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

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A summary of our clustering results for the real data appears in Table 4.17. In general,the tMMDR algorithm gives higher ARI values but requires more features than itsGMMDR analogue. This is to be expected since there are more parameters to be estimatedin a t-mixture model than in a Gaussian mixture model, hence more informationis required.Table 4.17: Summary of the results for the real data.tMMDRGMMDRData Clust. Vars. Model Clust. Feat. ARI Model Clust. Feat. ARICoffee 2 13 CIUU 2 5 1 E 2 1 1Wine 3 13 CUCC 3 4 0.9309 VEV 3 5 0.85Crabs 4 5 CUCC 4 4 0.8617 EEV 4 3 0.8195Banknotes 2 6 CICU 3 6 0.8603 EEI 4 3 0.6739Diabetes 3 5 UUUC 3 4 0.7020 VEV 3 3 0.653640
Chapter 5ConclusionThis thesis introduces the idea of dimension reduction for model-based clustering withinthe multivariate t-distribution framework (tMMDR). This approach is based on existingwork for reducing dimensionality in the case of finite Gaussian mixtures (GMMDR).Work focused on identifying the smallest subspace of the data which captured theinherent cluster structure. This information was gathered by looking at how the groupmeans and group covariances varied, using the eigen-decomposition of the kernel matrix.The elements of the subspace consisted of linear combinations of the original data, whichwere ordered by importance via the associated eigenvalues. Observations were thenprojected onto the subspace and the resulting set of variables captured most of theclustering structure available in the data.The tMMDR approach was illustrated using simulated and real data and its performancecompared to that of the GMMDR method. The evaluation was done usingthe ARI. In the case of synthetic data, tMMDR produced better results generally andspecifically for the scenarios with higher levels of complexity (i.e., more data points andvariables). For the real data, the tMMDR algorithm showed higher ARI values than itsGMMDR analogue, but the former selected a slightly bigger set of features (usually onemore feature than GMMDR).Overall, using dimension reduction via mixtures of t-distributions shows great potentialfor model-based clustering.In terms of future work, the application of tMMDR and GMMDR to mixtures of factoranalyzers could be investigated. More generally, the method of dimension reductioncould be extended to other model-based paradigms such as classification.41
Page 1: Dimension Reduction for Model-based
Page 5 and 6: List of Tables2.1 Nomenclature for
Page 7: Chapter 1IntroductionClustering alg
Page 10 and 11: 2.1 Gaussian Mixture ModelsFrom McL
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: Table 4.16: A classification table
Page 49 and 50: Forina, M., C. Armanino, M. Castino
Page 51: Meng, X. L. and D. B. Rubin (1993).

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