Reinhard Furrer's Projects at NCAR

Statistical evaluation of climate model output

Spatial Hierarchical Bayes Model for AOGCM Climate Projections
In collaboration with Nychka, D., NCAR, Wigley, T. M. L., NCAR and Sain, S. R., University of Colorado, Denver.

Numerical experiments based on atmospheric-ocean general circulation models (AOGCMs) are one of the primary tools in deriving projections for future climate change. However, each model has its strengths and weaknesses within local and global scales. This motivates climate projections synthesized from results of several AOGCMs' output weighted according to model bias and convergence. We combine present day observations, present day and future climate projections in a single hierarchical Bayes model for which the posterior distributions are obtained with computer-intensive MCMC simulations. The novelty of our approach is that we use gridded, high resolution data within a spatial framework.

The proposed method can be developed further in order to be able to consider observations of fundamentally different nature (precipitation, temperature, and min/max thereof). We propose a multivariate approach and the consideration of heavy-tailed error distributions.

Statistics for large datasets

Fitting Large-Scale Spatial Models with Applications to Microarray Data Analysis
In collaboration with Sain, S. R., University of Colorado, Denver.

A single microarray includes over 400,000 individual observations, too much data for classical analysis techniques. We apply covariance tapering to a very general type of mixed model that has a random spatial component. Then, taking advantage of the sparse nature of such tapered covariance matrices, backfitting is used to estimate the fixed and random model parameters. Results are demonstrated on an experiment using microarrays to build a profile of differentially expressed genes relating to cerebral vascular malformations, an important cause of hemorrhagic stroke and seizures.

The taper technique is of general nature and can be applied to many other problems in the environmental and biological sciences. This requires more flexibility in the tapering technique. A potential approach is to taper directly the Cholesky factor instead of the covariance matrix itself.

Ensemble Kalman Filter

Approximation of Forecast Covariances in Kalman Filter Variants
In collaboration with Bengtsson, T., University of California, Berkeley.

Many modern geophysical problems are characterized by extremely high-dimensional systems and pose difficult challenges for real-time assimilation of system information and observations. Recent focus in the atmospheric sciences has been on representing the knowledge of the atmospheric state using a probability density function, and various sample based techniques have been developed to address the problem of real-time updating and forecasting for high-dimensional systems. We study the effects of matrix sample variability for two Monte-Carlo based Kalman filter variants, the ensemble Kalman filter and the square root filter.

For the time being we obtained some good but theoretical results for which we do not know their impact in practice. We envision to proceed with the application and validation of the method in large scale problems such as operational numerical weather prediction.

Extreme value theory

U-Statistics and PWM in Modeling Extremes
In collaboration with Naveau, P., University of Colorado, Boulder/Laboratoire des Sciences du Climat et de l'Environnement (LSCE-IPSL), Gif-sur-Yvette, France

The generalized Pareto distribution is a key ingredient in modeling the distribution of the excess of observations over large thresholds. The parameters can be estimated with maximum likelihood, conventional methods of moments or with probability weighted moments (PWM). We discuss PWM as a particular U-estimator with which we can derive exact variances and covariances of the estimator and extend its limit distribution to alpha-stability.

Among other reasons PWMs are criticized by statisticians because they are not as easily adaptable to the non iid case. We are currently generalizing the theory to the dependent case for which we assume some type of mixing behavior and to bi- and multivariate random variates.

Robustness in geostatistics

Robust Prediction for Contaminated Random Fields
In collaboration with Fournier, B., EPF Lausanne, Switzerland.

Interpolation of a spatially correlated random processes is used in many scientific domains. The best unbiased linear predictor (BLUP), often called kriging predictor in geostatistical science, is sensitive to outliers. Although there have been a few attempts to robustify the kriging predictor, none of them is completely satisfactory. We introduce a new robust linear predictor for a substitutive error model. First, we derive a BLUP, which is computationally very expensive even for moderate sample sizes. The exact solution is approximated by a simple linear predictor, which is robust with respect to substitutive errors.

Model assumptions in the considered model could still be weakened resulting in more flexibility with respect to possible structures. Natural extensions of the method are to non-Gaussian fields and correlated contamination scenarios in the substitutive error model.