Nonnegative FLASH example

Introduction

Nonnegative matrix factorization is straightforward in FLASH; we need only put a class of nonnegative priors on the factors and loadings. In ASH, the +uniform priors constitute such a class (and, at present, this is the only such class in ASH).

As an example of nonnegative matrix factorization via FLASH, I analyze the GTEx donation matrix. This matrix consists of 44 rows, each of which corresponds to a tissue or cell type (liver, lung, cortex, whole blood, etc.), and 571 columns, each of which corresponds to a donor. Note that a given donor can contribute multiple samples of a given tissue.

In this analysis, I ignore the fact that “errors” are not, as FLASH assumes, Gaussian. I will address this simplification in subsequent analyses (see here and here).

Data

First I set up the donation matrix. As in previous analyses, I use GTEx v6 data rather than the more recent v7 or v8 data.

raw <- read.csv("https://storage.googleapis.com/gtex_analysis_v6/annotations/GTEx_Data_V6_Annotations_SampleAttributesDS.txt",
                header=TRUE, sep='\t')

data <- raw[, c("SAMPID", "SMTSD")] # sample ID, tissue type
# Extract donor ID:
tmp <- strsplit(as.character(data$SAMPID), "-")
data$SAMPID <- as.factor(sapply(tmp, function(x) {x[[2]]})) 
names(data) <- c("DonorID", "TissueType")

data <- suppressMessages(reshape2::acast(data, TissueType ~ DonorID))

missing.tissues <- c(1, 8, 9, 20, 21, 24, 26, 27, 33, 36, 39)
data <- data[-missing.tissues, ]

gtex.colors <- read.table("https://github.com/stephenslab/gtexresults/blob/master/data/GTExColors.txt?raw=TRUE",
                          sep = '\t', comment.char = '')
gtex.colors <- gtex.colors[-c(7, 8, 19, 20, 24, 25, 31, 34, 37), 2]
gtex.colors <- as.character(gtex.colors)

As a preliminary step, it is useful to visualize the correlation matrix. I use Kushal Dey’s CorShrink package, which outputs the empirical correlation matrix and shrunken estimates of the “true” correlations.

tmp <- capture.output(
  CorShrink::CorShrinkData(t(data), sd_boot = TRUE, image = "both",
                           image.control = list(tl.cex = 0.25))
)

Using sd_boot as value column: use value.var to override.

Initialization function

To obtain nonnegative factor/loading pairs using FLASH, we can likely do better than to use the default SVD-type initialization. Here I use the nonnegative matrix factorization implemented in package NNLM.

udv_nn = function(Y, K = 1) {
  tmp = NNLM::nnmf(Y, K, verbose = FALSE)
  return(list(d = rep(1, K), u = tmp$W, v = t(tmp$H)))
}

Greedy loadings

One round of greedily adding factors and then backfitting produces five factor/loading pairs, but repeated rounds yield additional factor/loadings. Here I do three rounds of fitting. I obtain the following loadings:

devtools::load_all("~/GitHub/flashr")

Loading flashr

devtools::load_all("~/GitHub/ebnm")

Loading ebnm

ebnm_fn = "ebnm_ash"
ebnm_param = list(mixcompdist = "+uniform", warmstart = TRUE)

run_flash_once <- function(f_init) {
  flash(data, f_init = f_init, 
        ebnm_fn = ebnm_fn, ebnm_param = ebnm_param,
        var_type="constant", init_fn = udv_nn, 
        backfit = TRUE, verbose = FALSE)
}

fl_g <- run_flash_once(f_init = NULL)
fl_g <- run_flash_once(f_init = fl_g)
fl_g <- run_flash_once(f_init = fl_g)
fl_g$objective

[1] -15156.65

plot(fl_g, plot_loadings = TRUE, loading_colors = gtex.colors,
     loading_legend_size = 3, 
     plot_grid_nrow = 4, plot_grid_ncol = 3)

