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library(mashr)Loading required package: ashrsource('../code/generateDataV.R')
source('../code/summary.R')We use EM algorithm to estimate V.
B is the \(n\times R\) true value matrix. \(\mathbf{z}\) is a length n vector.
\[ p(\hat{\mathbf{B}}, \mathbf{B}, \mathbf{z}) h(\boldsymbol{\pi}) = \prod_{j=1}^{J} \prod_{p = 1}^{P}\left[\pi_{p} N_{R}(\hat{\mathbf{b}}_{j}; \mathbf{b}_{j}, \mathbf{S}_{j}\mathbf{V}\mathbf{S}_{j})N_{R}(\mathbf{b}_{j}; \mathbf{0}, \Sigma_{p})\right]^{\mathbb{I}(z_{j} = p)} \prod_{p=1}^{P} \pi_{p}^{\lambda_{p}-1} \]
\[ \begin{align*} P(z_{j}=p, \mathbf{b}_{j}|\hat{\mathbf{b}}_{j}) &= \frac{P(z_{j}=p, \mathbf{b}_{j},\hat{\mathbf{b}}_{j})}{P(\hat{\mathbf{b}}_{j})} = \frac{P(\hat{\mathbf{b}}_{j}|\mathbf{b}_{j})P(\mathbf{b}_{j}|z_{j}=p) P(z_{j}=p)}{P(\hat{\mathbf{b}}_{j})} \\ &= \frac{\pi_{p} N_{R}(\hat{\mathbf{b}}_{j}; \mathbf{b}_{j}, \mathbf{S}_{j}\mathbf{V}\mathbf{S}_{j})N_{R}(\mathbf{b}_{j}; \mathbf{0}, \Sigma_{p})}{\sum_{p'}\pi_{p'} N_{R}(\hat{\mathbf{b}}_{j}; \mathbf{0}, \mathbf{S}_{j}\mathbf{V}\mathbf{S}_{j} + \Sigma_{p'})} \\ &= \frac{\pi_{p} N_{R}(\hat{\mathbf{b}}_{j}; \mathbf{0}, \mathbf{S}_{j}\mathbf{V}\mathbf{S}_{j} + \Sigma_{p})}{\sum_{p'}\pi_{p'} N_{R}(\hat{\mathbf{b}}_{j}; \mathbf{0}, \mathbf{S}_{j}\mathbf{V}\mathbf{S}_{j} + \Sigma_{p'})} \frac{N_{R}(\hat{\mathbf{b}}_{j}; \mathbf{b}_{j}, \mathbf{S}_{j}\mathbf{V}\mathbf{S}_{j})N_{R}(\mathbf{b}_{j}; \mathbf{0}, \Sigma_{p})}{N_{R}(\hat{\mathbf{b}}_{j}; \mathbf{0}, \mathbf{S}_{j}\mathbf{V}\mathbf{S}_{j} + \Sigma_{p})} \\ &= \gamma_{jp} P(\mathbf{b}_{j}|z_{j}=p, \hat{\mathbf{b}}_{j}) \\ &= P(z_{j}=p|\hat{\mathbf{b}}_{j}) P(\mathbf{b}_{j}|z_{j}=p, \hat{\mathbf{b}}_{j}) \end{align*} \]
E step: \[ \begin{align*} \mathbb{E}_{\mathbf{z}, \mathbf{B}|\hat{\mathbf{B}}}\log p(\hat{\mathbf{B}}, \mathbf{B}, \mathbf{z}) h(\boldsymbol{\pi}) &= \mathbb{E}_{\mathbf{z}, \mathbf{B}|\hat{\mathbf{B}}} \{ \sum_{j=1}^{J}\sum_{p = 1}^{P} \mathbb{I}(z_{j} = p)\left[\log \pi_{p} + \log N_{R}(\hat{\mathbf{b}}_{j}; \mathbf{b}_{j}, \mathbf{S}_{j}\mathbf{V}\mathbf{S}_{j}) + \log N_{R}(\mathbf{b}_{j}; \mathbf{0}, \Sigma_{p})\right] + \sum_{p=1}^{P} (\lambda_{p}-1) \log \pi_{p} \} \\ &= \sum_{j=1}^{J} \sum_{p=1}^{P} \gamma_{jp} \left[\log \pi_{p} - \frac{1}{2}\log |\mathbf{V}| - \log |\mathbf{S}_{j}| - \frac{1}{2}\mathbb{E}_{\mathbf{b}_{j}|\hat{\mathbf{b}}_{j}, z_{j}=p}\left((\hat{\mathbf{b}}_{j}-\mathbf{b}_{j})^{T}\mathbf{S}_{j}^{-1}\mathbf{V}^{-1}\mathbf{S}_{j}^{-1}(\hat{\mathbf{b}}_{j}-\mathbf{b}_{j})\right) - \frac{1}{2}\log |\Sigma_{p}| - \frac{1}{2}\mathbb{E}_{\mathbf{b}_{j}|\hat{\mathbf{b}}_{j}, z_{j}=p}\left(\mathbf{b}_{j}^{T}\Sigma_{p}^{-1}\mathbf{b}_{j} \right) \right] + \sum_{p=1}^{P} (\lambda_{p}-1)\log \pi_{p} \end{align*} \]
We have constraint on V, the diagonal of V must be 1. Let \(V = DCD\), C is the covariance matrix, D = \(diag(1/sqrt(C_{jj}))\).
\[ \begin{align*} f(\mathbf{C}) &= \sum_{j=1}^{J} \sum_{p=1}^{P} \gamma_{jp} \left[- \frac{1}{2}\log |\mathbf{D}\mathbf{C}\mathbf{D}| - \frac{1}{2}\mathbb{E}_{\mathbf{b}_{j}|\hat{\mathbf{b}}_{j}, z_{j}=p}\left((\hat{\mathbf{b}}_{j}-\mathbf{b}_{j})^{T}\mathbf{S}_{j}^{-1}\mathbf{D}^{-1}\mathbf{C}^{-1}\mathbf{D}^{-1}\mathbf{S}_{j}^{-1}(\hat{\mathbf{b}}_{j}-\mathbf{b}_{j})\right) \right] \\ &= \sum_{j=1}^{J} \sum_{p=1}^{P} \gamma_{jp} \left[- \frac{1}{2}\log |\mathbf{C}| - \log |\mathbf{D}|- \frac{1}{2}\mathbb{E}_{\mathbf{b}_{j}|\hat{\mathbf{b}}_{j}, z_{j}=p}\left((\hat{\mathbf{b}}_{j}-\mathbf{b}_{j})^{T}\mathbf{S}_{j}^{-1}\mathbf{D}^{-1}\mathbf{C}^{-1}\mathbf{D}^{-1}\mathbf{S}_{j}^{-1}(\hat{\mathbf{b}}_{j}-\mathbf{b}_{j})\right) \right] \\ f(\mathbf{C})' &= \sum_{j=1}^{J} \sum_{p=1}^{P} \gamma_{jp}\left[ -\frac{1}{2} \mathbf{C}^{-1} + \frac{1}{2} \mathbf{C}^{-1} \mathbf{D}^{-1}\mathbf{S}_{j}^{-1}\mathbb{E}\left((\hat{\mathbf{b}}_{j}-\mathbf{b}_{j}) (\hat{\mathbf{b}}_{j}-\mathbf{b}_{j})^{T}|\hat{\mathbf{b}}_{j}, z_{j} = p \right)\mathbf{S}_{j}^{-1}\mathbf{D}^{-1} \mathbf{C}^{-1} \right] = 0 \\ \mathbf{C} &= \frac{1}{J} \sum_{j=1}^{J} \sum_{p=1}^{P} \gamma_{jp}\mathbf{D}^{-1}\mathbf{S}_{j}^{-1}\mathbb{E}\left((\hat{\mathbf{b}}_{j}-\mathbf{b}_{j}) (\hat{\mathbf{b}}_{j}-\mathbf{b}_{j})^{T}|\hat{\mathbf{b}}_{j}, z_{j} = p \right)\mathbf{S}_{j}^{-1}\mathbf{D}^{-1} \\ &= \frac{1}{J} \mathbf{D}^{-1}\sum_{j=1}^{J} \mathbf{S}_{j}^{-1}\mathbb{E}\left((\hat{\mathbf{b}}_{j}-\mathbf{b}_{j}) (\hat{\mathbf{b}}_{j}-\mathbf{b}_{j})^{T}|\hat{\mathbf{b}}_{j}\right)\mathbf{S}_{j}^{-1}\mathbf{D}^{-1} \end{align*} \] We