CURRENT AND RECENT RESEARCH ACTIVITY

Thoughts on Research

Donald E. Myers

1. VALID MODELS FOR SPACE-TIME VARIOGRAMS

The variogram is used (especially in geostatistics) to quantify spatial correlation, i.e., similarity or dis-similarity in a statistical sense as a function of separation distance and direction. The variograms is analogous to the (auto)covariance function or (auto) correlation function except that it exists under weaker conditions. For a function defined on n-dimensional Euclidean space to be valid variogram it must satisfy certain conditions, e.g., the growth rate must be less than quadratic and it must be conditionally negative definite. The second condition is not an easy one to check for a particular function hence the practice is to use known valid models or positive linear combinations (the class of valid models is closed under positive linear combinations). When extending the variogram into space-time there are two general approaches that might be used, one is to treat space-time as simply a higher dimensional Euclidean space and the second is to “separate” space and time. The dis-advantage to the first approach is that it means one must have a metric or norm on space-time, which essentially means that time as a “dimension” is not really different than other Euclidean dimensions which contradicts some of the usual perceptions of time. The second approach is essentially the same as that of constructing a valid model in n-dimensional space from two models, one valid on k-dimensional space and the other on –k dimensional space (for space-time use k=1). This problem has already been addressed in part in a series of papers by Myers jointly with one or more of De Ioca, Posa, De Cesare.

The second part of the problem is using data in space-time to estimate and model a space-time variogram. This problem has already been addressed in part in a series of papers by Myers jointly with one or more of De Ioca, Posa, De Cesare. However neither of these problems can be considered completely resolved and work continues. In particular it is important to see links between the methods and constructions that have been used and those implicit in the Kalman filter.

As a possible application of the work on space-time variograms referenced above, a current project with EPA is looking at the use of space-time variograms to detect vegetation change in a ten year period using data from the Oregon Pilot Study area. The project will use NDVI (Normalized Vegetation Index) satellite data.

Robin Harris, P.I, Donald E. Myers and Mary Kay O’Rourke Co-PI’s

This project is funded by NIH through the Arizona Cancer Center