# Simulation with non-canonical matrices

#### Matthew Stephens

#### 2020-06-17

## Goal

To try out some simulations that don’t match the canonical covariance matrices and illustrate how the data driven matrices help.

## Simple simulation

Here the function `simple_sims_2` simulates data in five conditions with just two types of effect:

1. shared effects only in the first two conditions; and

2. shared effects only in the last three conditions.

    library(ashr)
    library(mashr)
    set.seed(1)
    simdata = simple_sims2(1000,1)
    true.U1 = cbind(c(1,1,0,0,0),c(1,1,0,0,0),rep(0,5),rep(0,5),rep(0,5))
    true.U2 = cbind(rep(0,5),rep(0,5),c(0,0,1,1,1),c(0,0,1,1,1),c(0,0,1,1,1))
    U.true  = list(true.U1 = true.U1, true.U2 = true.U2)

## Simple simulation

Run 1-by-1 to add the strong signals and ED covariances.

    data   = mash_set_data(simdata$Bhat, simdata$Shat)
    m.1by1 = mash_1by1(data)
    strong = get_significant_results(m.1by1)
    U.c    = cov_canonical(data)
    U.pca  = cov_pca(data,5,strong)
    U.ed   = cov_ed(data,U.pca,strong)

    # Computes covariance matrices based on extreme deconvolution,
    # initialized from PCA.
    m.c    = mash(data, U.c)
    m.ed   = mash(data, U.ed)
    m.c.ed = mash(data, c(U.c,U.ed))
    m.true = mash(data, U.true)

    print(get_loglik(m.c),digits = 10)
    print(get_loglik(m.ed),digits = 10)
    print(get_loglik(m.c.ed),digits = 10)
    print(get_loglik(m.true),digits = 10)

The log-likelihood is much better from data-driven than canonical covariances. This is good! Indeed, here the data-driven fit is very slightly better fit than the true matrices, but only very slightly.