Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalized) linear models, (generalized) additive models, smoothing spline models, state space models, semiparametric regression, spatial and spatiotemporal models, log-Gaussian Cox processes and geostatistical and geoadditive models. We consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form owing to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, in terms of both convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo sampling is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations is computational: where Markov chain Monte Carlo algorithms need hours or days to run, our approximations provide more precise estimates in seconds or minutes. Another advantage with our approach is its generality, which makes it possible to perform Bayesian analysis in an automatic, streamlined way, and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged.

/pdf/approximate-bayesian-inference-for-latent-gaussian-models-by-2m3x22eic2.pdf

Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations

The MaxEnt software package is one of the most popular tools for species distribution and environmental niche modeling, with over 1000 published applications since 2006. Its popularity is likely for two reasons: 1) MaxEnt typically outperforms other methods based on predictive accuracy and 2) the software is particularly easy to use. MaxEnt users must make a number of decisions about how they should select their input data and choose from a wide variety of settings in the software package to build models from these data. The underlying basis for making these decisions is unclear in many studies, and default settings are apparently chosen, even though alternative settings are often more appropriate. In this paper, we provide a detailed explanation of how MaxEnt works and a prospectus on modeling options to enable users to make informed decisions when preparing data, choosing settings and interpreting output. We explain how the choice of background samples reflects prior assumptions, how nonlinear functions of environmental variables (features) are created and selected, how to account for environmentally biased sampling, the interpretation of the various types of model output and the challenges for model evaluation. We demonstrate MaxEnt’s calculations using both simplified simulated data and occurrence data from South Africa on species of the flowering plant family Proteaceae. Throughout, we show how MaxEnt’s outputs vary in response to different settings to highlight the need for making biologically motivated modeling decisions.

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1600-0587.2013.07872.x

A practical guide to MaxEnt for modeling species' distributions: what it does, and why inputs and settings matter

Under the hood Issues in the specification and interpretation of spatial regression models

With scientific data available at geocoded locations, investigators are increasingly turning to spatial process models for carrying out statistical inference. Over the last decade, hierarchical models implemented through Markov chain Monte Carlo methods have become especially popular for spatial modelling, given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics. However, fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations, rendering such models infeasible for large spatial data sets. This computational burden is exacerbated in multivariate settings with several spatially dependent response variables. It is also aggravated when data are collected at frequent time points and spatiotemporal process models are used. With regard to this challenge, our contribution is to work with what we call predictive process models for spatial and spatiotemporal data. Every spatial (or spatiotemporal) process induces a predictive process model (in fact, arbitrarily many of them). The latter models project process realizations of the former to a lower dimensional subspace, thereby reducing the computational burden. Hence, we achieve the flexibility to accommodate non-stationary, non-Gaussian, possibly multivariate, possibly spatiotemporal processes in the context of large data sets. We discuss attractive theoretical properties of these predictive processes. We also provide a computational template encompassing these diverse settings. Finally, we illustrate the approach with simulated and real data sets.

/pdf/gaussian-predictive-process-models-for-large-spatial-data-25ay8hc7gs.pdf

Gaussian predictive process models for large spatial data sets

Response surface methods have been used for a variety of applications in aerospace engineering, particularly in multidisciplinary design optimization. We investigate the use of kriging models as alternatives to traditional second-order polynomial response surfaces for constructing global approximations for use in a real aerospace engineering application, namely, the design of an aerospike nozzle. Our objective is to examine the difeculties in building and using kriging models to create accurate global approximations to facilitate multidisciplinary design optimization. Error analysis of the response surface and kriging models is performed, along with a graphical comparison of the approximations. Four optimization problems are also formulated and solved using both sets of approximation models to gain insight into their use for multidisciplinary design optimization. We end that the kriging models, which use only a constant “global” model and a Gaussian correlation function, yield global approximations that are slightly more accurate than the response surface models.

Noel A. C. Cressie

Papers

Statistics for Spatial Data: Cressie/Statistics