Simulation 1: Inference of the Unknown Periodicity
Ziang Zhang
2025-04-15
Last updated: 2025-04-16
Checks: 7 0
Knit directory: BOSS_website/
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | 8418236 | Ziang Zhang | 2025-04-16 | workflowr::wflow_publish("analysis/sim1.Rmd") |
| Rmd | fd5cfc6 | Ziang Zhang | 2025-04-15 | update the code |
| html | d6fe802 | Ziang Zhang | 2025-04-15 | Build site. |
| Rmd | 54c40ce | Ziang Zhang | 2025-04-15 | workflowr::wflow_publish("analysis/sim1.Rmd") |
| Rmd | c92efe8 | Ziang Zhang | 2025-04-15 | add result |
| 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"))Simulate the data:
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 |
Define the objective function:
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)
}
# Apply smoothing spline to smooth the grid result
surrogate_ss <- function(xvalue, data_to_smooth){
predict(ss(x = data_to_smooth$x, y = data_to_smooth$y, df = length(unique(data_to_smooth$x)), m = 2, all.knots = TRUE), x = xvalue)$y
}
surrogate_gp <- 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
)$mean
}Exact Grid Implementation
Set up a dense grid:
x_vals <- seq(lower, upper, by = 0.005)Compute the objective function on the grid:
begin_time <- Sys.time()
total <- length(x_vals)
pb <- txtProgressBar(min = 0, max = total, style = 3)
exact_vals <- c()
for (i in 1:total) {
xi <- x_vals[i]
exact_vals <- c(exact_vals, eval_once(xi))
setTxtProgressBar(pb, i)
}
close(pb)
exact_grid_result <- data.frame(x = x_vals, exact_vals = exact_vals)
exact_grid_result$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"))Take a quick look at the result:
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
Let’s assess the performance of the BOSS algorithm with different choices of \(B\).
eval_num <- seq(from = 10, to = 80, by = 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, optim.n = 10,
update_step = 5, max_iter = (eval_number - 5),
opt.lengthscale.grid = 100,
opt.grid = nrow(exact_grid_result),
# turning off AGHQ check
AGHQ_iter_check = Inf, AGHQ_eps = 0,
delta = 0.01, noise_var = noise_var,
lower = lower, upper = upper,
initial_design = 5)
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$fn <- function(x) as.numeric(surrogate_ss(x, data_to_smooth = data_to_smooth))
ff$fn <- function(x) as.numeric(surrogate_gp(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"))Visualize the results:
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)
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$pos))
}
# show some plots
plot_list[[2]]
plot_list[[5]]
plot_list[[15]]
Use KL and KS statistics to assess the result:
#### Compute the KL distance:
Compute_KL <- function(x, qx, px){
to_kept <- which(px > 0)
x <- x[to_kept]
qx <- qx[to_kept]
px <- px[to_kept]
# px <- px + .Machine$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)
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