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Local Classification Methods for Heterogeneous Classes Julia Schiffner and Claus Weihs Department of Statistics, Dortmund University of Technology SFB 475 Complexity Reduction in Multivariate Data Structures August 13, 2008 J. Schiffner

  1. Local Classification Methods for Heterogeneous Classes Julia Schiffner and Claus Weihs Department of Statistics, Dortmund University of Technology SFB 475 ‘Complexity Reduction in Multivariate Data Structures’ August 13, 2008 J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  2. Outline 1 Introduction – Heterogeneous Classes 2 Three Classification Methods Based on Mixture Models 3 Local Fisher Discriminant Analysis – LFDA 4 Summary & Outlook J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  3. Introduction – Heterogeneous Classes package klaR : miscellaneous functions for classification and visualization classification into K given classes c 1 , . . . , c K underlying assumption for many classification methods: random feature x homogeneous within the classes and heterogeneous across the classes problem: heterogeneous classes J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  4. Introduction – Heterogeneous Classes package klaR : miscellaneous functions for classification and visualization classification into K given classes c 1 , . . . , c K underlying assumption for many classification methods: random feature x homogeneous within the classes and heterogeneous across the classes problem: heterogeneous classes J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  5. Introduction – Heterogeneous Classes package klaR : miscellaneous functions for classification and visualization classification into K given classes c 1 , . . . , c K underlying assumption for many classification methods: random feature x homogeneous within the classes and heterogeneous across the classes problem: heterogeneous classes J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  6. Introduction – Heterogeneous Classes problem: heterogeneous classes 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 2 1 1 1 11 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 2 2 1 2 2 2 2 1 22 1 1 2 1 2 2 2 1 1 1 2 1 2 2 2 2 22 2 2 2 2 1 2 2 1 1 2 1 2 2 2 22 2 2 2 2 2 1 1 1 22 2 2 2 2 2 2 2 2 2 2 2 22 2 2 1 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  7. Introduction – Heterogeneous Classes problem: heterogeneous classes 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 11 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 1 1 1 2 1 2 1 2 2 2 22 1 1 1 2 2 1 2 2 1 1 1 2 2 1 2 2 2 2 1 2 22 2 2 1 1 2 2 2 2 1 2 2 2 22 2 2 2 2 1 1 1 22 2 2 2 2 2 2 2 2 2 2 22 2 2 2 1 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 way out: local methods 2 2 classification methods based on mixture models , e. g. mixture discriminant analysis (MDA) other prototype methods: K-means, learning vector quantization (LVQ) k-nearest-neighbor classifier (kNN) local likelihood methods: localized logistic regression, localized LDA (LLDA, in klaR ) local Fisher discriminant analysis (LFDA) tree-based methods: CART, random forests J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  8. Introduction – Heterogeneous Classes problem: heterogeneous classes 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 11 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 1 1 1 2 1 2 1 2 2 2 22 1 1 1 2 2 1 2 2 1 1 1 2 2 1 2 2 2 2 1 2 22 2 2 1 1 2 2 2 2 1 2 2 2 22 2 2 2 2 1 1 1 22 2 2 2 2 2 2 2 2 2 2 22 2 2 2 1 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 way out: local methods 2 2 classification methods based on mixture models , e. g. mixture discriminant analysis (MDA) other prototype methods: K-means, learning vector quantization (LVQ) k-nearest-neighbor classifier (kNN) local likelihood methods: localized logistic regression, localized LDA (LLDA, in klaR ) local Fisher discriminant analysis (LFDA) tree-based methods: CART, random forests J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  9. Mixture Models in