Multinomial Logistic Regression with Heavy-Tailed Priors
Longhai Li and Steven Liu
2021-05-03
Source:vignettes/simu.Rmd
simu.RmdData Generation
Load the necessary libraries:
library(HTLR)
library(bayesplot)
#> This is bayesplot version 1.8.0
#> - Online documentation and vignettes at mc-stan.org/bayesplot
#> - bayesplot theme set to bayesplot::theme_default()
#> * Does _not_ affect other ggplot2 plots
#> * See ?bayesplot_theme_set for details on theme settingThe description of the dataset generating scheme is found from Li and Yao (2018).
There are 4 groups of features:
feature #1: marginally related feature
feature #2: marginally unrelated feature, but feature #2 is correlated with feature #1
feature #3 - #10: marginally related features and also internally correlated
feature #11 - #2000: noise features without relationship with the y
SEED <- 1234
n <- 510
p <- 2000
means <- rbind(
c(0, 1, 0),
c(0, 0, 0),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1)
) * 2
means <- rbind(means, matrix(0, p - 10, 3))
A <- diag(1, p)
A[1:10, 1:3] <-
rbind(
c(1, 0, 0),
c(2, 1, 0),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1),
c(0, 0, 1)
)
set.seed(SEED)
dat <- gendata_FAM(n, means, A, sd_g = 0.5, stdx = TRUE)
str(dat)
#> List of 4
#> $ X : num [1:510, 1:2000] -1.423 -0.358 -1.204 -0.556 0.83 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
#> $ muj: num [1:2000, 1:3] -0.456 0 -0.456 -0.376 -0.376 ...
#> $ SGM: num [1:2000, 1:2000] 0.584 0.597 0 0 0 ...
#> $ y : int [1:510] 1 2 3 1 2 3 1 2 3 1 ...Look at the correlation between features:
Split the data into training and testing sets:
set.seed(SEED)
dat <- split_data(dat$X, dat$y, n.train = 500)
str(dat)
#> List of 4
#> $ x.tr: num [1:500, 1:2000] 0.889 -0.329 1.58 0.213 0.214 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
#> $ y.tr: int [1:500] 2 3 2 1 2 3 3 3 1 2 ...
#> $ x.te: num [1:10, 1:2000] 0.83 -0.555 1.041 -1.267 1.15 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
#> $ y.te: int [1:10] 2 3 2 1 2 2 2 1 2 3Model Fitting
Fit a HTLR model with all default settings:
set.seed(SEED)
system.time(
fit.t <- htlr(dat$x.tr, dat$y.tr)
)
#> user system elapsed
#> 107.473 0.116 19.695
print(fit.t)
#> Fitted HTLR model
#>
#> Data:
#>
#> response: 3-class
#> observations: 500
#> predictors: 2001 (w/ intercept)
#> standardised: TRUE
#>
#> Model:
#>
#> prior dist: t (df = 1, log(w) = -10.0)
#> init state: lasso
#> burn-in: 1000
#> sample: 1000 (posterior sample size)
#>
#> Estimates:
#>
#> model size: 4 (w/ intercept)
#> coefficients: see help('summary.htlr.fit')With another configuration:
set.seed(SEED)
system.time(
fit.t2 <- htlr(X = dat$x.tr, y = dat$y.tr,
prior = htlr_prior("t", df = 1, logw = -20, sigmab0 = 1500),
iter = 4000, init = "bcbc", keep.warmup.hist = T)
)
#> user system elapsed
#> 173.580 0.638 31.048
print(fit.t2)
#> Fitted HTLR model
#>
#> Data:
#>
#> response: 3-class
#> observations: 500
#> predictors: 2001 (w/ intercept)
#> standardised: TRUE
#>
#> Model:
#>
#> prior dist: t (df = 1, log(w) = -20.0)
#> init state: bcbc
#> burn-in: 2000
#> sample: 2000 (posterior sample size)
