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Nonparametric approaches face problems of computational complexity Semiparametric approaches combine the advantages of both parametric and nonparametric approaches and are more popular Semiparametric approaches are computationally ef cient and provide locally optimal solutions A popular semiparametric density estimation approach is the expectation maximization (EM) algorithm The EM algorithm uses Gaussian mixture model and the maximum likelihood approach to estimate the pdf for a given data set The EM algorithm is well known to converge ef ciently to the local optimum Researchers from statistics and economics communities have recently shown interest in using maximum entropy (ME) measures for density estimation problems For example, it has been shown that an encoding and decoding approach based stochastic gradient ascent method for entropy maximization can be used to estimate probability densities It has also been proposed that a sequential updating procedure be used to compute maximum entropy densities subject to known moment constraints We may use a ME density approach to estimate error distribution in regression models All researchers have reported good results with the use of ME measures Others argue that ME densities have simple function forms that nest most of the commonly used pdfs, including the normal distribution, which is considered a special case as opposed to a limiting case All the procedures proposed and used in these studies converge to the local optimum and suffer from similar computational issues as that of the EM algorithm While the ME approach for density estimation has shown promise, current approaches cannot be applied directly to clustering problems in data mining These approaches assume either knowledge of data distribution or prior knowledge of movement constraints Since a typical application of density estimation in the data mining literature is likely to be an unsupervised learning problem, such prior knowledge may not be available In this case, we use a global search heuristic genetic algorithm for density estimation Genetic algorithms (GAs) are population based parallel search techniques that are likely to nd heuristic optimal solutions to optimization problems Unlike gradient search algorithms used in EM and ME density estimation, which are likely to get stuck in a local optimum, GAs are likely to provide solutions that are close to the global optimum We use GAs to estimate pdfs on a simulated data set using both the maximum likelihood (ML) formulation and the ME formulation Semiparametric Density Estimation Typical semiparametric density estimation consists of a nite mixture of the density models of k probability distributions, where each of the k distributions represents a cluster The value of k is usually less than the number of data points We use a Gaussian mixture model, where true pdf is considered to be a linear combination of k basis functions We use GAS for optimizing the nontrivial ML and ME optimization problems GAs are general purpose, stochastic, parallel search approaches that are used for optimization problems [8] We use oating point representation for our research
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All the optimization variables are represented as genes in a population member We use single-point crossover and a single gene mutation Simulated Data and Experiments We generate a two-attribute simulated data set using simulations of three different normal distributions We generate our data using three normal distributions with means of 1, 0, and 1 The standard deviations for all the distributions are considered equal to one These three distributions represent three clusters Figures 31 and 32 illustrate the data distributions and their contour plots We generated 20 data points for each distribution with a total of 60 data points We test the accuracy of our ML and ME optimization on our simulated data set Since the problem is that of cluster analysis, our data input vector contained the tuple < x1, x2> and we speci ed the value of k 3 in our experiments We use the GA procedure to optimize the ML and ME functions The GA parameters were set after initial experimentation as follows Mutation rate was set to 01, crossover rate was set to 03, and the terminating iteration condition was set to 1000 learning generations The cluster assignment for each data point in the sample was conducted after the optimization of the function and nding the maximum value of the likelihood Figure 31 shows the cluster means and standard deviations obtained for ME and ML objective functions The cluster assignments from the algorithms were compared with the actual distribution from which a data point was generated to compute correct classi cation The preliminary results of our experiments indicate that ME approach, when compared to ML, appears to fare well The standard deviations of the ML approach are lower than or equal to the ME approach and it appears that the ML approach is overly conservative
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