The Piemonte dataset example

Abstract

In this vignette we illustrate how to fit some of the spacetime models in Lindgren et al. (2023), for the dataset analysed in Cameletti et al. (2013). To perform this we will use the Bayesian paradigm with theINLA package, using the features provided by the inlabru package to facilitate the coding.

Introduction

The packages and setup

We start loading the required packages and those for doing the visualizations, the ggplot2 and patchwork packages.

library(ggplot2)
library(patchwork)
library(INLA)
#> Loading required package: Matrix
#> Loading required package: sp
#> This is INLA_24.10.02 built 2024-10-02 18:19:47 UTC.
#>  - See www.r-inla.org/contact-us for how to get help.
#>  - List available models/likelihoods/etc with inla.list.models()
#>  - Use inla.doc(<NAME>) to access documentation
library(INLAspacetime)
#> Loading required package: fmesher
library(inlabru)

We will ask it to return the WAIC, DIC and CPO

ctrc <- list(
  waic = TRUE,
  dic = TRUE,
  cpo = TRUE)

Getting the dataset

We will use the dataset analysed in Cameletti et al. (2013), that can be downloaded as follows. First, we set the filenames

u0 <- paste0(
    "http://inla.r-inla-download.org/",
    "r-inla.org/case-studies/Cameletti2012/")
coofl <- "coordinates.csv"
datafl <- "Piemonte_data_byday.csv"
bordersfl <- "Piemonte_borders.csv"

Download and read the borders file

### get the domain borders
if(!file.exists(bordersfl))
    download.file(paste0(u0, bordersfl), bordersfl)
dim(pborders <- read.csv(bordersfl))
#> [1] 27821     2

Download and read the coordinates file

### get the coordinates
if(!file.exists(coofl))
    download.file(paste0(u0, coofl), coofl)
dim(locs <- read.csv(coofl))
#> [1] 24  3

Download and read the dataset

### get the dataset
if(!file.exists(datafl))
    download.file(paste0(u0, datafl), datafl)
dim(pdata <- read.csv(datafl))
#> [1] 4368   11

Inspect the dataset

head(pdata)
#>   Station.ID     Date     A   UTMX    UTMY   WS   TEMP   HMIX PREC   EMI PM10
#> 1          1 01/10/05  95.2 469.45 4972.85 0.90 288.81 1294.6    0 26.05   28
#> 2          2 01/10/05 164.1 423.48 4950.69 0.82 288.67 1139.8    0 18.74   22
#> 3          3 01/10/05 242.9 490.71 4948.86 0.96 287.44 1404.0    0  6.28   17
#> 4          4 01/10/05 149.9 437.36 4973.34 1.17 288.63 1042.4    0 29.35   25
#> 5          5 01/10/05 405.0 426.44 5045.66 0.60 287.63 1038.7    0 32.19   20
#> 6          6 01/10/05 257.5 394.60 5001.18 1.02 288.59 1048.3    0 34.24   41

Prepare the time to be used

range(pdata$Date <- as.Date(pdata$Date, "%d/%m/%y"))
#> [1] "2005-10-01" "2006-03-31"
pdata$time <- as.integer(difftime(
    pdata$Date, min(pdata$Date), units = "days")) + 1

Standardize the covariates that will be used in the data analysis and define a dataset including the needed information where the outcome is the log of PM10, as used in Cameletti et al. (2013).

### prepare the covariates
xnames <- c("A", "WS", "TEMP", "HMIX", "PREC", "EMI")
xmean <- colMeans(pdata[, xnames])
xsd <- sapply(pdata[xnames], sd)

### prepare the data (st loc, scale covariates and log PM10)
dataf <- data.frame(pdata[c("UTMX", "UTMY", "time")],
                    scale(pdata[xnames], xmean, xsd),
                    y = log(pdata$PM10))
str(dataf)
#> 'data.frame':    4368 obs. of  10 variables:
#>  $ UTMX: num  469 423 491 437 426 ...
#>  $ UTMY: num  4973 4951 4949 4973 5046 ...
#>  $ time: num  1 1 1 1 1 1 1 1 1 1 ...
#>  $ A   : num  -1.3956 -0.7564 -0.0254 -0.8881 1.4785 ...
#>  $ WS  : num  -0.0777 -0.2319 0.038 0.4429 -0.6561 ...
#>  $ TEMP: num  2.1 2.07 1.82 2.06 1.86 ...
#>  $ HMIX: num  2.18 1.69 2.53 1.38 1.37 ...
#>  $ PREC: num  -0.29 -0.29 -0.29 -0.29 -0.29 ...
#>  $ EMI : num  -0.1753 -0.3454 -0.6353 -0.0985 -0.0324 ...
#>  $ y   : num  3.33 3.09 2.83 3.22 3 ...

