Gaussian Mixtures Vol.2 - Introduction to - Alexandre Devert PDF
Preview Gaussian Mixtures Vol.2 - Introduction to - Alexandre Devert
UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Gaussian Mixtures Vol.2 Introduction to probabilistic models Devert Alexandre SchoolofSoftwareEngineeringofUSTC December13,2012—Slide1/29 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Table of Contents Multivariate Gaussian distribution 1 Multivariate mixtures 2 Definition Parameter estimation EM clustering 3 Clustering EM clustering vs. kmeans DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—GaussianMixturesVol.2—Slide2/29 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Multivariate Gaussian distribution The multivariate Gaussian distribution is N dimensions equivalent of a Gaussian distribution DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—GaussianMixturesVol.2—Slide3/29 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Multivariate Gaussian distribution To build a multivariate Gaussian distribution, step 1 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.006 4 2 0 2 4 6 Take a single variate distribution with µ = 0 and σ = 1 DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—GaussianMixturesVol.2—Slide4/29 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Multivariate Gaussian distribution To build a multivariate Gaussian distribution, step 2 probability density samples Pull it to N dimension with a central symmetry DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—GaussianMixturesVol.2—Slide5/29 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Multivariate Gaussian distribution To build a multivariate Gaussian distribution, step 3 probability density samples Scale its along the axes DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—GaussianMixturesVol.2—Slide6/29 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Multivariate Gaussian distribution To build a multivariate Gaussian distribution, step 4 probability density samples Rotate it DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—GaussianMixturesVol.2—Slide7/29 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Multivariate Gaussian distribution To build a multivariate Gaussian distribution, step 5 probability density samples Translate it DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—GaussianMixturesVol.2—Slide8/29 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Probability density Probability density for a multivariate Gaussian distribution in dimension n 1 Φ(x) = e−1(x−µ)TΣ−1(x−µ) 2 (2π)k|Σ|1 2 2 • µ is a vector ⇒ the mean of the distribution • σ is a positive definite matrix ⇒ the covariance of the the distribution DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—GaussianMixturesVol.2—Slide9/29 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA SCHOOL OF SOFTWARE ENGINEERING OF USTC Parameters estimations The sample mean for a multivariate Gaussian distribution n 1 (cid:88) X¯ = X i n i=1 You can compute it as the mean of the samples coordinates per coordinates, as introduced in the univariate case. DevertAlexandre(SchoolofSoftwareEngineeringofUSTC)—GaussianMixturesVol.2—Slide10/29
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