arXiv subjects

Luca Scrucca

Publications and source records attributed to Luca Scrucca.

At least 19 recordsLinked to original sources

arXiv2024

A Model-Based Clustering Approach for Bounded Data Using Transformation-Based Gaussian Mixture Models

The clustering of bounded data presents unique challenges in statistical analysis due to the constraints imposed on the data values. This paper introduces a novel method for model-based clustering specifically designed for bounded data. Building on the transformation-based approach to Gaussian mixture density estimation introduced by Scrucca (2019), we extend this framework to develop a probabilistic clustering algorithm for data with bounded support that allows for accurate clustering while respecting the natural bounds of the variables. In our proposal, a flexible range-power transformation is employed to map the data from its bounded domain to the unrestricted real space, hence enabling the estimation of Gaussian mixture models in the transformed space. Despite the close connection to density estimation, the behavior of this approach has not been previously investigated in the literature. Furthermore, we introduce a novel measure of clustering uncertainty, the Normalized Classification Entropy (NCE), which provides a general and interpretable measure of classification uncertainty. The performance of the proposed method is evaluated through real-world data applications involving both fully and partially bounded data, in both univariate and multivariate settings, showing improved cluster recovery and interpretability. Overall, the empirical results demonstrate the effectiveness and advantages of our approach over traditional and advanced model-based clustering techniques that rely on distributions with bounded support.

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arXiv2024

Assessing uncertainty in Gaussian mixtures-based entropy estimation

Entropy estimation plays a crucial role in various fields, such as information theory, statistical data science, and machine learning. However, traditional entropy estimation methods often struggle with complex data distributions. Mixture-based estimation of entropy has been recently proposed and gained attention due to its ease of use and accuracy. This paper presents a novel approach to quantify the uncertainty associated with this mixture-based entropy estimation method using weighted likelihood bootstrap. Unlike standard methods, our approach leverages the underlying mixture structure by assigning random weights to observations in a weighted likelihood bootstrap procedure, leading to more accurate uncertainty estimation. The generation of weights is also investigated, leading to the proposal of using weights obtained from a specific Dirichlet distribution which, in conjunction with centered percentile intervals, yields the optimal setting to ensure empirical coverage closer to the nominal level. Extensive simulation studies comparing different resampling strategies are presented and results discussed. The proposed approach is illustrated by analyzing the log-returns of daily Gold prices at COMEX for the years 2014--2022, and the Net Rating scores, an advanced statistic used in basketball analytics, for NBA teams with reference to the 2022/23 regular season.

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arXiv2024

A Model-Based Approach to Shot Charts Estimation in Basketball

Shot charts in basketball analytics provide an indispensable tool for evaluating players' shooting performance by visually representing the distribution of field goal attempts across different court locations. However, conventional methods often overlook the bounded nature of the basketball court, leading to inaccurate representations, particularly along the boundaries and corners. In this paper, we propose a novel model-based approach to shot chart estimation and visualization that explicitly considers the physical boundaries of the basketball court. By employing Gaussian mixtures for bounded data, our methodology allows to obtain more accurate estimation of shot density distributions for both made and missed shots. Bayes' rule is then applied to derive estimates for the probability of successful shooting from any given locations, and to identify the regions with the highest expected scores. To illustrate the efficacy of our proposal, we apply it to data from the 2022-23 NBA regular season, showing its usefulness through detailed analyses of shot patterns for two prominent players.

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arXiv2023

An introduction and tutorial to model-based clustering in education via Gaussian mixture modelling

Heterogeneity has been a hot topic in recent educational literature. Several calls have been voiced to adopt methods that capture different patterns or subgroups within students behavior or functioning. Assuming that there is an average pattern that represents the entirety of student populations requires the measured construct to have the same causal mechanism, same development pattern, and affect students in exactly the same way. Using a person-centered method (Finite Gaussian mixture model or latent profile analysis), the present tutorial shows how to uncover the heterogeneity within engagement data by identifying three latent or unobserved clusters. This chapter offers an introduction to the model-based clustering that includes the principles of the methods, a guide to choice of number of clusters, evaluation of clustering results and a detailed guide with code and a real-life dataset. The discussion elaborates on the interpretation of the results, the advantages of model-based clustering as well as how it compares with other methods.

