Last updated: 2023-09-12

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Rmd 7be2eae DongyueXie 2023-09-12 wflow_publish("analysis/multiplicative_additive.Rmd")

Introduction

I want to find a simple example to illustrate the difference between multiplicative and additive methods.

The multiplicative effects come with the log link in Poisson model can be understood as follows:

Let \(\boldsymbol{x}_i\) denote the a data vector, then we can write \(\mathbb{E}(\boldsymbol{x}_i) = \exp(\boldsymbol{\mu} + \sum_kl_{ik}\boldsymbol{f}_k)=\exp(\boldsymbol{\mu})\prod_k \exp(l_{ik}\boldsymbol{f}_k).\)

Here I start with a very simple example where the two factors look like:

\(\boldsymbol{f}_1 = (1,1,1,1,0,0,0,0), \boldsymbol{f}_2=(1,1,0,0,1,1,0,0)\).

So for the \(i\)th sample, its mean is \(background\times \exp(l_{i1}\boldsymbol{f}_1)\times \exp(l_{i2}\boldsymbol{f}_2)\).

WLOG let’s set background to be all 1’s. The observations(elements) of \(i\)th sample would be \(\exp(l_{i1})\exp(l_{i2}),\exp(l_{i1}),\exp(l_{i2}),1\).

Let’s set the following loadings:

  1. \(l_1=l_2=0\); 2. \(l_1=2, l_2=0\); 3. \(l_1=0,l_2=2\); 4. \(l_1=4,l_2=2\).

The corresponding Poisson rates are

\((1,1,1,1),(\exp(2),\exp(2),1,1),(\exp(2),1,\exp(2),1), (\exp(6),\exp(4),\exp(2),1)\).

N = 100
Ftrue = cbind(c(1,1,1,1,0,0,0,0),c(1,1,0,0,1,1,0,0))
set.seed(12345)
#Ltrue = matrix(rexp(N*2,1),nrow=N)
Ltrue = cbind(rep(c(0,2,0,4),each=N/4),rep(c(0,0,2,2),each=N/4))
mu = tcrossprod(Ltrue,Ftrue)
lambda = exp(mu)
Y = matrix(rpois(N*nrow(Ftrue),lambda),nrow=N)

Ftrue
     [,1] [,2]
[1,]    1    1
[2,]    1    1
[3,]    1    0
[4,]    1    0
[5,]    0    1
[6,]    0    1
[7,]    0    0
[8,]    0    0
plot(Ltrue[,1])

plot(Ltrue[,2])

Can NMF recover Ltrue and Ftrue, if we directly apply NMF on \(LF\)?

nmf_fit = NNLM::nnmf(mu,k=2,loss='mse')
plot(nmf_fit$H[1,])

plot(nmf_fit$H[2,])

Looks good.

How about running EBNMF on \(LF\)? It is not working because “The data matrix must not have any rows or columns whose entries are either identically zero or all missing.”

ebnmf_fit = flashier::flash(mu,ebnm_fn = c(ebnm::ebnm_unimodal_nonnegative,ebnm::ebnm_unimodal_nonnegative),var_type = 0,backfit = T)

Now we fit NMF to the data matrix Y:

Let’s start with k=2

fit0 = fastTopics::fit_poisson_nmf(Y,2)
Initializing factors using Topic SCORE algorithm.
Initializing loadings by running 10 SCD updates.
Fitting rank-2 Poisson NMF to 100 x 8 dense matrix.
Running 100 SCD updates, without extrapolation (fastTopics 0.6-142).
plot(fit0$F[,1])

plot(fit0$F[,2])

plot(fit0$L[,1])

plot(fit0$L[,2])

Then k = 3

fit0 = fastTopics::fit_poisson_nmf(Y,3)
Initializing factors using Topic SCORE algorithm.
Warning in value[[3L]](cond): Topic SCORE failure occurred; falling back to
init.method == "random"
Topic SCORE failure occurred; using random initialization instead.
Fitting rank-3 Poisson NMF to 100 x 8 dense matrix.
Running 100 SCD updates, without extrapolation (fastTopics 0.6-142).
plot(fit0$F[,1])

plot(fit0$F[,2])

plot(fit0$F[,3])

plot(fit0$L[,1])

plot(fit0$L[,2])

plot(fit0$L[,3])

Now we try to fit ebnmf on log transformed data:

fit2 = flashier::flash(log(1+Y),ebnm_fn = c(ebnm::ebnm_point_exponential,ebnm::ebnm_point_exponential),var_type = 0,backfit = T)
Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Adding factor 4 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 3 factors (tolerance: 1.19e-05)...
  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...
  Difference between iterations is within 1.0e-04...
  Difference between iterations is within 1.0e-05...
Wrapping up...
Done.
Nullchecking 3 factors...
Done.
plot(fit2$F_pm[,1])

plot(fit2$F_pm[,2])

plot(fit2$F_pm[,3])

plot(fit2$L_pm[,1])

plot(fit2$L_pm[,2])

plot(fit2$L_pm[,3])

fit2 = flashier::flash(log(0.5+Y),ebnm_fn = c(ebnm::ebnm_point_exponential,ebnm::ebnm_point_exponential),var_type = 0,backfit = T)
Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 2 factors (tolerance: 1.19e-05)...
  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...
  Difference between iterations is within 1.0e-04...
  Difference between iterations is within 1.0e-05...
Wrapping up...
Done.
Nullchecking 2 factors...
Done.
plot(fit2$F_pm[,1])

