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Rmd 170c0bc Matthew Stephens 2020-10-09 workflowr::wflow_publish(“nmf_simple_examples.Rmd”) 
library(NNLM)
library("flashier")
library("magrittr")

Introduction

I wanted to try some simple non-negative covariance examples, to assess challenges of getting convergence.

Simple 3-factor case

I set up a covariance matrix with 3 factors (columns of L). I add some very small noise.

set.seed(123)
L=  matrix(0,nrow=100,ncol=3)
L[1:50,1] = 1
L[51:100,2] = 1
L[26:75,3] = 1
S = L %*% t(L) + rnorm(100*100,0,0.01)
image(S)

Version Author Date
ef937bb Matthew Stephens 2020-10-09

SVD

Start with svd: you can see the PCs kind of pick up the three factors, although not exactly of course (SVD is not non-negative….) So this should be a relatively easy case.

S.svd = svd(S)
par(mfcol=c(1,3))
plot(S.svd$u[,1],main='first eigenvector')
plot(S.svd$u[,2],main='second eigenvector')
plot(S.svd$u[,3],main='third eigenvector')

Version Author Date
ef937bb Matthew Stephens 2020-10-09

NMF

Try non-negative matrix factorization. It works well here.

S.nnmf = nnmf(S,k=3)
par(mfcol=c(1,3))
plot(S.nnmf$W[,1])
plot(S.nnmf$W[,2])
plot(S.nnmf$W[,3])

Version Author Date
ef937bb Matthew Stephens 2020-10-09

Harder 9-factor case

Negligible noise

Now I do 9 non-negative factors, each having 20 positive entries (out of 100).

K=9
set.seed(1)
L2 = matrix(0,nrow=100,ncol=K)
for(i in 1:K){L2[sample(1:100,20),i]=1}
S2 = L2 %*% t(L2) +rnorm(100*100,0,0.01)
image(S2)

Version Author Date
ef937bb Matthew Stephens 2020-10-09

SVD

Do svd. We see the rank 10 structure clearly, but the actual factors are clearly now all mixed up among the PCs.

S2.svd = svd(S2)
plot(S2.svd$d[1:30],main="eigenvalues")

Version Author Date
ef937bb Matthew Stephens 2020-10-09 
par(mfcol=c(1,3))
plot(S2.svd$u[,1],main='first eigenvector')
plot(S2.svd$u[,2],main='second eigenvector')
plot(S2.svd$u[,3],main='third eigenvector')

Version Author Date
ef937bb Matthew Stephens 2020-10-09

NMF

NMF – slightly suprisingly to me it looks great! (I did give it the right K)

S2.nnmf = nnmf(S2,k=K)
# for each column of W find the best matching column in L
get_bestmatch = function(L,W){
  LW.c = (cor(L,W)) # finds correlation between columns of L and W
  bestmatch = rep(0, ncol(W))
  for(i in 1:ncol(W)){
    bestmatch[i] = which.max(LW.c[,i])
  }
  return(bestmatch)
}

bm = get_bestmatch(L2,S2.nnmf$W)
par(mfcol=c(3,3),mai=rep(0.25,4))
for(i in 1:K){
  plot(L2[,bm[i]],S2.nnmf$W[,i], main="True L vs Estimate")
}

Version Author Date
ef937bb Matthew Stephens 2020-10-09

Higher noise (NMF)

Try adding some noise and running NMF. Now the results are (as expected) less clean.

S2n = S2+rnorm(100*100,0,1)
S2n.nnmf = nnmf(S2n,k=K)

bm = get_bestmatch(L2,S2n.nnmf$W)
par(mfcol=c(3,3),mai=rep(0.25,4))
for(i in 1:K){
  plot(L2[,bm[i]],S2n.nnmf$W[,i], main="True L vs Estimate")
}

Version Author Date
ef937bb Matthew Stephens 2020-10-09

Here I see if I can improve performance by using EB shrinkage methods. (Could also try penalties?). In this one I just use a non-negative prior (not 0-1).