#>
#> Estimates:
#>
#> model size: 4 (w/ intercept)
#> coefficients: see help('summary.htlr.fit')Model Inspection
Look at the point summaries of posterior of selected parameters:
summary(fit.t2, features = c(1:10, 100, 200, 1000, 2000), method = median)
#> class 2 class 3
#> Intercept -3.4331981899 -1.017781e+00
#> V1 10.7498173904 -5.771760e-02
#> V2 -6.7851857040 -2.069727e-02
#> V3 -0.0906662106 3.071093e+00
#> V4 0.0006032815 6.441610e-03
#> V5 -0.0015691931 8.762247e-04
#> V6 0.0006750577 1.555408e-02
#> V7 -0.0049129322 1.180563e-01
#> V8 -0.0105900490 -6.650417e-05
#> V9 0.0058307368 5.258013e-03
#> V10 0.0641186369 6.134679e-01
#> V100 0.0040313044 -1.092937e-02
#> V200 -0.0070023972 -5.321507e-03
#> V1000 0.0105899311 1.261966e-02
#> V2000 -0.0009776419 5.754737e-03
#> attr(,"stats")
#> [1] "median"Plot interval estimates from posterior draws using bayesplot:
post.t <- as.matrix(fit.t2, k = 2)
## signal parameters
mcmc_intervals(post.t, pars = c("Intercept", "V1", "V2", "V3", "V1000"))
Trace plot of MCMC draws:
as.matrix(fit.t2, k = 2, include.warmup = T) %>%
mcmc_trace(c("V1", "V1000"), facet_args = list("nrow" = 2), n_warmup = 2000)
The coefficient of unrelated features (noise) are not updated during some iterations due to restricted Gibbs sampling Li and Yao (2018), hence the computational cost is greatly reduced.
Make Predictions
A glance at the prediction accuracy:
y.class <- predict(fit.t, dat$x.te, type = "class")
y.class
#> y.pred
#> [1,] 2
#> [2,] 2
#> [3,] 2
#> [4,] 3
#> [5,] 2
#> [6,] 2
#> [7,] 2
#> [8,] 3
#> [9,] 2
#> [10,] 3
print(paste0("prediction accuracy of model 1 = ",
sum(y.class == dat$y.te) / length(y.class)))
#> [1] "prediction accuracy of model 1 = 0.7"
y.class2 <- predict(fit.t2, dat$x.te, type = "class")
print(paste0("prediction accuracy of model 2 = ",
sum(y.class2 == dat$y.te) / length(y.class)))
#> [1] "prediction accuracy of model 2 = 0.7"More details about the prediction result:
predict(fit.t, dat$x.te, type = "response") %>%
evaluate_pred(y.true = dat$y.te)
#> $prob_at_truelabels
#> [1] 0.97878658 0.21548397 0.96428770 0.01749886 0.99980780 0.71747816
#> [7] 0.99999093 0.07889146 0.99191520 0.98473633
#>
#> $table_eval
#> Case ID True Label Pred. Prob 1 Pred. Prob 2 Pred. Prob 3 Wrong?
#> 1 1 2 2.119554e-02 9.787866e-01 1.788045e-05 0
#> 2 2 3 2.321169e-01 5.523992e-01 2.154840e-01 1
#> 3 3 2 3.569284e-02 9.642877e-01 1.946075e-05 0
#> 4 4 1 1.749886e-02 2.982570e-10 9.825011e-01 1
#> 5 5 2 1.041303e-04 9.998078e-01 8.807380e-05 0
#> 6 6 2 2.496472e-01 7.174782e-01 3.287462e-02 0
#> 7 7 2 2.085443e-06 9.999909e-01 6.981432e-06 0
#> 8 8 1 7.889146e-02 3.926230e-04 9.207159e-01 1
#> 9 9 2 8.056565e-03 9.919152e-01 2.823952e-05 0
#> 10 10 3 1.526062e-02 3.045530e-06 9.847363e-01 0
#>
#> $amlp
#> [1] 0.8533691
#>
#> $err_rate
#> [1] 0.3
#>
#> $which.wrong
#> [1] 2 4 8Li, Longhai, and Weixin Yao. 2018. “Fully Bayesian Logistic Regression with Hyper-Lasso Priors for High-Dimensional Feature Selection.” Journal of Statistical Computation and Simulation 88 (14). Taylor & Francis: 2827–51.