The problem of complex data analysis is a central topic of modern statistical science and learning systems and is becoming of broader interest with the increasing prevalence of high-dimensional data. The challenge is to develop statistical models and autonomous algorithms that are able to acquire knowledge from raw data for exploratory analysis, which can be achieved through clustering techniques or to make predictions of future data via classification (i.e., discriminant analysis) techniques. Latent data models, including mixture model-based approaches are one of the most popular and successful approaches in both the unsupervised context (i.e., clustering) and the supervised one (i.e, classification or discrimination). Although traditionally tools of multivariate analysis, they are growing in popularity when considered in the framework of functional data analysis (FDA). FDA is the data analysis paradigm in which the individual data units are functions (e.g., curves, surfaces), rather than simple vectors. In many areas of application, the analyzed data are indeed often available in the form of discretized values of functions or curves (e.g., time series, waveforms) and surfaces (e.g., 2d-images, spatio-temporal data). This functional aspect of the data adds additional difficulties compared to the case of a classical multivariate (non-functional) data analysis. We review and present approaches for model-based clustering and classification of functional data. We derive well-established statistical models along with efficient algorithmic tools to address problems regarding the clustering and the classification of these high-dimensional data, including their heterogeneity, missing information, and dynamical hidden structure. The presented models and algorithms are illustrated on real-world functional data analysis problems from several application area.

Regression mixture models are widely studied in statistics, machine learning and data analysis. Fitting regression mixtures is challenging and is usually performed by maximum likelihood by using the expectation-maximization (EM) algorithm. However, it is well-known that the initialization is crucial for EM. If the initialization is inappropriately performed, the EM algorithm may lead to unsatisfactory results. The EM algorithm also requires the number of clusters to be given a priori; the problem of selecting the number of mixture components requires using model selection criteria to choose one from a set of pre-estimated candidate models. We propose a new fully unsupervised algorithm to learn regression mixture models with unknown number of components. The developed unsupervised learning approach consists in a penalized maximum likelihood estimation carried out by a robust expectation-maximization (EM) algorithm for fitting polynomial, spline and B-spline regressions mixtures. The proposed learning approach is fully unsupervised: 1) it simultaneously infers the model parameters and the optimal number of the regression mixture components from the data as the learning proceeds, rather than in a two-fold scheme as in standard model-based clustering using afterward model selection criteria, and 2) it does not require accurate initialization unlike the standard EM for regression mixtures. The developed approach is applied to curve clustering problems. Numerical experiments on simulated data show that the proposed robust EM algorithm performs well and provides accurate results in terms of robustness with regard initialization and retrieving the optimal partition with the actual number of clusters. An application to real data in the framework of functional data clustering, confirms the benefit of the proposed approach for practical applications.

This paper introduces a novel mixture model-based approach for simultaneous clustering and optimal segmentation of functional data which are curves presenting regime changes. The proposed model consists in a finite mixture of piecewise polynomial regression models. Each piecewise polynomial regression model is associated with a cluster, and within each cluster, each piecewise polynomial component is associated with a regime (i.e., a segment). We derive two approaches for learning the model parameters. The former is an estimation approach and consists in maximizing the observed-data likelihood via a dedicated expectation-maximization (EM) algorithm. A fuzzy partition of the curves in K clusters is then obtained at convergence by maximizing the posterior cluster probabilities. The latter however is a classification approach and optimizes a specific classification likelihood criterion through a dedicated classification expectation-maximization (CEM) algorithm. The optimal curve segmentation is performed by using dynamic programming. In the classification approach, both the curve clustering and the optimal segmentation are performed simultaneously as the CEM learning proceeds. We show that the classification approach is the probabilistic version that generalizes the deterministic K-means-like algorithm proposed in H\'ebrail et al. (2010). The proposed approach is evaluated using simulated curves and real-world curves. Comparisons with alternatives including regression mixture models and the K-means like algorithm for piecewise regression demonstrate the effectiveness of the proposed approach.

