A simulation study comparing the wavelet shrinkage procedure SMASH (Xing and Stephens) and other popular wavelet denoising procedures

For a general introduction to DSCs, see here.

In this simulation study, we aim to compare several methods for performing nonparametric regression in the Gaussian case. The problem is of the form Y_i=\mu_i+\epsilon_i, i=1,...,n, where \mu is the underlying mean function and assumed to be "smooth", and \epsilon_i's are independent Gaussian noise with mean 0 and variance \sigma_i^2. The goal is to recover \mu as accurately as possible given Y.

The simulation schemes cover a wide range of different mean and variance functions (\mu and \sigma), as well as different signal to noise ratios (SNRs). The various methods are run and the resulting estimate of \mu is scored using the mean integrated squared error (MISE), which is simply the standard mean squared error rescaled appropriately.

install.packages("devtools")
library(devtools)
install_github("zrxing/dscr-smash")

• Clone/Fork+Clone this repository.
• Make sure to have the following packages/libraries installed
  • R packages: smash (from ashwave repo; see below), dscr, ashr (see below), dependencies in ashr & smash (e.g. wavethresh, EbayesThresh).
  • Matlab libraries: WaveLab (http://statweb.stanford.edu/~wavelab/), WavDen (http://www-ljk.imag.fr/SMS/software/GaussianWaveDen/down.html).
• Add path to the Matlab bin directory to R's path variable using add_path().
• Add path to dscr-smash/methods folder in Matlab.
• Run the R script run_dsc.R, either in batch mode or interactively.
• Upon completion results will be saved as both as an R object (res.Robj) and an R image file (res.RData).
• To obtain the interactive RShiny plot, open graphs.Rmd in RStudio and click on "Ruin Document". Additional description is contained within the graphs.Rmd.

• Add in the relevant test functions in datamakers/datamaker.R.
• Modify scenarios.R by adding the functions to meanfn or varfn as appropriate. Add function names to meanfn.short or varfn.short where needed.
• If a new SNR is desired, add a new SNR to rsnr.

• Write a new R script in methods/ directory.
• The method must be able to take as input the observed data, the true standard deviation function, as well as an estimated standard deviation function.
• The method must return a vector of estimated means as the output.
• Add the method to the list of methods in the methods.R file by adding the line add_method(dsc_smash, [name_of_method] [method], [method_arguments]).

• Add a new score function (name it something other than score) that takes as input the vector of estimated means, and returns a scalar indicating performance.
• Add the line add_score(dsc_smash, [score_function], "[name_of_score_function]").

This DSC uses the following formats:

input: list(x [vector], sig.true [vector], sig.est [vector]) #x is the vector of observations. sig.true contains the true values of \sigma_i, and sig.est is an estimate of \sigma under the assumption that all \sigma_i's are equal.

meta: list(mu [vector]) #mu contains the true values of \mu_i as defined above

output: an estimate of \mu [vector]

The performance of a method is scored by the quantity 10000*sum((\mu_i-\mu-hat_i)^2)/sum(\mu_i^2), where \mu is the true mean function and \mu-hat is the estimated mean function

See score.R.

(NB: results run with the following versions of the softwares: ashr commit: bf9cca351d7cf804b67a56552f8057b3af7c11cb; ashwave commit: e59bdebb61ae06a6459c9248a5167178fe0150df)