Simulation 3: Two-dimensional Non-linear Regression
Dayi Li
2025-04-19
Last updated: 2025-05-16
Checks: 7 0
Knit directory: BOSS_website/
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Data
library(tidyverse)
library(tikzDevice)
library(rstan)
library(INLA)
library(inlabru)
library(modeest)
function_path <- "./code"
output_path <- "./output/sim3"
data_path <- "./data/sim3"
source(paste0(function_path, "/00_BOSS.R"))Consider the following non-linear regression model:
\[\begin{align*} y_i \mid \log(\rho_i) &\overset{ind}{\sim}\mathcal{N}(\log\rho(r_i), \sigma^2), \\ \log\rho(r_i) &= \log\rho_0 - \gamma\log\left\{1 + (r_i/R)^\beta\right\}. \end{align*}\]
We simulate \(n = 200\) data points based on the above model with \(\rho_0 = 10\), \(R = 2\), \(\beta = 2\), \(\gamma = -2.5\), and \(\sigma = 0.5\). The inferential goal is the nuisance parameters \(R\) and \(\beta\).
r <- seq(0, 20, length.out = 50)
beta <- 10
a <- 2
b <- 2
c <- -2.5
set.seed(1234)
Ir <- beta*(1 + (r/a)^b)^c
lr <- log(Ir) + rnorm(length(r), 0, 0.5)
data <- data.frame(r, lr)
ggplot(data, aes(r, lr)) + geom_point() + ylab('y')
inlabru
We first run inlabru to to fit the model. We set the
following priors for the parameters:
\[\begin{align*} \rho_0 \sim \mathcal{N}(0, 1000), \ & R \sim \mathrm{Unif}(0.1, 5), \\ \beta \sim \mathrm{Unif}(0.1, 4), \ \gamma \sim \mathcal{N}(0, & 1000), \ \sigma^2 \sim \mathrm{Inv-Gamma}(1, 10^{-5}). \end{align*}\]
a_fun <- function(u){
qunif(pnorm(u), 0.1, 5)
}
b_fun <- function(u){
qunif(pnorm(u), 0.1, 4)
}
cmp <- ~ a(1, model="linear", mean.linear=0, prec.linear=1) +
b(1, model="linear", mean.linear=0, prec.linear=1) +
c(1) + Intercept(1)
form <- lr ~ Intercept + c*log(1 + (r/a_fun(a))^b_fun(b))
fit <- bru(cmp, formula = form, data = data, family = 'gaussian')BOSS
Now let’s run BOSS.
BOSS-modal
We first specify the (unnormalized) log-posterior for \((R,\beta)\). Note that for this specific problem, the unnormalized log-posterior has a closed-form expression:
# specify the objective function for BOSS: unnormalized log posterior of (R, beta)
eval_func <- function(par, x = r, y = lr){
a <- par[1]
b <- par[2]
n <- length(r)
X <- matrix(cbind(rep(1, n), log(1 + (r/a)^b)), ncol = 2)
Vb <- solve(t(X) %*% X + diag(1/1000, 2))
P <- diag(n) - X %*% Vb %*% t(X)
mlik <- log(det(Vb))/2 - log(1000) + lgamma((n+1)/2) - (n+1)/2*log(1e-5 + t(y) %*% P %*% y/2) -
n/2*log(pi) -5*log(10)
return(mlik)
}BOSS-modal
Next, we run the BOSS algorithm where the stopping criteria is based on the convergence of the posterior mode. Specifically, we check the modal convergence every BO iteration, and consider the convergence statistics of the average \(5\) nearest neighbor distance around the current mode.
set.seed(1234)
res_opt_modal <- BOSS(eval_func, criterion = 'modal', update_step = 5, max_iter = 100, D = 2,
lower = rep(0.1, 2), upper = c(5, 4),
noise_var = 1e-6,
modal_iter_check = 1, modal_check_warmup = 20, modal_k.nn = 5,
modal_eps = 0.01,
initial_design = 5, delta = 0.01^2,
optim.n = 5, optim.max.iter = 100)
save(res_opt_modal, file = paste0(output_path, "/BOSS_modal_sim3.rda"))The above BOSS algorithm with modal convergence
criterion converged in \(91\)
iterations.
