A Guide to the SpatialGEV Package

Introduction to the GEV-GP Model

The generalized extreme value (GEV) distribution is often used to analyze sequences of maxima within non-overlapping time periods. An example of this type of data is the monthly maximum rainfall levels recorded over years at a weather station. Since there are typically a large number of weather stations within a country or a state, it is more ideal to have a model that can borrow information from nearby weather stations to increase inference and prediction accuracy. Such spatial information is often pooled using the Gaussian process.

The GEV-GP model is a hierarchical model with a data layer and a spatial random effects layer. Let \({\boldsymbol{x}}_1, \ldots, {\boldsymbol{x}}_n \in \mathbb{R}^2\) denote the geographical coordinates of \(n\) locations, and let \(y_{ik}\) denote the extreme value measurement \(k\) at location \(i\), for \(k = 1, \ldots, n_i\). The data layer specifies that each observation \(y_{ik}\) has a generalized extreme value distribution, denoted by \(y \sim \mathrm{GEV}(a, b, s)\), whose CDF is given by \[\begin{equation} F(y\mid a, b, s) = \begin{cases} \exp\left\{-\left(1+s\frac{y-a}{b}\right)^{-\frac{1}{s}}\right\} \ \ &s\neq 0,\\ \exp\left\{-\exp\left(-\frac{y-a}{b}\right)\right\} \ \ &s=0, \end{cases} \label{eqn:gev-distn} \end{equation}\] where \(a\in\mathbb{R}\), \(b>0\), and \(s\in\mathbb{R}\) are location, scale, and shape parameters, respectively. The support of the GEV distribution depends on the parameter values: \(y\) is bounded below by \(a-b/s\) when \(s>0\), bounded above by \(a-b/s\) when \(s<0\), and unbounded when \(s=0\). To capture the spatial dependence in the data, we assume some or all of the GEV parameters in the data layer are spatially varying. Thus they are introduced as random effects in the model.

A Gaussian process \(z({\boldsymbol{x}})\sim \mathcal{GP}(\mu({\boldsymbol{x}}_{cov}), k({\boldsymbol{x}}, {\boldsymbol{x}}'))\) is fully characterized by its mean \(\mu({\boldsymbol{x}}_{cov})\) and kernel function \(k({\boldsymbol{x}}, {\boldsymbol{x}}') = {\mathrm{Cov}}( z({\boldsymbol{x}}), z({\boldsymbol{x}}') )\), which captures the strength of the spatial correlation between locations. The mean is a function of parameters \(\boldsymbol{\beta}\) and covariates \({\boldsymbol{x}}_{cov}=(x_1,\ldots,x_p)'\). We assume that given the locations, the data follow independent GEV distributions each with their own parameters. The complete GEV-GP hierarchical model then becomes \[\begin{equation} \begin{aligned} y_{ik} \mid a({\boldsymbol{x}}_i), b({\boldsymbol{x}}_i), s & {\overset{ind}{\sim}}\mathrm{GEV}\big( a({\boldsymbol{x}}_i), \exp( b({\boldsymbol{x}}_i) ), \exp(s)\big)\\ a({\boldsymbol{x}}) \mid \boldsymbol{\beta}_a, {\boldsymbol{\theta}}_a &\sim \mathcal{GP}\big( {\boldsymbol{X}}_a\boldsymbol{\beta}_a, k({\boldsymbol{x}}, {\boldsymbol{x}}' \mid {\boldsymbol{\theta}}_a) \big)\\ log(b)({\boldsymbol{x}}) \mid \boldsymbol{\beta}_b, {\boldsymbol{\theta}}_b &\sim \mathcal{GP}\big( {\boldsymbol{X}}_b\boldsymbol{\beta}_b, k({\boldsymbol{x}}, {\boldsymbol{x}}' \mid {\boldsymbol{\theta}}_b) \big)\\ s({\boldsymbol{x}}) \mid \boldsymbol{\beta}_s, {\boldsymbol{\theta}}_s &\sim \mathcal{GP}\big( {\boldsymbol{X}}_s\boldsymbol{\beta}_s, k({\boldsymbol{x}}, {\boldsymbol{x}}' \mid {\boldsymbol{\theta}}_a) \big). \end{aligned} \end{equation}\] In this package, a uniform prior \(\pi({\boldsymbol{\theta}}) \propto 1\) is specified on the fixed effect and hyperparameters \[{\boldsymbol{\theta}}=(s, \boldsymbol{\beta}_a, \boldsymbol{\beta}_b, \boldsymbol{\beta}_s, {\boldsymbol{\theta}}_a, {\boldsymbol{\theta}}_b, {\boldsymbol{\theta}}_s).\]

What Does SpatialGEV Do?

The package provides an interface to estimate the approximate joint posterior distribution of the spatial random effects \(a\) and \(b\) in the GEV-GP model. The main functionalities of the package are:

Details about the approximate posterior inference can be found in Chen, Ramezan, and Lysy (2021).