can update \(\mathbf{C}\) and \(\mathbf{V}\) as \[ \hat{\mathbf{C}}_{(t+1)} = \hat{\mathbf{D}}^{-1}_{(t)}\frac{1}{J} \left[\sum_{j=1}^{J} \mathbf{S}_{j}^{-1}\mathbb{E}\left[ (\hat{\mathbf{b}}_{j} - \mathbf{b}_{j})(\hat{\mathbf{b}}_{j} - \mathbf{b}_{j})^{T} | \hat{\mathbf{b}}_{j}\right]\mathbf{S}_{j}^{-1} \right] \hat{\mathbf{D}}^{-1}_{(t)} \\ \hat{\mathbf{D}}_{(t+1)} = diag(1/\sqrt{\hat{\mathbf{C}}_{(t+1)jj}}) \\ \hat{\mathbf{V}}_{(t+1)} = \hat{\mathbf{D}}_{(t+1)}\hat{\mathbf{C}}_{(t+1)}\hat{\mathbf{D}}_{(t+1)} \] The resulting \(\hat{\mathbf{V}}_{(t+1)}\) is equivalent as \[ \hat{\mathbf{C}}_{(t+1)} =\frac{1}{J} \left[\sum_{j=1}^{J} \mathbf{S}_{j}^{-1}\mathbb{E}\left[ (\hat{\mathbf{b}}_{j} - \mathbf{b}_{j})(\hat{\mathbf{b}}_{j} - \mathbf{b}_{j})^{T} | \hat{\mathbf{b}}_{j}\right]\mathbf{S}_{j}^{-1} \right] \\ \hat{\mathbf{D}}_{(t+1)} = diag(1/\sqrt{\hat{\mathbf{C}}_{(t+1)jj}}) \\ \hat{\mathbf{V}}_{(t+1)} = \hat{\mathbf{D}}_{(t+1)}\hat{\mathbf{C}}_{(t+1)}\hat{\mathbf{D}}_{(t+1)} \]
Algorithm:
Input: X, Ulist, init_V
Given V, estimate pi by max loglikelihood (convex problem)
Compute loglikelihood
delta = 1
while delta > tol
M step: update C
Convert to V
Given V, estimate pi by max loglikelihood (convex problem)
Compute loglikelihood
Update deltapenalty <- function(prior, pi_s){
subset <- (prior != 1.0)
sum((prior-1)[subset]*log(pi_s[subset]))
}
mixture.MV <- function(mash.data, Ulist, init_V=diag(ncol(mash.data$Bhat)), max_iter = 500, tol=1e-5, prior = c('nullbiased', 'uniform'), cor = TRUE, track_fit = FALSE){
prior <- match.arg(prior)
tracking = list()
m.model = fit_mash_V(mash.data, Ulist, V = init_V, prior=prior)
pi_s = get_estimated_pi(m.model, dimension = 'all')
prior.v <- mashr:::set_prior(length(pi_s), prior)
# compute loglikelihood
log_liks <- numeric(max_iter+1)
log_liks[1] <- get_loglik(m.model)+penalty(prior.v, pi_s)
V = init_V
result = list(V = V, logliks = log_liks[1], mash.model = m.model)
for(i in 1:max_iter){
if(track_fit){
tracking[[i]] = result
}
# max_V
V = E_V(mash.data, m.model)
if(cor){
V = cov2cor(V)
}
m.model = fit_mash_V(mash.data, Ulist, V, prior=prior)
pi_s = get_estimated_pi(m.model, dimension = 'all')
log_liks[i+1] <- get_loglik(m.model)+penalty(prior.v, pi_s)
