Last updated: 2021-08-29

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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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Introduction

Suppose now we are given count matrix \(X_{ij}\) and subject indicator \(u_i\): the expression of gene \(j\) in cell \(i\) is \(x_{ij}\) and we know cell \(i\) is from subject \(u_i\). It’s reasonable to assume that cells from the same subject \(i\) are correlated, and that correlation corrupt the biologically meaningful low-rank structure in the mean matrix. Below we model it using background model.

Here the model is

\[\begin{align} & X_{ij} \sim \text{Pois}(\lambda_{ij})\\ & \lambda_{ij} = m_{j u_i} \sum_k l_{ik} f_{jk} \end{align}\]

Here \(m_{jd}\) is the background frequency of gene \(j\) in subject \(d\). I implemented MLE of this model in ebpmf.alpha::pmf_subject using EM algorithm. I also implemented a more complicated empirical bayes version (using the MLE algorithm for its initialization):

\[\begin{align} & X_{ij} \sim \text{Pois}(\lambda_{ij})\\ & \lambda_{ij} = m_{j u_i} \sum_k w_k l_{ik} f_{jk}\\ & m_{jd} \sim \text{Ga}(\alpha^m_{d}, \beta^m_{d})\\ & l_{ik} \sim \text{Ga}(\alpha^l_{k}, \beta^l_{k})\\ & f_{jk} \sim \sum_c \pi^f_{kc} \text{Ga}(a_c, a_c) \end{align}\]

Example

knitr::opts_chunk$set(warning = FALSE, message = FALSE) 
library(ebpmf.alpha)
set.seed(123)
n = 100
p = 50
K = 3

## simulate data
L = matrix(100*runif(n*K), ncol = K)
F = matrix(runif(p*K), ncol = K)
m = matrix(runif(p*2), nrow = p)
u = c(replicate(n/2, 1), replicate(n/2, 2))
lam = L %*% t(F)
lam[u == 1,] = t(t(lam[u == 1,]) * m[,1])
lam[u == 2,] = t(t(lam[u == 2,]) * m[,2])
X = matrix(rpois(n = n*p, lambda = lam), nrow = n)

Fit with MLE

fit = ebpmf.alpha::pmf_subject(X = X, u = u, K = K, maxiter = 100)

Fit with EB

init = ebpmf.alpha::initialize_wbg2_subject_from_LFm(L = fit$L, F = fit$F, m = fit$m)
fit.eb = ebpmf.alpha::ebpmf_wbg2_subject(X = X, u = u, K = K, init = init, maxiter = 500)
par(mfrow = c(1,2))
plot(m, fit$m, ylab = "mhat", main = "MLE")
plot(m, fit.eb$m$mean, ylab = "mhat", main = "EB")

Version Author Date
88f9feb zihao12 2021-08-29 
par(mfrow = c(1,2))
plot(fit$log_liks, main = "MLE")
plot(fit.eb$ELBO, main = "EB")

Version Author Date
88f9feb zihao12 2021-08-29 

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] ebpmf.alpha_0.6.0 workflowr_1.6.2  

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.5         knitr_1.28         whisker_0.3-2      magrittr_1.5      
 [5] ebpm_0.0.1.3       lattice_0.20-38    R6_2.4.1           rlang_0.4.5       
 [9] stringr_1.4.0      tools_3.5.1        grid_3.5.1         xfun_0.8          
[13] irlba_2.3.3        git2r_0.26.1       matrixStats_0.54.0 htmltools_0.5.0   
[17] yaml_2.2.0         digest_0.6.25      rprojroot_1.3-2    mixsqp_0.3-46     
[21] Matrix_1.2-17      later_1.1.0.1      promises_1.1.1     fs_1.3.1          
[25] glue_1.4.1         evaluate_0.14      rmarkdown_2.1      stringi_1.6.2     
[29] compiler_3.5.1     backports_1.1.7    httpuv_1.5.4