mmultinom1
Zihao
2021-07-05
Last updated: 2021-07-07
Checks: 7 0
Knit directory: ebpmf_data_analysis/
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Untracked: output/fastTopics_fit/fit_droplet_simulated_fastTopics_k9.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k10.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k10.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k11.Rds
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Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k13.Rds
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Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k14.Rds
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Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k15.Rds
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Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k16.Rds
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Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k17.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k17.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k18.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k18.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k19.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k19.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k2.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k2.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k20.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k20.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k3.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k3.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k4.Rds
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Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k5.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k5.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k6.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k6.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k7.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k7.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k8.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k8.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k9.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_fastTopics_k9.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k10.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k10.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k11.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k11.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k12.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k12.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k13.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k13.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k14.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k14.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k15.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k15.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k16.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k16.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k17.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k17.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k18.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k18.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k19.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k19.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k2.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k2.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k20.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k20.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k3.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k3.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k4.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k4.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k5.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k5.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k6.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k6.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k7.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k7.Rout
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k8.Rds
Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k8.Rout
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Untracked: output/fastTopics_fit/fit_pbmc_68k_simulated_fastTopics_k9.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k10.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k10.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k100.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k100.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k11.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k11.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k12.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k12.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k13.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k13.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k14.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k14.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k15.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k15.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k16.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k16.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k17.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k17.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k18.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k18.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k19.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k19.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k2.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k2.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k20.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k20.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k25.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k25.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k3.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k3.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k30.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k30.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k35.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k35.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k4.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k4.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k40.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k40.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k45.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k45.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k5.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k5.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k50.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k50.Rout
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Untracked: output/fastTopics_fit/fit_sla_fastTopics_k7.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k7.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k70.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k70.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k8.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k8.Rout
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Untracked: output/fastTopics_fit/fit_sla_fastTopics_k80.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k9.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k9.Rout
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k90.Rds
Untracked: output/fastTopics_fit/fit_sla_fastTopics_k90.Rout
Untracked: output/fastTopics_fit/fit_sla_sim_fastTopics_evals.Rds
Untracked: output/fastTopics_fit/fit_sla_sim_fastTopics_k10.Rds
Untracked: output/fastTopics_fit/fit_sla_sim_fastTopics_k10.Rout
Untracked: output/fastTopics_fit/fit_sla_sim_fastTopics_k11.Rds
Untracked: output/fastTopics_fit/fit_sla_sim_fastTopics_k11.Rout
Untracked: output/fastTopics_fit/fit_sla_sim_fastTopics_k12.Rds
Untracked: output/fastTopics_fit/fit_sla_sim_fastTopics_k12.Rout
Untracked: output/fastTopics_fit/fit_sla_sim_fastTopics_k13.Rds
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | 09ed5d3 | Zihao | 2021-07-07 | update mmultinom1 |
| html | 5fc47d4 | Zihao | 2021-07-05 | Build site. |
| Rmd | 5c67270 | Zihao | 2021-07-05 | mmultinom1.Rmd |
The mmultinom subproblem
In PMF using EM, to estimate \(F\) we need to solve the subproblem (except we use \(E(Z_{ijk})\) instead of \(Z_{ijk}\)):
If we do some scaling so that \(\sum_j f_{jk} = 1\), and denote \(m_{jk} := \sum_i Z_{ijk}, \ n_k := \sum_i l_{ik}, \ \beta_{jk} := f_{jk}\), we have can map it to the mmultinom problem defined below (not equivalent, but 1-1 between the MLEs)
The mmultinom problem becomes “hard” in high-dimensional setting (when \(n_k < p\), say), and I expect MLE performs poorly in this setting. This scenario occurs in PMF when we use large \(K\), or the total number of words is too small compared to \(p\) (the data cannot support good estimation of desired number of topics)
The quality of the single MLE estimate \(\hat{\beta}_{jk}\) only concerns the true \(\beta_{jk}\), and \(n_k\), as \(n_k \hat{\beta}_{jk} \sim \mathrm{Bin}(n_k, \beta_{jk})\). We have \(\mathrm{E}(\frac{\hat{\beta}_{jk}}{\beta_{jk}}) = 1\) & \(\mathrm{Var}(\frac{\hat{\beta}_{jk}}{\beta_{jk}}) = \frac{1 - \beta_{jk}}{n_k \beta_{jk}} \approx \frac{1}{n_k \beta_{jk}}\). Therefore, I expect the estimate to be good as long as \(n_k \beta_{jk}\) is large, and bad when it’s small.
