Last updated: 2018-08-20

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

We want to estimate \(\rho\) \[ \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) | \left(\begin{matrix} x \\ y \end{matrix} \right) \sim N(\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) ; \left(\begin{matrix} x \\ y \end{matrix} \right), \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right)) \] \[ \left(\begin{matrix} x \\ y \end{matrix} \right) \sim \sum_{k=0}^{K} \pi_{k} N( \left(\begin{matrix} x \\ y \end{matrix} \right); 0, U_{k} ) \] \(\Rightarrow\) \[ \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) \sim \sum_{k=0}^{K} \pi_{k} N( \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right); 0, \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + U_{k} ) \] \[ \Sigma_{k} = \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + U_{k} = \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix} \right) + \left( \begin{matrix} u_{k11} & u_{k12} \\ u_{k21} & u_{k22} \end{matrix} \right) = \left( \begin{matrix} 1+u_{k11} & \rho+u_{k12} \\ \rho+u_{k21} & 1+u_{k22} \end{matrix} \right) \] Let \(\sigma_{k11} = \sqrt{1+u_{k11}}\), \(\sigma_{k22} = \sqrt{1+u_{k22}}\), \(\phi_{k}=\frac{\rho+u_{k12}}{\sigma_{k11}\sigma_{k22}}\)

MLE

The loglikelihood is (with penalty) \[ l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{k=0}^{K} \pi_{k}N(x_{i}; 0, \Sigma_{k}) + \sum_{k=0}^{K} (\lambda_{k}-1) \log \pi_{k} \]

The penalty on \(\pi\) encourages over-estimation of \(\pi_{0}\), \(\lambda_{k}\geq 1\).

\[ l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{k=0}^{K} \pi_{k}\frac{1}{2\pi\sigma_{k11}\sigma_{k22}\sqrt{1-\phi_{k}^2}} \exp\left( -\frac{1}{2(1-\phi_{k}^2)}\left[ \frac{x_{i}^2}{\sigma_{k11}^2} + \frac{y_{i}^2}{\sigma_{k22}^2} - \frac{2\phi_{k}x_{i}y_{i}}{\sigma_{k11}\sigma_{k22}}\right] \right) + \sum_{k=0}^{K} (\lambda_{k}-1) \log \pi_{k} \]

Note: This probelm is convex with respect to \(\pi\). In terms of \(\rho\), the covenxity depends on the data.

Algorithm:

Input: X, init_pi, init_rho, Ulist
Compute loglikelihood
delta = 1
while delta > tol
  Given pi, estimate rho by max loglikelihood (optim function)
  Given rho, estimate pi by max loglikelihood (convex problem)
  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 <- optimize_pi_rho_times(data$Bhat, xUlist, init_rho = c(-0.7,0,0.7))
Warning in REBayes::KWDual(A, rep(1, k), normalize(w), control = control): estimated mixing distribution has some negative values:
               consider reducing rtol
Warning in mixIP(matrix_lik = structure(c(0.0627889120852815,
0.0114523005348735, : Optimization step yields mixture weights that are
either too small, or negative; weights have been corrected and renormalized
after the optimization.
plot(result[[1]]$loglik)

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Version Author Date
568fbe6 zouyuxin 2018-08-13
faa5b1a zouyuxin 2018-07-26

The estimated \(\rho\) is 0.6124039.

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

The log likelihood is -1.23017710^{4}. There are 55 significant samples, 2 false positives. The RRMSE is 0.5964282.

The estimated pi is

barplot(get_estimated_pi(m.mle), las=2, cex.names = 0.7, main='MLE', ylim=c(0,0.8))

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bf28e18 zouyuxin 2018-08-14
568fbe6 zouyuxin 2018-08-13

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$B, m.correct)
m.mle.seq = ROC.table(data$B, m.mle)

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Version Author Date
568fbe6 zouyuxin 2018-08-13

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] knitr_1.20       kableExtra_0.9.0 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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