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Mixture models: datapoint has nonzero probs to belong to multiple k distributions

Explore mixture models to estimate data distribution, with emphasis on hidden variables and Gaussian hypotheses. Learn to estimate posteriors and maximize results through analytical methods. Utilize techniques like Vector Quantization and Self-Organizing Maps for data visualization and analysis.

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Mixture models: datapoint has nonzero probs to belong to multiple k distributions

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  1. EM • Mixture models: datapoint has nonzero probs to belong to multiple k distributions • So Hidden Var in each datapoint: e.g for k=2 <xi zi1 zi2> • hypothesis h about parameters of ,say, k Gaussians. • Estimateposterior: E (P(D|h’) | x, h) • Maximize : argmax E(P(D|h’)|x, h) • Result: priors on distribution, mean and variance binary membership var, hidden Analytical for multivariate normal distr

  2. VQ • vector quantization • map a set of M coding vectors (red) to a cloud of N data points (not shown) : Q(xi)=cj • ..using neighboring relations (Euclidian distance) • http://www.data-compression.com/vq.html

  3. SOM • M “neurons”= M coding vectors (cfr. VQ) • But.. neurons are CONNECTED, so they form an “elastic net”. • Elasticity or mutual influence determined by a kernel function that is defined on a 2D grid • the coding vectors and their “projection (visualization)” on the 2D grid constitute the SOM

  4. Useful Code • sDpima = som_read_data('d:\\patrick\\projects\\oefn_johan\\PIMADATAsorted2.txt'); • sDpima=som_normalize(sDpima, 'var') • sMap=som_make(…) • sMap=som_autolabel(sMap, sDpima, 'vote') • som_show(sMap,'norm','d') % basic visualization • som_show(sMap,'umat','all','empty','Labels') % UMatrix • som_show_add('label',sMap,'Textsize',8,'TextColor','r','Subplot',2) %labels • SORT DATASET (file) according to labels with e.g. spreadsheet program • % h1 = som_hits(sMap,sDpima.data(1:500,:));  -1’s • % h2 = som_hits(sMap,sDpima.data(501:768,:));  +1’s • % som_show_add('hit',[h1, h2],'MarkerColor',[1 0 0; 0 1 0],'Subplot',1)

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