rmad",
                   method.type = "tithresh",
                   mise        = hetero.data.tithresh.rmad.2$mise),
        data.frame(method      = "tithresh.smash",
                   method.type = "tithresh",
                   mise        = hetero.data.tithresh.smash.2$mise),
        data.frame(method      = "tithresh.true",
                   method.type = "tithresh",
                   mise        = hetero.data.tithresh.true.2$mise),
        data.frame(method      = "ebayesthresh",
                   method.type = "homo",
                   mise        = hetero.data.ebayes.2$mise),
        data.frame(method      = "smash.true",
                   method.type = "true",
                   mise        = hetero.data.smash.true.2$mise))
pdat <-
  transform(pdat,
            method = factor(method,
                            names(sort(tapply(pdat$mise,pdat$method,mean),
                                       decreasing = TRUE))))

Create the combined boxplot and violin plot using ggplot2.

p <- ggplot(pdat,aes(x = method,y = mise,fill = method.type)) +
     geom_violin(fill = "skyblue",color = "skyblue") +
     geom_boxplot(width = 0.15,outlier.shape = NA) +
     scale_fill_manual(values=c("darkorange","dodgerblue","limegreen","gold"),
                       guide = FALSE) +
     coord_flip() +
     scale_y_continuous(breaks = seq(1,5)) +
     labs(x = "",y = "MISE") +
     theme(axis.line = element_blank(),
           axis.ticks.y = element_blank())
print(p)

Expand here to see past versions of plot-3-create-1.png:

Version	Author	Date
05684ba	Peter Carbonetto	2018-12-04

Similar to the “Spikes” scenario, we see that the SMASH method, when allowing for heteroskedastic variances, outperforms both the TI thresholding and EbayesThresh approaches.

Session information

sessionInfo()
# R version 3.4.3 (2017-11-30)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS High Sierra 10.13.6
# 
# Matrix products: default
# BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.4/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_0.9.3 ggplot2_3.1.0
# 
# loaded via a namespace (and not attached):
#  [1] Rcpp_1.0.0        later_0.7.2       dscr_0.1-8       
#  [4] compiler_3.4.3    pillar_1.2.1      git2r_0.23.0     
#  [7] plyr_1.8.4        workflowr_1.1.1   bindr_0.1.1      
# [10] R.methodsS3_1.7.1 R.utils_2.6.0     tools_3.4.3      
# [13] digest_0.6.17     evaluate_0.11     tibble_1.4.2     
# [16] gtable_0.2.0      pkgconfig_2.0.2   rlang_0.2.2      
# [19] shiny_1.1.0       yaml_2.2.0        bindrcpp_0.2.2   
# [22] withr_2.1.2       stringr_1.3.1     dplyr_0.7.6      
# [25] knitr_1.20        rprojroot_1.3-2   grid_3.4.3       
# [28] tidyselect_0.2.4  glue_1.3.0        R6_2.2.2         
# [31] rmarkdown_1.10    purrr_0.2.5       magrittr_1.5     
# [34] whisker_0.3-2     promises_1.0.1    backports_1.1.2  
# [37] scales_0.5.0      htmltools_0.3.6   assertthat_0.2.0 
# [40] xtable_1.8-2      mime_0.5          colorspace_1.4-0 
# [43] httpuv_1.4.3      stringi_1.2.4     lazyeval_0.2.1   
# [46] munsell_0.4.3     R.oo_1.21.0

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Gaussian mean estimation in simulated data sets

Zhengrong Xing, Peter Carbonetto and Matthew Stephens

Set up environment

Load results

Simulated data with constant variances

Simulated data with heteroskedastic errors: “Spikes” mean signal and “Clipped Blocks” variance

Simulated data with heteroskedastic errors: “Corner” mean signal and “Doppler” variance

Session information