Van Lieshout, M.N.M. and Molchanov, I.S. and Zuev, S. (2001) Clustering methods based on variational analysis in the space of measures. Biometrika, 88 (4). pp. 10211033. ISSN 14643510

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Abstract
We formulate clustering as a minimisation problem in the space of measures by modelling the cluster centres as a Poisson process with unknown intensity function.We derive a Wardtype clustering criterion which, under the Poisson assumption, can easily be evaluated explicitly in terms of the intensity function. We show that asymptotically, i.e. for increasing total intensity, the optimal intensity function is proportional to a dimensiondependent power of the density of the observations. For fixed finite total intensity, no explicit solution seems available. However, the Wardtype criterion to be minimised is convex in the intensity function, so that the steepest descent method of Molchanov and Zuyev (2001) can be used to approximate the global minimum. It turns out that the gradient is similar in form to the functional to be optimised. If we discretise over a grid, the steepest descent algorithm at each iteration step increases the current intensity function at those points where the gradient is minimal at the expense of regions with a large gradient value. The algorithm is applied to a toy onedimensional example, a simulation from a popular spatial cluster model and a reallife dataset from Strauss (1975) concerning the positions of redwood seedlings. Finally, we discuss the relative merits of our approach compared to classical hierarchical and partition clustering techniques as well as to modern model based clustering methods using Markov point processes and mixture distributions.
Item type:  Article 

ID code:  4606 
Keywords:  cluster analysis, poisson point process, steepest descent, statistics, modelling science, biometrics, Probabilities. Mathematical statistics, Agricultural and Biological Sciences(all), Applied Mathematics, Statistics and Probability, Statistics, Probability and Uncertainty, Mathematics(all), Agricultural and Biological Sciences (miscellaneous) 
Subjects:  Science > Mathematics > Probabilities. Mathematical statistics 
Department:  Faculty of Science > Mathematics and Statistics > Statistics and Modelling Science 
Depositing user:  Strathprints Administrator 
Date Deposited:  06 Nov 2007 
Last modified:  27 Mar 2015 08:55 
URI:  http://strathprints.strath.ac.uk/id/eprint/4606 
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