Penalized model-based clustering for three-way data structures

Recently, there has been an increasing interest in developing statistical methods able to find groups in matrix-valued data. To this extent, matrix Gaussian mixture models (MGMM) provide a natural extension to the popular model-based clustering based on Normal mixtures. Unfortunately, the overparametrization issue, already affecting the vector-variate framework, is further exacerbated when it comes to MGMM, since the number of parameters scales quadratically with both row and column dimensions. In order to overcome this limitation, the present paper introduces a sparse model-based clustering approach for three-way data structures. By means of penalized estimation, our methodology shrinks the estimates towards zero, achieving more stable and parsimonious clustering in high dimensional scenarios. An application to satellite images underlines the benefits of the proposed method

An adaptive functional regression framework for spatially heterogeneous signals in spectroscopy

The attention towards food products characteristics, such as nutritional properties and traceability, has risen substantially in the recent years. Consequently, we are witnessing an increased demand for the development of modern tools to monitor, analyse and assess food quality and authenticity. Within this framework, an essential set of data collection techniques is provided by vibrational spectroscopy. In fact, methods such as Fourier near infrared and mid infrared spectroscopy have been often exploited to analyze different foodstuffs. Nonetheless, existing statistical methods often struggle to deal with the challenges presented by spectral data, such as their high dimensionality, paired with strong relationships among the wavelengths. Therefore, the definition of proper statistical procedures accounting for the peculiarities of spectroscopy data is paramount. In this work, motivated by two dairy science applications, we propose an adaptive functional regression framework for spectroscopy data. The method stems from the trend filtering literature, allowing the definition of a highly flexible and adaptive estimator able to handle different degrees of smoothness. We provide a fast optimization procedure that is suitable for both Gaussian and non Gaussian scalar responses, and allows for the inclusion of scalar covariates. Moreover, we develop inferential procedures for both the functional and the scalar component thus enhancing not only the interpretability of the results, but also their usability in real world scenarios. The method is applied to two sets of MIR spectroscopy data, providing excellent results when predicting milk chemical composition and cows' dietary treatments. Moreover, the developed inferential routine provides relevant insights, potentially paving the way for a richer interpretation and a better understanding of the impact of specific wavelengths on milk features.Comment: 20 pages, 3 figure

Group-Wise Shrinkage Estimation in Penalized Model-Based Clustering

Finite Gaussian mixture models provide a powerful and widely employed probabilistic approach for clustering multivariate continuous data. However, the practical usefulness of these models is jeopardized in high-dimensional spaces, where they tend to be over-parameterized. As a consequence, different solutions have been proposed, often relying on matrix decompositions or variable selection strategies. Recently, a methodological link between Gaussian graphical models and finite mixtures has been established, paving the way for penalized model-based clustering in the presence of large precision matrices. Notwithstanding, current methodologies implicitly assume similar levels of sparsity across the classes, not accounting for different degrees of association between the variables across groups. We overcome this limitation by deriving group-wise penalty factors, which automatically enforce under or over-connectivity in the estimated graphs. The approach is entirely data-driven and does not require additional hyper-parameter specification. Analyses on synthetic and real data showcase the validity of our proposal

Sparse model-based clustering of three-way data via lasso-type penalties

Mixtures of matrix Gaussian distributions provide a probabilistic framework for clustering continuous matrix-variate data, which are becoming increasingly prevalent in various fields. Despite its widespread adoption and successful application, this approach suffers from over-parameterization issues, making it less suitable even for matrix-variate data of moderate size. To overcome this drawback, we introduce a sparse model-based clustering approach for three-way data. Our approach assumes that the matrix mixture parameters are sparse and have different degree of sparsity across clusters, allowing to induce parsimony in a flexible manner. Estimation of the model relies on the maximization of a penalized likelihood, with specifically tailored group and graphical lasso penalties. These penalties enable the selection of the most informative features for clustering three-way data where variables are recorded over multiple occasions and allow to capture cluster-specific association structures. The proposed methodology is tested extensively on synthetic data and its validity is demonstrated in application to time-dependent crime patterns in different US cities

Modal clustering asymptotics with applications to bandwidth selection

Density-based clustering relies on the idea of linking groups to some specific features of the probability distribution underlying the data. The reference to a true, yet unknown, population structure allows to frame the clustering problem in a standard inferential setting, where the concept of ideal population clustering is defined as the partition induced by the true density function. The nonparametric formulation of this approach, known as modal clustering, draws a correspondence between the groups and the domains of attraction of the density modes. Operationally, a nonparametric density estimate is required and a proper selection of the amount of smoothing, governing the shape of the density and hence possibly the modal structure, is crucial to identify the final partition. In this work, we address the issue of density estimation for modal clustering from an asymptotic perspective. A natural and easy to interpret metric to measure the distance between density-based partitions is discussed, its asymptotic approximation explored, and employed to study the problem of bandwidth selection for nonparametric modal clustering

Organismo e organizzazione: sulla dimensione (meta)politica della filosofia cosmica di Maria Montessori