The data model definition

We consider the following linear mixed model for the outcome $\mathbf{y} = \mathbf{W}\mathbf{\beta} + \mathbf{A}\mathbf{u} + \mathbf{e}$ where $\beta$ are fixed effects, or regression coefficients including the intercept, for the matrix of covariates $\mathbf{W}$ , $\mathbf{u}$ is the spatio-temporal random effect having the matrix $\mathbf{A}$ the projector matrix from the discretized domain to the data. The spatio-temporal random effect $\mathbf{u}$ is defined in a continuous spacetime domain being discretized considering meshes over time and space. The difference from Cameletti et al. (2013) is that we now use the models in Lindgren et al. (2023) for $\mathbf{u}$ .

Define a temporal mesh, with each knot spaced by h, where h = 1 means one per day.

nt <- max(pdata$time)
h <- 1
tmesh <- inla.mesh.1d(
  loc = seq(1, nt + h/2, h), 
  degree = 1)
tmesh$n
#> [1] 182

Define a spatial mesh, the same used in Cameletti et al. (2013).

smesh <- inla.mesh.2d(
    cbind(locs[,2], locs[,3]),
    loc.domain = pborders,
    max.edge = c(50, 300),
    offset = c(10, 140),
    cutoff = 5,
    min.angle = c(26, 21))
smesh$n
#> [1] 142

Visualize the spatial mesh, the border and the locations.

par(mfrow = c(1,1), mar = c(0,0,1,0))
plot(smesh, asp = 1)
lines(pborders, lwd = 2, col = "green4")
points(locs[, 2:3], pch = 19, col = "blue")

We set the prior for the likelihood precision considering a PC-prior, Simpson et al. (2017), through the following probabilistic statements: P( $\sigma_e > U_{\sigma_e}$ ) = $\alpha_{\sigma_e}$ , using $U_{\sigma_e}$ = 1 and $\alpha_{\sigma_e} = 0.05$ .

lkprec <- list(
    prec = list(prior = "pcprec", param = c(1, 0.05)))

With inlabru we can define the likelihood model with the like() function and use it for fitting models with different linear predictors later.

lhood <- like(
  formula = y ~ .,
  family = "gaussian",
  control.family = list(
    hyper = lkprec),
  data = dataf)

The linear predictor, the right-rand side of the formula, can be defined using the same expression for of the both models that we are going to fit and is

M <- ~ -1 + Intercept(1) + A + WS + TEMP + HMIX + PREC + EMI +
    field(list(space = cbind(UTMX, UTMY), 
               time = time),
          model = stmodel)

The spacetime models

The implementation of the spacetime model uses the cgeneric interface in INLA, see its documentation for details. Therefore we have a C code to mainly build the precision matrix and compute the model parameter priors and compiled as static library. We have this code included in the INLAspacetime package but it is also being copied to the INLA package and compiled with the same compilers in order to avoid possible mismatches. In order to use it, we have to define the matrices and vectors needed, including the prior parameter definitions.

The class of models in Lindgren et al. (2023) have the spatial range, temporal range and marginal standard deviation as parameters. We consider the PC-prior, as in Fuglstad et al. (2017), for these parameters defined from the probability statements: P( $r_s<U_{r_s}$ )= $\alpha_{r_s}$ , P( $r_t<U_{r_t}$ )= $\alpha_{r_t}$ and P( $\sigma<U_{\sigma}$ )= $\alpha_{\sigma}$ . We consider $U_{r_s}=100$ , $U_{r_t}=5$ and $U_{\sigma}=2$ . $\alpha_{r_s}=\alpha_{r_t}=\alpha_{\sigma}=0.05$

The selection of one of the models in Lindgren et al. (2023) is by chosing the $\alpha_t$ , $\alpha_s$ and $\alpha_e$ as integer numbers. We will start considering the model $\alpha_t=1$ , $\alpha_s=0$ and $\alpha_t=2$ , which is a model with separable spatio-temporal covariance, and then we fit some of the other models later.

