Simulation 1: Inference of the Unknown Periodicity
Ziang Zhang
2025-04-15
Last updated: 2025-05-09
Checks: 7 0
Knit directory: BOSS_website/
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|---|---|---|---|---|
| Rmd | 8aba532 | Ziang Zhang | 2025-05-09 | workflowr::wflow_publish("analysis/sim1.Rmd") |
| Rmd | 88e29b3 | Ziang Zhang | 2025-04-30 | update boss_kl |
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Data
library(BayesGP)
library(tidyverse)
library(npreg)
function_path <- "./code"
output_path <- "./output/sim1"
data_path <- "./data/sim1"
source(paste0(function_path, "/00_BOSS.R"))We simulate \(n = 100\) data with true periodicity \(a = 1.5\): \[ \begin{aligned} y_i|\mu_i &\overset{ind}{\sim} \text{Poisson}(\mu_i), \\ \log(\mu_i) =& 1 + 0.5 \cos\bigg(\frac{2\pi x_i}{1.5}\bigg) - 1.3 \sin\bigg(\frac{2\pi x_i}{1.5}\bigg) + \\ & 1.1 \cos\bigg(\frac{4\pi x_i}{1.5}\bigg) + 0.3 \sin\bigg(\frac{4\pi x_i}{1.5}\bigg) + \epsilon_i, \\ \epsilon_i &\overset{iid}{\sim} \mathcal{N}(0,4). \end{aligned} \] The covariates \(\boldsymbol{x} = \{x_i\}_{i=1}^n\) are independently simulated from \(\text{Unif}[0,10]\).
lower = 0.5; upper = 4.5; a = 1.5; noise_var = 1e-6
### Simulate data:
set.seed(123)
n <- 100
x <- runif(n = n, min = 0, max = 10)
true_func <- function(x, true_alpha = 1.5){1 +
0.5 * cos((2*pi*x)/true_alpha) - 1.3 * sin((2*pi*x)/true_alpha) +
1.1 * cos((4*pi*x)/true_alpha) + 0.3 * sin((4*pi*x)/true_alpha)}
log_mu <- true_func(x) + rnorm(n, sd = 2)
y <- rpois(n = n, lambda = exp(log_mu))
data <- data.frame(y = y, x = x, indx = 1:n, log_mu = log_mu)
plot(y ~ x, type = "p", data = arrange(data, x))
| Version | Author | Date |
|---|---|---|
| facb4d4 | Ziang Zhang | 2025-04-15 |
Assume the prior is \(\alpha \sim N(3, 0.5^2)\), we then define the objective function needed for BOSS:
log_prior <- function(alpha){
dnorm(x = alpha, mean = 3, log = T, sd = 0.5)
}
eval_once <- function(alpha){
a_fit <- (2*pi)/alpha
x <- data$x
data$cosx <- cos(a_fit * x)
data$sinx <- sin(a_fit * x)
data$cos2x <- cos(2*a_fit * x)
data$sin2x <- sin(2*a_fit * x)
mod <- model_fit(formula = y ~ cosx + sinx + cos2x + sin2x + f(x = indx, model = "IID",
sd.prior = list(param = 1)),
data = data, method = "aghq", family = "Poisson", aghq_k = 4
)
(mod$mod$normalized_posterior$lognormconst) + log_prior(alpha)
}
surrogate <- function(xvalue, data_to_smooth, choice_cov) {
predict_gp(
data = data_to_smooth,
x_pred = matrix(xvalue, ncol = 1),
choice_cov = choice_cov,
noise_var = noise_var
)$mean
}Exact Grid Implementation
First, as an oracle approach, we set up a dense grid on \([0.5,4.5]\):
x_vals <- seq(lower, upper, by = 0.005)Compute the objective function on the grid:
begin_time <- Sys.time()
total <- length(x_vals)
pb <- txtProgressBar(min = 0, max = total, style = 3)
exact_vals <- c()
for (i in 1:total) {
xi <- x_vals[i]
exact_vals <- c(exact_vals, eval_once(xi))
setTxtProgressBar(pb, i)
}
close(pb)
exact_grid_result <- data.frame(x = x_vals, exact_vals = exact_vals)
exact_grid_result$exact_vals <- exact_grid_result$exact_vals - max(exact_grid_result$exact_vals)
exact_grid_result$fx <- exp(exact_grid_result$exact_vals)
end_time <- Sys.time()
end_time - begin_time
# Calculate the differences between adjacent x values
dx <- diff(exact_grid_result$x)
# Compute the trapezoidal areas and sum them up
integral_approx <- sum(0.5 * (exact_grid_result$fx[-1] + exact_grid_result$fx[-length(exact_grid_result$fx)]) * dx)
exact_grid_result$pos <- exact_grid_result$fx / integral_approx
plot(exact_grid_result$x, exact_grid_result$pos, type = "l", col = "red", xlab = "x (0-10)", ylab = "density", main = "Posterior")
abline(v = a, col = "purple")
grid()
save(exact_grid_result, file = paste0(output_path, "/exact_grid_result.rda"))We can take a quick look at the posterior density obtained from the exact grid. Because of the strong prior centered at \(\alpha = 3\), the posterior density is not exactly unimodal at the true value \(1.5\).