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arXiv2022

Mixture-based estimation of entropy

The entropy is a measure of uncertainty that plays a central role in information theory. When the distribution of the data is unknown, an estimate of the entropy needs be obtained from the data sample itself. We propose a semi-parametric estimate, based on a mixture model approximation of the distribution of interest. The estimate can rely on any type of mixture, but we focus on Gaussian mixture model to demonstrate its accuracy and versatility. Performance of the proposed approach is assessed through a series of simulation studies. We also illustrate its use on two real-life data examples.

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arXiv2021

A COVINDEX based on a GAM beta regression model with an application to the COVID-19 pandemic in Italy

Detecting changes in COVID-19 disease transmission over time is a key indicator of epidemic growth.Near real-time monitoring of the pandemic growth is crucial for policy makers and public health officials who need to make informed decisions about whether to enforce lockdowns or allow certain activities. The effective reproduction number Rt is the standard index used in many countries for this goal. However, it is known that due to the delays between infection and case registration, its use for decision making is somewhat limited. In this paper a near real-time COVINDEX is proposed for monitoring the evolution of the pandemic. The index is computed from predictions obtained from a GAM beta regression for modelling the test positive rate as a function of time. The proposal is illustrated using data on COVID-19 pandemic in Italy and compared with Rt. A simple chart is also proposed for monitoring local and national outbreaks by policy makers and public health officials.

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arXiv2021

Modal clustering on PPGMMGA projection subspace

PPGMMGA is a Projection Pursuit (PP) algorithm aimed at detecting and visualizing clustering structures in multivariate data. The algorithm uses the negentropy as PP index obtained by fitting Gaussian Mixture Models (GMMs) for density estimation, and then optimized using Genetic Algorithms (GAs). Since the PPGMMGA algorithm is a dimension reduction technique specifically introduced for visualization purposes, cluster memberships are not explicitly provided. In this paper a modal clustering approach is proposed for estimating clusters of projected data points. In particular, a modal EM algorithm is employed to estimate the modes corresponding to the local maxima in the projection subspace of the underlying density estimated using parsimonious GMMs. Data points are then clustered according to the domain of attraction of the identified modes. Simulated and real data are discussed to illustrate the proposed method and evaluate the clustering performance.

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arXiv2020

A fast and efficient Modal EM algorithm for Gaussian mixtures

In the modal approach to clustering, clusters are defined as the local maxima of the underlying probability density function, where the latter can be estimated either non-parametrically or using finite mixture models. Thus, clusters are closely related to certain regions around the density modes, and every cluster corresponds to a bump of the density. The Modal EM algorithm is an iterative procedure that can identify the local maxima of any density function. In this contribution, we propose a fast and efficient Modal EM algorithm to be used when the density function is estimated through a finite mixture of Gaussian distributions with parsimonious component-covariance structures. After describing the procedure, we apply the proposed Modal EM algorithm on both simulated and real data examples, showing its high flexibility in several contexts.

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arXiv2020

Better than the best? Answers via model ensemble in density-based clustering

With the recent growth in data availability and complexity, and the associated outburst of elaborate modelling approaches, model selection tools have become a lifeline, providing objective criteria to deal with this increasingly challenging landscape. In fact, basing predictions and inference on a single model may be limiting if not harmful; ensemble approaches, which combine different models, have been proposed to overcome the selection step, and proven fruitful especially in the supervised learning framework. Conversely, these approaches have been scantily explored in the unsupervised setting. In this work we focus on the model-based clustering formulation, where a plethora of mixture models, with different number of components and parametrizations, is typically estimated. We propose an ensemble clustering approach that circumvents the single best model paradigm, while improving stability and robustness of the partitions. A new density estimator, being a convex linear combination of the density estimates in the ensemble, is introduced and exploited for group assignment. As opposed to the standard case, where clusters are typically associated to the components of the selected mixture model, we define partitions by borrowing the modal, or nonparametric, formulation of the clustering problem, where groups are linked with high-density regions. Staying in the density-based realm we thus show how blending together parametric and nonparametric approaches may be beneficial from a clustering perspective.

