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mix - Mixture Models Mixture Models Jia Li Department of...

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Mixture Models Mixture Models Jia Li Department of Statistics The Pennsylvania State University Email: [email protected] http://www.stat.psu.edu/ jiali Jia Li http://www.stat.psu.edu/ jiali
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Mixture Models Clustering by Mixture Models General background on clustering Example method: k-means Mixture model based clustering Model estimation Jia Li http://www.stat.psu.edu/ jiali
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Mixture Models Clustering A basic tool in data mining/pattern recognition: Divide a set of data into groups. Samples in one cluster are close and clusters are far apart. Jia Li http://www.stat.psu.edu/ jiali
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Mixture Models Motivations: Discover classes of data in an unsupervised way (unsupervised learning). E cient representation of data: fast retrieval, data complexity reduction. Various engineering purposes: tightly linked with pattern recognition. Jia Li http://www.stat.psu.edu/ jiali
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Mixture Models Approaches to Clustering Represent samples by feature vectors. Define a distance measure to assess the closeness between data. “Closeness” can be measured in many ways. Define distance based on various norms. For gene expression levels in a set of micro-array data, “closeness” between genes may be measured by the Euclidean distance between the gene profile vectors, or by correlation . Jia Li http://www.stat.psu.edu/ jiali
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Mixture Models Approaches: Define an objective function to assess the quality of clustering and optimize the objective function ( purely computational ). Clustering can be performed based merely on pair-wise distances. How each sample is represented does not come into the picture. Statistical model based clustering . Jia Li http://www.stat.psu.edu/ jiali
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Mixture Models K-means Assume there are M clusters with centroids Z = { z 1 , z 2 , ..., z M } . Each training sample is assigned to one of the clusters. Denote the assignment function by η ( · ). Then η ( i ) = j means the i th training sample is assigned to the j th cluster. Goal: minimize the total mean squared error between the training samples and their representative cluster centroids, that is, the trace of the pooled within cluster covariance matrix . arg min Z , η N i =1 x i z η ( i ) 2 Jia Li http://www.stat.psu.edu/ jiali
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Mixture Models Denote the objective function by L ( Z , η ) = N i =1 x i z η ( i ) 2 . Intuition: training samples are tightly clustered around the centroids. Hence, the centroids serve as a compact representation for the training data. Jia Li http://www.stat.psu.edu/ jiali
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Mixture Models Necessary Conditions If Z is fixed, the optimal assignment function η ( · ) should follow the nearest neighbor rule, that is, η ( i ) = arg min j { 1 , 2 ,..., M } x i z j . If η ( · ) is fixed, the cluster centroid z j should be the average of all the samples assigned to the j th cluster: z j = i : η ( i )= j x i N j .
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