Possible improvement by borrowing information
Perhaps in reality most words are not active in any of those topics (the same logic behind removing some words before applying models). Therefore, we can assume \(\beta_{jk} = \beta_{j0} d_{jk}\). We can estimate \(\beta_{j0}\) with all \(\sum_k n_k\) occurences of words, and then estimate each \(d_{jk}\) with \(n_k\) occurences but assuming sparsity on \(|d_{jk} - 1|\).
I haven’t figured out a prior for \(d_{jk}\) yet, but it’s much easier if we use an “equivalent” Poisson model.
“Realistic” cases
I use the fitted \(L, F\) (fastTopics on SLA) to generate the
mmultinomproblem below. (dictionary size \(p \approx 9000\)). I change the difficulty of the problem by changing the document length and number of documentsI find even in the most difficult case, MLE is sufficient for those “top words”: whose corresponding \(\beta_{jk} > 100/p\) (around \(0.01\) in this example) (also implies they are the top 1% words). It gets bad for those with smaller \(\beta_{jk}\).
I look at the other document data and speculate MLE would be fine for \(K = 100\) for top 1% words.
rm(list = ls())
library(fastTopics)
source("code/mmultinom.R")
source("code/mmultinom_util.R")
Attaching package: 'dplyr'The following objects are masked from 'package:stats':
filter, lagThe following objects are masked from 'package:base':
intersect, setdiff, setequal, unionsource("code/util.R")
set.seed(123)Load data
dataname = "sla"
K = 20
fitted = readRDS(sprintf("output/fastTopics_fit/fit_%s_fastTopics_k%d.Rds",
dataname, K))
m = nrow(fitted$L)Look at distribution of \(\beta_{jk}\)
par(mfrow = c(2, 1))
#plot(ecdf(fitted$F), xlab = "beta_jk")
quantile(fitted$F, probs = seq(0.9, 1, 0.01)) 90% 91% 92% 93% 94%
0.0001831250 0.0002159829 0.0002533175 0.0003011958 0.0003630245
95% 96% 97% 98% 99%
0.0004488902 0.0005765570 0.0007854040 0.0011595225 0.0020953976
100%
0.0830365615 hist(log10(fitted$F), xlab = "log10(beta_jk)", probability = TRUE)
hist(log10(fitted$F), xlab = "log10(beta_jk)", probability = TRUE,xlim = c(-5, 0), breaks = 30)
| Version | Author | Date |
|---|---|---|
| 5fc47d4 | Zihao | 2021-07-05 |
80% of \(\beta_{jk}\) are just \(0\) (set to a small value \(< 10^{-15}\)), as a result of MLE fit
Use all documents, full document length
model = fitted
data_sim = TM2mmultinom(fit = model)
fit_mle = mmultinom.mle(M = data_sim$M, n = data_sim$n)
par(mfrow = c(K/4, 4))
ce = replicate(K, 0)
for(k in 1:K){
p1 = data_sim$beta[k,]
p2 = fit_mle[k, ]
ce[k] = cross_entropy(p1 = p1, p2 = p2)
idx <- (log10(p1) > -5)
plot(p1[idx], log(p1[idx]) - log(p2[idx]),
ylab = "log(beta) - log(betahat)", xlab = "beta",
main = sprintf("topic %d; n = %d", k, round(data_sim$n[k])))
abline(h = 0, col = "blue")
# plot(log10(p1), log10(p1) - log10(p2),
# ylab = "log10(beta) - log10(betahat)", xlab = "log10(beta)",
# main = sprintf("topic %d; n = %d", k, round(data_sim$n[k])))
# abline(h = 0, col = "blue")
}
| Version | Author | Date |
|---|---|---|
| 5fc47d4 | Zihao | 2021-07-05 |
round(ce, digits = 3) [1] 0.713 1.122 0.566 0.642 0.714 1.216 0.483 1.110 0.967 0.564 0.337
[12] 0.840 0.605 0.671 0.452 0.133 0.939 0.497 0.673 0.534sum(ce)[1] 13.77742Use less documents
Suppose we only observe 20% of the documents
model = fitted
idx = sample(x = 1:m, size = round(0.2*m))
model$L <- model$L[idx, ]
model$s <- model$s[idx]