Let’s check the convergence diagnostic plot:
load(paste0(output_path, "/BOSS_modal_sim3.rda"))
plot(res_opt_modal$modal_result$modal_max_diff ~ res_opt_modal$modal_result$i, type = 'o', xlab = 'Iteration', ylab = 'Modal Stats')
MCMC
Lastly, we implement the MCMC-based method using stan to
obtain the oracle.
set.seed(1234)
MCMC_fit <- stan(
file = "code/nlreg.stan", # Stan program
data = list(x = r, y = lr, N = length(r)), # named list of data
chains = 4, # number of Markov chains
warmup = 3000, # number of warmup iterations per chain
iter = 30000, # total number of iterations per chain
cores = 4, # number of cores (could use one per chain)
algorithm = 'NUTS')
# thin the samples fo plotting
MCMC_samp <- as.data.frame(MCMC_fit)
#MCMC_samp_thin <- MCMC_samp[seq(1, 76000, by = 8),]
save(MCMC_samp, file = paste0(output_path, "/MCMC_sim3.rda"))Results Comparison
Marginal Posterior Distributions
We first compare the marginal posterior distributions of \(R\) and \(\beta\) from inlabru, BOSS,
and MCMC.
# inlabru marginal samples
set.seed(1234)
inla.samples.a <- a_fun(inla.rmarginal(108000, fit$marginals.fixed$a))
inla.samples.b <- b_fun(inla.rmarginal(108000, fit$marginals.fixed$b))# BOSS-modal marginal samples
data_to_smooth <- list()
unique_data <- unique(data.frame(x = res_opt_modal$result$x, y = res_opt_modal$result$y))
data_to_smooth$x <- as.matrix(dplyr::select(unique_data, -y))
data_to_smooth$y <- (unique_data$y - mean(unique_data$y))
square_exp_cov <- square_exp_cov_generator_nd(length_scale = res_opt_modal$length_scale, signal_var = res_opt_modal$signal_var)
surrogate <- function(xvalue, data_to_smooth, cov){
predict_gp(data_to_smooth, x_pred = xvalue, choice_cov = cov, noise_var = 1e-6)$mean
}
ff <- list()
ff$fn <- function(x) as.numeric(surrogate(x, data_to_smooth = data_to_smooth, cov = square_exp_cov))
x.1 <- (seq(from = 0.1, to = 5, length.out = 500) - 0.1)/4.9
x.2 <- (seq(from = 0.1, to = 4, length.out = 500) - 0.1)/3.9
x_vals <- expand.grid(x.1, x.2)
names(x_vals) <- c('x.1','x.2')
x_original <- t(t(x_vals)*(c(5, 4) - c(0.1, 0.1)) + c(0.1, 0.1))
fn_vals <- apply(x_vals, 1, function(x) ff$fn(x = matrix(x, ncol = 2))) + mean(unique_data$y)
fn_true <- apply(x_original, 1, function(x) eval_func(x))
# normalize
lognormal_const <- log(sum(exp(fn_vals))*0.0098*0.0078)
lognormal_const_true <- log(sum(exp(fn_true))*0.0098*0.0078)
post_x_modal <- data.frame(x_original, pos = exp(fn_vals - lognormal_const))
post_x_true <- data.frame(x_original, pos = exp(fn_true - lognormal_const_true))
save(post_x_modal, file = paste0(output_path, "/BOSS_post.rda"))
