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Rmd | 38c2d39 | Matthew Stephens | 2023-10-21 | wflow_publish("flashier_log1p.Rmd") |
library(flashier)
Loading required package: ebnm
Loading required package: magrittr
When applied to (log1p-transformed) Poisson data, flashier often overestimates the rank (too many factors). However this is usually in simulations where the underlying low-rank structure is the log-mean (or maybe the mean). In practice we don’t know if the low-rank structure is in the log mean or some other function of the mean - for example, in the log1p(mean). Motivated by this, I look at how (nonnegative) flashier behaves on log1p transformed data when the underlying log1p(mu) is itself low rank with non-negative factors.
I want to start by studying the mean and variance of log(1+X)
By Taylor series of \(f(x) = log(1+x)\) we have \[\log(1+X) \approx \log(1+\mu) + (X-\mu)f'(\mu) + 0.5 (X-\mu)^2 f''(\mu)\] so \[E(\log(1+X)) = \log(1+\mu) - 0.5 \mu/(1+\mu)^2\] For small \(\mu\) this is approximately \(\mu/2\) and for large \(\mu\) it is \(\log(1+\mu)\).
Rearranging the above gives \[(log(1+X) - log(1+\mu))^2 \approx (X-\mu)^2 f'(\mu)^2\] so \[var(log(1+X)) \approx \mu/(1+\mu)^2.\]
We can check the accuracy by simulation. It matches pretty well. Also, I note that the standard deviation is not that variable for mu in the range exp(-5) to exp(3). (a factor of just under 10 variation in standard deviation, so not entirely negligible, but it could be worse). This might suggest we might get away without taking account of variation in standard error, which is practically very convenient.
mm = exp(seq(-5,3,length=9))
m = v =mm
for(i in 1:length(mm)){
x = rpois(100000,mm[i])
m[i] = mean(log(1+x))
v[i] = var(log(1+x))
}
plot(mm,m)
lines(mm, log(1+mm) - 0.5*mm/(1+mm)^2 )
plot(mm,sqrt(v))
lines(mm, sqrt(mm)/(1+mm))
abline(v=1)
We can extend this to \(log(c+X)\). By Taylor series we have: \[E(log(c+X)) = log(c+\mu) - 0.5 \mu/(c+\mu)^2\] and \[Var(log(c+X)) \approx \mu/(c+\mu)^2.\]
c=0.1
mm = exp(seq(-5,3,length=9))
mc = vc =mm
for(i in 1:length(mm)){
x = rpois(100000,mm[i])
mc[i] = mean(log(c+x))
vc[i] = var(log(c+x))
}
plot(mm, log(c+mm) - 0.5*mm/(c+mm)^2,type="l" )
points(mm,mc)
plot(mm,sqrt(vc))
lines(mm, sqrt(mm)/(c+mm))
Interesting that (unless I made a mistake) the mean is quite off for small mu and the variance is also off. I need to understand this better. However, I do note again that the standard deviation doesn’t vary too much with mm.
I do a simple simulation where \(log(1+\mu) =LF'\) where \(L\) and \(F\) are non-negative and \(LF'\) is rank 4..
set.seed(1)
n= 1000
p = 200
K = 4
LL = matrix(runif(n*K),nrow=n)
FF = matrix(runif(p*K),nrow=p)
mu = matrix(exp(LL %*% t(FF))-1,ncol=p, nrow=n)
X= matrix(rpois(n*p,mu), ncol=p, nrow=n)
hist(mu)
mean(X>0)
[1] 0.744955
Try flashier
fit.nn.1 = flash(log(X+1),ebnm_fn = ebnm_point_exponential,var_type=2,backfit=TRUE)
Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Adding factor 4 to flash object...
Adding factor 5 to flash object...
Adding factor 6 to flash object...
Adding factor 7 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 6 factors (tolerance: 2.98e-03)...
Difference between iterations is within 1.0e+02...