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arXiv2020

Handling missing data in model-based clustering

Gaussian Mixture models (GMMs) are a powerful tool for clustering, classification and density estimation when clustering structures are embedded in the data. The presence of missing values can largely impact the GMMs estimation process, thus handling missing data turns out to be a crucial point in clustering, classification and density estimation. Several techniques have been developed to impute the missing values before model estimation. Among these, multiple imputation is a simple and useful general approach to handle missing data. In this paper we propose two different methods to fit Gaussian mixtures in the presence of missing data. Both methods use a variant of the Monte Carlo Expectation-Maximisation (MCEM) algorithm for data augmentation. Thus, multiple imputations are performed during the E-step, followed by the standard M-step for a given eigen-decomposed component-covariance matrix. We show that the proposed methods outperform the multiple imputation approach, both in terms of clusters identification and density estimation.

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arXiv2019

Projection pursuit based on Gaussian mixtures and evolutionary algorithms

We propose a projection pursuit (PP) algorithm based on Gaussian mixture models (GMMs). The negentropy obtained from a multivariate density estimated by GMMs is adopted as the PP index to be maximised. For a fixed dimension of the projection subspace, the GMM-based density estimation is projected onto that subspace, where an approximation of the negentropy for Gaussian mixtures is computed. Then, Genetic Algorithms (GAs) are used to find the optimal, orthogonal projection basis by maximising the former approximation. We show that this semi-parametric approach to PP is flexible and allows highly informative structures to be detected, by projecting multivariate datasets onto a subspace, where the data can be feasibly visualised. The performance of the proposed approach is shown on both artificial and real datasets.

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arXiv2019

Investigation of Parameter Uncertainty in Clustering Using a Gaussian Mixture Model Via Jackknife, Bootstrap and Weighted Likelihood Bootstrap

Mixture models are a popular tool in model-based clustering. Such a model is often fitted by a procedure that maximizes the likelihood, such as the EM algorithm. At convergence, the maximum likelihood parameter estimates are typically reported, but in most cases little emphasis is placed on the variability associated with these estimates. In part this may be due to the fact that standard errors are not directly calculated in the model-fitting algorithm, either because they are not required to fit the model, or because they are difficult to compute. The examination of standard errors in model-based clustering is therefore typically neglected. The widely used R package mclust has recently introduced bootstrap and weighted likelihood bootstrap methods to facilitate standard error estimation. This paper provides an empirical comparison of these methods (along with the jackknife method) for producing standard errors and confidence intervals for mixture parameters. These methods are illustrated and contrasted in both a simulation study and in the traditional Old Faithful data set and Thyroid data set.

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arXiv2018

A transformation-based approach to Gaussian mixture density estimation for bounded data

Finite mixture of Gaussian distributions provide a flexible semi-parametric methodology for density estimation when the variables under investigation have no boundaries. However, in practical applications variables may be partially bounded (e.g. taking non-negative values) or completely bounded (e.g. taking values in the unit interval). In this case the standard Gaussian finite mixture model assigns non-zero densities to any possible values, even to those outside the ranges where the variables are defined, hence resulting in severe bias. In this paper we propose a transformation-based approach for Gaussian mixture modelling in case of bounded variables. The basic idea is to carry out density estimation not on the original data but on appropriately transformed data. Then, the density for the original data can be obtained by a change of variables. Both the transformation parameters and the parameters of the Gaussian mixture are jointly estimated by the Expectation-Maximisation (EM) algorithm. The methodology for partially and completely bounded data is illustrated using both simulated data and real data applications.