save(post_x_true, file = paste0(output_path, "/true_post.rda"))load(paste0(output_path, "/BOSS_post.rda"))
load(paste0(output_path, "/true_post.rda"))
dx <- unique(post_x_modal$x.1)[2] - unique(post_x_modal$x.1)[1]
dy <- unique(post_x_modal$x.2)[2] - unique(post_x_modal$x.2)[1]
set.seed(123456)
sample_idx <- rmultinom(1:250000, size = 108000, prob = post_x_modal$pos)
sample_x_modal <- data.frame(post_x_modal, n = sample_idx)
sample_idx_true <- rmultinom(1:250000, size = 108000, prob = post_x_true$pos)
sample_x_true <- data.frame(post_x_true, n = sample_idx_true)
samples_BOSS_modal <- data.frame(do.call(rbind, apply(sample_x_modal, 1, function(x) cbind(runif(x[4], x[1], x[1]+dx), runif(x[4], x[2], x[2] + dy)))))
samples_true <- data.frame(do.call(rbind, apply(sample_x_true, 1, function(x) cbind(runif(x[4], x[1], x[1]+dx), runif(x[4], x[2], x[2] + dy)))))
# MCMC marginal samples
load(paste0(output_path, "/MCMC_sim3.rda"))
# Combine all samples together
R_marginal <- data.frame(R = c(inla.samples.a, samples_BOSS_modal[,1], MCMC_samp$a,
samples_true[,1]),
method = rep(c('inlabru', 'BOSS', 'MCMC', 'Grid (Oracle)'),
c(length(inla.samples.a),
length(samples_BOSS_modal[,1]),
length(MCMC_samp$a), length(samples_true[,1]))))
beta_marginal <- data.frame(beta = c(inla.samples.b, samples_BOSS_modal[,2], MCMC_samp$b,
samples_true[,2]),
method = rep(c('inlabru', 'BOSS', 'MCMC', 'Grid (Oracle)'),
c(length(inla.samples.b),
length(samples_BOSS_modal[,2]),
length(MCMC_samp$b), length(samples_true[,2]))))Plot the marginal posterior densities
#tikz(file = "R_marginal.tex", standAlone=T, width = 4, height = 3)
ggplot(R_marginal, aes(R)) + geom_density(aes(color = method), show_guide = F) +
stat_density(aes(x = R, colour = method),
geom="line",position="identity") + theme_minimal() + xlab('$R$')Warning: The `show_guide` argument of `layer()` is deprecated as of ggplot2 2.0.0.
ℹ Please use the `show.legend` argument instead.
This warning is displayed once every 8 hours.
Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
generated.
#dev.off()
#system('pdflatex R_marginal.tex')#tikz(file = "beta_marginal.tex", standAlone=T, width = 4, height = 3)
ggplot(beta_marginal, aes(beta)) + geom_density(aes(color = method), show_guide = F) +
stat_density(aes(x = beta, colour = method),
geom="line",position="identity") + theme_minimal() + xlab('$\\beta$')
#dev.off()
#system('pdflatex beta_marginal.tex')Joint Posterior Distribution
We now compare the results of the posterior distributions from
inlabru, modal-based BOSS, and AGHQ-based BOSS, and
MCMC.
