Last updated: 2021-10-21
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| Rmd | 9f10528 | Matthew Stephens | 2021-10-21 | workflowr::wflow_publish(“flash_f1_size.Rmd”) |
Following up on some results from Joonsuk Kang, I wanted to investigate how the “size” (PVE) of the first factor affects ability to detect a second factor in a simple situation.
I’ll simulate two factors, both random normal:
library("flashr")
n = 100
p = 100
k = 2
LL = matrix(rnorm(p*k), nrow=n, ncol=k)
FF = matrix(rnorm(p*k), nrow=p, ncol=k)
E = matrix(rnorm(n*p),nrow=n,ncol=p)
Y = LL %*% t(FF) + E
svd(Y)$d[1:5] # singular values[1] 100.15732 89.47766 20.06658 18.85328 18.39188fit1 = flashr::flash(Y,verbose=FALSE)
cor(flash_get_ldf(fit1)$l,LL) [,1] [,2]
[1,] -0.4315995 -0.8948275
[2,] 0.8991089 -0.4269508Actually you see the non-identifiability here: because both factors are a similar size they have similar eigenvalues so you get the rotation-invariance issue.
Now make first factor stronger. This makes the two factors identifiable and results are more accurate:
FF[,1] = FF[,1]* 10
Y = LL %*% t(FF) + E
svd(Y)$d[1:5] # singular values[1] 927.71117 95.06672 20.07628 18.84830 18.40176fit1 = flashr::flash(Y, verbose=FALSE)
cor(flash_get_ldf(fit1)$l,LL) [,1] [,2]
[1,] 0.999896879 0.008274971
[2,] -0.002016449 0.994142020Now make first factor much stronger. It still seems to work….
FF[,1] = FF[,1]* 1000
Y = LL %*% t(FF) + E
svd(Y)$d[1:5] # singular values[1] 926873.75463 95.07521 20.07720 18.84767 18.40268fit1 = flashr::flash(Y,verbose=F)
cor(flash_get_ldf(fit1)$l,LL) [,1] [,2]
[1,] 1.000000000 -0.0009850985
[2,] 0.006629118 0.9941076381
sessionInfo()R version 4.1.0 Patched (2021-07-20 r80657)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Big Sur 11.6
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/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] flashr_0.6-8
loaded via a namespace (and not attached):
[1] softImpute_1.4-1 tidyselect_1.1.1 xfun_0.24 ashr_2.2-47
[5] purrr_0.3.4 reshape2_1.4.4 splines_4.1.0 lattice_0.20-44
[9] colorspace_2.0-2 vctrs_0.3.8 generics_0.1.0 htmltools_0.5.1.1
[13] yaml_2.2.1 utf8_1.2.2 rlang_0.4.11 mixsqp_0.3-43
[17] later_1.2.0 pillar_1.6.3 glue_1.4.2 DBI_1.1.1
[21] REBayes_2.2 trust_0.1-8 lifecycle_1.0.1 plyr_1.8.6
[25] stringr_1.4.0 munsell_0.5.0 gtable_0.3.0 workflowr_1.6.2
[29] evaluate_0.14 knitr_1.33 httpuv_1.6.1 invgamma_1.1
[33] irlba_2.3.3 fansi_0.5.0 Rcpp_1.0.7 promises_1.2.0.1
[37] scales_1.1.1 horseshoe_0.2.0 truncnorm_1.0-8 fs_1.5.0
[41] deconvolveR_1.2-1 ggplot2_3.3.5 digest_0.6.27 stringi_1.7.3
[45] dplyr_1.0.7 ebnm_0.1-50 grid_4.1.0 rprojroot_2.0.2
[49] tools_4.1.0 magrittr_2.0.1 tibble_3.1.4 crayon_1.4.1
[53] whisker_0.4 pkgconfig_2.0.3 ellipsis_0.3.2 Matrix_1.3-4
[57] SQUAREM_2021.1 assertthat_0.2.1 rmarkdown_2.9 R6_2.5.1
[61] git2r_0.28.0 compiler_4.1.0