This paper aims to offer the first results of a research, still in progress, on the philosophical dimension of Maria Montessori's perspective, to broaden the understanding of the assumptions and objectives she set out to achieve through her Method. In particular, the article will be devoted to the exposition and recomposition of the main elements of Montessori's reflection on the socio-political dimension, during the last thirty years of her life, known as the ‘new horizon’. After exposing the teleological assumptions of the ‘Cosmic Plan’, derived from the rethinking of biological and geological notions, it will be shown that Montessori applies to nature’s ‘unity of method’ both to the process of psychophysical development of the human individual and to the configurations of social organization, orienting both towards ever greater and firmer cohesion. Starting from the discourses on the human creation of a ‘supernature’, which reveals the interdependence between all peoples (indeed, united in a ‘Unique Nation’), the essay will analyse the features of the new morality which, according to Montessori, will mark the ‘new world’ created by the child, normally developed through a free and natural adaptation to the new social conditions, in the context of technological civilization, the ‘new man will recognize the connection with all the beings of ‘creation’ and he will assume his responsibility for the care of fellow human beings and of the whole planet, fulfill the mission the cosmic plan has assigned to humanity.Este artículo presentará los primeros resultados de una investigación, aún en curso sobre la dimensión filosófica de la perspectiva de María Montessori, con el fin de ampliar la comprensión de los supuestos y objetivos que ella buscaba alcanzar a través de su Método. En particular, el artículo se dedicará a la exposición y recomposición de los principales elementos de la reflexión de Montessori respecto a la dimensión sociopolítica durante los últimos treinta años de su vida, relacionados a los “nuevos horizontes”. Tras exponer los principios teleológicos, derivados de la reelaboración de las nociones biológicas y geológicas, del “Plan Cósmico”, se mostrará cómo Montessori aplica la “unidad de método” de la naturaleza tanto al proceso de desarrollo psicofísico del individuo humano como a las configuraciones de la organización social, orientando ambos a una cohesión más y más amplia y sólida. A partir de los discursos sobre la creación humana de una “supernaturaleza”, que revela la interdependencia entre todos los pueblos (unidos, de hecho, en una “Nación Única”), se analizarán los rasgos  de la nueva moral que, según Montessori, marcará el "nuevo mundo" creado por el niño. Este desarrollo se da normalmente a través de una adaptación libre y natural a las nuevas condiciones sociales, en el contexto de la civilización tecnológica, pues el “hombre nuevo” reconocerá la conexión con todos los seres de la “creación”, y asumirá la responsabilidad del cuidado de sus semejantes y de todo el planeta, para cumplir la misión que el plan cósmico ha asignado a la humanidad.  Questo saggio intende offrire i primi risultati di una ricerca, ancora in corso, sulla dimensione filosofica della prospettiva di Maria Montessori, al fine di ampliare la comprensione dei presupposti e degli obiettivi che ella si proponeva di raggiungere attraverso il suo Metodo. In particolare, l’articolo sarà dedicato all'esposizione e alla ricomposizione degli elementi principali della riflessione montessoriana sulla dimensione socio-politica, durante gli ultimi trent'anni della sua vita, detti dei “nuovi orizzonti”. Dopo aver esposto i presupposti teleologici del “Piano Cosmico”, derivati dal ripensamento di nozioni biologiche e geologiche, si mostrerà come Montessori applichi l’“unità di metodo” della natura sia al processo di sviluppo psicofisico dell'individuo umano sia alle configurazioni dell'organizzazione sociale, orientando entrambi verso una sempre maggiore e più solida coesione. Partendo dai discorsi sulla creazione umana di una “supernatura”, che rivela l’interdipendenza tra tutti i popoli (uniti in una “Nazione Unica”), il saggio analizzerà le caratteristiche della nuova moralità che, secondo Montessori, avrebbe contrassegnato il “nuovo mondo” edificato dal bambino. Normalmente sviluppato attraverso un libero e naturale adattamento alle nuove condizioni sociali, nel contesto della civiltà tecnologica, l’“uomo nuovo” avrebbe riconosciuto il legame con tutti gli esseri del “creato” e si sarebbe assunto la responsabilità della cura dei propri simili e dell’intero pianeta, per adempiere alla missione che il piano cosmico ha assegnato all’umanità.  Organismo e organização: sobre a dimensão (meta)política da filosofia cósmica de Maria Montessori O ensaio apresentará os primeiros resultados de uma investigação, ainda em curso, sobre a dimensão filosófica da perspectiva de Maria Montessori, a fim de alargar a compreensão dos pressupostos e objectivos que ela se propôs alcançar através do seu Método. Em particular, o artigo será dedicado à exposição e recomposição dos principais elementos da reflexão de Montessori sobre a dimensão sócio-política durante os últimos trinta anos da sua vida, conhecidos como os dos "novos horizontes". Depois de expor os pressupostos teleológicos, derivados da reformulação de noções biológicas e geológicas, do "Plano Cósmico", será mostrado como Montessori aplica a "unidade de método" da natureza tanto ao processo de desenvolvimento psicofísico do indivíduo humano como às configurações de organização social, dirigindo-os para uma coesão cada vez maior e constante. Com base nas declarações sobre a criação humana de uma "supernatureza", que revela a interdependência entre todos os povos (unidos, de facto, numa "Nação única"), serão analisados os traços da nova moralidade que, segundo Montessori, marcará o "novo mundo" criado pela criança. Tendo-se desenvolvido normalmente através de uma adaptação livre e natural às novas condições sociais, no contexto da civilização tecnológica, o "novo homem" reconhecerá a ligação com todos os seres da "criação", e assumirá a responsabilidade pelo cuidado dos seus semelhantes e de todo o planeta, de modo a cumprir à missão que o plano cósmico atribuiu à humanidade