inlabru joint posterior distribution:
# get joint posterior of (R, beta) from inlabru
joint_samp <- inla.posterior.sample(10000, fit, selection = list(a = 1, b = 1), seed = 12345)
joint_samp <- do.call('rbind', lapply(joint_samp, function(x) matrix(x$latent, ncol = 2)))
inla.joint.samps <- data.frame(a = a_fun(joint_samp[,1]), b = b_fun(joint_samp[,2]))
# plot joint posterior of (R, beta) from inlabru
#tikz(file = "joint_post_R_beta_inlabru.tex", standAlone=T, width = 4, height = 3)
ggplot(inla.joint.samps, aes(a, b)) + stat_density_2d(
geom = "raster",
aes(fill = after_stat(density)), n = 300,
contour = FALSE) +
geom_point(data = data.frame(a = a_fun(fit$summary.fixed$mode[1]), b = b_fun(fit$summary.fixed$mode[2])), color = 'red', shape = 1, size =0.5) +
geom_point(data = data.frame(a = 2, b = 2), color = 'orange', size =0.5) +
coord_fixed() + scale_fill_viridis_c(name = 'Density') + theme_minimal() + xlab('$R$') + ylab('$\\beta$') + xlim(c(0.1, 5)) + ylim(c(0.1, 4))
#dev.off()
#system('pdflatex joint_post_R_beta_inlabru.tex')BOSS joint posterior distribution:
# plot joint posterior of (R, beta) from BOSS
#tikz(file = "joint_post_R_beta.tex", standAlone=T, width = 4, height = 3)
ggplot(post_x_modal, aes(x.1,x.2)) + geom_raster(aes(fill = (pos))) +
geom_point(data = data.frame(x.1 = post_x_modal$x.1[which.max(post_x_modal$pos)], x.2 = post_x_modal$x.2[which.max(post_x_modal$pos)]), color = 'red', shape = 1, size =0.5) +
geom_point(data = data.frame(x.1 = 2, x.2 = 2), color = 'orange', size =0.5) + coord_fixed() + scale_fill_viridis_c(name = 'Density') + theme_minimal() + xlab('$R$') + ylab('$\\beta$')
#dev.off()
#system('pdflatex joint_post_R_beta.tex')MCMC joint posterior distribution:
#tikz(file = "joint_post_R_beta_MCMC.tex", standAlone=T, width = 4, height = 3)
ggplot(MCMC_samp, aes(a, b)) + stat_density_2d(
geom = "raster",
aes(fill = after_stat(density)), n = 300,
contour = FALSE) +
geom_point(data = data.frame(a = post_x_modal$x.1[which.max(post_x_modal$pos)], b = post_x_modal$x.2[which.max(post_x_modal$pos)]), color = 'red', shape = 1, size =0.5) +
geom_point(data = data.frame(a = 2, b = 2), color = 'orange', size =0.5) + coord_fixed() + scale_fill_viridis_c(name = 'Density') + theme_minimal() + xlab('$R$') + ylab('$\\beta$') + xlim(c(0.1, 5)) + ylim(c(0.1, 4))
| Version | Author | Date |
|---|---|---|
| 2cfedcf | david.li | 2025-05-13 |
#dev.off()
#system('pdflatex joint_post_R_beta_MCMC.tex')True joint posterior distribution (grid):
# plot joint posterior of (R, beta) from BOSS
#tikz(file = "joint_post_R_beta_oracle.tex", standAlone=T, width = 4, height = 3)
ggplot(post_x_true, aes(x.1,x.2)) + geom_raster(aes(fill = (pos))) +
geom_point(data = data.frame(x.1 = post_x_true$x.1[which.max(post_x_true$pos)], x.2 = post_x_true$x.2[which.max(post_x_true$pos)]), color = 'red', shape = 1, size =0.5) +
geom_point(data = data.frame(x.1 = 2, x.2 = 2), color = 'orange', size =0.5) + coord_fixed() + scale_fill_viridis_c(name = 'Density') + theme_minimal() + xlab('$R$') + ylab('$\\beta$')
#dev.off()
#system('pdflatex joint_post_R_beta_oracle.tex')From the above results, it is clear that BOSS is much better at
depicting the joint posterior distribution than inlabru.
The joint distribution from inlabru is simply the product
of the marginal distribution, which completely ignores the more complex
structures in the joint posterior. Interestingly though, BOSS is also
much better than HMC. The posterior samples from HMC is significantly
oversampling from the tail regions. This is likely due to highly warped
geometry of the posterior and the un-optimized step number and step size
used in the NUTS sampler.
Does Starting Design Points Matter?