Defining a particular model

We define an object with the needed use the function stModel.define() where the model is selected considering the values for $\alpha_t$ , $\alpha_s$ and $\alpha_e$ collapsed. In order to illustrate how it is done, we can set an overall integrate-to-zero constraint, which is not need but helps model components identification. It uses the weights based on the mesh node volumes, from both the temporal and spatial meshes. This can be set automatically when defining the model by adding constr = TRUE.

model <- "102"
stmodel <- stModel.define(
    smesh, tmesh, model,
    control.priors = list(
        prs = c(150, 0.05),
        prt = c(10, 0.05),
        psigma = c(5, 0.05)),
    constr = TRUE)

Initial values for the hyper-parameters help to fit the models in less computing time. It is also important to consider in the light that each dataset has its own parameter scale. For example, we have to consider that the spatial domain within a box of around $203.7$ by $266.5$ kilometers, which we already did when building the mesh and setting the prior for or $r_s$ .

We can set initial values for the log of the parameters so that it would take less iterations to converge:

theta.ini <- c(4, 7, 7, 1)

The code to fit the model through inlabru is

fit102 <- 
    bru(M,
        lhood,
        options = list(
            control.mode = list(theta = theta.ini, restart = TRUE),
            control.compute = ctrc))

Summary of the posterior marginal distributions for the fixed effects

fit102$summary.fixed[, c(1, 2, 3, 5)]
#>                  mean          sd   0.025quant   0.975quant
#> Intercept  3.73691317 0.246417220  3.250673748  4.222816213
#> A         -0.17938800 0.050034883 -0.278564843 -0.081325443
#> WS        -0.06043228 0.008441944 -0.076976227 -0.043866739
#> TEMP      -0.12241124 0.035188677 -0.191484232 -0.053473095
#> HMIX      -0.02373414 0.013205041 -0.049590657  0.002200081
#> PREC      -0.05355502 0.008607154 -0.070418495 -0.036660669
#> EMI        0.03628956 0.015196090  0.006127752  0.065808877

For the hyperparameters, we transform the posterior marginal distributions for the model hyperparameters from the ones computed in internal scale, $\log(1/\sigma^2_e)$ , $\log(r_s)$ , $\log(r_t)$ and $\log(\sigma)$ , to the user scale parametrization, $\sigma_e$ , $r_s$ , $r_t$ and $\sigma$ , respectivelly.

post.h <- list(
  sigma_e = inla.tmarginal(function(x) exp(-x/2), 
                           fit102$internal.marginals.hyperpar[[1]]),
  range_s = inla.tmarginal(function(x) exp(x), 
                           fit102$internal.marginals.hyperpar[[2]]),
  range_t = inla.tmarginal(function(x) exp(x), 
                           fit102$internal.marginals.hyperpar[[3]]),
  sigma_u = inla.tmarginal(function(x) exp(x), 
                           fit102$internal.marginals.hyperpar[[4]])
)

Then we compute and show the summary of it

shyper <- t(sapply(post.h, function(m) 
  unlist(inla.zmarginal(m, silent = TRUE))))
shyper[, c(1, 2, 3, 7)]
#>                mean           sd  quant0.025  quant0.975
#> sigma_e   0.1810305  0.003764679   0.1737495   0.1885346
#> range_s 280.5995231 17.003754692 248.8020388 315.5644017
#> range_t  49.8800970  8.106482692  36.0764477  67.8434453
#> sigma_u   1.1368871  0.086879523   0.9781079   1.3191272

However, it is better to look at the posterior marginal itself, and we will visualize it later.

The model fitted in Cameletti et al. (2013) includes two more covariates and setup a model for discrete temporal domain where the temporal correlation is modeled as a first order autoregression with parameter $\rho$ . In the fitted model here is defined considering continuous temporal domain with the range parameter $r_s$ . However, the first order autocorrelation could be taken as $\rho = \exp(-h\sqrt{8\nu}/r_s)$ , where $h$ is the temporal resolution used in the temporal mesh and $\nu$ is equal $0.5$ for the fitted model. We can compare ou results with Table 3 in Cameletti et al. (2013) with

c(shyper[c(1, 4, 2), 1], 
  a = exp(-h * sqrt(8 * 0.5) / shyper[3, 1]))
#>     sigma_e     sigma_u     range_s           a 
#>   0.1810305   1.1368871 280.5995231   0.9606971