plot(fit2$F_pm[,2])

fit1 = ebpmf::ebpmf_log(Y,l0=0,f0=0,var_type = 'constant',
                        flash_control = list(fix_l0=T,fix_f0=T,ebnm.fn=c(ebnm::ebnm_unimodal_nonnegative,ebnm::ebnm_unimodal_nonnegative),
                                                                         loadings_sign=1,factors_sign =1))
Initializing
Solving VGA constant...For large matrix this may require large memory usage
Running initial EBMF fit
Running iterations...
iter 10, avg elbo=-2.6956, K=4
iter 20, avg elbo=-2.65927, K=4
iter 30, avg elbo=-2.6475, K=4
iter 40, avg elbo=-2.63747, K=4
iter 50, avg elbo=-2.62896, K=4

sessionInfo()
R version 4.1.0 (2021-05-18)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: CentOS Linux 7 (Core)

Matrix products: default
BLAS:   /software/R-4.1.0-no-openblas-el7-x86_64/lib64/R/lib/libRblas.so
LAPACK: /software/R-4.1.0-no-openblas-el7-x86_64/lib64/R/lib/libRlapack.so

locale:
 [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C         LC_TIME=C           
 [4] LC_COLLATE=C         LC_MONETARY=C        LC_MESSAGES=C       
 [7] LC_PAPER=C           LC_NAME=C            LC_ADDRESS=C        
[10] LC_TELEPHONE=C       LC_MEASUREMENT=C     LC_IDENTIFICATION=C 

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] workflowr_1.6.2

loaded via a namespace (and not attached):
  [1] mcmc_0.9-7         bitops_1.0-7       matrixStats_0.59.0
  [4] fs_1.5.0           progress_1.2.2     httr_1.4.5        
  [7] rprojroot_2.0.2    tools_4.1.0        bslib_0.4.2       
 [10] utf8_1.2.3         R6_2.5.1           irlba_2.3.5.1     
 [13] uwot_0.1.14        lazyeval_0.2.2     colorspace_2.1-0  
 [16] withr_2.5.0        wavethresh_4.7.2   tidyselect_1.2.0  
 [19] prettyunits_1.1.1  ebpm_0.0.1.3       compiler_4.1.0    
 [22] git2r_0.28.0       glmnet_4.1-2       cli_3.6.1         
 [25] quantreg_5.94      SparseM_1.81       plotly_4.10.1     
 [28] horseshoe_0.2.0    sass_0.4.0         smashrgen_1.2.5   
 [31] caTools_1.18.2     flashier_0.2.51    scales_1.2.1      
 [34] mvtnorm_1.1-2      SQUAREM_2021.1     quadprog_1.5-8    
 [37] pbapply_1.7-0      mixsqp_0.3-48      stringr_1.5.0     
 [40] digest_0.6.31      rmarkdown_2.9      MCMCpack_1.6-3    
 [43] deconvolveR_1.2-1  vebpm_0.4.8        pkgconfig_2.0.3   
 [46] htmltools_0.5.4    ebpmf_2.3.2        fastTopics_0.6-142
 [49] fastmap_1.1.0      invgamma_1.1       highr_0.9         
 [52] htmlwidgets_1.6.1  rlang_1.1.1        rstudioapi_0.13   
 [55] shape_1.4.6        jquerylib_0.1.4    generics_0.1.3    
 [58] jsonlite_1.8.4     dplyr_1.1.0        magrittr_2.0.3    
 [61] smashr_1.3-6       Matrix_1.5-3       Rcpp_1.0.10       
 [64] munsell_0.5.0      fansi_1.0.4        RcppZiggurat_0.1.6
 [67] lifecycle_1.0.3    stringi_1.6.2      whisker_0.4       
 [70] yaml_2.3.7         MASS_7.3-54        Rtsne_0.16        
 [73] grid_4.1.0         parallel_4.1.0     promises_1.2.0.1  
 [76] ggrepel_0.9.3      crayon_1.5.2       lattice_0.20-44   
 [79] cowplot_1.1.1      splines_4.1.0      hms_1.1.2         
 [82] knitr_1.33         pillar_1.8.1       softImpute_1.4-1  
 [85] codetools_0.2-18   glue_1.6.2         evaluate_0.14     
 [88] trust_0.1-8        data.table_1.14.8  RcppParallel_5.1.7
 [91] foreach_1.5.1      nloptr_1.2.2.2     vctrs_0.6.2       
 [94] httpuv_1.6.1       MatrixModels_0.5-1 gtable_0.3.1      
 [97] purrr_1.0.1        ebnm_1.0-54        tidyr_1.3.0       
[100] ashr_2.2-54        cachem_1.0.5       ggplot2_3.4.1     
[103] xfun_0.24          Rfast_2.0.7        NNLM_0.4.4        
[106] coda_0.19-4        later_1.3.0        mr.ash_0.1-87     
[109] survival_3.2-11    viridisLite_0.4.1  truncnorm_1.0-8   
[112] tibble_3.2.1       iterators_1.0.13   ellipsis_0.3.2