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Orthonormal balances resulting from a sequential binary partition (SBP) are one of the preferred tools for transforming compositional data in real space coordinates. The interpretability of this approach, however, greatly depends on the relevance of the SBP. SBPs can be chosen with the help of expert knowledge or with data-oriented methods, such as Principal Balances analysis. This results in an SBP whose balances maximize the explained variance in a subsequent manner. Principal balances can be calculated in an exact way or in an approximate fashion by using methods based on PCA for compositional data. In this work a method for the approximation of principal balances in the more complex case of three-way compositions is proposed. Here the additional difficulty given by the introduction of third mode variability is dealt with. In particular an algorithm based on the Tucker3 model is used which allows to keep the variability of the third dimension separate in the definition of principal balances.

Both the scientific and political communities agree that significant reductions in CO2 emissions are necessary to limit the magnitude and extent of climate change and of course the energy efficiency is one of the most interesting issues analyzed by economists and policy makers within this debate. Different measures of energy efficiency in manufacturing can be defined but broadly this is the ratio of the production output to the energy input, usually disaggregated by industry. We create a global data set of energy intensity in manufacturing and analyze its structure by country, time and industry applying parallel factor analysis (CP). Since we are interested in the structure of the energy intensity, the absolute values are no more relevant for the analysis and the nature of this data set is compositional which requires specific adaptation of the methodology and suitable software.

The PARAFAC-ALS procedure for estimating CP parameters on tridimen-sional tensors is sensitive to data collinearity. This inefficiency is especially problematic if collinearity is paired with other issues such as data of large dimensions and difficulties in establishing correct model rank. When dealing with compositional data, i.e. positive values with a covariance bias, multicollinearity is inherent by definition, and it is preserved also if the data is transformed in log-ratios by means of the clr function. For this reason, alternative estimating procedures may be considered, such as INT and INT-2. These dual-step methods use the properties of the SWATLD and ATLD algorithms during initialization to overcome ALS inefficiency while still providing least squares results. Their comparative performance is tested in an extensive simulation study on collinear data.

The accuracy of the ALS procedure for fitting the CP decomposition is affected by incorrect model selection due to the properties of its objective function. A study on ALS performance is presented in order to show that for large data sets this occasional shortcoming becomes prevalent. This deficiency should warn off researchers from employing ALS unless the rank of the underlying trilinear structure can be established in advance. Multi-optimization provides a possible solution: ALS can be initialized with procedures insensitive to over-factoring such as SWATLD and ATLD. In this manner it is possible to overcome factorization issues and provide a gain in efficiency without relying on computationally expensive model selection procedures.

Compositional data are quantitative descriptions of the parts of some whole, conveying relative information. The relationship between two sets of compositional descriptors can be explored by use of Canonical Correlation analysis with a procedure based on Partial Least Squares (PLS). This method offers a way to deal with matrix singularity in an efficient fashion and presents the further advantage of being easy to interpret. In order to fully explore the potential of PLS for analyzing the relationships between two sets of compositions, the performances of the NIPALS, SIMPLS and Kernel algorithms are compared on simulated data.

The PARAFAC-ALS algorithm is the most widely used procedure for approximating arrays with a trilinear structure because it provides least squares solutions and delivers consistent outputs. Nonetheless, it is particularly slow at converging especially under challenging conditions, i.e. data multicollinearity, high factors’ congruence and over-factoring. This shortcoming can be quite problematic when dealing with three-way arrays of large dimensions. More efficient procedures can be employed, such as ATLD, however they are far less reliable. As an alternative, ATLD and ALS can be combined in a multi-optimization procedure in order to increase efficiency without reducing accuracy. This novel approach has been carried out and tested on artificial and real data

he National Agency for the Evaluation of Universities and Research (ANVUR) has for some decades defined the criteria for systematically evaluating student satisfaction. The analysis of these data presents various difficulties both in terms of data collection and analysis. The aim of this work is to propose Cande- comp/Parafac for a compositional analysis, which is able to capture the multidi- mensional aspects of the phenomenon taking into account its ordinal nature and the temporal characteristics of data collection.

