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library(BayesGP)
library(tidyverse)
library(parallel)
source("code/functions.R")

Introduction

In this tutorial, we introduce the basic steps to fit a sGP model using the seasonal B-spline approximation introduced in Zhang et al, 2024.

For illustration, we use one of the synthetic datasets described in the main paper. The dataset and its corresponding true function are shown in the plots below.

n <- 500
location_of_interest <- seq(0, 10, length.out = 500)
true_f <- function(x){
  if(x < 2){
    return(2*sin(2 * 2 * pi * x) * (3-x))
  } else if (x > 2 && x < 4){
    return(2*sin(2 * 2 * pi * x))
  } else{
    return(2*sin(2 * 2 * pi * x) * (log(x-3) + 1))
  }
}
true_f <- Vectorize(true_f)

set.seed(123)
data <- simulate_data_poisson(func = true_f, n = n, sigma = 0.5, region = c(0,10), offset = 0)

par(mfrow = c(1,2))
plot(data$x, data$y, type = "p", col = "black", 
     pch = 20, cex = 0.5,
     ylab = "y", xlab = "x")
lines(location_of_interest, exp(true_f(location_of_interest)), col = "red", lwd = 2)
plot(location_of_interest, true_f(location_of_interest),
     type = "l", col = "black",
     pch = 20, cex = 0.5,
     ylab = "y", xlab = "x")

Version Author Date
c4bd829 Ziang Zhang 2024-11-21 
par(mfrow = c(1,1))

The hierarchical model we consider is as follows:

\[\begin{equation} \begin{aligned} Y_i|x_i,\xi_i &\sim \text{Poisson}(\lambda_i), \\ \text{log}(\lambda_i) &= g(x_i) + \xi_i, \\ g(x) &\sim \text{sGP}(\sigma_x), \\ \xi_i &\sim \text{N}(0, \sigma_\xi^2). \end{aligned} \end{equation}\]

We assume the sGP prior has a frequency of \(2\) (\(\alpha = 4\pi\)), and use an Exponential prior for the one-step PSD \(\sigma_x(1)\) defined as: \[ \text{P}(\sigma_x(1) > 2) = 0.1. \]

The standard deviation \(\sigma_\xi\) of the observation-level random intercept follows an Exponential prior with a median of \(1\).

To make the computation more efficient, we will use \(10\) equally spaced knots to define the B-spline basis, which will then be used to approximate the sGP prior.

Inference

To make approximate Bayesian inference of the above model, we make use of the BayesGP package:

mod <- BayesGP::model_fit(
    y ~ f(
      x,
      model = "sgp",
      region = c(0,10),
      freq = 2,
      k = 10, # number of knots
      sd.prior = list(param = list(u = 2, alpha = 0.1), h = 1)
    ) +
      f(index, model = "iid", sd.prior = 1),
    data = data,
    family = "Poisson"
  )

We can take a quick look at the posterior summary:

summary(mod)
Here are some posterior/prior summaries for the parameters: 
        name median q0.025 q0.975       prior prior:P1 prior:P2
1  intercept  0.106 -0.055  0.274      Normal        0    1e+03
2    x (PSD)  0.890  0.636  1.342 Exponential        2    1e-01
3 index (SD)  0.485  0.423  0.556 Exponential        1    5e-01
For Normal prior, P1 is its mean and P2 is its variance. 
For Exponential prior, prior is specified as P(theta > P1) = P2.

We can also obtain the posterior of \(g\) at any location of interest:

post_g <- predict(mod, newdata = data.frame(x = location_of_interest), variable = "x")
head(post_g)
           x     q0.025      q0.5   q0.975      mean
1 0.00000000 -0.1867575 0.2723952 0.711317 0.2688137
2 0.02004008  1.2278211 1.6493109 2.052013 1.6462239
3 0.04008016  2.5478229 2.9261264 3.291498 2.9251860
4 0.06012024  3.6668587 4.0243567 4.373277 4.0238335
5 0.08016032  4.5179810 4.8739401 5.217268 4.8721472
6 0.10020040  5.0675664 5.4185435 5.772141 5.4165004