load(paste0(output_path, "/exact_grid_result.rda"))
plot(x = exact_grid_result$x, y = exact_grid_result$pos, col = "black", cex = 0.5, type = "l",
xlab = "x", ylab = "density", main = "Posterior Density", lwd = 2)
abline(v = exact_grid_result$x[which.max(exact_grid_result$exact_vals)], col = "green", lty = "dashed")
abline(v = exact_grid_result$x[which.max(exact_grid_result$exact_vals)], col = "blue", lty = "dashed")
abline(v = a, col = "purple", lty = "dashed")
grid()
| Version | Author | Date |
|---|---|---|
| facb4d4 | Ziang Zhang | 2025-04-15 |
BOSS Implementation
Now, let’s assess the performance of the BOSS algorithm with different choices of \(B\), ranging from \(10\) to \(80\).
eval_num <- seq(10, 80, by = 5)
# Initialize BOSS with 3 equally spaced design points
initial_design <- 5Running the BOSS algorithm at each \(B\):
set.seed(123)
n_grid <- nrow(exact_grid_result); optim.n = 20
objective_func <- eval_once
result_ad <- BOSS(
func = objective_func,
update_step = 5, max_iter = (max(eval_num) - initial_design),
optim.n = optim.n,
delta = 0.01, noise_var = noise_var,
lower = lower, upper = upper,
# Checking KL convergence
criterion = "KL", KL.grid = 3000, KL_check_warmup = 5, KL_iter_check = 5, KL_eps = 0,
initial_design = initial_design
)
BO_result_list <- list()
BO_result_original_list <- list()
for (i in 1:length(eval_num)) {
eval_number <- eval_num[i]
result_ad_selected <- list(x = result_ad$result$x[1:eval_number, , drop = F],
x_original = result_ad$result$x_original[1:eval_number, , drop = F],
y = result_ad$result$y[1:eval_number])
# Optimize the hyperparameter
opt_hyper <- opt_hyperparameters(result_ad_selected, optim.n = optim.n)
result_ad_selected$length_scale <- opt_hyper$length_scale
result_ad_selected$signal_var <- opt_hyper$signal_var
# Compute final surrogate
data_to_smooth <- result_ad_selected
data_to_smooth$y <- data_to_smooth$y - mean(data_to_smooth$y)
BO_result_original_list[[i]] <- data_to_smooth
ff <- list()
ff$fn <- function(x) as.numeric(surrogate(x, data_to_smooth = data_to_smooth, choice_cov = square_exp_cov_generator_nd(length_scale = result_ad_selected$length_scale, signal_var = result_ad_selected$signal_var)))
x_vals <- (seq(from = lower, to = upper, length.out = n_grid) - lower)/(upper - lower)
fn_vals <- sapply(x_vals, ff$fn)
obj <- function(x) {exp(ff$fn(x))}
lognormal_const <- log(integrate(obj, lower = 0, upper = 1, subdivisions = 1000)$value)
post_x <- data.frame(y = x_vals, pos = exp(fn_vals - lognormal_const))
BO_result_list[[i]] <- data.frame(x = (lower + x_vals*(upper - lower)), pos = post_x$pos /(upper - lower))
}
save(BO_result_list, file = paste0(output_path, "/BO_result_list.rda"))
save(BO_result_original_list, file = paste0(output_path, "/BO_result_original_list.rda"))
save(result_ad, file = paste0(output_path, "/result_ad.rda"))Take a look at the convergence diagnostic of the BOSS algorithm:
load(paste0(output_path, "/result_ad.rda"))
plot(result_ad$KL_result$KL ~ result_ad$KL_result$i, type = "o", xlab = "iteration", ylab = "KL Stats")
The KL statistics appears to be stable after \(60\) iterations of BO, indicating that the BOSS algorithm likely converges.
