Abstract
Clusters of different objects are of great interest in many fields, such as agriculture and ecology. One kind of clustering methods is very different from the traditional statistical clustering analysis, which is based on discrete data points. This method of clustering defines clusters as the connected areas where a well-defined spatial random field is above certain threshold. The statistical properties, especially the distributional properties, of the defined clusters are vital for the studies of related fields. However, the available statistical techniques for analyzing clustering models are not applicable to these problems. We study the distribution properties of the clusters by defining a distribution function of the clusters rigorously and providing methods to estimate the spatial distribution function. Our results are illustrated by numerical experiments and an application to a real world problem.
Highlights
Analyses of clusters of soil, water and species have been of great interest in agriculture, ecology and hydrology; see, for example, Asnera and Warner (2003); Wootton (2001); Martin and Goldenfeld (2006); Sole (2007)
The analyses of clusters often focus on the spatial properties, such as the size of an individual cluster and the locations of the centers of clusters, assuming the object of interest is modeled by a continuous random process y(s, ω), where s is the parameter of space and ω is some sample point
More examples will be given after the Empirical Distribution Function (EDF) is defined
Summary
Analyses of clusters of soil, water and species have been of great interest in agriculture, ecology and hydrology; see, for example, Asnera and Warner (2003); Wootton (2001); Martin and Goldenfeld (2006); Sole (2007). The mathematical and statistical definition of the size of individual cluster has not been well defined and studied, though samples can be collected from images obtained from many different ways, such as remote sensors. Given a well-defined spatial statistical model, it is generally difficult to define a random variable as the size of an individual cluster and study its distribution. Without a well defined distribution function of the clusters, it is difficult for researchers to study the statistical properties of the data of image clustering and perform efficient statistical inferences.
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