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library(tidyverse)
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ ggplot2   3.5.1     ✔ tibble    3.2.1
✔ lubridate 1.9.3     ✔ tidyr     1.3.1
✔ purrr     1.0.2     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(BayesGP)

Introduction

One important factor that affects the inference of the sGP model is the choice of the prior on its SD parameter \(\sigma\). The suggested way to construct prior on \(\sigma\) is through an Exponential prior on its \(h\)-step predictive SD \(\sigma(h)\), with the form of \[ \text{P}[\sigma(h)>u] = 0.5, \] where \(u\) is prior median.

In this tutorial, we will examine the sensitivity of the sGP model to the choice of the prior on \(\sigma\) by fitting the model to the Lynx dataset with different priors on \(\sigma\).

data <- data.frame(year = seq(1821, 1934, by = 1), logy = log(as.numeric(lynx)), y = as.numeric(lynx))
data$x <- data$year - min(data$year)
x <- data$x
y <- data$y
data_reduced <- data[1:80,]
test_data <- data[-c(1:80),]
### Region of prediction
region_lynx <- c(1821,1960)

Varying the threshold

First, we write a function that takes in the prior median u and fits the sGP model with the corresponding prior on \(\sigma(50)\), and then returns the posterior summary of the fitted model.

fit_once <- function(u, alpha = 0.5){
  pred_SD <- list(u = u, alpha = alpha)
  results_sGP <- BayesGP::model_fit(
    formula = y ~ f(x = year, model = "sgp", k = 30,
                    period = 10,
                    sd.prior = list(param = pred_SD, h = 50), 
                    initial_location = "left", region = region_lynx) +
      f(x = x, model = "IID", sd.prior = list(param = list(u = 1, alpha = 0.5))),
    data = data_reduced,
    family = "poisson")
  pred_g1 <- predict(results_sGP, newdata = data.frame(x = x, year = data$year), variable = "year", include.intercept = T, quantiles = c(0.1,0.9))
  return(pred_g1)
}

Let’s try different values of \(u\):

alpha = 0.5
u_vec = c(0.01, 0.03, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1, 1.5, 2)
pred_summary <- lapply(u_vec, fit_once, alpha = alpha)
pred_means <- do.call(rbind, lapply(pred_summary, function(x) x$mean))

Let’s plot the posterior mean

# Create a color palette from light to dark
color_palette <- colorRampPalette(c("lightblue", "blue"))

# Generate colors for the number of `u_vec`
colors <- color_palette(length(u_vec))

plot(data$year, data$logy, type = "p", col = "black", lwd = 2, xlab = "Year", ylab = "Lynx count", cex = 0.1)
matlines(data$year, t(pred_means), col = colors, lty = 1)
legend("topleft", legend = paste("u =", u_vec), col = colors, lty = 1, cex = 1)

Version Author Date
814c6ba Ziang Zhang 2024-11-26
0218bdf Ziang Zhang 2024-11-21

We could also plot the MSE of the posterior mean for different values of \(u\).

MSEs <- apply(pred_means, 1, function(x) mean((x - data$logy)^2))
plot(u_vec, MSEs, type = "o", col = "blue", lwd = 2, xlab = "u", ylab = "MSE")

Version Author Date
814c6ba Ziang Zhang 2024-11-26
0218bdf Ziang Zhang 2024-11-21

Overall, unless the value of the prior median \(u\) is too small, the MSE is not sensitive to \(u\).

We could similar check how the coverage of the 80% credible interval changes with different values of \(u\).

coverage <- lapply(pred_summary, function(x) mean(data$logy > x$q0.1 & data$logy < x$q0.9))
plot(u_vec, coverage, type = "o", col = "blue", lwd = 2, xlab = "u", ylab = "Coverage", ylim = c(0,1))
abline(h = 0.8, col = "red", lty = "dashed")

Version Author Date
814c6ba Ziang Zhang 2024-11-26
0218bdf 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] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] BayesGP_0.1.3   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 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         highr_0.11         
[19] cluster_2.1.6       pkgconfig_2.0.3     fds_1.8            
[22] Matrix_1.6-4        KernSmooth_2.23-24  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