Peter Guttorp is Professor of Statistics, Quantitative Ecology & Resource Management, and Urban Design & Planning at the University of Washington, Professor at the Norwegian Computing Center in Oslo, Norway, and Adjunct Professor of Statistics and Actuarial Science at Simon Fraser University, Burnaby, Canada. He is an invited member of the ISI, and a fellow of the ASA. He has served as president of the International Environmetric Society, is co-editor of Environmetrics, and has served on the editorial boards of many journals. His research interests include stochastic models in biological and earth sciences. Recently his focus has been on statistical climatology, but he also has a long-standing interest in point processes and the history of statistics. Peter will present his IMS Medallion Lecture at the JSM in Montreal on Tuesday, August 6, 2013, at 10:30am in room 710b.

Pointing in new directions

Over the last decade or two, much of point process analysis has focused on Markovian point process models and log-Gaussian doubly stochastic Poisson processes. While these are flexible models, they are of the black box type, in which the modeling is not directly driven by understanding of the underlying scientific phenomenon. Much of the analysis of spatial patterns has been done using second order parameters, such as Ripley’s K-function, that are essentially designed for isotropic patterns, with dependence structure depending only on distance. Modern data tend to be collected in streams, and so the patterns should have some dynamic aspects to them. While a pattern of large tree locations may not be changing much over time (with the exception of the occasional tree death), the pattern of eye movements in a person viewing a painting is inherently dynamic, with temporal as well as spatial nonstationarity evident in the data.

The history of point processes goes back to 1767, at least, when Michell studied the probability of observing two stars close together “upon the supposition that they had been scattered by mere chance,” in other words, using a homogeneous Poisson process. The same type of calculation was done by Clausius in 1857 in the molecular kinetic theory of heat, and by Abbe in 1879 to describe the distribution of blood cells over a microscope slide. New point process models have often come from scientific applications. Neyman described the distribution of larvae on a potato field to introduce cluster processes in 1939, while Le Cam described a doubly stochastic precipitation mechanism in 1949.

Epidermic nerve fibers and self-avoiding clusters

The nerve fiber bundles that go from deep skin to the epidermic layer can be damaged by high blood sugar, and are therefore thought to have potential diagnostic value for diseases such as diabetes. The availability of realizable techniques for obtaining data on these nerve fibers is relatively recent. While the process really is three-dimensional, a two-dimensional projection is all that is needed when looking at the locations of the nerve fibers. This process is a cluster process, with cluster centers being the roots of the fibers, and cluster points being nerve fiber ends. For physiological reasons, the clusters tend not to overlap, and thus models in which clusters are not independent but in essence avoiding each others are needed. The cluster centers have different characteristics depending on what body part the sample is taken from. Thus, a more general approach to cluster processes is needed to analyze these data. Furthermore, the data sets originating in a neurophysiological laboratory at the University of Minnesota, can be used as a laboratory for looking at separation of model assumptions.

Art and the eye

The first major experiment in tracking eye movements when watching art was published in 1935 by Buswell, using film recording with a time resolution of 30 frames per second. Since then the technology of course has improved, and the subjects no longer need to have their heads held fixed in a frame. The time resolution is now a millisecond.

In an experiment in Finland, the eye movement of ten subjects with some experience in watching art, and ten novices, were recorded for different paintings. The basic structure of eye movement when viewing a painting or other structured image is to fixate the pupil in a spot for some amount of time, and then very quickly move the eye to a different spot. The movement is called a saccade. Previous research indicates that especially experienced viewers would fixate the most important parts of the painting (if there are people in the picture, they tend to have high frequency of fixations, for example), and then follow the structural construction of the painter to the next point of importance. On the other hand, novices tend to focus in different areas from experienced viewers, and follow less important structural elements.

A simple model puts together independent fixation durations and saccades between fixation locations. Temporal nonstationarity can be modeled through time dependent, intensity-based new locations, and differences between novices and non-novices modeled using different distributions to draw the fixation and saccade durations from.