Comparing different models

We now fit the model $121$ for $u$ as well, we use the same code for building the model matrices

model <- "121"
stmodel <- stModel.define(
    smesh, tmesh, model,
    control.priors = list(
        prs = c(150, 0.05),
        prt = c(10, 0.05),
        psigma = c(5, 0.05)),
    constr = TRUE)

and use the same code for fitting as follows

fit121 <- 
    bru(M,
        lhood,
        options = list(
            control.mode = list(theta = theta.ini, restart = TRUE),
            control.compute = ctrc))

We will join these fits into a list object to make it easier working with it

results <- list("u102" = fit102, "u121" = fit121)

The computing time for each model fit

sapply(results, function(r) r$cpu.used)
#>                u102       u121
#> Pre       0.5084863   0.339206
#> Running 131.6539879 248.460999
#> Post      4.6370063   3.047057
#> Total   136.7994804 251.847262

and the number of fn-calls during the optimization are

sapply(results, function(r) r$misc$nfunc)
#> u102 u121 
#>  296  358

The posterior mode for each parameter in each model (in internal scale) are

sapply(results, function(r) r$mode$theta)
#>                                                  u102      u121
#> Log precision for the Gaussian observations 3.4187391  3.593241
#> Theta1 for field                            5.6340554  7.292386
#> Theta2 for field                            3.8864037 10.984899
#> Theta3 for field                            0.1210238  2.156164

We compute the posterior marginal distribution for the hyper-parameters in the user-interpretable scale, like we did before for the first model, with

posts.h2 <- lapply(1:2, function(m) vector("list", 4L))
for(m in 1:2) {
    posts.h2[[m]]$sigma_e =  
      data.frame(
        parameter = "sigma_e", 
        inla.tmarginal(
          function(x) exp(-x/2), 
          results[[m]]$internal.marginals.hyperpar[[1]]))
    for(p in 2:4) {
      posts.h2[[m]][[p]] <-   
      data.frame(
        parameter = c(NA, "range_s", "range_t", "sigma_u")[p], 
        inla.tmarginal(
          function(x) exp(x), 
          results[[m]]$internal.marginals.hyperpar[[p]])
      )
    }
}

Join these all to make visualization easier

posts.df <- rbind(
  data.frame(model = "102", do.call(rbind, posts.h2[[1]])),
  data.frame(model = "121", do.call(rbind, posts.h2[[2]]))
)

ggplot(posts.df) +
  geom_line(aes(x = x, y = y, group = model, color = model)) +
  ylab("Density") + xlab("") + 
  facet_wrap(~parameter, scales = "free")

The comparison of the model parameters of $\mathbf{u}$ for different models have to be done in light with the covariance functions as illustrated in Lindgren et al. (2023). The fitted $\sigma_e$ by the different models are comparable and we can see that when considering model $121$ for $\mathbf{u}$ , its posterior marginal are concentrated in values lower than when considering model $102$ .

We can look at the posterior mean of $u$ from both models and see that under model ‘121’ there is a wider spread.

par(mfrow = c(1, 1), mar = c(3, 3, 0, 0.0), mgp = c(2, 1, 0))
uu.hist <- lapply(results, function(r)
    hist(r$summary.random$field$mean,
         -60:60/20, plot = FALSE))
ylm <- range(uu.hist[[1]]$count, uu.hist[[2]]$count)
plot(uu.hist[[1]], ylim = ylm,
     col = rgb(1, 0.1, 0.1, 1.0), border = FALSE, 
     xlab = "u", main = "")
plot(uu.hist[[2]], add = TRUE, col = rgb(0.1, 0.1, 1, 0.5), border = FALSE)
legend("topleft", c("separable", "non-separable"), 
       fill = rgb(c(1,0.1), 0.1, c(0.1, 1), c(1, 0.5)), 
       border = 'transparent', bty = "n")

We can also check fitting statistics such as DIC, WAIC, the negative of the log of the probability ordinates (LPO) and its cross-validated version (LCPO), summarized as the mean.