Difference between iterations is within 1.0e+01...
--Estimate of factor 6 is numerically zero!
--Estimate of factor 6 is numerically zero!
--Estimate of factor 4 is numerically zero!
Difference between iterations is within 1.0e+00...
Difference between iterations is within 1.0e-01...
Wrapping up...
Done.
Nullchecking 6 factors...
One factor is identically zero.
Wrapping up...
Removed one factor.
Done.
The greedy approach fits 6 factors, but the backfitting ends up with 5 factors. And one of the factors is capturing a single row.
plot(fit.nn.1$F_pm[,4])
which.max(fit.nn.1$F_pm[,4])
[1] 55
plot(fit.nn.1$L_pm[,4])
plot(X[,54])
fit.nn.1$flash_fit$tau[54]
[1] 4.274929
So it seems the extra factor here is due to flash converging to a solution that picks out one column, and sets its variance close to 0 (tau is very big). And the null check does not work because the flash objective goes to infinity as tau goes to infinity with a perfect fit to the column. We could probably avoid this by setting a minimum tau somehow (or regularizing it).
After experimenting I found that S=0.5 fixed it (but smaller values of S do not).
fit.nn.1 = flash(log(X+1),ebnm_fn = ebnm_point_exponential,var_type=2,backfit=TRUE, S=0.5)
Adding factor 1 to flash object...
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Adding factor 4 to flash object...
Adding factor 5 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 4 factors (tolerance: 2.98e-03)...
Difference between iterations is within 1.0e+02...
Difference between iterations is within 1.0e+01...
Difference between iterations is within 1.0e+00...
Difference between iterations is within 1.0e-01...
Difference between iterations is within 1.0e-02...
Wrapping up...
Done.
Nullchecking 4 factors...
Done.
We need more investigation, but maybe flash can do OK with estimating the rank if the true log(1+mu) is low rank and we fix the issue with outlying factors capturing a single column.
sessionInfo()
R version 4.2.1 (2022-06-23)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur ... 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.2/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] flashier_1.0.7 magrittr_2.0.3 ebnm_1.0-55
loaded via a namespace (and not attached):
[1] softImpute_1.4-1 tidyselect_1.2.0 xfun_0.37 bslib_0.4.2
[5] purrr_1.0.2 ashr_2.2-63 splines_4.2.1 lattice_0.20-45
[9] colorspace_2.1-0 vctrs_0.6.4 generics_0.1.3 htmltools_0.5.4
[13] yaml_2.3.7 utf8_1.2.3 rlang_1.1.1 mixsqp_0.3-48
[17] jquerylib_0.1.4 later_1.3.0 pillar_1.9.0 glue_1.6.2
[21] trust_0.1-8 lifecycle_1.0.3 stringr_1.5.0 munsell_0.5.0
[25] gtable_0.3.4 workflowr_1.7.0 evaluate_0.22 knitr_1.42
[29] fastmap_1.1.1 httpuv_1.6.9 invgamma_1.1 parallel_4.2.1
[33] irlba_2.3.5.1 fansi_1.0.5 highr_0.10 Rcpp_1.0.11
[37] promises_1.2.0.1 scales_1.2.1 cachem_1.0.7 horseshoe_0.2.0
[41] jsonlite_1.8.7 truncnorm_1.0-9 fs_1.6.3 deconvolveR_1.2-1
[45] ggplot2_3.4.3 digest_0.6.33 stringi_1.7.12 dplyr_1.1.3
[49] rprojroot_2.0.3 grid_4.2.1 cli_3.6.1 tools_4.2.1
[53] sass_0.4.5 tibble_3.2.1 tidyr_1.3.0 whisker_0.4.1
[57] pkgconfig_2.0.3 Matrix_1.5-3 SQUAREM_2021.1 rmarkdown_2.20
[61] rstudioapi_0.14 R6_2.5.1 git2r_0.31.0 compiler_4.2.1