Let’s compare the results from the BOSS algorithm with the exact grid result:
load(paste0(output_path, "/BO_result_list.rda"))
load(paste0(output_path, "/BO_result_original_list.rda"))
plot_list <- list()
for (i in 1:length(eval_num)) {
plot_list[[i]] <- ggplot() +
geom_line(data = BO_result_list[[i]], aes(x = x, y = pos), color = "red", size = 1) +
geom_line(data = exact_grid_result, aes(x = x, y = pos), color = "black", size = 1, linetype = "dashed") +
ggtitle(paste0("Comparison Posterior Density: B = ", eval_num[i])) +
xlab("Value") +
ylab("Density") +
theme_minimal() +
theme(text = element_text(size = 10), axis.text = element_text(size = 15)) + # only change the lab and axis text size
coord_cartesian(ylim = c(0, max(exact_grid_result$pos)))
}B = 10
plot_list[[1]]
B = 30
plot_list[[5]]
Comparison of KL and KS statistics
To assess the accuracy of BOSS, we will compute the KL and KS statistics comparing to the posterior from oracle approach:
#### Compute the KL distance:
KL_vec <- c()
for (i in 1:length(eval_num)) {
KL_vec[i] <- Compute_KL(x = exact_grid_result$x, px = exact_grid_result$pos, qx = BO_result_list[[i]]$pos)
}
plot((KL_vec) ~ eval_num, type = "o", ylab = "KL", xlab = "eval number: B", cex.lab = 1, cex.axis = 1)
| Version | Author | Date |
|---|---|---|
| 761276f | Ziang Zhang | 2025-04-28 |
#### Compute the KS distance:
KS_vec <- c()
for (i in 1:length(eval_num)) {
KS_vec[i] <- Compute_KS(x = exact_grid_result$x, px = exact_grid_result$pos, qx = BO_result_list[[i]]$pos)
}
plot((KS_vec) ~ eval_num, type = "o", ylab = "KS", xlab = "eval number: B", cex.lab = 1, cex.axis = 1)
| Version | Author | Date |
|---|---|---|
| 761276f | Ziang Zhang | 2025-04-28 |
This is the KL and KS distance between BOSS and the exact grid result for this particular replication. To more robustly assess the performance, let’s consider the KL and KS distance over \(10\) independent replications.
