Fitting sGP Models to the Lynx Dataset: An Intermediate Tutorial
Ziang Zhang
2024-11-20
Last updated: 2024-11-21
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Knit directory: online_tut/
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Introduction:
In this tutorial, we will use the lynx dataset as an
example, which can be accessed directly from R. Let’s load the dataset
and visualize it:
data <- data.frame(year = seq(1821, 1934, by = 1), y = as.numeric(lynx))
data$x <- data$year - min(data$year)
plot(lynx)
| Version | Author | Date |
|---|---|---|
| 8e9829b | Ziang Zhang | 2024-11-20 |
Based on a visual examination of the dataset, we can observe an obvious 10-year quasi-periodic behavior in the lynx count with evolving amplitudes over time. Therefore, we will consider fitting the following model:
\[ \begin{equation} \begin{aligned} y_i|\lambda_i &\sim \text{Poisson}(\lambda_i) ,\\ \log(\lambda_i) &= \eta_i = \beta_0 + g(x_i) + \xi_i,\\ g &\sim \text{sGP}_{\alpha} \big(\sigma\big), \ \alpha = \frac{2\pi}{10}, \\ \xi_i &\sim N(0,\sigma_\xi). \end{aligned} \end{equation} \] Here, each \(y_i\) represents the lynx count, \(x_i\) represents the number of years since 1821, and \(\xi_i\) is an observation-level random intercept to account for overdispersion effect.
Preparation:
To carry out our inference in a fully Bayesian manner, we will load the following required packages:
require(sGPfit)
require(BayesGP)
require(aghq)
require(TMB)
require(tidyverse)
require(Matrix)If you haven’t installed the prototype package sGPfit yet, you can
download it from GitHub using the following command:
devtools::install_git("https://github.com/AgueroZZ/sGPfit").
In this example, we will also use the aghq package for posterior approximation. To utilize it, we need to compile a C++ template for the model and load the output. Execute the following code to compile the C++ file and load the resulting library:
compile(file = "code/tut.cpp")
dyn.load(dynlib("code/tut"))Once the C++ file is compiled and the library is loaded, we can proceed with the remaining inference procedures.
Prior Elicitation:
To specify the priors for the sGP boundary conditions and the intercept parameter, we assume independent normal priors with mean 0 and variance 1000. For the overdispersion parameter \(\sigma_\xi\), we assign an exponential prior with \(P(\sigma_\xi > 1) = 0.01\).
To determine the prior for the standard deviation parameter \(\sigma\) of the sGP, we employ the concept
of predictive standard deviation (PSD). We start with an exponential
prior on the 50 years PSD: \[P(\sigma(50)>1) = 0.01.\] To convert
this prior to the original standard deviation parameter \(\sigma\), we use the
compute_d_step_sGPsd function from the sGPfit
package:
prior_PSD <- list(u = 1, alpha = 0.01)
correction_factor <- sGPfit::compute_d_step_sGPsd(d = 50, a = 2*pi/10)
prior_SD <- list(u = prior_PSD$u/correction_factor, alpha = prior_PSD$alpha)
prior_SD$u
[1] 0.1256637
$alpha
[1] 0.01Based on the above computation, the corresponding exponential prior on the original SD parameter \(\sigma\) is: \[P(\sigma>0.126) = 0.01.\]
Inference of the quasi-periodic function:
First, we need to set up the design matrix X for the two
boundary conditions and the intercept parameter:
x <- data$x
a <- 2*pi/10
X <- as(cbind(cos(a*x),sin(a*x),1), "dgTMatrix")To perform inference for the unknown function \(g\) using the sGP, we have two possible approaches:
- This approach is most efficient when the locations
xare equally spaced. For large datasets with irregularly spaced locations, the second approach using the seasonal B-spline (sB-spline) approximation is more computationally efficient. - The sGP is approximated as \(\tilde{g}k(x) = \sum{i=1}^k w_i \psi_i(x)\), where \({\psi_i, i \in [k]}\) is a set of sB-spline basis functions, and \(\boldsymbol{w} = {w_i, i \in [k]}\) is a set of Gaussian weights.
Inference with the State-Space approach:
To use the state-space representation, we need to create two
additional matrices: \(B\) and \(Q\). The matrix \(B\) is the design matrix of each value of
\([g(x_i),g'(x_i)]\), which will be
(mostly) a diagonal matrix in this case. The matrix \(Q\) is the precision matrix of \([g(x),g'(x)]\), which can be
constructed using the joint_prec_construct function.
