Fitting a real-data covariance matrix

Last updated: 2020-09-13

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File	Version	Author	Date	Message
Rmd	3fca558	Jason Willwerscheid	2020-09-13	wflow_publish(“analysis/covmat_1kg.Rmd”)
html	d1220b2	Jason Willwerscheid	2020-09-13	Build site.
Rmd	8633525	Jason Willwerscheid	2020-09-13	wflow_publish(“analysis/covmat_1kg.Rmd”)
html	c77198e	Jason Willwerscheid	2020-09-12	Build site.
Rmd	c433e17	Jason Willwerscheid	2020-09-12	wflow_publish(“analysis/covmat_1kg.Rmd”)
html	ca2258e	Jason Willwerscheid	2020-09-11	Build site.
Rmd	f11bdf5	Jason Willwerscheid	2020-09-11	wflow_publish(“analysis/covmat_1kg.Rmd”)

suppressMessages({
  library(flashier)
  library(drift.alpha)
  library(tidyverse)
})

I fit flash to the covariance matrix of the 1kg_phase3_derived dataset using point-Laplace priors. I show results from the greedy fit, backfit, and LL' + D fit.

covmat <- readRDS("../data/datasets/1kg_phase3_derived/1kg_phase3_derived_covmat.rds")
meta <- readRDS("../data/datasets/1kg_phase3_derived/1kg_phase3_derived_meta.rds")

plot_fl <- function(LL) {
  df <- data.frame(LL)
  colnames(df) <- paste0("Factor ", formatC(1:ncol(LL), width = 2, flag = "0"))
  df$subpop <- meta$pop
  df$superpop <- meta$super_pop
  df <- df %>% arrange(superpop)
  df$idx <- 1:nrow(df)
  gath_df <- df %>% 
    gather(K, value, -subpop, -idx, -superpop) %>%
    mutate(K = factor(K))
  med_gath_df <- gath_df %>% 
    group_by(subpop, K) %>% 
    summarise(value=median(value), idx=median(idx))
  
  p <- ggplot(gath_df, aes(x=idx, y=value, color=superpop)) + 
    geom_point() +
    facet_wrap(~K) + 
    geom_hline(yintercept = 0, linetype = "dashed") +
    theme(axis.title.x=element_blank(),
          axis.text.x=element_blank(),
          axis.ticks.x=element_blank()) +
    labs(color="superpop")
  return(p)
}

Greedy fit

fl_g <- flash.init(covmat) %>%
  flash.set.verbose(0) %>%
  flash.add.greedy(Kmax = 20, 
                   prior.family = prior.point.laplace())

plot(plot_fl(fl_g$loadings.pm[[1]] %*% diag(sqrt(fl_g$loadings.scale))))

Version	Author	Date
c77198e	Jason Willwerscheid	2020-09-12
ca2258e	Jason Willwerscheid	2020-09-11

Backfit

fl_bf <- fl_g %>% flash.backfit()

plot(plot_fl(fl_bf$loadings.pm[[1]] %*% diag(sqrt(fl_bf$loadings.scale))))

Version	Author	Date
c77198e	Jason Willwerscheid	2020-09-12
ca2258e	Jason Willwerscheid	2020-09-11

Additional diagonal variance

n <- nrow(covmat)

fl <- fl_bf
diag_S2 <- mean(diag(covmat)^2 
                - 2 * diag(covmat) * rowSums(fl$flash.fit$EF[[1]] * fl$flash.fit$EF[[2]])
                + rowSums(fl$flash.fit$EF2[[1]] * fl$flash.fit$EF2[[2]])
                - rowSums(fl$flash.fit$EF[[1]]^2 * fl$flash.fit$EF[[2]]^2))
diag_S2 <- diag_S2 + sum(crossprod(fl$flash.fit$EF[[1]] * fl$flash.fit$EF[[2]])) / n
diag_S2 <- diag_S2 - 1 / fl$flash.fit$tau
  
elbo_diff <- Inf
while (elbo_diff > 0.1) {
  old_elbo <- fl$elbo
  fl <- flash.init(covmat, S = diag(rep(sqrt(diag_S2), n)), var.type = 0) %>%
    flash.set.verbose(0) %>%
    flash.init.factors(EF = fl$flash.fit$EF, EF2 = fl$flash.fit$EF2,
                       prior.family = prior.point.laplace()) %>%
    flash.backfit()
  cat("SD (diagonal):", formatC(sqrt(diag_S2), format = "e", digits = 2),
      " SD (non-diag):", formatC(sqrt(1 / fl$flash.fit$tau[1, 2]), format = "e", digits = 2),
      " ELBO:", fl$elbo, "\n")
  elbo_diff <- fl$elbo - old_elbo
  
  diag_S2 <- mean(diag(covmat)^2 
                  - 2 * diag(covmat) * rowSums(fl$flash.fit$EF[[1]] * fl$flash.fit$EF[[2]])
                  + rowSums(fl$flash.fit$EF2[[1]] * fl$flash.fit$EF2[[2]])
                  - rowSums(fl$flash.fit$EF[[1]]^2 * fl$flash.fit$EF[[2]]^2))
  diag_S2 <- diag_S2 + sum(crossprod(fl$flash.fit$EF[[1]] * fl$flash.fit$EF[[2]])) / n
  diag_S2 <- diag_S2 - 1 / fl$flash.fit$tau[1, 2]
}

