Multinomial Logistic Regression with Heavy-Tailed Priors
Longhai Li and Steven Liu
2020-09-08
1 Data Generation
Load the necessary libraries:
library(HTLR)
library(bayesplot)
#> This is bayesplot version 1.7.1
#> - 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:
# require(corrplot)
cor(dat$X[ , 1:11]) %>% corrplot::corrplot(tl.pos = "n")
Split the data into training and testing sets:
set.seed(SEED)
dat <- split_data(dat$X, dat$y, n.train = 500)
str(dat)
#> List of 4
#> $ x.tr: num [1:500, 1:2000] 0.9157 0.0218 -0.6693 0.4797 0.4862 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
#> $ y.tr: int [1:500] 1 2 1 2 1 3 2 3 2 3 ...
#> $ x.te: num [1:10, 1:2000] 1.031 -0.55 -1.208 -0.858 -1.035 ...
#> ..- attr(*, "dimnames")=List of 2
#> .. ..$ : NULL
#> .. ..$ : chr [1:2000] "V1" "V2" "V3" "V4" ...
#> $ y.te: int [1:10] 2 1 3 1 3 1 1 1 1 12 Model 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
#> 103.543 0.072 18.435
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
#> 180.828 0.363 31.796
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')3 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 -3.2057501605 -0.7572061810
#> V1 10.7863600649 0.3382765258
#> V2 -6.7701256982 -0.3134671887
#> V3 -0.1955792631 3.1621270554
#> V4 -0.0038819957 -0.0089674488
#> V5 0.0008969358 0.0054764104
#> V6 0.0061773755 0.0797936181
#> V7 0.0140647886 0.0186878519
#> V8 0.0002888387 -0.0023150566
#> V9 -0.0038139108 0.0092575854
#> V10 0.0198427504 0.0153933706
#> V100 -0.0131670180 -0.0039492715
#> V200 0.0069327003 -0.0009540371
#> V1000 0.0080409405 -0.0022844461
#> V2000 0.0014780032 0.0024727814
#> 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.
4 Make Predictions
A glance at the prediction accuracy:
y.class <- predict(fit.t, dat$x.te, type = "class")
y.class
#> y.pred
#> [1,] 2
#> [2,] 3
#> [3,] 3
#> [4,] 1
#> [5,] 3
#> [6,] 1
#> [7,] 1
#> [8,] 1
#> [9,] 1
#> [10,] 1
print(paste0("prediction accuracy of model 1 = ",
sum(y.class == dat$y.te) / length(y.class)))
#> [1] "prediction accuracy of model 1 = 0.9"
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 = 0.9"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] 0.979194928 0.008449285 0.859655156 0.967135714 0.995201056 0.903743061
#> [7] 0.974073783 0.523414557 0.674799377 0.755160714
#>
#> $table_eval
#> Case ID True Label Pred. Prob 1 Pred. Prob 2 Pred. Prob 3 Wrong?
#> 1 1 2 0.006681587 9.791949e-01 0.01412348 0
#> 2 2 1 0.008449285 6.275159e-03 0.98527556 1
#> 3 3 3 0.140344799 4.517124e-08 0.85965516 0
#> 4 4 1 0.967135714 9.119180e-04 0.03195237 0
#> 5 5 3 0.004798943 1.613052e-09 0.99520106 0
#> 6 6 1 0.903743061 5.933941e-08 0.09625688 0
#> 7 7 1 0.974073783 5.548914e-06 0.02592067 0
#> 8 8 1 0.523414557 2.844161e-01 0.19216930 0
#> 9 9 1 0.674799377 2.853111e-01 0.03988953 0
#> 10 10 1 0.755160714 7.700381e-05 0.24476228 0
#>
#> $amlp
#> [1] 0.6433173
#>
#> $err_rate
#> [1] 0.1
#>
#> $which.wrong
#> [1] 2Li, 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.