Colloquium

Department of Statistics

Dissertation Proposal
April 20 (Fri) 11 a.m.
Alavi Commons Room, 6625 Everett Tower

Estimation and Inference for Multivariate Spatial and Spatio-Temporal Mixed Effects Models

Casey Jelsema
Department of Statistics
Western Michigan University

Right skewed data are commonly found in the environmental sciences. Lognormal kriging equations provide optimal estimates for such data. A common problem in lognormal kriging is that of block kriging, which seeks the average prediction over a region of nite area. Lognormal kriging and block kriging equations have been previously derived, but these needed to be extended to the multivariable setting. Following this, we applied lognormal block kriging method to a multivariable spatial dataset. Reduced rank approaches were inevitable as our datasets are large, in the order of 60,000 observations for each spatial process under consideration. We used Fixed Rank Kriging (FRK) to develop a spatial mixed effects (SME) model, also extending this to the multivariable setting. After setting up the multivariable spatial mixed effects model, we estimated the covariance matrix of the processes. There are multiple methods to accomplish this, we explored two of them: the EM algorithm and the method of moments (MoM).

The EM algorithm depends heavily on the likelihood assumption, but for large datasets, validating a perfect normality assumption is a daunting task. In the exploratory data analysis, quantile plots revealed many outliers, indicating that the data was not perfectly lognormal. This called for distribution-free methods of covariance estimation. We used the relatively well-known MoM estimation, but even the recommended MOM estimation of the covariance is susceptible to outliers. As a result we have begun to explore a way in which we can robustify the covariance estimator produced through the MOM. We will also be exploring robust alternatives to the Frobenius norm as a fitting stage for covariance estimation.

Another line of research we are exploring is the problem of kriging on spatial processes which exhibit different distributions. This requires first fitting a distribution to each process, and then creating a model incorporating this information before estimation. Since the dataset we use is large, this presents similar challenges as the other research and requires a spatial mixed effects model for efficient computation.

Finally, spatial processes often depend on time as well as space, but purely spatial models will fail to account for this. Current methods for space-time versions of FRK suffer from the same susceptibility to outliers as the spatial-only case. We will be investigate methods of estimation for space-time models which are robust out outliers.

All statistics students are expected to attend.

Past colloquiums