We here check if BOSS is robust towards the starting design points. We run BOSS for \(20\) times with different initial design points and run until the modal statistics have reached below \(\epsilon =0.01\).
res_list <- vector('list', 20)
for (i in 1:20) {
set.seed(i)
res_list[[i]] <- BOSS(eval_func, criterion = 'modal', update_step = 5, max_iter = 100, D = 2,
lower = rep(0.1, 2), upper = c(5, 4),
noise_var = 1e-6,
modal_iter_check = 1, modal_check_warmup = 20, modal_k.nn = 5,
modal_eps = 0.01,
initial_design = 5, delta = 0.01^2,
optim.n = 5, optim.max.iter = 100)
}
save(res_list, file = paste0(output_path, "/BOSS_robust_sim3.rda"))load(paste0(output_path, "/BOSS_robust_sim3.rda"))
sample_list <- vector('list', 20)
for(i in 1:20){
# BOSS-modal marginal samples
data_to_smooth <- list()
unique_data <- unique(data.frame(x = res_list[[i]]$result$x, y = res_list[[i]]$result$y))
data_to_smooth$x <- as.matrix(dplyr::select(unique_data, -y))
data_to_smooth$y <- (unique_data$y - mean(unique_data$y))
square_exp_cov <- square_exp_cov_generator_nd(length_scale = res_list[[i]]$length_scale, signal_var = res_list[[i]]$signal_var)
surrogate <- function(xvalue, data_to_smooth, cov){
predict_gp(data_to_smooth, x_pred = xvalue, choice_cov = cov, noise_var = 1e-6)$mean
}
ff <- list()
ff$fn <- function(x) as.numeric(surrogate(x, data_to_smooth = data_to_smooth, cov = square_exp_cov))
x.1 <- (seq(from = 0.1, to = 5, length.out = 300) - 0.1)/4.9
x.2 <- (seq(from = 0.1, to = 4, length.out = 300) - 0.1)/3.9
x_vals <- expand.grid(x.1, x.2)
names(x_vals) <- c('x.1','x.2')
x_original <- t(t(x_vals)*(c(5, 4) - c(0.1, 0.1)) + c(0.1, 0.1))
fn_vals <- apply(x_vals, 1, function(x) ff$fn(x = matrix(x, ncol = 2))) + mean(unique_data$y)
# normalize
lognormal_const <- log(sum(exp(fn_vals))*0.0098*0.0078*25/9)
post_x_modal <- data.frame(x_original, pos = exp(fn_vals - lognormal_const))
dx <- unique(post_x_modal$x.1)[2] - unique(post_x_modal$x.1)[1]
dy <- unique(post_x_modal$x.2)[2] - unique(post_x_modal$x.2)[1]
set.seed(123456)
sample_idx <- rmultinom(1:90000, size = 49500, prob = post_x_modal$pos)
sample_x_modal <- data.frame(post_x_modal, n = sample_idx)
sample_list[[i]] <- data.frame(do.call(rbind, apply(sample_x_modal, 1, function(x) cbind(runif(x[4], x[1], x[1]+dx), runif(x[4], x[2], x[2] + dy)))))
}
save(sample_list, file = paste0(output_path, "/BOSS_sample_small_n_sim3.rda"))load(paste0(output_path, "/BOSS_sample_small_n_sim3.rda"))
p_R <- ggplot(samples_true, aes(x = X1)) + theme_minimal() + xlab('$R$')
for (i in 1:20) {
p_R <- p_R + geom_density(data = sample_list[[i]], aes(x = X1), color = 'red', alpha = 0.1)
}
p_R <- p_R + geom_density(color = 'black', size = 1.5) Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.
This warning is displayed once every 8 hours.
Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
generated.#tikz(file = "marginal_R_robust.tex", standAlone=T, width = 4, height = 3)
p_R
| Version | Author | Date |
|---|---|---|
| 2cfedcf | david.li | 2025-05-13 |
#dev.off()
#system('pdflatex marginal_R_robust.tex')
p_b <- ggplot(samples_true, aes(x = X2)) + theme_minimal() + xlab('$\\beta$')
for (i in 1:20) {
p_b <- p_b + geom_density(data = sample_list[[i]], aes(x = X2), color = 'red', alpha = 0.1)
}
p_b <- p_b + geom_density(color = 'black', size = 1.5)
#tikz(file = "marginal_beta_robust.tex", standAlone=T, width = 4, height = 3)
p_b
| Version | Author | Date |
|---|---|---|
| 2cfedcf | david.li | 2025-05-13 |
#dev.off()
#system('pdflatex marginal_beta_robust.tex')In the above, black density comes from the grid-based oracle posterior while red ones come from BOSS. We can see that BOSS is highly robust to the initial design points selection.
sessionInfo()R version 4.4.1 (2024-06-14)
Platform: aarch64-apple-darwin20
Running under: macOS 15.0
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/Toronto
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] modeest_2.4.0 inlabru_2.11.1 fmesher_0.1.7
[4] INLA_24.06.27 sp_2.1-4 Matrix_1.7-0
[7] rstan_2.32.6 StanHeaders_2.32.10 tikzDevice_0.12.6
[10] lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1
[13] dplyr_1.1.4 purrr_1.0.2 readr_2.1.5
[16] tidyr_1.3.1 tibble_3.2.1 ggplot2_3.5.1
[19] tidyverse_2.0.0 workflowr_1.7.1
loaded via a namespace (and not attached):
[1] mnormt_2.1.1 DBI_1.2.3 gridExtra_2.3
[4] inline_0.3.19 rlang_1.1.4 magrittr_2.0.3
[7] clue_0.3-65 git2r_0.33.0 matrixStats_1.4.1
[10] e1071_1.7-16 compiler_4.4.1 getPass_0.2-4
[13] loo_2.8.0 callr_3.7.6 vctrs_0.6.5
[16] rmutil_1.1.10 crayon_1.5.3 pkgconfig_2.0.3
[19] fastmap_1.2.0 labeling_0.4.3 utf8_1.2.4
[22] promises_1.3.0 rmarkdown_2.28 tzdb_0.4.0
[25] ps_1.8.0 MatrixModels_0.5-3 xfun_0.47
[28] cachem_1.1.0 jsonlite_1.8.9 highr_0.11
[31] later_1.3.2 parallel_4.4.1 cluster_2.1.6
[34] R6_2.5.1 bslib_0.8.0 stringi_1.8.4
[37] rpart_4.1.23 numDeriv_2016.8-1.1 jquerylib_0.1.4
[40] Rcpp_1.0.13 knitr_1.48 filehash_2.4-6
[43] httpuv_1.6.15 splines_4.4.1 timechange_0.3.0
[46] tidyselect_1.2.1 rstudioapi_0.16.0 yaml_2.3.10
[49] timeDate_4041.110 codetools_0.2-20 processx_3.8.4
[52] pkgbuild_1.4.4 lattice_0.22-6 plyr_1.8.9
[55] withr_3.0.1 evaluate_1.0.0 stable_1.1.6
[58] sf_1.0-19 units_0.8-5 proxy_0.4-27
[61] RcppParallel_5.1.10 pillar_1.9.0 whisker_0.4.1
[64] KernSmooth_2.23-24 stats4_4.4.1 sn_2.1.1
[67] generics_0.1.3 rprojroot_2.0.4 hms_1.1.3
[70] munsell_0.5.1 scales_1.3.0 timeSeries_4041.111
[73] class_7.3-22 glue_1.7.0 statip_0.2.3
[76] tools_4.4.1 spatial_7.3-17 fBasics_4041.97
[79] fs_1.6.4 grid_4.4.1 QuickJSR_1.6.0
[82] colorspace_2.1-1 cli_3.6.3 fansi_1.0.6
[85] viridisLite_0.4.2 gtable_0.3.5 stabledist_0.7-2
[88] sass_0.4.9 digest_0.6.37 classInt_0.4-10
[91] farver_2.1.2 htmltools_0.5.8.1 lifecycle_1.0.4
[94] httr_1.4.7 MASS_7.3-61