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The CP decomposition is the most appropriate tool for modeling data arrays with a trilinear structure. Model fitting can be hindered by several issues, including computational inefficiency, bad initialization, excessive modeled noise, sensitivity to over-factoring and collinearity. Many algorithms have been proposed for parameter estimation, each with specific strengths and weaknesses. Fast procedures tend to be less stable and vice-versa. Stability is usually prioritized by preferring the least-square approach ALS, albeit slow and sensitive to excess factors. As a solution integrated methods have been proposed in the literature. First, estimation is initialized with a fast procedure to ensure competitive speed then results are refined with ALS to improve precision. In this work, we implement a novel integrated algorithm called INT-3 where ASD steps are concatenated with ALS. ASD was selected because of its remarkable speed and low memory consumption requirements. INT-3 performance is tested against ALS on artificial data.

Different techniques exist to analyze multi-way data but PARAFAC is one of the most popular. The usual way of parameter estimation in PARAFAC is an alternating least squares (ALS) procedure which yields least-squares solutions and provides consistent outcomes. Together with these desirable features, the ALS procedure suffers several major flaws which might be particularly problematic for large-scale problems: slow convergence and sensitiveness to degeneracy conditions such as over-factoring, collinearity, bad initialization and local minima. Furthermore, it is well-known that algorithms which rely on least squares easily break down in the presence of outliers. The issue of nonrobustness of the ALS procedure was addressed by [1] and software for computing the robust PARAFAC model is available in the R package rrcov3way (see [4]). The other issues were addressed in a number of works proposing algorithms more efficient than ALS. However, often these do not provide stable results because the increased speed might come at the expense of accuracy. An integrated algorithm was proposed by [2] and [3] which seems to combine improved speed and stability. The purpose of this work is to elaborate this algorithm by adding capabilities for dealing with outliers which might be present in the data. The idea of a robust version of ALS is to identify enough ”good“ observations and to perform the classical ALS on these observations. This is repeated until no significant change is observed. Finally, a reweighting step is carried out to improve the efficiency of the estimators. In order to identify the ”good” observations a robust version of principal component analysis on the unfolded array is used. It is obvious that the robust procedure will be much more time consuming than the classical one, repeating many times the ALS optimization. Therefore, any improvement of the performance of the parameter estimation procedure will contribute to the improvement of the performance of the complete robust procedure. We combine the robust PARAFAC procedure proposed by [1] with the highly efficient estimation algorithm INT2 proposed by [2] in order to obtain a fast estimation and robust to outliers PARAFAC modeling technique which we call R-INT2. As before it starts by robust principal components to identify any outlying points and then iterates using the INT2 algorithm until no significant change is observed. After convergence a reweighting step with INT2 is conducted which produces the final solution. The performance of the newly proposed algorithm R-INT2 for robust estimation of trilinear PARAFAC models is demonstrated in a brief simulation study comparing classical ALS, the robust version based on ALS ([1]) and the new R-INT2. First of, all we want to verify that R-INT2 works well on data sets with and without contamination by identifying the outliers at least as good as the robust ALS algorithm retrieving solutions with good statistical quality. At the same time we want to verify that the convergence is improved significantly and thus the computational time is reduced. Future work should bring a thorough investigation of the properties of the algorithm, comparison to the many existing fast alternatives and studying the possibilities for combination with other computational algorithms.

The usual way of parameter estimation in CANDECOM/PARAFAC (CP) is an alternating least squares (ALS) procedure that yields least-squares solutions and provides consistent outcomes but at the same time has several deficiencies, like sensitivity to the presence of outliers in the data, slow convergence, and susceptibility to degeneracy conditions. A number of works have addressed these weaknesses, but to our knowledge, there is no outlier-robust procedure that is highly computationally efficient at the same time, especially for large data sets. We propose a robust procedure based on an integrated estimation algorithm, alternative to ALS, which guards against outliers and is computationally efficient at the same time.

In the last years, the use of composite indicators has consistently increased, and the necessity to build model-based composite indicators with a strong methodological statistical approach becomes more and more important for reasons of trustworthiness. In this paper, we propose to build a composite indicators system able to measure different levels of relations among (group of) variables according to an ultrametric form which detects a hierarchical structure upon (group of) variables. Each dimension is measured as a specific composite indicator which reflects a subset of variables. In order to show its potential and applicability, the methodology is employed to analyze a dataset which contains variables about separated waste collection in Italy taking into consideration both its performance and its costs

In questo capitolo s’illustrano alcuni strumenti utili a descrivere e confrontare le distribuzioni di differenti variabili. Per iniziare, s’introducono gli indici di disuguaglianza, valori caratteristici che misurano l’attitudine di un fenomeno ad assumere modalità differenti. I diversi indici sono analizzati nel dettaglio e classificati in base al tipo di variabile cui sono applicabili. S’include anche una misura specifica per caratteri trasferibili: la concentrazione. S’introduce, poi, il box plot, un importante grafico per la rappresentazione delle principali caratteristiche di una distribuzione. Successivamente, si passa alla definizione dei concetti di simmetria e curtosi per valutare la forma di una distribuzione. Il capitolo si conclude con alcune tecniche per operare confronti basilari tra variabili: rapporti statistici, variazioni e numeri indice. Obiettivo del capitolo è far comprendere l’importanza che l’analisi della variabilità ha sulla sintesi della distribuzione di una variabile, indirizzando alla scelta della migliore tecnica.