Take a look at the plot of them:

plot(location_of_interest, true_f(location_of_interest),
     type = "l", col = "black",
     pch = 20, cex = 0.5,
     ylab = "y", xlab = "x")
lines(x = location_of_interest, y = (post_g$mean), col = "blue", lwd = 1, lty = 2)
polygon(c(location_of_interest, rev(location_of_interest)),
        c(post_g$q0.025, rev(post_g$q0.975)),
        col = adjustcolor("blue", alpha.f = 0.2), border = NA)
legend("topright", legend = c("True function", "Posterior mean"),
       col = c("black", "blue"), lty = c(1, 2), lwd = c(1, 1))

Version Author Date
c4bd829 Ziang Zhang 2024-11-21

We can also just obtain the raw samples of \(g\) at these locations:

post_g_raw <- predict(mod, newdata = data.frame(x = location_of_interest), variable = "x", only.samples = TRUE)
plot(location_of_interest, true_f(location_of_interest),
     type = "l", col = "black",
     pch = 20, cex = 0.5,
     ylab = "y", xlab = "x")
matlines(location_of_interest, post_g_raw[,2:12], col = "pink", lty = 2, lwd = 0.5)

Version Author Date
c4bd829 Ziang Zhang 2024-11-21 

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] parallel  stats     graphics  grDevices utils     datasets  methods  
[8] base     

other attached packages:
 [1] lubridate_1.9.3 forcats_1.0.0   stringr_1.5.1   dplyr_1.1.4    
 [5] purrr_1.0.2     readr_2.1.5     tidyr_1.3.1     tibble_3.2.1   
 [9] ggplot2_3.5.1   tidyverse_2.0.0 BayesGP_0.1.3   workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] gtable_0.3.6        TMB_1.9.15          xfun_0.48          
 [4] bslib_0.8.0         ks_1.14.3           processx_3.8.4     
 [7] lattice_0.22-6      numDeriv_2016.8-1.1 callr_3.7.6        
[10] tzdb_0.4.0          bitops_1.0-9        vctrs_0.6.5        
[13] tools_4.3.1         ps_1.8.0            generics_0.1.3     
[16] aghq_0.4.1          fansi_1.0.6         cluster_2.1.6      
[19] highr_0.11          pkgconfig_2.0.3     fds_1.8            
[22] KernSmooth_2.23-24  Matrix_1.6-4        data.table_1.16.2  
[25] lifecycle_1.0.4     compiler_4.3.1      git2r_0.33.0       
[28] statmod_1.5.0       munsell_0.5.1       getPass_0.2-4      
[31] mvQuad_1.0-8        httpuv_1.6.15       htmltools_0.5.8.1  
[34] rainbow_3.8         sass_0.4.9          RCurl_1.98-1.16    
[37] yaml_2.3.10         pracma_2.4.4        later_1.3.2        
[40] pillar_1.9.0        jquerylib_0.1.4     whisker_0.4.1      
[43] MASS_7.3-60         cachem_1.1.0        mclust_6.1.1       
[46] tidyselect_1.2.1    digest_0.6.37       mvtnorm_1.3-1      
[49] stringi_1.8.4       splines_4.3.1       pcaPP_2.0-5        
[52] rprojroot_2.0.4     fastmap_1.2.0       grid_4.3.1         
[55] colorspace_2.1-1    cli_3.6.3           magrittr_2.0.3     
[58] utf8_1.2.4          withr_3.0.2         scales_1.3.0       
[61] promises_1.3.0      timechange_0.3.0    rmarkdown_2.28     
[64] httr_1.4.7          deSolve_1.40        hms_1.1.3          
[67] evaluate_1.0.1      knitr_1.48          rlang_1.1.4        
[70] Rcpp_1.0.13-1       hdrcde_3.4          glue_1.8.0         
[73] fda_6.2.0           rstudioapi_0.16.0   jsonlite_1.8.9     
[76] R6_2.5.1            fs_1.6.4