simulate_once <- function(seed){
set.seed(seed)
n <- 100; x <- runif(n = n, min = 0, max = 10)
true_func <- function(x, true_alpha = 1.5) {
1 +
0.5 * cos((2 * pi * x) / true_alpha) - 1.3 * sin((2 * pi * x) / true_alpha) +
1.1 * cos((4 * pi * x) / true_alpha) + 0.3 * sin((4 * pi * x) / true_alpha)
}
log_mu <- true_func(x) + rnorm(n, sd = 2)
y <- rpois(n = n, lambda = exp(log_mu))
data <- data.frame(
y = y,
x = x,
indx = 1:n,
log_mu = log_mu
)
# Define the objective function
eval_once <- function(alpha){
a_fit <- (2*pi)/alpha
x <- data$x
data$cosx <- cos(a_fit * x)
data$sinx <- sin(a_fit * x)
data$cos2x <- cos(2*a_fit * x)
data$sin2x <- sin(2*a_fit * x)
mod <- model_fit(formula = y ~ cosx + sinx + cos2x + sin2x + f(x = indx, model = "IID",
sd.prior = list(param = 1)),
data = data, method = "aghq", family = "Poisson", aghq_k = 4
)
(mod$mod$normalized_posterior$lognormconst) + log_prior(alpha)
}
#################################
#################################
######### Fitting Grid: ########
#################################
#################################
#################################
exact_vals <- c()
total <- length(x_vals)
pb <- txtProgressBar(min = 0, max = total, style = 3)
for (i in 1:total) {
xi <- x_vals[i]
exact_vals <- c(exact_vals, eval_once(xi))
setTxtProgressBar(pb, i)
}
close(pb)
exact_grid_result <- data.frame(x = x_vals, exact_vals = exact_vals)
exact_grid_result$exact_vals <- exact_grid_result$exact_vals - max(exact_grid_result$exact_vals)
exact_grid_result$fx <- exp(exact_grid_result$exact_vals)
# Calculate the differences between adjacent x values
dx <- diff(exact_grid_result$x)
# Compute the trapezoidal areas and sum them up
integral_approx <- sum(0.5 * (exact_grid_result$fx[-1] + exact_grid_result$fx[-length(exact_grid_result$fx)]) * dx)
exact_grid_result$pos <- exact_grid_result$fx / integral_approx
#################################
#################################
######### Fitting BOSS: ########
#################################
#################################
#################################
n_grid <- nrow(exact_grid_result); optim.n = 20
objective_func <- eval_once
result_ad <- BOSS(
func = objective_func,
update_step = 5, max_iter = (max(eval_num) - initial_design),
optim.n = optim.n,
delta = 0.01, noise_var = noise_var,
lower = lower, upper = upper,
# Checking KL convergence
criterion = "KL", KL.grid = 3000, KL_check_warmup = 5, KL_iter_check = 5, KL_eps = 0,
initial_design = initial_design
)
BO_result_list <- list()
BO_result_original_list <- list()
for (i in 1:length(eval_num)) {
eval_number <- eval_num[i]
result_ad_selected <- list(x = result_ad$result$x[1:eval_number, , drop = F],
x_original = result_ad$result$x_original[1:eval_number, , drop = F],
y = result_ad$result$y[1:eval_number])
# Optimize the hyperparameter
opt_hyper <- opt_hyperparameters(result_ad_selected, optim.n = optim.n)
result_ad_selected$length_scale <- opt_hyper$length_scale
result_ad_selected$signal_var <- opt_hyper$signal_var
# Compute final surrogate
data_to_smooth <- result_ad_selected
data_to_smooth$y <- data_to_smooth$y - mean(data_to_smooth$y)
BO_result_original_list[[i]] <- data_to_smooth
ff <- list()
ff$fn <- function(x) as.numeric(surrogate(x, data_to_smooth = data_to_smooth, choice_cov = square_exp_cov_generator_nd(length_scale = result_ad_selected$length_scale, signal_var = result_ad_selected$signal_var)))
x_vals <- (seq(from = lower, to = upper, length.out = n_grid) - lower)/(upper - lower)
fn_vals <- sapply(x_vals, ff$fn)
obj <- function(x) {exp(ff$fn(x))}
lognormal_const <- log(integrate(obj, lower = 0, upper = 1, subdivisions = 1000)$value)