S2n.f <- flash.init(S2n) %>% flash.init.factors(list(S2n.nnmf$W,t(S2n.nnmf$H)),prior.family = prior.nonnegative()) %>% flash.backfit()
Backfitting 9 factors (tolerance: 1.49e-04)...
  Difference between iterations is within 1.0e+01...
  Difference between iterations is within 1.0e+00...
  Difference between iterations is within 1.0e-01...
  Difference between iterations is within 1.0e-02...
  Difference between iterations is within 1.0e-03...
Wrapping up...
Done.
for(i in 1:K){
  plot(L2[,bm[i]],S2n.f$loadings.pm[[1]][,i], main="True L vs Estimate")
}

Version Author Date
ef937bb Matthew Stephens 2020-10-09

Although the plots do not look so different, the EB shrinkage ddoes consistently improves the correlations between the true values and the estimates:

cors = matrix(nrow=K, ncol=2)
colnames(cors) = c("flash","nnmf")
for(i in 1:K){
  cors[i,] = (c(cor(L2[,bm[i]],S2n.f$loadings.pm[[1]][,i]), cor(L2[,bm[i]],S2n.nnmf$W[,i])))
}
print(cors,digits=2)
      flash nnmf
 [1,]  0.96 0.91
 [2,]  0.95 0.92
 [3,]  0.95 0.90
 [4,]  0.98 0.94
 [5,]  0.95 0.90
 [6,]  0.95 0.89
 [7,]  0.96 0.92
 [8,]  0.96 0.93
 [9,]  0.93 0.88

Tree-like case

We have found some challenges with tree-like case, so we try that here. This simulates a symmetric 4-tip tree (6 branches total), with a factor for each branch.

set.seed(1)
L3 = matrix(0,nrow=100,ncol=6)
L3[1:50,1] = 1 #top split L
L3[51:100,2] = 1 # top split R
L3[1:25,3]  = 1
L3[26:50,4] = 1
L3[51:75,5] = 1
L3[76:100,6] = 1
S3 = L3 %*% t(L3) +rnorm(100*100,0,0.01)
image(S3)

The results confirm that finding the representation that we want (in which each factor represents a branch) is not achieved by off-the-shelf methods. This is essentially because many factors (in the branch represenation) are linearly dependent with one another.

Here, many of the estimated factors from NMF correlate most with the top split (factor 1 or 2) and have partial memberships that capture some of the lower splits.

S3.nnmf = nnmf(S3,k=7)
bm = get_bestmatch(L3,S3.nnmf$W)
par(mfcol=c(3,3),mai=rep(0.25,4))
for(i in 1:7){
  plot(L3[,bm[i]],S3.nnmf$W[,i], main=paste0("True L (",bm[i], ") vs Estimate"))
}


sessionInfo()
R version 3.6.0 (2019-04-26)
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/3.6/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] magrittr_1.5   flashier_0.2.7 NNLM_0.4.4    

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.5       pillar_1.4.6     compiler_3.6.0   later_1.1.0.1   
 [5] git2r_0.27.1     workflowr_1.6.2  tools_3.6.0      digest_0.6.25   
 [9] evaluate_0.14    lifecycle_0.2.0  tibble_3.0.3     lattice_0.20-41 
[13] pkgconfig_2.0.3  rlang_0.4.7      Matrix_1.2-18    rstudioapi_0.11 
[17] parallel_3.6.0   yaml_2.2.1       ebnm_0.1-24      xfun_0.16       
[21] invgamma_1.1     stringr_1.4.0    knitr_1.29       fs_1.4.2        
[25] vctrs_0.3.4      rprojroot_1.3-2  grid_3.6.0       glue_1.4.2      
[29] R6_2.4.1         rmarkdown_2.3    mixsqp_0.3-43    irlba_2.3.3     
[33] ashr_2.2-51      whisker_0.4      backports_1.1.10 promises_1.1.1  
[37] ellipsis_0.3.1   htmltools_0.5.0  httpuv_1.5.4     stringi_1.4.6   
[41] truncnorm_1.0-8  SQUAREM_2020.3   crayon_1.3.4