Robust inference for parsimonious model-based clustering

Dotto, Francesco; Farcomeni, Alessio

doi:10.1080/00949655.2018.1554659

“…However, the use of the PLR decomposition of an orthogonal matrix is not limited to CPCA, and other statistical models may benefit from its use. Indeed, the PLR decomposition may be used to simplify the ML estimation of the orthogonal matrix related, only to mention a few, to: CPCA based on further non-normal distributions for the groups, other multiple group models allowing for common covariance struc-tures (Flury 1986a;Greselin and Punzo 2013), parsimonious model-based clustering, classification and discriminant analysis (Banfield and Raftery 1993;Flury et al 1994;Celeux and Govaert 1995;Fraley and Raftery 2002;Andrews and McNicholas 2012;Bagnato et al 2014;Lin 2014;Vrbik and McNicholas 2014;Dang et al 2015;Punzo et al 2018;Dotto and Farcomeni 2019), and sophisticated multivariate distributions (Forbes and Wraith 2014;Punzo and Tortora 2019). We pursue to handle these possibilities in future works.…”

Section: Discussionmentioning

confidence: 99%

Unconstrained representation of orthogonal matrices with application to common principal components

Bagnato

¹

,

Punzo

²

2020

View full text Add to dashboard Cite

Many statistical problems involve the estimation of a $$\left( d\times d\right) $$ d × d orthogonal matrix $$\varvec{Q}$$ Q . Such an estimation is often challenging due to the orthonormality constraints on $$\varvec{Q}$$ Q . To cope with this problem, we use the well-known PLU decomposition, which factorizes any invertible $$\left( d\times d\right) $$ d × d matrix as the product of a $$\left( d\times d\right) $$ d × d permutation matrix $$\varvec{P}$$ P , a $$\left( d\times d\right) $$ d × d unit lower triangular matrix $$\varvec{L}$$ L , and a $$\left( d\times d\right) $$ d × d upper triangular matrix $$\varvec{U}$$ U . Thanks to the QR decomposition, we find the formulation of $$\varvec{U}$$ U when the PLU decomposition is applied to $$\varvec{Q}$$ Q . We call the result as PLR decomposition; it produces a one-to-one correspondence between $$\varvec{Q}$$ Q and the $$d\left( d-1\right) /2$$ d d - 1 / 2 entries below the diagonal of $$\varvec{L}$$ L , which are advantageously unconstrained real values. Thus, once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to $$\varvec{L}$$ L . For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of $$\varvec{Q}$$ Q in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses.

show abstract

“…However, the use of the PLR decomposition of an orthogonal matrix is not limited to CPCA, and other statistical models may benefit from its use. Indeed, the PLR decomposition may be used to simplify the ML estimation of the orthogonal matrix related, only to mention a few, to: CPCA based on further non-normal distributions for the groups, other multiple group models allowing for common covariance structures (Flury, 1986a andPunzo, 2013), parsimonious model-based clustering, classification and discriminant analysis (Banfield and Raftery, 1993, Flury et al, 1994, Celeux and Govaert, 1995, Fraley and Raftery, 2002, Andrews and McNicholas, 2012, Bagnato et al, 2014, Lin, 2014, Vrbik and McNicholas, 2014, Dang et al, 2015, Dotto and Farcomeni, 2019, extensions of hidden Markov models Maruotti, 2016 and, and so-phisticated multivariate distributions (Forbes andWraith, 2014 andTortora, 2018). We pursue to handle these possibilities in future works.…”

Section: Discussionmentioning

confidence: 99%

Unconstrained representation of orthogonal matrices with application to common principle components

Bagnato¹,

Punzo²

2019

Preprint

0

View full text Add to dashboard Cite

Many statistical problems involve the estimation of a (d × d) orthogonal matrix Q. Such an estimation is often challenging due to the orthonormality constraints on Q. To cope with this problem, we propose a very simple decomposition for orthogonal matrices which we abbreviate as PLR decomposition. It produces a one-to-one correspondence between Q and a (d × d) unit lower triangular matrix L whose d (d − 1) /2 entries below the diagonal are unconstrained real values. Once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to L. For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of Q in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses.

show abstract

“…Notice that the constraint in ( 8) is still needed whenever either the shape or the volume is free to vary across components (García-Escudero et al, 2017), that is for all models in Table 1 that present "Required" entry in the ER column. The considered approach is the (semi)supervised version of the methodology proposed in Dotto and Farcomeni (2019), which is framed in a completely unsupervised scenario. Feasible and computationally efficient algorithms for enforcing the eigen-ratio constraint for different patterned models are reported in the Appendix C.…”

Section: Model Formulationmentioning

confidence: 99%

A robust approach to model-based classification based on trimming and constraints

Cappozzo

¹

,

Greselin

²

,

Murphy

³

2019

Adv Data Anal Classif

View full text Add to dashboard Cite

In a standard classification framework a set of trustworthy learning data are employed to build a decision rule, with the final aim of classifying unlabelled units belonging to the test set. Therefore, unreliable labelled observations, namely outliers and data with incorrect labels, can strongly undermine the classifier performance, especially if the training size is small. The present work introduces a robust modification to the Model-Based Classification framework, employing impartial trimming and constraints on the ratio between the maximum and the minimum eigenvalue of the group scatter matrices. The proposed method effectively handles noise presence in both response and exploratory variables, providing reliable classification even when dealing with contaminated datasets. A robust information criterion is proposed for model selection. Experiments on real and simulated data, artificially adulterated, are provided to underline the benefits of the proposed method.

show abstract

Robust inference for parsimonious model-based clustering

Cited by 9 publications

References 41 publications

Unconstrained representation of orthogonal matrices with application to common principal components

Unconstrained representation of orthogonal matrices with application to common principal components

Unconstrained representation of orthogonal matrices with application to common principle components

A robust approach to model-based classification based on trimming and constraints

Contact Info

Product

Resources

About