The eValuation of Research Quality (VQR) is one of the most important assessment processes achieved by the National Agency for the Evaluation of Universities and Research Institutes. Its main task is to provide information on the status of the Italian research system by assessing the performance of universities in various scientific areas. The basic evaluation criteria were defined by panels of experts according to the specific characteristics of each subject area and through a synthetic statement on the products submitted by researchers. With the aim of studying this phenomenon in depth, DINDSCAL (Direct Individual Differences Scaling) model is proposed for a compositional analysis of VQR dataset.

The eValuation of Research Quality (VQR) is one the most important assessment process achieved by the National Agency for the Evaluation of Universities and Research Institutes (ANVUR). Its main task is to provide information on the status of the Italian research system assessing the performance of universities in various scientific areas. The entities measured are made up of researchers, assistants, first and second band professors, fixed-term professors and researchers, technology and research executives. For the purposes, ”research products” as journal contributions, volume contributions, and other types of scientific products are considered. The basic evaluation criteria were defined by groups of experts (GEV) according to the specific characteristics of each subject area and through a synthetic statement on the products. In this framework differences between GEV groups on a differential set of quality judgment should be explained in terms of compositional dissimilarity matrices. In literature the INDSCAL (Individual Differences Scaling) model is used to study the individual differences in three-way data by doubly centered a set of matrices of squared dissimilarity measures between a range of stimuli. A direct approach is here preferred, defined DINDSCAL (Direct INDividual Differences SCALing), in order to directly analyze simultaneous slices of dissimilarity matrices organized as compositional data. The compositional aspect of data allow to understand, at a first glance, which is the research product with the highest assessment compared to the remaining ones, irrespective of the role and the type of institutions to which researchers belong. Additionally, the DINDSCAL algorithm underlines the main divergencies made by each GEV group in terms of research output classification.

The CANDECOMP/PARAFAC model is an extension of bilinear PCA and has been designed to model three-way data by preserving their multidimensional configuration. The Alternating Least Squares (ALS) procedure is the preferred estimating algorithm for this model because it guarantees stable results. It can, however, be slow at converging and sensitive to collinearity and over-factoring. Dealing with these issues is even more pressing when data are compositional and thus collinear by definition. In this talk the solution proposed is based on a multistage approach. Here parameters are optimized with procedures that work better for collinearity and over-factoring, namely ATLD and SWATLD, and then results are refined with ALS.

Multidimensional compositional arrays require special analytical tools to be modeled. Specifically, the variation of the data can be captured by linear combinations of a defined number of parameters, capable of describing the complexity of the data. Usually these models are described as generalizations of Principal Component Analysis to higher order cases. Here the Candecomp/Parafac (CP) model is defined for compositional data contaminated with extreme observations by using a novel integrated SWATLD-ALS algorithm. Since the new procedure does not find a solution in the least square sense, it is expected to have a better performance in terms of sensitivity to outliers than ALS. However, due to the instability of its loss function, it should not be used alone.

The Tucker3 model is one of the most widely used tools for factorial analysis of three-way data arrays. When orthogonal factors are extracted this model can be seen as a three-way PCA (principal component analysis). The Tucker3 model is characterized by extreme flexibility as it allows for the use of a different number of factors in each mode and it yields non-unique results. When this model is applied to vectors of non-negative values with a sum constraint all problems connected with the statistical analysis of compositions must be taken into consideration. Like other standard statistical techniques, this model cannot be directly applied. The aim of this paper is to present the theory behind the Tucker3 model on compositional data and to describe the TUCKALS3 algorithm.

