Last updated: 2021-07-01

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Knit directory: ebpmf_data_analysis/

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    Modified:   script/fit_kos_NMF_F.R
    Modified:   topicView-app/app.R

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Model

\[\begin{align} & X_{ij} \sim \mathrm{Pois}(\lambda_{ij})\\ & \lambda_{ij} = l_{i0} f_{j0} \sum_k w_k l_{ik} f_{jk}\\ & l_{ik} \sim g_{L, k}(.), f_{jk} \sim g_{F, k}(.)\\ & g_{L, k}(.), g_{F,k}(.) \in \mathcal{G} \end{align}\]

  • If we introduce hidden variable \(Z\): \(X_{ij} = \sum_k Z_{ijk}\), we can interpret the mean of \(Z_{ijk}\) as the product of the background frequency (\((l_{i0} f_{j0})\)) and the deviation from that (\(\sum_k w_k l_{ik} f_{jk}\)).

  • If \(w_k = 1/K\) and \(l_{ik} = f_{jk} = 1\) (the \(g\)’s put most mass around \(1\)), we have \(X_{ij} \sim \mathrm{Pois}(l_{i0} f_{j0})\). This is just rank-1 PMF with solution \(l_{i0} \propto \sum_{j} X_{ij} & f_{j0} \propto \sum_{i} X_{ij}\) and \((\sum_i l_{i0}) * (\sum_j f_{j0}) = \sum_{i, j} X_{ij}\).

  • We assume another version of “sparsity” here: for each \(k\), majority of the deviation \(l_{ik}\) would be close to 1; only a small proportion of top documents have larger deviation from background. Same assumption for factors.

  • Interpretation: if cell/document \(i\) is activated in topic/biological process \(k\), then \(l_{ik}\) will be larger than \(1\); if it’s de-activated then \(l_{ik}\) will be smaller than \(1\). Similarly, if word/gene \(j\) is overexpressed in topic/biological process \(k\), then \(f_{jk}\) will be larger than \(1\); if it’s less-expressed then \(f_{jk}\) will be smaller than \(1\).

  • We impose this assumption through \(\mathcal{G}\), dicussed in the next section

The algorithm for solving the model can be seen algorithm

Prior family \(\mathcal{G}\)

\[\begin{align} \mathcal{G} = \{g: g(.) = \sum_{l} \pi_l \mathrm{Ga}(.; 1/\phi_l, 1/\phi_l)\} \end{align}\] For a random variable \(X \sim \mathrm{Ga}(.; 1/\phi_l, 1/\phi_l)\), we have \[\begin{align} & \mathrm{E}(X) = 1\\ & \mathrm{Var}(X) = \phi\\ & \mathrm{Mode}(X) = max(0, 1 - \phi) \end{align}\]

Below I show the density for different \(\phi\)s. Note that the first one is a bit deceptive about the tail.

rm(list = ls())
par(las=1)
phis <- c(0.01, 0.9, 1, 10)
cols <- c("black", "red", "blue","green")

for(i in 1:length(phis)){
  if(i != 1){
    par(new = TRUE)
    plot(function(x) (dgamma(x, shape = 1/phis[i], rate = 1/phis[i])), 
     from = 0.001, to = 10, ylab = "dgamma", col = cols[i], axes = FALSE )
  }else{
    plot(function(x) (dgamma(x, shape = 1/phis[i], rate = 1/phis[i])), 
     from = 0.001, to = 10, ylab = "dgamma", col = cols[i])
  }
}
legend("topright", legend = paste("phi = ", phis), col = cols, lty=1, cex=0.8)

Version Author Date
4e731f3 zihao12 2020-10-26

look at the tail

for(i in 1:length(phis)){
  if(i != 1){
    par(new = TRUE)
    plot(function(x) (dgamma(x, shape = 1/phis[i], rate = 1/phis[i])), 
     from = 5, to = 20, ylab = "dgamma", col = cols[i], axes = FALSE )
  }else{
    plot(function(x) (dgamma(x, shape = 1/phis[i], rate = 1/phis[i])), 
     from = 5, to = 20, ylab = "dgamma", col = cols[i])
  }
}
legend("topright", legend = paste("phi = ", phis), col = cols, lty=1, cex=0.8)

Version Author Date
4e731f3 zihao12 2020-10-26
  • when \(\phi < 1\) it is concentrated near 1;
  • when \(\phi > 1\) it favors points near \(0\), as well as larger points (compared to other smaller \(\phi1\)).

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.15.7

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] workflowr_1.6.2

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.5      rprojroot_1.3-2 digest_0.6.25   later_1.1.0.1  
 [5] R6_2.4.1        backports_1.1.7 git2r_0.26.1    magrittr_1.5   
 [9] evaluate_0.14   stringi_1.6.2   rlang_0.4.5     fs_1.3.1       
[13] promises_1.1.1  whisker_0.3-2   rmarkdown_2.1   tools_3.5.1    
[17] stringr_1.4.0   glue_1.4.1      httpuv_1.5.4    xfun_0.8       
[21] yaml_2.2.0      compiler_3.5.1  htmltools_0.5.0 knitr_1.28