Athanasios Christou Micheas

Associate Professor of Statistics

Research Interests

General Research Interests

Multivariate Analysis, Model Selection and Checking, Bayesian Computation, Bayesian Robustness, Statistical Shape Analysis, Stochastic Geometry, Spatial Temporal Statistics, Point Process Theory, Random Set Theory.

Selected Research Interests

Modelling Objects

We often encounter objects that are relatively difficult to measure and describe in quantitative terms. Tree foliage, clouds, and automobiles passing through a busy intersection are examples of such objects. In certain applications, we observe large numbers of points in three-dimensions, where different subsets of points correspond to the different objects. One approach to quantifying these objects is to use the points collected to develop three-dimensional models that approximate the objects. When the objects of interest are widely spaced, it is much easier to identify them, but if they are close to each other, the membership of some sampled points to specific objects becomes unclear. In either case, the creation of statistical models that describe each object is crucial to perform inference.

Growth or Evolution Models

Growth, or change (evolution) in objects over time, is ubiquitous in nature. From observing tumor growth to assess the effectiveness of a medical treatment, studying the propagation and development of storm systems from radar images, to the investigation of an epidemic as it spreads through a populated area, defining and studying growth/evolution models has become an integral component of scientific research over the last century.

Growth can be modeled in a number of different ways. Many approaches have been deterministic, for example involving differential equations. Such models require a complete understanding of the phenomenon of interest, which is often unattainable. Only in the past few decades has there been a serious attempt to model growth of spatial objects stochastically and, more importantly, take the spatial component into consideration in the model. The use of set models, such as the Boolean model (e.g., see Cressie, 1993), was a major step towards understanding the evolution of an object, but introduced major mathematical and computational difficulties. Moreover, most important applications are not one-dimensional and in two and higher dimensions there is no clear way of specifying a stochastic mechanism to govern a random set. Defining a random variable, vector or matrix is a well understood concept that leaves little room for questioning. This is not the case with random sets.

Modelling and Theory of Point Processes

Except for random sets, which include any possible type of data as special cases, there are perhaps no more complicated data structures than observing realizations of point processes (point patterns). They are the natural generalization, from random variables and vectors, where we observe now a collection of points in R^{d} and require a model of the stochastic mechanism they arise from. We study the most commonly used model, that of a Non-homogeneous Poisson Point Process (NHPP) and its extensions, mainly from the Bayesian perspective.

Shape Analysis

The study of shapes that appear in experiments, has captured the attention of researchers in a variety of scientific contexts. From image analysis of Magnetic Resonance Images(MRI) to genetics, biology and a host of others, new methodologies are sorely needed in order to provide a better understanding of the shape of an object as well as help predict the shape of an object. Moreover, studying shape differences, either in average shape or shape variability, can provide medical doctors for example, with a new way of diagnosis and treatment of a certain disease, or paleontologists with a new method of categorizing specimens in certain age periods, and so forth.

Our goal is to create new methodologies that address these questions. We describe shapes by using "landmarks". Landmarks are points of correspondence on each shape, that match between and within populations of shapes. Then describing these landmarks stochastically, using some "shape distribution", allows us to infer about the average(modal) shape of a population of shapes, as well as assert shape differences between two or more populations of shapes. Bayesian as well as classical methods of estimation are being developed.