The monitoring of production processes on a planar surface typically involves sampling network to gather information about the status of the process. In order to save time and money, when the process goes into a stable status it might be appropriate to reduce the dimension of the sampling grid. In some cases, the allocation of a new network of smaller dimension is not free of constraints and it might be necessary the selection of a subgrid extracted from the original network. Discussion is focused on some recent methods used to achieve this aim. Possible extensions to consider jointly tabu search algorithm and co-kriging models is reported.

Il contributo analizza il ruolo delle skills richieste nel mercato del lavoro italiano usando un nuovo dataset creato combinando dati provenienti da fonti aziendali ed esterne. Da un punto di vista metodologico, una Conjoint analysis e` stata implementata ai dati per stimare le utilita parziali legate alle competenze pi ` u` richieste definendone la migliore combinazione desiderata in un’ottica di reclutamento. In particolare, i profili dei lavoratori reclutati da The Adecco Group in Italia nel periodo 2016-2021 sono stati analizzati alla ricerca di trend e dinamiche negli ultimi anni In this paper, the roles of the most requested skills for getting a job in Italian Labour market are investigated using a new data set created by combining information from business and external sources. From a methodological point of view, a Conjoint analysis is performed to estimate the partial utilities for the most requested skills and their importance in defining the best combinations of skills to match job requirements. In particular, the profiles of workers recruited by The Adecco Group in Italy in the period 2016-2021 have been analysed detecting dynamics and movements during last years.

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.

This study focuses on the manager’s professional work and its evolution in the last years. In particular, t he main focus is to detect the possible new dynamics in the managerial behaviour able to define the new professional figure of the ‘open manager’ , based on some evidence derived from a survey conducted by interviewing a set of managers of Italian companies using a structured questionnaire. By using an agglomerative hierarchical cluster procedure with the Ward’s method, results six different groups of managers with similar behaviours were defined, based on the responses to the questionnaire items.

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.

Small and medium enterprises (SMEs) contribution to the European Union economy has always been relevant, for both value added and the creation of jobs. On the same time, SMEs are more fragile and likely to default as compared to larger firms so to induce public policies targeted on them. Specific default predic- tion models, accounting for SMEs idyiosincratic traits, are based on several types of data, mainly accounting indicators. We explore the possibility of complementing accounting information with data scraped from the firms’ websites.

The statistical literature proposed many contributions about survival analysis in medical research, in this work this approach is proposed in a business context. The aim of this paper is to control the mortality of the users belonging to an e-mail subscribers list for a company operating in the healthcare information sector. Having available the survival times for each subscriber, the choice was oriented to survival models to evaluate the abandon of the customers. A survival analysis was conducted through a Cox model considering some risk factors of the subscriber. The selected Cox model carried to the identification of risk profiles representing different situations in terms of probability of abandon.

Three-way rating data on student satisfaction contain the scores assigned by students to a set of items measuring different aspects of educational quality at different time points. Such data provide information on the magnitude of satisfaction as well as information on how aspects vary with respect to each other and how they contribute to the total satisfaction of each student. Data magnitude is predominant in determining variability patterns, thus, any standard tool applied to these arrays only yields a one-dimensional solution measuring scale differences. The relative changes among items go completely undetected unless a compositional approach is used in combination with a multilinear tool. A case study on student satisfaction is presented to demonstrate that this method provides an insightful analysis of the role played by each aspect in generating satisfaction throughout faculties and years.

CP decomposition of large third-order tensors can be computationally challenging. Parameters are typically estimated by means of the ALS procedure because it yields least-squares solutions and provides consistent outcomes. Nevertheless, ALS presents two major flaws which are particularly problematic for large-scale problems: slow convergence and sensitiveness to degeneracy conditions such as over-factoring, collinearity, bad initialization and local minima. More efficient algorithms have been proposed in the literature. They are, however, much less dependable than ALS in delivering stable results because the increased speed often comes at the expense of accuracy. In particular, the ATLD procedure is one of the fastest alternatives, but it is hardly employed because of the unreliable nature of its convergence. As a solution, multi-optimization is proposed. ATLD and ALS steps are concatenated in an integrated procedure with the purpose of increasing efficiency without a significant loss in precision. This methodology has been implemented and tested under realistic conditions on simulated data sets.