data_sim = TM2mmultinom(fit = model)
fit_mle = mmultinom.mle(M = data_sim$M, n = data_sim$n)
par(mfrow = c(K/4, 4))
ce = replicate(K, NA)
for(k in 1:K){
p1 = data_sim$beta[k,]
p2 = fit_mle[k, ]
ce[k] = cross_entropy(p1 = p1, p2 = p2)
idx <- (log10(p1) > -5)
plot(p1[idx], log(p1[idx]) - log(p2[idx]),
ylab = "log(beta) - log(betahat)", xlab = "beta",
main = sprintf("topic %d; n = %d", k, round(data_sim$n[k])))
abline(h = 0, col = "blue")
# plot(log10(p1), log10(p1) - log10(p2),
# ylab = "log10(beta) - log10(betahat)", xlab = "log10(beta)",
# main = sprintf("topic %d; n = %d", k, round(data_sim$n[k])))
# abline(h = 0, col = "blue")
}
| Version | Author | Date |
|---|---|---|
| 5fc47d4 | Zihao | 2021-07-05 |
round(ce, digits = 3) [1] 5.221 5.472 3.807 4.352 4.329 7.062 3.612 6.823 6.175 3.489 2.209
[12] 4.686 3.563 4.158 2.714 0.979 5.411 2.657 4.141 4.033sum(ce)[1] 84.89279use shorter & less documents
I use half of the documents and 1/5 document length
model = fitted
idx = sample(x = 1:m, size = round(0.5*m))
model$L <- model$L[idx, ]
model$s <- model$s[idx]/5
data_sim = TM2mmultinom(fit = model)
fit_mle = mmultinom.mle(M = data_sim$M, n = data_sim$n)
par(mfrow = c(K/4, 4))
ce = replicate(K, 0)
for(k in 1:K){
p1 = data_sim$beta[k,]
p2 = fit_mle[k, ]
ce[k] = cross_entropy(p1 = p1, p2 = p2)
idx <- (log10(p1) > -5)
plot(p1[idx], log(p1[idx]) - log(p2[idx]),
ylab = "log(beta) - log(betahat)", xlab = "beta",
main = sprintf("topic %d; n = %d", k, round(data_sim$n[k])))
abline(h = 0, col = "blue")
# plot(log10(p1), log10(p1) - log10(p2),
# ylab = "log10(beta) - log10(betahat)", xlab = "log10(beta)",
# main = sprintf("topic %d; n = %d", k, round(data_sim$n[k])))
# abline(h = 0, col = "blue")
}
round(ce, digits = 3) [1] 8.407 10.991 5.788 7.206 6.437 12.731 5.199 8.643 8.910 5.562
[11] 4.305 8.768 7.329 6.528 4.833 1.981 8.200 5.609 6.239 6.033sum(ce)[1] 139.6989
sessionInfo()R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Scientific Linux 7.4 (Nitrogen)
Matrix products: default
BLAS/LAPACK: /software/openblas-0.2.19-el7-x86_64/lib/libopenblas_haswellp-r0.2.19.so
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] dplyr_0.8.3 fastTopics_0.5-53 workflowr_1.6.2
loaded via a namespace (and not attached):
[1] ggrepel_0.9.1 Rcpp_1.0.5 invgamma_1.1
[4] lattice_0.20-38 tidyr_1.1.0 prettyunits_1.0.2
[7] assertthat_0.2.1 rprojroot_1.3-2 digest_0.6.20
[10] truncnorm_1.0-8 R6_2.4.0 backports_1.1.4
[13] MatrixModels_0.4-1 evaluate_0.14 coda_0.19-3
[16] httr_1.4.1 ggplot2_3.3.3 pillar_1.4.2
[19] rlang_0.4.10 progress_1.2.2 lazyeval_0.2.2
[22] data.table_1.13.2 irlba_2.3.3 SparseM_1.77
[25] whisker_0.3-2 Matrix_1.2-18 rmarkdown_1.13
[28] Rtsne_0.15 stringr_1.4.0 htmlwidgets_1.3
[31] uwot_0.1.5 munsell_0.5.0 mixsqp_0.3-43
[34] compiler_3.6.1 httpuv_1.5.1 xfun_0.8
[37] pkgconfig_2.0.2 SQUAREM_2017.10-1 mcmc_0.9-6
[40] htmltools_0.3.6 tidyselect_1.1.0 tibble_2.1.3
[43] quadprog_1.5-7 viridisLite_0.3.0 crayon_1.3.4
[46] later_0.8.0 MASS_7.3-51.4 grid_3.6.1
[49] jsonlite_1.6 gtable_0.3.0 lifecycle_0.1.0
[52] git2r_0.26.1 magrittr_1.5 scales_1.1.0
[55] RcppParallel_4.4.3 stringi_1.4.3 fs_1.3.1
[58] promises_1.0.1 vctrs_0.3.1 cowplot_1.0.0
[61] tools_3.6.1 glue_1.3.1 purrr_0.3.4
[64] hms_0.5.3 yaml_2.2.0 colorspace_1.4-1
[67] ashr_2.2-47 plotly_4.9.0 knitr_1.23
[70] quantreg_5.41 MCMCpack_1.4-5