An adequacy approach for deciding the number of clusters for OTRIMLE robust Gaussian mixture-based clustering

We introduce a new approach to deciding the number of clusters. The approach is applied to Optimally Tuned Robust Improper Maximum Likelihood Estimation (OTRIMLE; Coretto & Hennig, Journal of the American Statistical Association111, 1648-1659) of a Gaussian mixture model allowing for observations to be classified as 'noise', but it can be applied to other clustering methods as well. The quality of a clustering is assessed by a statistic Q that measures how close the within-cluster distributions are to elliptical unimodal distributions that have the only mode in the mean. This non-parametric measure allows for non-Gaussian clusters as long as they have a good quality according to Q. The simplicity of a model is assessed by a measure S that prefers a smaller number of clusters unless additional clusters can reduce the estimated noise proportion substantially. The simplest model is then chosen that is adequate for the data in the sense that its observed value of Q is not significantly larger than what is expected for data truly generated from the fitted model, as can be assessed by parametric bootstrap. The approach is compared with model-based clustering using the Bayesian information criterion (BIC) and the integrated complete likelihood (ICL) in a simulation study and on real two data sets

A simulations study to compare robust clustering methods based on mixtures

Abstract The following mixture model-based clustering methods are compared in a simulation study with one-dimensional data, fixed number of clusters and a focus on outliers and uniform "noise": an ML-estimator (MLE) for Gaussian mixtures, an MLE for a mixture of Gaussians and a uniform distribution (interpreted as "noise component" to catch outliers), an MLE for a mixture of Gaussian distributions where a uniform distribution over the range of the data is fixed (Fraley and Raftery in Comput J 41:578-588, 199

Estimation and computations for Gaussian mixtures with uniform noise under separation constraints

In this paper we study a finite Gaussian mixture model with an additional uniform component that has the role to catch points in the tails of the data distribution. An adaptive constraint enforces a certain level of separation between the Gaussian mixture components and the uniform component representing noise and outliers in the tail of the distribution. The latter makes the proposed tool particularly useful for robust estimation and outlier identification. A constrained ML estimator is introduced for which existence and consistency is shown. One of the attractive features of the methodology is that the noise level is estimated from data. We also develop an EM-type algorithm with proven convergence. Based on numerical evidence we show how the methods developed in this paper are useful for several fundamental data analysis tasks: outlier identification, robust location-scale estimation, clustering, and density estimation

Identifiability for mixtures of distributions from a location-scale family with uniforms

In this paper we study the indentifiability of a class of mixture models where a finite number of one-dimensional location scale distributions is mixed with a finite number of uniform distributions on an interval. We define identifiability and we show that, under certain conditions, the afore-mentioned class of distributions is identifiable

Robust Improper Maximum Likelihood: Tuning, Computation, and a Comparison With Other Methods for Robust Gaussian Clustering

The two main topics of this paper are the introduction of the \u201coptimally tuned improper maximum likelihood estimator\u201d (OTRIMLE) for robust clustering based on the multivariate Gaussian model for clusters, and a comprehensive simulation study comparing the OTRIMLE to Maximum Likelihood in Gaussian mixtures with and without noise component, mixtures of t-distributions, and the TCLUST approach for trimmed clustering. The OTRIMLE uses an im- proper constant density for modelling outliers and noise. This can be chosen optimally so that the non-noise part of the data looks as close to a Gaussian mixture as possible. Some deviation from Gaussianity can be traded in for lowering the estimated noise proportion. Covariance ma- trix constraints and computation of the OTRIMLE are also treated. In the simulation study, all methods are confronted with setups in which their model assumptions are not exactly fulfilled, and in order to evaluate the experiments in a standardized way by misclassification rates, a new model-based definition of \u201ctrue clusters\u201d is introduced that deviates from the usual identifica- tion of mixture components with clusters. In the study, every method turns out to be superior for one or more setups, but the OTRIMLE achieves the most satisfactory overall performance. The methods are also applied to two real datasets, one without and one with known \u201ctrue\u201d clusters

An adequacy approach for deciding the number of clusters for OTRIMLE robust Gaussian mixture-based clustering

We introduce a new approach to deciding the number of clusters. The approach is applied to Optimally Tuned Robust Improper Maximum Likelihood Estimation (OTRIMLE; Coretto & Hennig, Journal of the American Statistical Association111, 1648–1659) of a Gaussian mixture model allowing for observations to be classified as ‘noise’, but it can be applied to other clustering methods as well. The quality of a clustering is assessed by a statistic Q that measures how close the within-cluster distributions are to elliptical unimodal distributions that have the only mode in the mean. This non-parametric measure allows for non-Gaussian clusters as long as they have a good quality according to Q. The simplicity of a model is assessed by a measure S that prefers a smaller number of clusters unless additional clusters can reduce the estimated noise proportion substantially. The simplest model is then chosen that is adequate for the data in the sense that its observed value of Q is not significantly larger than what is expected for data truly generated from the fitted model, as can be assessed by parametric bootstrap. The approach is compared with model-based clustering using the Bayesian information criterion (BIC) and the integrated complete likelihood (ICL) in a simulation study and on real two data sets

I Velivoli che hanno Influenzato il Mercato Aeronautico Commerciale

Lo studio mira a identificare quali aeromobili hanno pi\uf9 inciso sul trasporto commerciale mondiale dal dopoguerra ad oggi. L\u2019indicatore scelto per fare questa analisi \ue8 stato il Revenue Passenger Kilometers, che ognuno di questi modelli ha realizzato dalla sua entrata in servizio sino alla fine del 2003. Per ottenere questo risultato sono stati considerati i dati relativi alle caratteristiche tecniche medie dei diversi aeromobili considerando un ciclo operativo medio tipico. I risultati finali di questo studio hanno collocato al primo posto il Boeing 747 con un significativo margine rispetto agli altri modelli; infine \ue8 stato sottolineato il netto vantaggio, in termini di RPK complessivi, che la Boeing sino ad ora ha rispetto al consorzio Airbus

DeepGUM: Learning Deep Robust Regression with a Gaussian-Uniform Mixture Model

International audienceIn this paper we address the problem of how to robustly train a Con-vNet for regression, or deep robust regression. Traditionally, deep regression employ the L2 loss function, known to be sensitive to outliers, i.e. samples that either lie at an abnormal distance away from the majority of the training samples, or that correspond to wrongly annotated targets. This means that, during back-propagation, outliers may bias the training process due to the high magnitude of their gradient. In this paper, we propose DeepGUM: a deep regression model that is robust to outliers thanks to the use of a Gaussian-uniform mixture model. We derive an optimization algorithm that alternates between the unsupervised detection of outliers using expectation-maximization, and the supervised training with cleaned samples using stochastic gradient descent. DeepGUM is able to adapt to a continuously evolving outlier distribution, avoiding to manually impose any threshold on the proportion of outliers in the training set. Extensive experimental evaluations on four different tasks (facial and fashion landmark detection, age and head pose estimation) lead us to conclude that our novel robust technique provides reliability in the presence of various types of noise and protection against a high percentage of outliers