Compositional data with a tridimensional structure are not uncommon in social sciences. The CANDECOMP/PARAFAC model is one of the most adequate techniques for modeling these arrays without confusing modes variability. Estimating parameters in this setting can be particularly difficult because compositional data are multicollinear by definition and because, in general, for socio-economic data the exact number of latent variables is harder to determine. The most used fitting procedure in the literature is the PARAFAC-ALS algorithm which, however, is sensitive to both the difficulties presented, namely it is sensitive to multicollinearity and to the use of the wrong number of factors. In this work an integrated PARAFAC-ALS algorithm initialized with SWATLD steps is proposed as an effective solution to these deficiencies. This approach is tested on simulated multicollinear data in comparison with standard ALS and proves capable of performing better in terms of robustness against over-factoring and temporary degeneracies, it is faster at converging even in case of collinearity and it still provides a least-squares solution.

An adaptation of the PARAFAC-ALS algorithm is implemented with the purpose of providing accurate and efficient estimates of CANDECOMP/PARAFAC parameters without being influenced by data characteristics and model specifications. This is a dual-optimization procedure in which values are first estimated by SWATLD and then refined through standard ALS steps. The use of an additional optimization phase is suggested with the purpose of canceling out specific ALS inefficiencies such as slow convergence and occurrence of over-factoring degeneracies. A complex simulation study is then implemented in order to identify the most adequate transition point between the stages of the proposed methodology and to show its advantages with respect to standard ALS.

Measuring academic educational quality presents three major difficulties, typical of all customer satisfaction and service quality studies: the use of subjective scales; the ordinal nature of the data; and the multifold structure of satisfaction. In order to solve these problems, principal component analysis (PCA) of compositional data is proposed in this work. The core idea behind this methodology is to analyze by PCA the relative information within the data rather than focusing on absolute scores. This approach is discussed in comparison with a widely used Item Response Theory method (the Partial Credit Model) in order to assess its merits, e.g. always identifying a coherent preference structure. Both procedures were, thus, carried out on a real dataset collected with the 2013/14 ANVUR questionnaire by L’Universita´ di Napoli-L’Orientale.

Mortality levels for subpopulations, such as countries in a region or provinces within a country, generally change in a similar fashion over time, as a result of common historical experiences in terms of health, culture, and economics. Forecasting mortality for such populations should consider the correlation between their mortality levels. In this perspective, we suggest using multilinear component techniques to identify a common time trend and then use it to forecast coherently the mortality of subpopulations. Moreover, this multiway approach is performed on life table deaths by referring to Compositional Data Analysis (CoDa) methodology. Compositional data are strictly positive values summing to a constant and represent part of a whole. Life table deaths are compositional by definition because they provide the age composition of deaths per year and sum to the life table radix. In bilinear models the use of life table deaths treated as compositions generally leads to less biased forecasts than other commonly used models by not assuming a constant rate of mortality improvement. As a consequence, an extension of this approach to multiway data is here presented. Specifically, a CoDa adaptation of the Tucker3 model is implemented for life table deaths arranged in three-dimensional arrays indexed by time, age, and population. The proposed procedure is used to forecast the mortality of Canadian provinces in a comparative study. The results show that the proposed model leads to coherent forecasts.

Studies on public social spending often fail to address the issues connected with budgetary constraints. Budget lines require public entities to partition resources among sectors of spending on the basis of preferred combinations and trade-offs. Standard exploratory tools do not allow to unveil this preference structure as they are hindered by the differences in budget scales and by the bounded nature of sector variability, i.e. an increase in one sector means a missed increase or a decrease in other sectors. In this work Italian public social spending is modeled with an alternative log-ratio methodology which allows to study relative variation patterns among sectors. It is also important to note that since the data is collected across time a three-way approach is recommended so that the variability of each mode is kept separate.