Structure of multidimensional patterns

The problem of describing the structure of multidimensional data is important in exploratory data analysis, statistical pattern recognition, and image processing. A data set is viewed as a collection of points embedded in a high dimensional space. The primary goal of this research is to determine if the data have any clustering structure; such a structure implies the presence of class information (categories) in the data. A statistical hypothesis is used in the decision making. To this end, data with no structure are defined as data following the uniform distribution over some compact convex set in K-dimensional space, called the sampling window. This thesis defines two new tests for uniformity along with various sampling window estimators. The first test is a volume-based test which captures density changes in the data. The second test compares a uniformly distributed sample to the data by using the minimal spanning tree (MST) of the polled samples. Sampling window estimators are provided for simple sampling windows and use the convex hull of the data as a general sampling window estimator. For both of the tests for uniformity, theoretical results are provided on their size, and study their size and power against clustered alternatives is studied. Simulation is also used to study the efficacy of the sampling window estimators