Canonical correlation analysis (CCA) is a useful tool for investigating the relationships between two sets of variables. If dispersion matrices can be inverted, canonical variates with maximal correlation are generally identified by means of singular value decomposition. However, when one or both variable groups are compositional, this classical approach cannot be followed. Compositional data are positive values which carry relative information describing the parts of a whole. In consequence they present a perfectly multicollinear structure and are characterized by singular dispersion matrices. As a solution to this issue which excludes a standard approach, an alternative way of computing canonical variates is proposed. Data are first transformed in log-ratio coordinates, then the Partial Least Squares approach is applied. This method provides a fast and easy way to deal with non-invertible dispersion matrices and, in addition, it yields results which are easy to interpret. The proposed methodology is assessed in an experimental study in which a comparison among alternative PLS algorithms is also provided, namely NIPALS, SIMPLS and Kernel.

Compositional Data Analysis can be useful for unveiling relative variability patterns among variables describing the parts of a phenomenon. Compositions are often represented as orthonormal balances associated with a sequential binary partition (SBP). Principal balances analysis (PBA) is a tool used to find a meaningful SBP by subsequently maximizing explained variability. The exact estimation of PBA is prohibitive for large datasets; therefore, algorithms providing an acceptable approximation are used instead. For compositional data of third-order, such exploratory search must account for third-mode variability. To this end, this work introduces a three-way adaptation of PBA in which estimation is carried out by Tucker3. A study on the composition of academic recruitment fields by Italian macro-region and gender/role is carried out to illustrate the merits of this procedure.

Multifold data structures are generally stored in high-dimensional objects defined as nth-order tensors. Generalization of trilinear decompositions such as the CANDECOMP/PARAFAC model can be used for modelling 4th order tensors. The application of these techniques is, however, quite limited due to procedural complexity and interpretational issues. These concerns increase when tensors contain data with a compositional structure. This work aims at addressing these difficulties through an application on Italian university staff.

OECD-PISA survey data include performance measurements, expressed as tables of plausible values, and a variety of socio-biographical information. Such data, if properly modeled, can provide useful insights on the causes of low performing students. A methodology which deals with multivariate sets of plausible values and investigates the effects of context variables without assumptions is required. Here Multiple Factor Analysis with External Information is proposed. Specifically, after defining context variable groupings, a partitioning of the variability structure of the data tables is carried out using projection operators than a simplified Multiple Factor Analysis with bootstrap is performed.

Will the extra years of life gained by the increase in life expectancy be lived in good or bad health? As forecasts support social, economic and medical decisions, as well as individuals' choices, there is a clear rationale for forecasting healthy life expectancy. However, only a limited number of models are available to forecast healthy life expectancy. Some are based on multistate modelling, which can be data-demanding and requires separate forecasts of transition rates for mortality within different health statuses and the incidence rate. We here suggest a less data demanding model to forecast mortality and health prevalence simultaneously. The model is based on the Sullivan method, which uses cross-sectional data, and Compositional Data Analysis. The method is applied to Swedish female mortality aged 65 and above. We show that deaths have been shifted towards older ages and not-limited, leading to an increase in both life expectancy and disability-free life expectancy.

The aim of this contribution is to identify the archetypal universities in the Italian students’ mobility network in terms of their attitude in attracting students. We define a set of networks according to the disciplinary groups by relying upon administrative data regarding students’ mobility between bachelor’s and master’s degrees. For each disciplinary group, a network has been defined by considering the universities as nodes and the flows of students moving between nodes as links. Then, in each network, the set of archetypal universities is based on several network centrality indexes. Finally, these archetypes are used as benchmarks to identify the main determinants of universities’ performances.

The latent structure of four-dimensional tensors can be investigated by means of the four-way CANDECOMP/PARAFAC model. This technique is seldom used because its estimating design is challenging from an algorithmic and interpretational standpoint. Parameter estimation with a least-squares approach can be computationally costly, especially under difficult conditions such as factor collinearity and model over-specification. In this work, we implement a 4th-order extension of the efficient trilinear procedure INT-2 to tackle estimating setbacks and test it in a simulation study.

This contribution has the dual goal of testing the usage of readability and lexical complexity tools in social media writing and analyzing the relationship between linguistic complexity, cultural topics, and social engagement. A corpus of tweets from the official Twitter page of the Italian Ministry of Culture has been stored together with available information on social engagement. once the tweets have been properly processed with standard Natural language processing tasks, linguistic complexity is measured with eight readability and lexical diversity indicators developed outside the social media context. Topic modeling is then carried out through the LDA technique. An explorative study of cause-effect relationships between lexical complexity, topic grouping, and engagement measures is performed by using a PlS-PM mixed approach.