SpatialGEV depends on the package TMB to perform the Laplace approximation. Make sure you have TMB installed following their instruction before installing SpatialGEV. Moreover, SpatialGEV uses several functions from the INLA package for approximating the Matérn covariance with the SPDE representation and for creating meshes on the spatial domain. If the user would like to use the SPDE approximation, please first install INLA. Since INLA is not on CRAN, it needs to be downloaded following their instruction here.

To install SpatialGEV, run the following:


Using the SpatialGEV Package

Exploratory analysis

We now demonstrate how to use this package through a simulation study. The simulated data used in this example comes with the package as a list variable simulatedData2, which contains the following:

a, logb and logs are simulated from Gaussian random fields using the R package SpatialExtremes. Using the simulated GEV parameters, we generate 50 to 70 observed data \({\boldsymbol{y}}_i\) at each location \(i, \ i=1,\ldots,400\).

Spatial variation of \(a\) and \(log(b)\) can be viewed by plotting them on regular lattices:

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Number of observations at each location is shown in the figure below.

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Below are histograms of observations at \(8\) randomly sampled locations.

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Model fitting

To fit the GEV-GP model to this simulated dataset, the first step is calling the spatialGEV_fit() function, for which several arguments must be provided:

There are other arguments which user might want to specify to override the defaults:

The code below fits a GEV-GP model to the simulated data. The model assumes that all spatial random effects follow Gaussian processes with the exponential kernel: \[k({\boldsymbol{x}}, {\boldsymbol{x}}') = \sigma^2\exp\left(-\frac{||{\boldsymbol{x}}-{\boldsymbol{x}}'||}{\ell}\right).\] No covariates are included in this model, so by default an intercept parameter \(\beta_0\) is estimated for the GP of each spatial random effect.

Posterior sampling

To obtain posterior samples of \(a\), \(b\), and \(s\), we pass the fitted model object fit to spatialGEV_sample(), which takes in three arguments:

The following line of code draws 2000 samples from the posterior distribution and the posterior predictive distribution:

Then use summary() to view the summary statistics of the posterior samples.

pos_summary <- summary(sam)
#>                    2.5%        25%       50%       75%      97.5%       mean
#> a1          60.37275783 61.5980928 62.210164 62.835360 63.9804803 62.2141060
#> a2          60.75855130 61.7379378 62.328065 62.859586 63.9297423 62.3079065
#> a3          60.81983443 61.9089171 62.431818 62.940149 63.9610251 62.4305312
#> a4          60.79951615 61.8344749 62.345769 62.881036 63.8523161 62.3527635
#> a5          60.74117629 61.7295232 62.198760 62.745477 63.7537897 62.2326125
#> log_b1       2.93220334  2.9719966  2.991997  3.014360  3.0553563  2.9930003
#> log_b2       2.93448172  2.9723931  2.992898  3.011996  3.0494946  2.9924037
#> log_b3       2.94528959  2.9782006  2.996280  3.014380  3.0515383  2.9964359
#> log_b4       2.94725158  2.9830574  2.999908  3.017881  3.0510366  2.9998425
#> log_b5       2.95743662  2.9905772  3.008068  3.025838  3.0566551  3.0079155
#> s1          -3.86044147 -3.2227490 -2.921353 -2.609795 -2.0274468 -2.9200414
#> s2          -3.80428531 -3.2219404 -2.931187 -2.610808 -2.0574936 -2.9274155
#> s3          -3.76864653 -3.2296526 -2.950027 -2.664731 -2.1404922 -2.9526005
#> s4          -3.71662682 -3.1967825 -2.940971 -2.666217 -2.1067600 -2.9334609
#> s5          -3.67186841 -3.1723053 -2.923136 -2.676004 -2.2098599 -2.9288787
#> s400        -3.45638367 -2.8673479 -2.571656 -2.296527 -1.7403831 -2.5842184
#> beta_a      57.52397467 58.9584185 59.670918 60.446429 61.9572194 59.7032630
#> beta_b       2.90031130  2.9653742  3.001447  3.035666  3.1069824  3.0004916
#> beta_s      -3.19469003 -2.7156268 -2.459331 -2.212726 -1.7199864 -2.4620612
#> log_sigma_a -0.48482824  0.4901036  1.081240  1.610590  2.6947695  1.0606942
#> log_ell_a    0.00258028  1.3207450  2.001316  2.668811  3.9966672  1.9918314
#> log_sigma_b -7.24392340 -6.0568859 -5.344071 -4.671646 -3.4885019 -5.3640991
#> log_ell_b   -0.22138081  1.4165251  2.310469  3.212735  4.8795805  2.3244647
#> log_sigma_s -1.83110995 -1.0777626 -0.689746 -0.312738  0.3793501 -0.7062119
#> log_ell_s    0.08662430  0.9420228  1.363870  1.809115  2.6334788  1.3711216
#>        2.5%      25%      50%      75%    97.5%     mean
#> y1 37.54193 57.15973 70.41631 89.40103 147.7703 76.44984
#> y2 37.37216 55.66691 69.70116 86.28537 143.6716 74.30785
#> y3 38.43884 56.23423 70.12602 89.56860 151.4831 76.44669
#> y4 37.20683 56.01811 70.36766 89.65431 144.2591 75.52356
#> y5 36.58043 55.96459 70.51883 89.12971 147.1605 75.66020

