Local statistical modeling by cluster-weighted

Noname manuscript No. (will be inserted by the editor) Local Statistical Modeling via Cluster -W eighted A pproach with Elliptical Distributions Salva tore Ingrassia · Simona C. Minotti · Giorgio V ittadini Recei ved: date / Accept ed: date Abstract Cluster W eighted Modeling (CWM) is a mixture approach regarding the mod- elisation of the joint probability of data coming from a heterogeneous population. Under Gaussian assumptions, we in vestigate statistical pro perties of CWM from both the theoreti- cal and numerical point of vie w; in particular , we s ho w t hat CWM includ es as special cases mixtures of distrib utions and mixtures of r egression s. Further , we introd uce CWM based o n Student- t dist rib utions providing more robust fi tting for groups of observ ations with longer than no rmal tails o r atypical o bserv ations. Theoretical results are illustrated using som e em- pirical studies, considering both real and simulated data. K eywo rds Cluster-W eighted Modeling, Mixture Models, Model-Based Clustering. Salv atore Ingrassia Diparti mento di Impresa, Cultu re e Societ ` a Uni versit ` a di Catani a Corso Itali a 55, - Catani a (Italy). E-mail: s.ingrassia@unict.i t Simona C. Minotti Diparti mento di Stati stica Uni versit ` a di Milano-B icocc a V ia Bicocca degli Arcimboldi 8 - 20126 Milano (Ital y). E-mail: simona.minotti@unimib .it Giorgi o V ittadini Diparti mento di Metodi Quantita tivi per l’Economia e le Scienz e Aziendal i Uni versit ` a di Milano-B icocc a V ia Bicocca degli Arcimboldi 8 - 20126 Milano (Ital y). E-mail: giorgio.vitt adini@unimib .it 0 10 20 30 40 50 0 50 100 150 200 250 300 CWM X Y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 2000 4000 6000 8000 0 500 1000 1500 CWM X Y 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 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 −15 −10 −5 0 5 10 15 40 60 80 100 CWM X Y 5 10 15 20 25 −40 −20 0 20 40 60 80 CWM X Y 0 10 20 30 40 50 0 50 100 150 200 250 300 FMRC X Y 0 10 20 30 40 50 0 50 100 150 200 250 300 FMR X Y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 2000 4000 6000 8000 0 500 1000 1500 FMR X Y 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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 −15 −10 −5 0 5 10 15 40 60 80 100 FMR X Y 5 10 15 20 25 −40 −20 0 20 40 60 80 FMR X Y 0 10 20 30 −50 0 50 100 0 10 20 30 40 50 −50 0 50 100 150 200 250 0 10 20 30 40 50 0 50 100 150 200 250 300 T rue distrib ution X Y 5 10 15 20 25 −40 −20 0 20 40 60 80 true distrib ution X Y || | | | | | | | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | || | || | | | | | | | | | | | | | | | | | | | | | || | BD 15 25 35 45 8 12 16 20 6 10 14 18 15 25 35 45 | || | | | | | || | || | | | | | | | | | || | | | | | | | | | | | | | | || | | | | | || | | | | | | | | | | | | | | | | | | || | | || | | | | | | | | | | | | | | | | | | | | | || | | | | | | CL | | | | | | | | | | | | | | | | | | || | | | | | | | | | | || | | | | | | | | | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | ||| | | | | | | | | | || | | | | | | | | | | CW 20 30 40 50 8 12 16 20 | ||| | | | | | || | | | | | | | | || || | | | | | | | | | | | | | | | | | | | | | | || | | | | | | || | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | FL 6 10 14 18 20 30 40 50 8 10 14 8 10 14 | | | | | | | | | || | | | | | || | | | | | | | | | | | | | | | | | | | | | | | | | | || | | | | | | | | | | | | | || | | | | | | | | | | | | | | | | | | | | | | | || | | | | | | | | | | | | | | RW 1 2 −6 −4 −2 0 2 4 6 −20 0 20 40 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 −10 −5 0 5 10 −40 −20 0 20 5 10 15 20 25 −40 −20 0 20 40 60 80 FMR X Y −10 0 10 20 30 −50 0 50 100 −10 0 10 20 30 40 50 0 50 100 150 200 250 300 x Density 0 2000 4000 6000 8000 10000 0e+00 1e−04 2e−04 3e−04 4e−04 −15 −10 −5 0 5 10 15 40 60 80 100 true distrib ution X Y