Last updated: 2018-08-20

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library(mashr)
Loading required package: ashr
library(knitr)
library(kableExtra)
source('../code/estimate_cor.R')
source('../code/generateDataV.R')
source('../code/summary.R')

EM

We use EM algorithm to update V.

B is the \(n\times p\) true value matrix. \(\mathbf{z}\) is a length n vector.

E step

\[ P(\hat{B},B,\mathbf{z}|\rho, \pi) = \prod_{i=1}^{n} \prod_{k=0}^{K}\left[\pi_{k}N(\hat{b}_{i}; b_{i}, V)N(b_{i}; 0, U_{k})\right]^{\mathbb{I}(z_{i}=k)} \]

\[ \mathbb{E}_{\mathbf{z},B|\hat{B}} \log P(\hat{B},B,\mathbf{z}|\rho, \pi) = \sum_{i=1}^{n} \sum_{k=0}^{K} P(z_{i}=k|\hat{b}_{i})\left[ \log \pi_{k} + \mathbb{E}_{B|\hat{B}}(\log N(\hat{b}_{i}; b_{i}, V)) + \mathbb{E}_{B|\hat{B}}(\log N(b_{i}; 0, U_{k})) \right] \]

\[ \begin{align*} \log N(\hat{b}_{i}; b_{i}, V) + \log N(b_{i}; 0, U_{k}) &= -\frac{p}{2}\log 2\pi -\frac{1}{2}\log |V| - \frac{1}{2}(\hat{b}_{i}-b_{i})^{T}V^{-1}(\hat{b}_{i}-b_{i}) -\frac{p}{2}\log 2\pi -\frac{1}{2}\log |U_{k}| - \frac{1}{2}b_{i}^{T}U_{k}^{-1}b_{i} \\ &= -p\log 2\pi -\frac{1}{2}\log |U_{k}| -\frac{1}{2}\log |V| - \frac{1}{2}\hat{b}_{i}^{T}V^{-1}\hat{b}_{i} + \hat{b}_{i}^{T}V^{-1}b_{i} -\frac{1}{2}b_{i}^{T}V^{-1}b_{i} - \frac{1}{2}b_{i}^{T}U_{k}^{-1}b_{i} \\ \mathbb{E}_{b_{i}|\hat{b}_{i}}\left[ \log N(\hat{b}_{i}; b_{i}, V) + \log N(b_{i}; 0, U_{k}) \right] &= -p\log 2\pi -\frac{1}{2}\log |U_{k}| -\frac{1}{2}\log |V| - \frac{1}{2}\hat{b}_{i}^{T}V^{-1}\hat{b}_{i} + \hat{b}_{i}^{T}V^{-1}\mathbb{E}(b_{i}|\hat{b}_{i}) -\frac{1}{2}tr\left(V^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right) - \frac{1}{2}tr\left(U_{k}^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right) \end{align*} \]

\[ \gamma_{z_{i}}(k) = P(z_{i}=k|X_{i}) = \frac{\pi_{k}N(x_{i}; 0, V+U_{k})}{\sum_{k'=0}^{K}\pi_{k'}N(x_{i}; 0, V + U_{k'})} \]

M step

\(V\): \[ \begin{align*} f(V^{-1}) = \sum_{i=1}^{n} \sum_{k=0}^{K} \gamma_{Z_{i}}(k)\left[ -p\log 2\pi -\frac{1}{2}\log |U_{k}| -\frac{1}{2}\log |V| - \frac{1}{2}\hat{b}_{i}^{T}V^{-1}\hat{b}_{i} + \hat{b}_{i}^{T}V^{-1}\mathbb{E}(b_{i}|\hat{b}_{i}) -\frac{1}{2}tr\left(V^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right) - \frac{1}{2}tr\left(U_{k}^{-1}\mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right) \right] \end{align*} \]

\[ \begin{align*} f(V^{-1})' &= \sum_{i=1}^{n} \sum_{k=0}^{K} \gamma_{Z_{i}}(k)\left[ \frac{1}{2}V - \frac{1}{2}\hat{b}_{i}\hat{b}_{i}^{T} + \mathbb{E}(b_{i}|\hat{b}_{i})\hat{b}_{i}^{T} - \frac{1}{2} \mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right] = 0 \\ \frac{1}{2}Vn &= \sum_{i=1}^{n} \sum_{k=0}^{K} \gamma_{Z_{i}}(k)\left[\frac{1}{2}\hat{b}_{i}\hat{b}_{i}^{T} - \mathbb{E}(b_{i}|\hat{b}_{i})\hat{b}_{i}^{T} + \frac{1}{2} \mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right] \\ \hat{V} &= \frac{1}{n} \sum_{i=1}^{n} \left[\hat{b}_{i}\hat{b}_{i}^{T} - 2\mathbb{E}(b_{i}|\hat{b}_{i})\hat{b}_{i}^{T} + \mathbb{E}(b_{i}b_{i}^{T}|\hat{b}_{i}) \right] \\ &= \frac{1}{n} \sum_{i=1}^{n} \mathbb{E}\left[ (\hat{b}_{i} - b_{i})(\hat{b}_{i} - b_{i})^{T} | \hat{b}_{i}\right] \end{align*} \]