#> SD (diagonal): 2.85e-01  SD (non-diag): 1.47e-03  ELBO: 31798035 
#> SD (diagonal): 2.86e-01  SD (non-diag): 1.47e-03  ELBO: 31798035 
#> SD (diagonal): 2.86e-01  SD (non-diag): 1.47e-03  ELBO: 31798036


plot(plot_fl(fl$loadings.pm[[1]] %*% diag(sqrt(fl$loadings.scale))))

Version	Author	Date
c77198e	Jason Willwerscheid	2020-09-12
ca2258e	Jason Willwerscheid	2020-09-11

Drift

LL <- fl$loadings.pm[[1]] %*% diag(sqrt(fl$loadings.scale))
LL <- cbind(abs(LL[, 1]), pmax(LL[, -1], 0), pmax(-LL[, -1], 0))
LL <- LL[, c(1, as.vector(sapply(2:12, `+`, c(0, 11))))]

LL_scale <- apply(LL, 2, max)
LL <- t(t(LL) / LL_scale)

which_k <- (LL_scale > .01)
LL <- LL[, which_k]
LL_scale <- LL_scale[which_k]

dr <- init_from_covmat(covmat, LL, p = 30000, prior.s2 = LL_scale^2)
dr <- drift(dr, maxiter = 200, verbose = FALSE)

plot(plot_fl(dr$EL %*% diag(sqrt(dr$prior_s2))))

Version	Author	Date
d1220b2	Jason Willwerscheid	2020-09-13
c77198e	Jason Willwerscheid	2020-09-12

sessionInfo()

#> R version 3.5.3 (2019-03-11)
#> Platform: x86_64-apple-darwin15.6.0 (64-bit)
#> Running under: macOS Mojave 10.14.6
#> 
#> 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] forcats_0.4.0      stringr_1.4.0      dplyr_0.8.0.1     
#>  [4] purrr_0.3.2        readr_1.3.1        tidyr_0.8.3       
#>  [7] tibble_2.1.1       ggplot2_3.2.0      tidyverse_1.2.1   
#> [10] drift.alpha_0.0.11 flashier_0.2.7    
#> 
#> loaded via a namespace (and not attached):
#>  [1] Rcpp_1.0.4.6      lubridate_1.7.4   invgamma_1.1     
#>  [4] lattice_0.20-38   assertthat_0.2.1  rprojroot_1.3-2  
#>  [7] digest_0.6.18     truncnorm_1.0-8   R6_2.4.0         
#> [10] cellranger_1.1.0  plyr_1.8.4        backports_1.1.3  
#> [13] evaluate_0.13     httr_1.4.0        pillar_1.3.1     
#> [16] rlang_0.4.2       lazyeval_0.2.2    readxl_1.3.1     
#> [19] rstudioapi_0.10   ebnm_0.1-21       irlba_2.3.3      
#> [22] whisker_0.3-2     Matrix_1.2-15     rmarkdown_1.12   
#> [25] labeling_0.3      munsell_0.5.0     mixsqp_0.3-40    
#> [28] broom_0.5.1       compiler_3.5.3    modelr_0.1.5     
#> [31] xfun_0.6          pkgconfig_2.0.2   SQUAREM_2017.10-1
#> [34] htmltools_0.3.6   tidyselect_0.2.5  workflowr_1.2.0  
#> [37] withr_2.1.2       crayon_1.3.4      grid_3.5.3       
#> [40] nlme_3.1-137      jsonlite_1.6      gtable_0.3.0     
#> [43] git2r_0.25.2      magrittr_1.5      scales_1.0.0     
#> [46] cli_1.1.0         stringi_1.4.3     reshape2_1.4.3   
#> [49] fs_1.2.7          xml2_1.2.0        generics_0.0.2   
#> [52] tools_3.5.3       glue_1.3.1        hms_0.4.2        
#> [55] parallel_3.5.3    yaml_2.2.0        colorspace_1.4-1 
#> [58] ashr_2.2-51       rvest_0.3.4       knitr_1.22       
#> [61] haven_2.1.1

Fitting a real-data covariance matrix

Jason Willwerscheid

9/11/2020

Greedy fit

Backfit

Additional diagonal variance

Drift