Last updated: 2019-11-20

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Rmd 83d15cc Jason Willwerscheid 2019-11-20 workflowr::wflow_publish(“analysis/misspec_k.Rmd”)

I run a series of simulations in which I simulate from the true EBMF model (with Gaussian noise) and then vary the noise model to see whether EBMF is still able to select the correct number of components.

The EBMF model is: \[ Y = LF' + E, \] where \(Y \in \mathbb{R}^{n \times n}\), \(L \in \mathbb{R}^{n \times k}\), \(F \in \mathbb{R}^{n \times k}\), and \(E \in \mathbb{R}^{n \times n}\). I fix \(k\) at 10, but I vary \(n\) from 50 to 5000.

For each trial and each choice of \(n\), I simulate factors and loadings from the following distributions: \[ \begin{aligned} L_{ik} &\sim \pi_{0\ell}^{(k)} \delta_0 + (1 - \pi_{0\ell}^{(k)}) N(0, \sigma_\ell^{(k)2}) \\ F_{jk} &\sim \pi_{0f}^{(k)} \delta_0 + (1 - \pi_{0f}^{(k)}) N(0, \sigma_f^{(k)2}) \end{aligned} \] \[ \begin{aligned} \pi_{0\ell}^{(k)}, \pi_{0f}^{(k)} &\sim \text{Beta}(0.5, 0.5) \\ \sigma_\ell^{(k)2}, \sigma_f^{(k)2} &\sim \text{Gamma}(1, 2) \end{aligned} \]

From each low-rank matrix \(LF'\), I generate four different data matrices using different error models:

  1. The Gaussian model (i.e., the EBMF model): \[ Y = LF' + E, \] where \[ E_{ij} \sim N(0, 0.5^2) \]
  2. A heavier-tailed model: \[ \begin{aligned} Y &= LF' + E \\ E_{ij} &\sim 0.5 * t_5 \end{aligned} \]
  3. A Poisson model: \[ X = \text{Poisson(exp(LF'))} \] In this case, I fit flash to the transformed data matrix \[ Y = \log(X + 1) \]
  4. A Poisson-lognormal model: \[ \begin{aligned} X &= \text{Poisson(exp(LF' + E))} \\ E_{ij} &\sim N(0, 0.5^2) \\ Y &= \log(X + 1) \end{aligned} \]

The code used to run the simulations can be viewed here.

Results

As expected, flash selects an approximately correct number of factors when the model is correctly specified (recall that I fix \(k = 10\)) and when there is a sufficient amount of data. Under model misspecification, however, it is unable to do so. The behavior under Poisson and Poisson-lognormal noise is similar: we see the number of factors continue to increase, apparently without bound, as the size of the dataset gets larger. The behavior under \(t\)-noise is more erratic.

library(flashier)
library(ggplot2)

res <- readRDS("./output/misspec_k/misspec_k.rds")
ggplot(res, aes(x = n, y = nfactors, col = n)) +
  geom_boxplot() + geom_hline(aes(yintercept = 10), linetype = "dashed") + 
  facet_wrap(~distn) + theme(legend.position = "none")


sessionInfo()
#> R version 3.5.3 (2019-03-11)
#> Platform: x86_64-apple-darwin15.6.0 (64-bit)
#> Running under: macOS Mojave 10.14.6
#> 
#> 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] ggplot2_3.2.0  flashier_0.2.2
#> 
#> loaded via a namespace (and not attached):
#>  [1] Rcpp_1.0.1        pillar_1.3.1      compiler_3.5.3   
#>  [4] git2r_0.25.2      workflowr_1.2.0   iterators_1.0.10 
#>  [7] tools_3.5.3       digest_0.6.18     tibble_2.1.1     
#> [10] evaluate_0.13     gtable_0.3.0      lattice_0.20-38  
#> [13] pkgconfig_2.0.2   rlang_0.3.1       Matrix_1.2-15    
#> [16] foreach_1.4.4     yaml_2.2.0        parallel_3.5.3   
#> [19] ebnm_0.1-24       xfun_0.6          withr_2.1.2      
#> [22] dplyr_0.8.0.1     stringr_1.4.0     knitr_1.22       
#> [25] fs_1.2.7          tidyselect_0.2.5  rprojroot_1.3-2  
#> [28] grid_3.5.3        glue_1.3.1        R6_2.4.0         
#> [31] rmarkdown_1.12    mixsqp_0.2-4      purrr_0.3.2      
#> [34] ashr_2.2-38       magrittr_1.5      whisker_0.3-2    
#> [37] backports_1.1.3   scales_1.0.0      codetools_0.2-16 
#> [40] htmltools_0.3.6   MASS_7.3-51.1     assertthat_0.2.1 
#> [43] colorspace_1.4-1  labeling_0.3      stringi_1.4.3    
#> [46] lazyeval_0.2.2    doParallel_1.0.14 pscl_1.5.2       
#> [49] munsell_0.5.0     truncnorm_1.0-8   SQUAREM_2017.10-1
#> [52] crayon_1.3.4