n = length(x)
B <- Matrix::Diagonal(n = 2*n)[,1:(2*n)]
B <- B[seq(1,2*n,by = 2),][, -c(1:2)]
Q <- joint_prec_construct(a = a, t_vec = x[-1], sd = 1)
Q <- as(as(Q, "matrix"),"dgTMatrix")With the matrices ready, the model can be fitted using
aghq as follows:
tmbdat <- list(
# Design matrix
B = B,
X = X,
# Precision matrix
P = Q,
logPdet = as.numeric(determinant(Q, logarithm = T)$modulus),
# Response
y = data$y,
# Prior
u = prior_SD$u,
alpha = prior_SD$alpha,
u_over = prior_PSD$u,
alpha_over = prior_PSD$alpha,
betaprec = 0.001
)
tmbparams <- list(
W = c(rep(0, (ncol(B) + ncol(X) + length(data$y)))),
theta = 0,
theta_over = 0
)
ff <- TMB::MakeADFun(
data = tmbdat,
parameters = tmbparams,
random = "W",
DLL = "tut",
silent = TRUE
)
ff$he <- function(w) numDeriv::jacobian(ff$gr,w)
fitted_mod <- aghq::marginal_laplace_tmb(ff,5,c(0,0))Inference with the seasonal-B spline approach:
To use the sB-spline approach, the sGP is approximated as \(\tilde{g}_k(x) = \sum_{i=1}^k w_i \psi_i(x)\), where \(\{\psi_i, i \in [k]\}\) is a set of sB-spline basis functions, and \(\boldsymbol{w} = \{w_i, i \in [k]\}\) is a set of Gaussian weights.
In this case, the design matrix \(B\) will be defined with element \(B_{ij} = \psi_j(x_i)\), which can be
constructed using the Compute_B_sB function. The matrix
\(Q\) will be the precision matrix of
the Gaussian weights \(\boldsymbol{w}\), which can be constructed
using the Compute_Q_sB function. As an example, let’s
consider \(k = 30\) sB-spline basis
functions constructed with equal spacing:
B2 <- Compute_B_sB(x = data$x, a = a, region = range(data$x), k = 30)
B2 <- as(B2,"dgTMatrix")
Q2 <- Compute_Q_sB(a = a, k = 30, region = range(data$x))
Q2 <- as(as(Q2, "matrix"),"dgTMatrix")Once these matrices are constructed, the inference can be carried out in a similar manner as before:
tmbdat <- list(
# Design matrix
B = B2,
X = X,
# Precision matrix
P = Q2,
logPdet = as.numeric(determinant(Q2, logarithm = T)$modulus),
# Response
y = data$y,
# Prior
u = prior_SD$u,
alpha = prior_SD$alpha,
u_over = prior_PSD$u,
alpha_over = prior_PSD$alpha,
betaprec = 0.001
)
tmbparams <- list(
W = c(rep(0, (ncol(B2) + ncol(X) + length(data$y)))),
theta = 0,
theta_over = 0
)
ff <- TMB::MakeADFun(
data = tmbdat,
parameters = tmbparams,
random = "W",
DLL = "tut",
silent = TRUE
)
ff$he <- function(w) numDeriv::jacobian(ff$gr,w)
fitted_mod_sB <- aghq::marginal_laplace_tmb(ff,5,c(0,0))Comparing the results from the two approachs
First, let’s obtain the posterior samples and summary using the state-space representation:
## Posterior samples:
samps1 <- sample_marginal(fitted_mod, M = 3000)
g_samps <- B %*% samps1$samps[1:ncol(B),] + X %*% samps1$samps[(ncol(B) + 1):(ncol(B) + ncol(X)),]## Posterior summary:
mean <- apply(as.matrix(g_samps), MARGIN = 1, mean)
upper <- apply(as.matrix(g_samps), MARGIN = 1, quantile, p = 0.975)
lower <- apply(as.matrix(g_samps), MARGIN = 1, quantile, p = 0.025)Next, let’s plot the posterior results obtained from the state-space representation:
## Plotting
plot(log(data$y) ~ data$x, xlab = "time", ylab = "Posterior of g(x)", ylim = c(3.1,9))
lines(upper ~ data$x, type = "l", col = "red", lty = "dashed")
lines(mean ~ data$x, type = "l", col = "blue")
lines(lower ~ data$x, type = "l", col = "red", lty = "dashed")
| Version | Author | Date |
|---|---|---|
| 8e9829b | Ziang Zhang | 2024-11-20 |
## Posterior of the SD parameter:
prec_marg <- fitted_mod$marginals[[1]]
logpostsigma <- compute_pdf_and_cdf(prec_marg,list(totheta = function(x) -2*log(x),fromtheta = function(x) exp(-x/2)),interpolation = 'spline')