post_x <- data.frame(y = x_vals, pos = exp(fn_vals - lognormal_const))
BO_result_list[[i]] <- data.frame(x = (lower + x_vals*(upper - lower)), pos = post_x$pos /(upper - lower))
}
#################################
#################################
######### Compute KS/KL: ########
#################################
#################################
#################################
#### Compute the KL distance:
KL_vec <- c()
for (i in 1:length(eval_num)) {
KL_vec[i] <- Compute_KL(x = exact_grid_result$x,
px = exact_grid_result$pos,
qx = BO_result_list[[i]]$pos)
}
#### Compute the KS distance:
KS_vec <- c()
for (i in 1:length(eval_num)) {
KS_vec[i] <- Compute_KS(x = exact_grid_result$x,
px = exact_grid_result$pos,
qx = BO_result_list[[i]]$pos)
}
return(data.frame(eval_num = eval_num, KL = KL_vec, KS = KS_vec))
}all_result <- data.frame(eval_num = eval_num,
KL = KL_vec,
KS = KS_vec)
seed_vec <- 1:10
for (i in 1:length(seed_vec)) {
seed <- seed_vec[i]
result <- simulate_once(seed = seed)
all_result <- rbind(all_result, result)
}
save(all_result, file = paste0(output_path, "/all_result.rda"))Plot the results across \(10\) independent replications:
load(paste0(output_path, "/all_result.rda"))
median_result <- all_result %>% group_by(eval_num) %>%
summarise(KL = median(KL), KS = median(KS))
lower_result <- all_result %>% group_by(eval_num) %>%
summarise(KL = quantile(KL, 0.1), KS = quantile(KS, 0.1))
upper_result <- all_result %>% group_by(eval_num) %>%
summarise(KL = quantile(KL, 0.9), KS = quantile(KS, 0.9))
plot(median_result$eval_num, median_result$KL, type = "o",
ylab = "KL", xlab = "Evaluation Budget: B",
cex.lab = 1.4, cex.axis = 1.2)
polygon(c(median_result$eval_num, rev(median_result$eval_num)),
c(lower_result$KL, rev(upper_result$KL)),
col = rgb(0.2, 0.2, 0.2, 0.2), border = NA)
| Version | Author | Date |
|---|---|---|
| 761276f | Ziang Zhang | 2025-04-28 |
plot(median_result$eval_num, median_result$KS, type = "o",
ylab = "KS", xlab = "Evaluation Budget: B",
cex.lab = 1.4, cex.axis = 1.2)
polygon(c(median_result$eval_num, rev(median_result$eval_num)),
c(lower_result$KS, rev(upper_result$KS)),
col = rgb(0.2, 0.2, 0.2, 0.2), border = NA)
| Version | Author | Date |
|---|---|---|
| 761276f | Ziang Zhang | 2025-04-28 |
sessionInfo()R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Monterey 12.7.4
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
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/Chicago
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] npreg_1.1.0 lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1
[5] dplyr_1.1.4 purrr_1.0.2 readr_2.1.5 tidyr_1.3.1
[9] tibble_3.2.1 ggplot2_3.5.1 tidyverse_2.0.0 BayesGP_0.1.3
[13] workflowr_1.7.1
loaded via a namespace (and not attached):
[1] sass_0.4.9 utf8_1.2.4 generics_0.1.3 stringi_1.8.4
[5] lattice_0.22-6 hms_1.1.3 digest_0.6.37 magrittr_2.0.3
[9] timechange_0.3.0 evaluate_1.0.1 grid_4.3.1 fastmap_1.2.0
[13] rprojroot_2.0.4 jsonlite_1.8.9 Matrix_1.6-4 processx_3.8.4
[17] whisker_0.4.1 ps_1.8.0 promises_1.3.0 httr_1.4.7
[21] fansi_1.0.6 scales_1.3.0 jquerylib_0.1.4 cli_3.6.3
[25] rlang_1.1.4 munsell_0.5.1 withr_3.0.2 cachem_1.1.0
[29] yaml_2.3.10 tools_4.3.1 tzdb_0.4.0 colorspace_2.1-1
[33] httpuv_1.6.15 vctrs_0.6.5 R6_2.5.1 lifecycle_1.0.4
[37] git2r_0.33.0 fs_1.6.4 pkgconfig_2.0.3 callr_3.7.6
[41] pillar_1.9.0 bslib_0.8.0 later_1.3.2 gtable_0.3.6
[45] glue_1.8.0 Rcpp_1.0.13-1 highr_0.11 xfun_0.48
[49] tidyselect_1.2.1 rstudioapi_0.16.0 knitr_1.48 farver_2.1.2
[53] htmltools_0.5.8.1 labeling_0.4.3 rmarkdown_2.28 compiler_4.3.1
[57] getPass_0.2-4