Model checking

Since we know the true values of \(a\), \(b\), and \(s\) in this simulation study, we are able to compare the posterior mean with the true values. The posterior means of \(a\), \(b\) and \(s\) at different locations are plotted against the true values below.

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Posterior prediction

Finally, we demonstrate how to make predictions at new locations. This is done using the spatialGEV_predict() function, which takes the following arguments:

We randomly sample 100 locations from the simulated dataset simulatedData as test locations which are left out. Data from the rest 300 training locations are used for model fitting.

In this dataset, the GEV parameters \(a\) and \(\log(b)\) are generated from the surfaces below, and the shape parameter is a constant \(\exp(-2)\) across space.

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We fit a GEV-GP model with random \(a\) and \(b\) and log-transformed \(s\) to the training dataset. The kernel used here is the SPDE approximation to the Matérn covariance function (Lindgren, Rue, and Lindström (2011)).

We see that using the SPDE approximation greatly speeds up model fitting:

The fitted model object is passed to spatialGEV_predict() for 2000 samples from the posterior predictive distributions. Note that this might take some time.

Then we call summary() on the pred object to obtain summary statistics of the posterior predictive samples at the test locations.

Since we have the true observations at the test locations, we can compare summary statistics of the true observations to those of the posterior predictive distributions. In the figures below, each circle represents a test location.

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Case study: Yearly maximum snowfall data in Ontario, Canada

In this section, we show how to use the SpatialGEV package to analyze a real dataset. The data used here are the 1987-2021 monthly total snowfall data (in cm) obtained from Environment and Natural Resources, Government of Canada. The link to download the raw data is https://climate-change.canada.ca/climate-data/#/monthly-climate-summaries. This dataset is automatically loaded with the package and is named ONsnow.

lon_range <- c(-96, -73)
lat_range <-  c(41.5, 55)
#>  Min.   :41.75   Min.   :-94.62   Length:63945       Length:63945       Min.   :1987   Min.   : 1.000   Min.   :  0.00  
#>  1st Qu.:43.58   1st Qu.:-81.50   Class :character   Class :character   1st Qu.:1991   1st Qu.: 3.000   1st Qu.:  0.00  
#>  Median :44.23   Median :-79.93   Mode  :character   Mode  :character   Median :1997   Median : 6.000   Median :  1.00  
#>  Mean   :44.96   Mean   :-80.88                                         Mean   :1999   Mean   : 6.469   Mean   : 15.23  
#>  3rd Qu.:45.50   3rd Qu.:-78.97                                         3rd Qu.:2005   3rd Qu.: 9.000   3rd Qu.: 23.60  
#>  Max.   :54.98   Max.   :-74.47                                         Max.   :2021   Max.   :12.000   Max.   :326.00
maps::map(xlim = lon_range, ylim = lat_range)

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Data preprocessing

We first grid the data using cells of length \(0.5^{\circ}\). By doing this, weather stations that are apart by less than \(0.5^{\circ}\) in longitude/latitude are grouped together in the same grid cell. From now on, we refer to each grid cell as a location.

For each location, we find the maximum snowfall amount each year and only keep locations where there are at least two years of records.

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Now we fit the GEV-GP model to the data using the exponential kernel function. Both \(a\) and \(b\) are treated as spatial random effects. \(s\) is constrained to be a positive constant. Note that here we have specified a \(\mathcal{N}(-5,5)\) prior on the log-transformed shape parameter. This is because we found that the shape parameter is estimated close to 0 and such a prior ensures model fitting procedure is numerically stable.

Next, 5000 samples are drawn from the joint posterior distribution of all parameters.

From the samples we can calculate the 5-year, 10-year, and 25-year return levels at all locations, which are the upper \(1/5\)%, \(1/10\)%, and \(1/25\)% quantiles of the extreme value distributions at these locations.

Plotted below are the return levels.

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Chen, Meixi, Reza Ramezan, and Martin Lysy. 2021. “Fast Approximate Inference for Spatial Extreme Value Models.” arXiv. https://arxiv.org/abs/2110.07051.

Lindgren, F. K., H. Rue, and J. Lindström. 2011. “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach.” Journal of the Royal Statistical Society, Series B 73: 423–98.