t(sapply(results, function(r) {
  c(DIC = mean(r$dic$local.dic, na.rm = TRUE),
    WAIC = mean(r$waic$local.waic, na.rm = TRUE),
    LPO = -mean(log(r$po$po), na.rm = TRUE), 
    LCPO = -mean(log(r$cpo$cpo), na.rm = TRUE))
}))
#>             DIC       WAIC        LPO       LCPO
#> u102 -0.3767400 -0.2595157 -0.4052380 -0.1173853
#> u121 -0.4110575 -0.3526168 -0.5079881 -0.1158159

The automatic group-leave-out cross validation

One may be interested in evaluating the model prediction. The leave-one-out strategy was already available in INLA since several years ago, see Held, Schrodle, and Rue (2010) for details. Recently, an automatic group cross validation strategy was implemented, see Liu and Rue (2023) for details.

g5cv <- lapply(
  results, inla.group.cv, num.level.sets = 5, 
  strategy = "posterior", size.max = 50)

We can inspect the detected observations that have the posterior linear predictor correlated with each one, including itself. For 100th observation under model “102” we have

g5cv$u102$group[[100]]
#> $idx
#> [1]  52  76  98 100 106 124
#> 
#> $corr
#> [1] 0.2580253 0.4298995 0.3006961 1.0000000 0.3300166 0.4298995

and for the result under model “121” we have

g5cv$u121$group[[100]]
#> $idx
#> [1]  52  76  98 100 124 148
#> 
#> $corr
#> [1] 0.1673704 0.3738149 0.1673704 1.0000000 0.3762592 0.1641839

which has intersection but are not the same, for the model setup used.

We can check which are these observations in the dataset

dataf[g5cv$u102$group[[100]]$idx, ]
#>       UTMX    UTMY time          A           WS     TEMP       HMIX       PREC
#> 52  437.36 4973.34    3 -0.8881265  0.982687762 1.077344 -0.6959771 1.94505331
#> 76  437.36 4973.34    4 -0.8881265  0.674216318 1.297701  0.9479287 3.70037355
#> 98  423.48 4950.69    5 -0.7563919 -0.578948923 1.387847 -0.4413098 0.08590275
#> 100 437.36 4973.34    5 -0.8881265  0.770613644 1.441935  0.2530248 0.16426526
#> 106 416.65 4985.65    5  0.3225334 -0.000564966 1.397863  1.1608209 0.63052220
#> 124 437.36 4973.34    6 -0.8881265  0.751334179 1.618220  1.0845756 0.03692618
#>             EMI        y
#> 52  -0.01821338 2.564949
#> 76  -0.01542101 2.708050
#> 98  -0.27115486 2.484907
#> 100  0.01622576 2.890372
#> 106 -0.61950205 2.708050
#> 124  0.02599903 2.995732

and found that most are at the same locations in nearby time.

We can compute the negative of the mean of the log score so that lower number is better

sapply(g5cv, function(r) -mean(log(r$cv), na.rm = TRUE))
#>       u102       u121 
#> 0.04101877 0.07005032

References

Cameletti, Michela, Finn Lindgren, Daniel Simpson, and Håvard Rue. 2013. “Spatio-Temporal Modeling of Particulate Matter Concentration Through the SPDE Approach.” AStA Advances in Statistical Analysis 97 (2): 109–31. https://doi.org/10.1007/s10182-012-0196-3.

Fuglstad, Geir-Arne, Daniel Simpson, Finn Lindgren, and Håvard Rue. 2017. “Constructing Priors That Penalize the Complexity of Gaussian Random Fields.” Journal of the American Statistical Association, no. 525: 445–52.

Held, Leonhard, Birgit Schrodle, and Håvard Rue. 2010. “Posterior and Cross-validatory Predictive Checks: A Comparison of MCMC and INLA.” In Statistical Modelling and Regression Structures, 111–31. Springer.

Lindgren, F., H. Bakka, D. Bolin, E. Krainski, and H. Rue. 2023. “A Diffusion-Based Spatio-Temporal Extension of Gaussian Matérn Fields.” ArXiv. https://arxiv.org/abs/2006.04917.

Liu, Z., and H. Rue. 2023. “Leave-Group-Out Cross-Validation for Latent Gaussian Models.” https://arxiv.org/abs/2210.04482.

Simpson, Daniel, Håvard Rue, Andrea Riebler, Thiago G Martins, and Sigrunn H Sørbye. 2017. “Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors.” Statistical Science 32 (1): 1–28.

Elias T Krainski

Started in 2022. Last update: Sun 06 Oct, 2024