Dimension Reduction for Model-based Clustering via Mixtures of ...

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2.1 Gaussian Mixture ModelsFrom McLachlan and Peel (2000), we know that the density of a multivariate Gaussianmixture model is given byf(x|ϑ) =G∑π g f N (x|µ g , Σ g ) ,g=1where f N (x|µ g , Σ g ) is the density of a multivariate Gaussian distribution with mean µ gand covariance matrix Σ g , i.e.f N (x|µ g , Σ g ) = exp{− 1(x − µ 2 g) ⊤ Σ −1g (x − µ g )}. (2.1)(2π) p 2 |Σ g | 1 2Figure 2.1 illustrates the contour and 2-dimensional plot for a bivariate normal distributionwith µ = [0, 0] ⊤ and Σ = diag(2). By varying Σ, one can obtain differentorientations, shapes and volumes for the distribution.2.2 Model-based Clustering with MCLUSTModel-based clustering is a method which assumes that the data are generated by amixture of underlying probability distributions in which each component represents adifferent group or cluster. The mixture likelihood is given byL M (θ 1 , . . . , θ G ; π 1 , . . . , π G |x) =n∏ G∑π g f N (x i |θ g ) , (2.2)i=1 g=1where x = {x 1 , . . . , x n } represents the data.For the Gaussian density in (2.1) clusters are ellipsoidal and centred at the meansµ g . Other geometric properties are determined by the covariances Σ g . Banfield andRaftery (1993) developed a model-based framework for clustering by using the followingeigenvalue decomposition of the covariance matrixΣ g = λ g D g A g D ⊤ g , (2.3)4
Figure 2.1: Contour and perspective plots for a bivariate Normal distribution withµ = [0, 0] ⊤ and Σ = diag(2).where• D g is the orthogonal matrix of eigenvectors which determines the orientation ofthe principal components of Σ g ;• A g is a diagonal matrix with elements proportional to the eigenvalues of Σ g whichdetermines the shape of the density contours;5
Page 1: Dimension Reduction for Model-based
Page 5 and 6: List of Tables2.1 Nomenclature for
Page 7: Chapter 1IntroductionClustering alg
Page 12 and 13: • λ g is a scalar which specifie
Page 14 and 15: In the M-step of the algorithm, the
Page 16 and 17: simple example. McLachlan and Peel
Page 18: For the tEIGEN models, parameter es
Page 21 and 22: onto the reduced subspace and their
Page 23 and 24: to the others. Specifically, we nee
Page 25 and 26: Definition 3.2 The tMMDR variables,
Page 27 and 28: • the forward step evaluates the
Page 29 and 30: Chapter 4ApplicationsWe now focus o
Page 31 and 32: Figure 4.2: Pairs plot for the thre
Page 33 and 34: Table 4.1: Average ARI (with standa
Page 35 and 36: The known classification of the cof
Page 37 and 38: Table 4.5: Variable description for
Page 39 and 40: 4.2.3 Australian Crabs DataCampbell
Page 41 and 42: Figure 4.6: Plots of tMMDR directio
Page 43 and 44: those notes. The results may indica
Page 45 and 46: Table 4.16: A classification table
Page 47 and 48: Chapter 5ConclusionThis thesis intr
Page 49 and 50: Forina, M., C. Armanino, M. Castino
Page 51: Meng, X. L. and D. B. Rubin (1993).

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