plot(pdf_transparam ~ transparam, data = logpostsigma, type = 'l', xlab = "SD", ylab = "Post")
| Version | Author | Date |
|---|---|---|
| 8e9829b | Ziang Zhang | 2024-11-20 |
## Posterior of the Overdispersion parameter:
prec_marg <- fitted_mod$marginals[[2]]
logpostsigma <- compute_pdf_and_cdf(prec_marg,list(totheta = function(x) -2*log(x),fromtheta = function(x) exp(-x/2)),interpolation = 'spline')
plot(pdf_transparam ~ transparam, data = logpostsigma, type = 'l', xlab = "Overdispersion", ylab = "Post")
| Version | Author | Date |
|---|---|---|
| 8e9829b | Ziang Zhang | 2024-11-20 |
Now, let’s obtain the posterior samples and summary using the seasonal B-spline approach:
## Posterior samples:
samps2 <- sample_marginal(fitted_mod_sB, M = 3000)
g_samps_2 <- B2 %*% samps2$samps[1:ncol(B2),] + X %*% samps2$samps[(ncol(B2) + 1):(ncol(B2) + ncol(X)),]## Posterior summary:
mean2 <- apply(as.matrix(g_samps_2), MARGIN = 1, mean)
upper2 <- apply(as.matrix(g_samps_2), MARGIN = 1, quantile, p = 0.975)
lower2 <- apply(as.matrix(g_samps_2), MARGIN = 1, quantile, p = 0.025)Finally, let’s plot the posterior results obtained from the seasonal B-spline approach:
## Plotting
plot(log(data$y) ~ data$x, xlab = "time", ylab = "Posterior of g(x)", ylim = c(3.1,9))
lines(upper2 ~ data$x, type = "l", col = "red", lty = "dashed")
lines(mean2 ~ data$x, type = "l", col = "blue")
lines(lower2 ~ data$x, type = "l", col = "red", lty = "dashed")
| Version | Author | Date |
|---|---|---|
| 8e9829b | Ziang Zhang | 2024-11-20 |
## Posterior of the SD parameter:
prec_marg <- fitted_mod_sB$marginals[[1]]
logpostsigma <- compute_pdf_and_cdf(prec_marg,list(totheta = function(x) -2*log(x),fromtheta = function(x) exp(-x/2)),interpolation = 'spline')
plot(pdf_transparam ~ transparam, data = logpostsigma, type = 'l', xlab = "SD", ylab = "Post")
| Version | Author | Date |
|---|---|---|
| 8e9829b | Ziang Zhang | 2024-11-20 |
## Posterior of the Overdispersion parameter:
prec_marg <- fitted_mod_sB$marginals[[2]]
logpostsigma <- compute_pdf_and_cdf(prec_marg,list(totheta = function(x) -2*log(x),fromtheta = function(x) exp(-x/2)),interpolation = 'spline')
plot(pdf_transparam ~ transparam, data = logpostsigma, type = 'l', xlab = "Overdispersion", ylab = "Post")
| Version | Author | Date |
|---|---|---|
| 8e9829b | Ziang Zhang | 2024-11-20 |
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] Matrix_1.6-4 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 TMB_1.9.15
[13] aghq_0.4.1 BayesGP_0.1.3 sGPfit_1.0.1 workflowr_1.7.1
loaded via a namespace (and not attached):
[1] gtable_0.3.6 xfun_0.48 bslib_0.8.0
[4] ks_1.14.3 processx_3.8.4 lattice_0.22-6
[7] callr_3.7.6 tzdb_0.4.0 numDeriv_2016.8-1.1
[10] bitops_1.0-9 vctrs_0.6.5 tools_4.3.1
[13] ps_1.8.0 generics_0.1.3 fansi_1.0.6
[16] cluster_2.1.6 highr_0.11 fds_1.8
[19] pkgconfig_2.0.3 KernSmooth_2.23-24 data.table_1.16.2
[22] lifecycle_1.0.4 compiler_4.3.1 git2r_0.33.0
[25] statmod_1.5.0 munsell_0.5.1 getPass_0.2-4
[28] mvQuad_1.0-8 httpuv_1.6.15 rainbow_3.8
[31] htmltools_0.5.8.1 sass_0.4.9 RCurl_1.98-1.16
[34] yaml_2.3.10 pracma_2.4.4 later_1.3.2
[37] pillar_1.9.0 jquerylib_0.1.4 whisker_0.4.1
[40] MASS_7.3-60 cachem_1.1.0 mclust_6.1.1
[43] tidyselect_1.2.1 digest_0.6.37 mvtnorm_1.3-1
[46] stringi_1.8.4 splines_4.3.1 pcaPP_2.0-5
[49] rprojroot_2.0.4 fastmap_1.2.0 grid_4.3.1
[52] colorspace_2.1-1 cli_3.6.3 magrittr_2.0.3
[55] utf8_1.2.4 withr_3.0.2 scales_1.3.0
[58] promises_1.3.0 timechange_0.3.0 rmarkdown_2.28
[61] httr_1.4.7 deSolve_1.40 hms_1.1.3
[64] evaluate_1.0.1 knitr_1.48 rlang_1.1.4
[67] Rcpp_1.0.13-1 hdrcde_3.4 glue_1.8.0
[70] fda_6.2.0 rstudioapi_0.16.0 jsonlite_1.8.9
[73] R6_2.5.1 fs_1.6.4