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arXiv2018

Model-based Clustering with Sparse Covariance Matrices

Finite Gaussian mixture models are widely used for model-based clustering of continuous data. Nevertheless, since the number of model parameters scales quadratically with the number of variables, these models can be easily over-parameterized. For this reason, parsimonious models have been developed via covariance matrix decompositions or assuming local independence. However, these remedies do not allow for direct estimation of sparse covariance matrices nor do they take into account that the structure of association among the variables can vary from one cluster to the other. To this end, we introduce mixtures of Gaussian covariance graph models for model-based clustering with sparse covariance matrices. A penalized likelihood approach is employed for estimation and a general penalty term on the graph configurations can be used to induce different levels of sparsity and incorporate prior knowledge. Model estimation is carried out using a structural-EM algorithm for parameters and graph structure estimation, where two alternative strategies based on a genetic algorithm and an efficient stepwise search are proposed for inference. With this approach, sparse component covariance matrices are directly obtained. The framework results in a parsimonious model-based clustering of the data via a flexible model for the within-group joint distribution of the variables. Extensive simulated data experiments and application to illustrative datasets show that the method attains good classification performance and model quality.

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arXiv2016

On some extensions to GA package: hybrid optimisation, parallelisation and islands evolution

Genetic algorithms are stochastic iterative algorithms in which a population of individuals evolve by emulating the process of biological evolution and natural selection. The R package GA provides a collection of general purpose functions for optimisation using genetic algorithms. This paper describes some enhancements recently introduced in version 3 of the package. In particular, hybrid GAs have been implemented by including the option to perform local searches during the evolution. This allows to combine the power of genetic algorithms with the speed of a local optimiser. Another major improvement is the provision of facilities for parallel computing. Parallelisation has been implemented using both the master-slave approach and the islands evolution model. Several examples of usage are presented, with both real-world data examples and benchmark functions, showing that often high-quality solutions can be obtained more efficiently.

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arXiv2015

Model-based SIR for dimension reduction

A new dimension reduction method based on Gaussian finite mixtures is proposed as an extension to sliced inverse regression (SIR). The model-based SIR (MSIR) approach allows the main limitation of SIR to be overcome, i.e., failure in the presence of regression symmetric relationships, without the need to impose further assumptions. Extensive numerical studies are presented to compare the new method with some of most popular dimension reduction methods, such as SIR, sliced average variance estimation, principal Hessian direction, and directional regression. MSIR appears sufficiently flexible to accommodate various regression functions, and its performance is comparable with or better, particularly as sample size grows, than other available methods. Lastly, MSIR is illustrated with two real data examples about ozone concentration regression, and hand-written digit classification.

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arXiv2015

Dimension reduction for model-based clustering

We introduce a dimension reduction method for visualizing the clustering structure obtained from a finite mixture of Gaussian densities. Information on the dimension reduction subspace is obtained from the variation on group means and, depending on the estimated mixture model, on the variation on group covariances. The proposed method aims at reducing the dimensionality by identifying a set of linear combinations, ordered by importance as quantified by the associated eigenvalues, of the original features which capture most of the cluster structure contained in the data. Observations may then be projected onto such a reduced subspace, thus providing summary plots which help to visualize the clustering structure. These plots can be particularly appealing in the case of high-dimensional data and noisy structure. The new constructed variables capture most of the clustering information available in the data, and they can be further reduced to improve clustering performance. We illustrate the approach on both simulated and real data sets.

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arXiv2015

Graphical tools for model-based mixture discriminant analysis

The paper introduces a methodology for visualizing on a dimension reduced subspace the classification structure and the geometric characteristics induced by an estimated Gaussian mixture model for discriminant analysis. In particular, we consider the case of mixture of mixture models with varying parametrization which allow for parsimonious models. The approach is an extension of an existing work on reducing dimensionality for model-based clustering based on Gaussian mixtures. Information on the dimension reduction subspace is provided by the variation on class locations and, depending on the estimated mixture model, on the variation on class dispersions. Projections along the estimated directions provide summary plots which help to visualize the structure of the classes and their characteristics. A suitable modification of the method allows us to recover the most discriminant directions, i.e., those that show maximal separation among classes. The approach is illustrated using simulated and real data.

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