Statistical approaches for Functional Data Analysis concern the paradigm for which the individuals are functions or curves rather than finite dimensional vectors. In this paper, we particularly focus on the modeling and the classification of functional data which are temporal curves presenting regime changes over time. More specifically, we propose a new mixture model-based discriminant analysis approach for functional data using a specific hidden process regression model. Our approach is particularly adapted to both handle the problem of complex-shaped classes of curves, where each class is composed of several sub-classes, and to deal with the regime changes within each homogeneous sub-class. The model explicitly integrates the heterogeneity of each class of curves via a mixture model formulation, and the regime changes within each sub-class through a hidden logistic process. The approach allows therefore for fitting flexible curve-models to each class of complex-shaped curves presenting regime changes through an unsupervised learning scheme, to automatically summarize it into a finite number of homogeneous clusters, each of them is decomposed into several regimes. The model parameters are learned by maximizing the observed-data log-likelihood for each class by using a dedicated expectation-maximization (EM) algorithm. Comparisons on simulated data and real data with alternative approaches, including functional linear discriminant analysis and functional mixture discriminant analysis with polynomial regression mixtures and spline regression mixtures, show that the proposed approach provides better results regarding the discrimination results and significantly improves the curves approximation.

We present a new mixture model-based discriminant analysis approach for functional data using a specific hidden process regression model. The approach allows for fitting flexible curve-models to each class of complex-shaped curves presenting regime changes. The model parameters are learned by maximizing the observed-data log-likelihood for each class by using a dedicated expectation-maximization (EM) algorithm. Comparisons on simulated data with alternative approaches show that the proposed approach provides better results.

In this paper, we study the modeling and the classification of functional data presenting regime changes over time. We propose a new model-based functional mixture discriminant analysis approach based on a specific hidden process regression model that governs the regime changes over time. Our approach is particularly adapted to handle the problem of complex-shaped classes of curves, where each class is potentially composed of several sub-classes, and to deal with the regime changes within each homogeneous sub-class. The proposed model explicitly integrates the heterogeneity of each class of curves via a mixture model formulation, and the regime changes within each sub-class through a hidden logistic process. Each class of complex-shaped curves is modeled by a finite number of homogeneous clusters, each of them being decomposed into several regimes. The model parameters of each class are learned by maximizing the observed-data log-likelihood by using a dedicated expectation-maximization (EM) algorithm. Comparisons are performed with alternative curve classification approaches, including functional linear discriminant analysis and functional mixture discriminant analysis with polynomial regression mixtures and spline regression mixtures. Results obtained on simulated data and real data show that the proposed approach outperforms the alternative approaches in terms of discrimination, and significantly improves the curves approximation.

Model-based clustering approaches concern the paradigm of exploratory data analysis relying on the finite mixture model to automatically find a latent structure governing observed data. They are one of the most popular and successful approaches in cluster analysis. The mixture density estimation is generally performed by maximizing the observed-data log-likelihood by using the expectation-maximization (EM) algorithm. However, it is well-known that the EM algorithm initialization is crucial. In addition, the standard EM algorithm requires the number of clusters to be known a priori. Some solutions have been provided in [31, 12] for model-based clustering with Gaussian mixture models for multivariate data. In this paper we focus on model-based curve clustering approaches, when the data are curves rather than vectorial data, based on regression mixtures. We propose a new robust EM algorithm for clustering curves. We extend the model-based clustering approach presented in [31] for Gaussian mixture models, to the case of curve clustering by regression mixtures, including polynomial regression mixtures as well as spline or B-spline regressions mixtures. Our approach both handles the problem of initialization and the one of choosing the optimal number of clusters as the EM learning proceeds, rather than in a two-fold scheme. This is achieved by optimizing a penalized log-likelihood criterion. A simulation study confirms the potential benefit of the proposed algorithm in terms of robustness regarding initialization and funding the actual number of clusters.