Algorithm:

Input: X, Ulist, init_V
Compute loglikelihood
delta = 1
while delta > tol
  Given rho, Estimate pi using convex method (current mash method)
  M step: update V --> cov2cor(V)
  Compute loglikelihood
  Update delta

Data

\[ \hat{\beta}|\beta \sim N_{2}(\hat{\beta}; \beta, \left(\begin{matrix} 1 & 0.5 \\ 0.5 & 1 \end{matrix}\right)) \]

\[ \beta \sim \frac{1}{4}\delta_{0} + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}\right)) \]

n = 4000

set.seed(1)
n = 4000; p = 2
Sigma = matrix(c(1,0.5,0.5,1),p,p)
U0 = matrix(0,2,2)
U1 = U0; U1[1,1] = 1
U2 = U0; U2[2,2] = 1
U3 = matrix(1,2,2)
Utrue = list(U0=U0, U1=U1, U2=U2, U3=U3)
data = generate_data(n, p, Sigma, Utrue)
m.data = mash_set_data(data$Bhat, data$Shat)
U.c = cov_canonical(m.data)
grid = mashr:::autoselect_grid(m.data, sqrt(2))
Ulist = mashr:::normalize_Ulist(U.c)
xUlist = mashr:::expand_cov(Ulist,grid,usepointmass =  TRUE)
result <- mixture.EMV.times(data$Bhat, xUlist, init_V = list(diag(ncol(data$Bhat)), matrix(c(1,-0.8,-0.8,1),2,2)), grid=1)
plot(result$result$log_liks)

Expand here to see past versions of unnamed-chunk-4-1.png:
Version Author Date
75fbfdb zouyuxin 2018-08-15

The estimated V is

result$result$V
          [,1]      [,2]
[1,] 1.0000000 0.4679947
[2,] 0.4679947 1.0000000
m.data.em = mash_set_data(data$Bhat, data$Shat, V = result$result$V)
U.c = cov_canonical(m.data.em)
m.em = mash(m.data.em, U.c, verbose= FALSE)
null.ind = which(apply(data$B,1,sum) == 0)

The log likelihood is -1.23036710^{4}. There are 24 significant samples, 0 false positives. The RRMSE is 0.58112.

The estimated pi is

barplot(get_estimated_pi(m.em), las=2, cex.names = 0.7, main='EM V', ylim=c(0,0.8))

Expand here to see past versions of unnamed-chunk-7-1.png:
Version Author Date
3612625 zouyuxin 2018-08-15

The ROC curve:

m.data.correct = mash_set_data(data$Bhat, data$Shat, V=Sigma)
m.correct = mash(m.data.correct, U.c, verbose = FALSE)
m.correct.seq = ROC.table(data$B, m.correct)
m.em.seq = ROC.table(data$B, m.em)

Expand here to see past versions of unnamed-chunk-9-1.png:
Version Author Date
3612625 zouyuxin 2018-08-15

Session information

sessionInfo()
R version 3.5.1 (2018-07-02)
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/3.5/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] kableExtra_0.9.0 knitr_1.20       mashr_0.2-11     ashr_2.2-10     

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.18      pillar_1.3.0      compiler_3.5.1   
 [4] git2r_0.23.0      plyr_1.8.4        workflowr_1.1.1  
 [7] R.methodsS3_1.7.1 R.utils_2.6.0     iterators_1.0.10 
[10] tools_3.5.1       digest_0.6.15     viridisLite_0.3.0
[13] tibble_1.4.2      evaluate_0.11     lattice_0.20-35  
[16] pkgconfig_2.0.1   rlang_0.2.1       Matrix_1.2-14    
[19] foreach_1.4.4     rstudioapi_0.7    yaml_2.2.0       
[22] parallel_3.5.1    mvtnorm_1.0-8     xml2_1.2.0       
[25] httr_1.3.1        stringr_1.3.1     REBayes_1.3      
[28] hms_0.4.2         rprojroot_1.3-2   grid_3.5.1       
[31] R6_2.2.2          rmarkdown_1.10    rmeta_3.0        
[34] readr_1.1.1       magrittr_1.5      whisker_0.3-2    
[37] scales_0.5.0      backports_1.1.2   codetools_0.2-15 
[40] htmltools_0.3.6   MASS_7.3-50       rvest_0.3.2      
[43] assertthat_0.2.0  colorspace_1.3-2  stringi_1.2.4    
[46] Rmosek_8.0.69     munsell_0.5.0     doParallel_1.0.11
[49] pscl_1.5.2        truncnorm_1.0-8   SQUAREM_2017.10-1
[52] crayon_1.3.4      R.oo_1.22.0

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