Stephan R. Sain

The North American Regional Climate Change Assessment Program: Overview of Phase I Results

The North American Regional Climate Change Assessment Program (NARCCAP) is an international effort designed to investigate the uncertainties in regional-scale projections of future climate and produce highresolution climate change scenarios using multiple regional climate models (RCMs) nested within atmosphere–ocean general circulation models (AOGCMs) forced with the Special Report on Emission Scenarios (SRES) A2 scenario, with a common domain covering the conterminous United States, northern Mexico, and most of Canada. The program also includes an evaluation component (phase I) wherein the participating RCMs, with a grid spacing of 50 km, are nested within 25 years of National Centers for Environmental Prediction–Department of Energy (NCEP–DOE) Reanalysis II. This paper provides an overview of evaluations of the phase I domain-wide simulations focusing on monthly and seasonal temperature and precipitation, as well as more detailed investigation of four subregions. The overall quality of the simulations i...

Cross-validation of multivariate densities

Abstract In recent years, the focus of study in smoothing parameter selection for kernel density estimation has been on the univariate case, while multivariate kernel density estimation has been largely neglected. In part, this may be due to the perception that calibrating multivariate densities is substantially more difficult. In this article, we explicitly derive and compare multivariate versions of the bootstrap method of Taylor, the least-squares cross-validation method developed by Bowman and Rudemo, and a biased cross-validation method similar to that of Scott and Terrell for multivariate kernel estimation using the product kernel estimator. The theoretical behavior of these cross-validation algorithms is shown to improve (surprisingly) as the dimension increases, approaching the best rate of O(n −1/2). Simulation studies suggest that the new biased cross-validation method performs quite well and with reasonable variability as compared to the other two methods. Bivariate examples with heart disease ...

On Locally Adaptive Density Estimation

Abstract Theoretical and practical aspects of the sample-point adaptive positive kernel density estimator are examined. A closed-form expression for the mean integrated squared error is obtained through the device of preprocessing the data by binning. With this expression, the exact behavior of the optimally adaptive smoothing parameter function is studied for the first time. The approach differs from most earlier techniques in that bias of the adaptive estimator remains O(h 2) and is not “improved” to the rate O(h 4). A practical algorithm is constructed using a modification of least squares cross-validation. Simulated and real examples are presented, including comparisons with a fixed bandwidth estimator and a fully automatic version of Abramsons adaptive estimator. The results are very promising.

Multivariate Bayesian analysis of atmosphere-ocean general circulation models

Numerical experiments based on atmosphere–ocean general circulation models (AOGCMs) are one of the primary tools in deriving projections for future climate change. Although each AOGCM has the same underlying partial differential equations modeling large scale effects, they have different small scale parameterizations and different discretizations to solve the equations, resulting in different climate projections. This motivates climate projections synthesized from results of several AOGCMs’ output. We combine present day observations, present day and future climate projections in a single highdimensional hierarchical Bayes model. The challenging aspect is the modeling of the spatial processes on the sphere, the number of parameters and the amount of data involved. We pursue a Bayesian hierarchical model that separates the spatial response into a large scale climate change signal and an isotropic process representing small scale variability among AOGCMs. Samples from the posterior distributions are obtained with computer-intensive MCMC simulations. The novelty of our approach is that we use gridded, high resolution data covering the entire sphere within a spatial hierarchical framework. The primary data source is provided by the Coupled Model Intercomparison Project (CMIP) and consists of 9 AOGCMs on a 2.8 by 2.8 degree grid under several different emission scenarios. In this article we consider mean seasonal surface temperature and precipitation as climate variables. Extensions to our model are also discussed.

Multivariate locally adaptive density estimation

Multivariate versions of variable bandwidth kernel density estimators can lead to improvement over kernel density estimators using global bandwidth choices. These estimators are more flexible and better able to model complex (multimodal) densities. In this work, two variable bandwidth estimators are discussed: the balloon estimator which varies the smoothing matrix with each estimation point and the sample point estimator which uses a different smoothing matrix for each data point. A binned version of the sample point estimator is developed that, for various situations in low to moderate dimensions, exhibits less error (MISE) than the fixed bandwidth estimator and the balloon estimator. A practical implementation of the sample point estimator is shown through simulation and example to do a better job at reconstructing features of the underlying density than fixed bandwidth estimators. Computational details, including parameterization of the smoothing matrix, are discussed throughout.

