Lecture'12:' SSMs;'Independent'Component'Analysis;' Canonical'Correla;on'Analysis'
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1 Lecture'12:' SSMs;'Independent'Component'Analysis;' Canonical'Correla;on'Analysis' Lester'Mackey' May'7,'2014' ' Stats'306B:'Unsupervised'Learning'
2 Beyond'linearity'in'state'space'modeling' Credit:'Alex'Simma' Weather'forecas,ng''''''' Pose'es,ma,on' Dynamics implicitly determined by geophysical simulations Observed image is a complex function of the 3D pose, other nearby objects & clutter, lighting conditions, camera calibration, etc.
3 Approximate'nonlinear'filters' Credit:'Alex'Simma' Typically'cannot'directly'represent'these' con,nuous'filtering'distribu,ons'or'determine'a' closed'form'for'the'predic;on'integral' A'wide'range'of'approximate'nonlinear'filters'have' thus'been'proposed,'including'! Histogramfilters! Extended&unscentedKalmanfilters! Par7clefilters!
4 Approximate'nonlinear'filtering'methods' Credit:'Alex'Simma' Histogram Filter:! Evaluateonfixeddiscre7za7ongrid! Replacescon7nuouslatentvariablebydiscretevariable! Onlyfeasibleinlowdimensions! Expensiveorinaccurate Extended/Unscented Kalman Filter:! ApproximateposteriorasGaussianvialineariza7on, quadrature,! Inaccurateformul7modalposteriordistribu7ons Particle Filter:! Approximatefilteringdistribu7onwithfixed setofpar7cles(deltamasses)! Par7cleloca7onsnotfixed;dynamically evaluatestateswithhighestprobability! MonteCarloapproxima7on
5 Independent'component'analysis'(ICA)' Courtesy:'Rob'Tibshirani' X = AS where X is a random p-vector representing multivariate input measurements. S is a latent source p-vector whose components are independently distributed random variables. A is p p mixing matrix. Given realizations x 1,x 2,...,x N of X, thegoalsoficaareto Estimate A Estimate the source distributions S j f Sj,j=1,...,p.
6 The'cocktail'party'problem' Courtesy:'Rob'Tibshirani' In a room there are p independent sources of sound, and p microphones placed around the room hear different mixtures. Source Signals Measured Signals PCA Solution ICA Solution Here each of the x ij = x j (t i )andrecoveredsourcesarea time-series sampled uniformly at times t i.
7 Independent'vs.'uncorrelated' Courtesy:'Rob'Tibshirani' WoLOG can assume that E(S) = 0 and Cov(S) = I, and hence Cov(X) =Cov(AS) =AA T. Suppose X = AS with S unit variance, uncorrelated Let R be any orthogonal p p matrix. Then and Cov(S )=I X = AS = AR T RS = A S It is not enough to find uncorrelated variables, as they are not unique under rotations. Hence methods based on second order moments, like principal components and Gaussian factor analysis, cannot recover A. ICA uses independence, and non-gaussianity of S, torecovera e.g. higher order moments.
8 Independent'vs.'uncorrelated' Source S Data X PCA Solution ICA Solution Principal components are uncorrelated linear combinations of X, chosen to successively maximize variance. Independent components are also uncorrelated linear combinations of X, chosen to be as independent as possible. Courtesy:'Rob'Tibshirani'
9 ICA'representa;on'of'natural'images' Courtesy:'Rob'Tibshirani' Pixel blocks are treated as vectors, and then the collection of such vectors for an image forms an image database. ICA can lead to a sparse coding for the image, using a natural basis. see Hoyer and Aapo Hyvärinen, Helsinki University of Technology)
10 ICA'and'electroencephalographic'(EEG)'data' 15'seconds'of'EEG'data'' 9'(of'100)'scalp'channels' 9'ICA'components'' Nearby'electrodes'record' nearly'iden;cal'mixtures' of'brain'and'nonxbrain' ac;vity;'ica'components' are'temporally'dis;nct.'' Pa;ent'blinking'(IC1/3)' IC12:'cardiac'pulse'' IC4/7'ac;vity'a[er'string' of'correct'responses' Colored'scalps'represent' ICA'unmixing'coefficients' a j 'as'heatmap,'showing' brain'or'scalp'loca;on'of' the'source.''
11 Approaches'to'ICA' Courtesy:'Rob'Tibshirani' ICA literature is HUGE. Recent book by Hyvärinen, Karhunen & Oja (Wiley, 2001) is a great source for learning about ICA, and some good computational tricks. Mutual Information and Entropy, maximizing non-gaussianity FastICA (HKO 2001), Infomax (Bell and Sejnowski, 1995) Likelihood methods ProdDenICA (Hastie +Tibshirani)- later Nonlinear decorrelation Y 1 independent Y 2 iff max g,f Corr[f(Y 1 ),g(y 2 )] = 0 (Hérault-Jutten, 1984), KernelICA (Bach and Jordan, 2001) Tensorial moment methods
12 Blackboard'discussion' " See'lecture'notes'
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