Classification marginal density: K � f ( x ) = p k f ( x | c k ) k = 1 model class conditional densities as mixtures data are generated by J sources s j hierarchical mixture model (Titsias & Likas, 2002) common components model (Titsias & Likas, 2001) J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  10. Mixture Models in Classification marginal density: K � f ( x ) = p k f ( x | c k ) k = 1 model class conditional densities as mixtures data are generated by J sources s j hierarchical mixture model (Titsias & Likas, 2002) common components model (Titsias & Likas, 2001) J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  11. Mixture Models in Classification marginal density: K � f ( x ) = p k f ( x | c k ) k = 1 model class conditional densities as mixtures data are generated by J sources s j hierarchical mixture model (Titsias & Likas, 2002) K J � � f ( x ) = p k π jk f ( x | c k , s j ) k = 1 j = 1 common components model (Titsias & Likas, 2001) J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  12. Mixture Models in Classification marginal density: K � f ( x ) = p k f ( x | c k ) k = 1 model class conditional densities as mixtures data are generated by J sources s j hierarchical mixture model (Titsias & Likas, 2002) J K � � f ( x | θ ) = p kj f ( x | µ kj , Σ kj ) π j j = 1 k = 1 common components model (Titsias & Likas, 2001) J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  13. Mixture Models in Classification marginal density: K � f ( x ) = p k f ( x | c k ) k = 1 model class conditional densities as mixtures data are generated by J sources s j hierarchical mixture model (Titsias & Likas, 2002) J K � � f ( x | θ ) = p kj f ( x | µ kj , Σ kj ) π j j = 1 k = 1 common components model (Titsias & Likas, 2001) K J � � f ( x ) = p k π jk f ( x | s j ) k = 1 j = 1 J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  14. Mixture Models in Classification marginal density: K � f ( x ) = p k f ( x | c k ) k = 1 model class conditional densities as mixtures data are generated by J sources s j hierarchical mixture model (Titsias & Likas, 2002) J K � � f ( x | θ ) = p kj f ( x | µ kj , Σ kj ) π j j = 1 k = 1 common components model (Titsias & Likas, 2001) J K J � � � f ( x | θ ) = p kj f ( x | µ j , Σ j ) = π j f ( x | µ j , Σ j ) π j j = 1 k = 1 j = 1 J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  15. Hierarchical Mixture Classifier class posterior estimation step 1 : estimate source posteriors assuming a simple mixture model (unsupervised, "hm1" ) J � f ( x | ϕ ) = π j f ( x | µ j , Σ j ) j = 1 EM algorithm ⇒ P ( s j | x , ˆ ϕ ) common components model (supervised, "hm2" ) J � f ( x | ϕ k ) = π jk f ( x | µ j , Σ j ) j = 1 EM algorithm ⇒ P ( s j | x , c ( x ) , ˆ ϕ c ( x ) ) step 2 : ML estimation of π j , p kj , µ kj , and Σ kj depending on x and the source posteriors J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  16. Hierarchical Mixture Classifier class posterior estimation step 1 : estimate source posteriors assuming a simple mixture model (unsupervised, "hm1" ) J � f ( x | ϕ ) = π j f ( x | µ j , Σ j ) j = 1 EM algorithm ⇒ P ( s j | x , ˆ ϕ ) common components model (supervised, "hm2" ) J � f ( x | ϕ k ) = π jk f ( x | µ j , Σ j ) j = 1 EM algorithm ⇒ P ( s j | x , c ( x ) , ˆ ϕ c ( x ) ) step 2 : ML estimation of π j , p kj , µ kj , and Σ kj depending on x and the source posteriors J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

  17. Hierarchical Mixture Classifier class posterior estimation step 1 : estimate source posteriors assuming a simple mixture model (unsupervised, "hm1" ) J � f ( x | ϕ ) = π j f ( x | µ j , Σ j ) j = 1 EM algorithm ⇒ P ( s j | x , ˆ ϕ ) common components model (supervised, "hm2" ) J � f ( x | ϕ k ) = π jk f ( x | µ j , Σ j ) j = 1 EM algorithm ⇒ P ( s j | x , c ( x ) , ˆ ϕ c ( x ) ) step 2 : ML estimation of π j , p kj , µ kj , and Σ kj depending on x and the source posteriors J. Schiffner and C. Weihs Local Classification Methods for Heterogeneous Classes

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