A multi-resolution Gaussian process model for the analysis of large spatial data sets

We develop a multiresolution model to predict two-dimensional spatial fields based on irregularly spaced observations. The radial basis functions at each level of resolution are constructed using a Wendland compactly supported correlation function with the nodes arranged on a rectangular grid. The grid at each finer level increases by a factor of two and the basis functions are scaled to have a constant overlap. The coefficients associated with the basis functions at each level of resolution are distributed according to a Gaussian Markov random field (GMRF) and take advantage of the fact that the basis is organized as a lattice. Several numerical examples and analytical results establish that this scheme gives a good approximation to standard covariance functions such as the Matérn and also has flexibility to fit more complicated shapes. The other important feature of this model is that it can be applied to statistical inference for large spatial datasets because key matrices in the computations are sparse. The computational efficiency applies to both the evaluation of the likelihood and spatial predictions.

Bayesian Functional ANOVA Modeling Using Gaussian Process Prior Distributions

Functional analysis of variance (ANOVA) models partition a func- tional response according to the main efiects and interactions of various factors. This article develops a general framework for functional ANOVA modeling from a Bayesian viewpoint, assigning Gaussian process prior distributions to each batch of functional efiects. We discuss the choices to be made in specifying such a model, advocating the treatment of levels within a given factor as dependent but exchangeable quantities, and we suggest weakly informative prior distributions for higher level parameters that may be appropriate in many situations. We discuss computationally e-cient strategies for posterior sampling using Markov Chain Monte Carlo algorithms, and we emphasize useful graphical summaries based on the posterior distribution of model-based analogues of traditional ANOVA decom- positions of variance. We illustrate this process of model speciflcation, posterior sampling, and graphical posterior summaries in two examples. The flrst consid- ers the efiect of geographic region on the temperature proflles at weather stations in Canada. The second example examines sources of variability in the output of regional climate models from a designed experiment.

Multidimensional Density Estimation

Abstract Modern data analysis requires a number of tools to undercover hidden structure. For initial exploration of data, animated scatter diagrams and nonparametric density estimation in many forms and varieties are the techniques of choice. This article focuses on the application of histograms and nonparametric kernel methods to explore data. The details of theory, computation, visualization, and presentation are all described.

A spatial analysis of multivariate output from regional climate models

Climate models have become an important tool in the study of climate and climate change, and ensemble experiments consisting of multiple climate-model runs are used in studying and quantifying the uncertainty in climate-model output. However, there are often only a limited number of model runs available for a particular experiment, and one of the statistical challenges is to characterize the distribution of the model output. To that end, we have developed a multivariate hierarchical approach, at the heart of which is a new representation of a multivariate Markov random field. This approach allows for flexible modeling of the multivariate spatial dependencies, including the cross-dependencies between variables. We demonstrate this statistical model on an ensemble arising from a regional-climate-model experiment over the western United States, and we focus on the projected change in seasonal temperature and precipitation over the next 50 years.

DOWNSCALING EXTREMES: A COMPARISON OF EXTREME VALUE DISTRIBUTIONS IN POINT-SOURCE AND GRIDDED PRECIPITATION DATA

There is substantial empirical and climatological evidence that precipitation extremes have become more extreme during the twentieth century, and that this trend is likely to continue as global warming becomes more intense. However, understanding these issues is limited by a fundamental issue of spatial scaling: most evidence of past trends comes from rain gauge data, whereas trends into the future are produced by climate models, which rely on gridded aggregates. To study this further, we fit the Generalized Extreme Value (GEV) distribution to the right tail of the distribution of both rain gauge and gridded events. The results of this modeling exercise confirm that return values computed from rain gauge data are typically higher than those computed from gridded data; however, the size of the difference is somewhat surprising, with the rain gauge data exhibiting return values sometimes two or three times that of the gridded data. The main contribution of this paper is the development of a family of regression relationships between the two sets of return values that also take spatial variations into account. Based on these results, we now believe it is possible to project future changes in precipitation extremes at the point-location level based on results from climate models.