Backfitted loadings

Next I add 12 factors at once using NNLM and then backfit. Note that the objective obtained using this method is much better than the above (by approximately 500).

fl_nnmf <- flash_add_factors_from_data(data, 12,
                                       init_fn = udv_nn, 
                                       backfit = FALSE)

fl_b <- flash_backfit(data, fl_nnmf,
                      ebnm_fn = ebnm_fn,
                      ebnm_param = ebnm_param,
                      var_type = "constant",
                      verbose = FALSE)
fl_b$objective

[1] -14666.36

plot(fl_b, plot_loadings = TRUE, loading_colors = gtex.colors,
     loading_legend_size = 3, 
     plot_grid_nrow = 4, plot_grid_ncol = 3)

NNLM loadings

The backfitted loadings can be compared with the loadings that are obtained by simply running NNLM. Results are quite similar.

plot(fl_nnmf, plot_loadings=TRUE, loading_colors = gtex.colors,
     loading_legend_size = 3, 
     plot_grid_nrow = 4, plot_grid_ncol = 3)

Discussion

The loading that accounts for the largest PVE (backfitted loading 3) puts a large weight on whole blood and a smaller weight on fibroblasts. In effect, there are many more whole blood samples than samples from any other tissue (1822 versus, on average, 224).

The loading with the second-largest PVE (loading 1) shares weights among brain tissues, with a smaller weight on pituitary tissues. FLASH finds a very similar loading in the GTEx summary data (loading 2 here, for example).

Loading 5 puts large weights on female reproductive tissues (ovary, uterus, and vagina), so it is most likely a sex-specific loading. Similarly, loading 4 puts its largest weights on testicular and prostate tissues. Both of these loadings put very similar weights on non-reproductive tissues (observe, in particular, the similarity of weights among gastro-intestinal tissues), so we should not make too much of correlations between reproductive and non-reproductive tissues. It is probably the case that the two sex-specific loadings are entangled with a third loading that is more properly weighted on gastro-intestinal tissues alone.

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] ebnm_0.1-15  flashr_0.6-2

loaded via a namespace (and not attached):
 [1] softImpute_1.4      NNLM_0.4.2          ashr_2.2-13        
 [4] reshape2_1.4.3      corrplot_0.84       lattice_0.20-35    
 [7] Rmosek_7.1.3        colorspace_1.3-2    testthat_2.0.0     
[10] htmltools_0.3.6     yaml_2.1.17         rlang_0.2.0        
[13] R.oo_1.21.0         pillar_1.2.1        withr_2.1.1.9000   
[16] R.utils_2.6.0       REBayes_1.2         foreach_1.4.4      
[19] plyr_1.8.4          stringr_1.3.0       munsell_0.4.3      
[22] commonmark_1.4      gtable_0.2.0        workflowr_1.0.1    
[25] R.methodsS3_1.7.1   devtools_1.13.4     codetools_0.2-15   
[28] memoise_1.1.0       evaluate_0.10.1     labeling_0.3       
[31] knitr_1.20          pscl_1.5.2          doParallel_1.0.11  
[34] parallel_3.4.3      Rcpp_0.12.18        corpcor_1.6.9      
[37] backports_1.1.2     scales_0.5.0        truncnorm_1.0-8    
[40] gridExtra_2.3       ggplot2_2.2.1       digest_0.6.15      
[43] stringi_1.1.6       grid_3.4.3          rprojroot_1.3-2    
[46] tools_3.4.3         magrittr_1.5        lazyeval_0.2.1     
[49] glmnet_2.0-13       CorShrink_0.1-6     tibble_1.4.2       
[52] whisker_0.3-2       MASS_7.3-48         Matrix_1.2-12      
[55] SQUAREM_2017.10-1   xml2_1.2.0          assertthat_0.2.0   
[58] rmarkdown_1.8       roxygen2_6.0.1.9000 iterators_1.0.9    
[61] R6_2.2.2            git2r_0.21.0        compiler_3.4.3