Oct. 4, 2012
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8:30am - 12:00pm, 2235 SAS Hall
Statistical Models and Methods for Spatial and Spatio-temporal Data with Strong Local Dependence
Instructor: Michael Stein, University of Chicago
For environmental processes observed densely in space and/or time, it is common to find that nearby observations have very similar values. This tutorial discusses various approaches to analyzing data with such structure, with a focus on Gaussian process models. Stationary Gaussian processes can be described in both the space-time and frequency domains; the relationships between these two descriptions provides key insights into the behavior of Gaussian processes. After summarizing these relationships, I will discuss how Gaussian process models can be used in practice, with a focus on the opportunities and challenges that occur for large datasets. As an example, I will examine total column ozone values measured by the satellite-based OMI (Ozone Monitoring Instrument) from a single orbit, yielding about 83,000 observations. Although 83,000 does not sound so large these days, the observations are not on a grid and thus the resulting 83,000 by 83,000 covariance matrix may not have any exploitable structure, making exact likelihood calculations under many natural models unfeasible. Low rank approximations to the covariance matrix greatly reduce the computational and memory burdens associated with using Gaussian process models. Furthermore, low rank approximations have been found useful for approximating matrices in a number of disciplines. However, in my experience, they generally provide a poor fit to spatial or space-time data when nearby observations are strongly dependent, at least if one uses likelihood-based statistical methods. I will consider alternative approaches to addressing the computational difficulties of analyzing large space-time datasets. I will conclude with some discussion of promising research directions in the statistical analysis of space-time environmental data.
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1:30pm - 5:00pm
, 2235 SAS Hall
Introduction to Analysis of Extremes: Univariate and Multivariate Cases
Instructor: Dan Cooley, Colorado State University
When assessing risk associated with extreme events, the questions that need to be answered require an accurate description of a distribution’s tail and may require the researcher to extrapolate beyond the range of the data in the tail. This course will introduce the ideas and techniques involved in the analysis of extremes. The first half of the course will be devoted to the analysis of univariate data and the second half to the analysis of multivariate data. Extreme value analyses are based on fundamental results from probability theory which provide distributions appropriate for modeling the tail. The initial portion of the course will be devoted to introducing these fundamental results, and this will be done via demonstrations and examples (rather than mathematical proofs) so that the attendees develop some intuition for the underlying theory. Attention will then turn to statistical analysis of extreme data and the techniques used to describe the tail. The multivariate portion of the course will largely focus on how dependence is described for extremes: via an angular measure rather than via correlation. The target audience is quantitative scientists and statisticians who are unfamiliar with these methods. Upper-division undergraduate mathematical literacy is required, and basic understanding of mathematical statistics is recommended.