Last updated: 2025-04-16

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Unstaged changes:
    Modified:   code/00_BOSS.R
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Rmd bdb322d Ziang Zhang 2025-04-16 workflowr::wflow_publish("analysis/sim1.Rmd")
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Rmd fd5cfc6 Ziang Zhang 2025-04-15 update the code
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html facb4d4 Ziang Zhang 2025-04-15 Build site.
Rmd 4d2b19c Ziang Zhang 2025-04-15 workflowr::wflow_publish("analysis/sim1.Rmd")

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 \(a = 1.5\):

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 <- c(15, 30, 45, 60, 80)
# Initialize BOSS with 3 equally spaced design points
initial_design <- 5

Running the BOSS algorithm at each \(B\):

objective_func <- eval_once
rel_runtime <- c()
BO_result_list <- list()
BO_result_original_list <- list()
for (i in 1:length(eval_num)) {
  n_grid <- nrow(exact_grid_result)
  eval_number <- eval_num[i]
  begin_time <- Sys.time()
  result_ad <- BOSS(func = objective_func, update_step = 5, max_iter = (eval_number - initial_design),
                       opt.lengthscale.grid = 100, opt.grid = nrow(exact_grid_result),
                       delta = 0.01, noise_var = noise_var,
                       lower = lower, upper = upper,
                       # turning off AGHQ check
                       AGHQ_iter_check = Inf, AGHQ_eps = 0,
                       initial_design = initial_design)

  end_time <- Sys.time()
  rel_runtime[i] <- as.numeric((end_time - begin_time), units = "mins")/1.344585
  
  data_to_smooth <- result_ad$result
  data_to_smooth$y <- data_to_smooth$y - mean(data_to_smooth$y)
  BO_result_original_list[[i]] <- data_to_smooth

  ff <- list()
  ff$fn <- function(x) as.numeric(surrogate(x, data_to_smooth = data_to_smooth, choice_cov = square_exp_cov_generator_nd(length_scale = result_ad$length_scale, signal_var = result_ad$signal_var)))
  x_vals <- (seq(from = lower, to = upper, length.out = n_grid) - lower)/(upper - lower)
  fn_vals <- sapply(x_vals, ff$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(rel_runtime, file = paste0(output_path, "/rel_runtime.rda"))

Let’s compare the first 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"))
load(paste0(output_path, "/rel_runtime.rda"))
plot(rel_runtime ~ eval_num, type = "o", ylab = "rel-runtime", xlab = "eval number: B", cex.lab = 1.0, cex.axis = 1.0)

Version Author Date
4f10511 Ziang Zhang 2025-04-16
d6fe802 Ziang Zhang 2025-04-15
facb4d4 Ziang Zhang 2025-04-15 
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
    lims(y = range(exact_grid_result$pos))
}

B = 15

plot_list[[1]]

B = 30

plot_list[[2]]

Version Author Date
4f10511 Ziang Zhang 2025-04-16
d6fe802 Ziang Zhang 2025-04-15
facb4d4 Ziang Zhang 2025-04-15

B = 80

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:
Compute_KL <- function(x, qx, px){
  to_kept <- which(px > 0)
  x <- x[to_kept]
  qx <- qx[to_kept]
  px <- px[to_kept]
  # px <- px + .Machine$double.eps
  # qx <- qx + .Machine$double.eps
  dx <- diff(x)
  left <- c(0,dx)
  right <- c(dx,0)
  0.5 * sum(left * log(px/qx) * px) + 0.5 * sum(right * log(px/qx) * px)
}
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)

#### Compute the KS distance:
Compute_KS <- function(x, qx, px){
  dx <- c(diff(x),0)
  max(abs(cumsum(qx * dx) - cumsum(px * dx)))
}
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)

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


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