result = list(V = V, logliks = log_liks[1:(i+1)], mash.model = m.model)
# Update delta
delta.ll <- log_liks[i+1] - log_liks[i]
if(delta.ll<=tol) break;
}
if(track_fit){
result$trace = tracking
}
return(result)
}
E_V = function(mash.data, m.model){
n = mashr:::n_effects(mash.data)
Z = mash.data$Bhat/mash.data$Shat
post.m.shat = m.model$result$PosteriorMean / mash.data$Shat
post.sec.shat = plyr::laply(1:n, function(i) (t(m.model$result$PosteriorCov[,,i]/mash.data$Shat[i,])/mash.data$Shat[i,]) + tcrossprod(post.m.shat[i,])) # nx2x2 array
temp1 = crossprod(Z)
temp2 = crossprod(post.m.shat, Z) + crossprod(Z, post.m.shat)
temp3 = unname(plyr::aaply(post.sec.shat, c(2,3), sum))
(temp1 - temp2 + temp3)/n
}
fit_mash_V <- function(mash.data, Ulist, V, prior=c('nullbiased', 'uniform')){
m.data = mashr::mash_set_data(Bhat=mash.data$Bhat, Shat=mash.data$Shat, V = V, alpha = mash.data$alpha)
m.model = mashr::mash(m.data, Ulist, prior=prior, verbose = FALSE, outputlevel = 3)
return(m.model)
}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)
result.mV <- mixture.MV(m.data, U.c)The estimated \(V\) is
result.mV$V [,1] [,2]
[1,] 1.0000000 0.5087511
[2,] 0.5087511 1.0000000m.mV = result.mV$mash.model
null.ind = which(apply(data$B,1,sum) == 0)The log likelihood is -12302.52. There are 26 significant samples, 0 false positives. The RRMSE is 0.5822265.
sessionInfo()R version 4.0.3 (2020-10-10)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/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] mashr_0.2.40 ashr_2.2-51 workflowr_1.6.2
loaded via a namespace (and not attached):
[1] Rcpp_1.0.5 plyr_1.8.6 pillar_1.4.7 compiler_4.0.3
[5] later_1.1.0.1 git2r_0.27.1 tools_4.0.3 digest_0.6.27
[9] evaluate_0.14 lifecycle_0.2.0 tibble_3.0.4 lattice_0.20-41
[13] pkgconfig_2.0.3 rlang_0.4.9 Matrix_1.2-18 rstudioapi_0.13
[17] yaml_2.2.1 mvtnorm_1.1-1 xfun_0.19 invgamma_1.1
[21] stringr_1.4.0 knitr_1.30 fs_1.5.0 vctrs_0.3.5
[25] rprojroot_2.0.2 grid_4.0.3 glue_1.4.2 R6_2.5.0
[29] rmarkdown_2.5 mixsqp_0.3-46 rmeta_3.0 irlba_2.3.3
[33] magrittr_2.0.1 whisker_0.4 MASS_7.3-53 promises_1.1.1
[37] ellipsis_0.3.1 htmltools_0.5.0 assertthat_0.2.1 abind_1.4-5
[41] httpuv_1.5.4 stringi_1.5.3 truncnorm_1.0-8 SQUAREM_2020.5
[45] crayon_1.3.4