arXiv subjects

Roberto Fontana

Publications and source records attributed to Roberto Fontana.

28 records · Page 2Linked to original sources

arXiv2016

Graphical models for studying museum networks: the Abbonamento Musei Torino Piemonte

Probabilistic graphical models are a powerful tool to represent real-word phenomena and to learn network structures starting from data. This paper applies graphical models in a new framework to study association rules driven by consumer choices in a network of museums. The network consists of the museums participating in the program of Abbonamento Musei Torino Piemonte, which is a yearly subscription managed by the Associazione Torino Città Capitale Europea. Consumers are card-holders, who are allowed to entry to all the museums in the network for one year. We employ graphical models to highlight associations among the museums driven by card-holder visiting behaviour. We use both undirected graphs to investigate the strength of the network and directed graphs to highlight asimmetry in the association rules.

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arXiv2015

Aberration in qualitative multilevel designs

Generalized Word Length Pattern (GWLP) is an important and widely-used tool for comparing fractional factorial designs. We consider qualitative factors, and we code their levels using the roots of the unity. We write the GWLP of a fraction ${\mathcal F}$ using the polynomial indicator function, whose coefficients encode many properties of the fraction. We show that the coefficient of a simple or interaction term can be written using the counts of its levels. This apparently simple remark leads to major consequence, including a convolution formula for the counts. We also show that the mean aberration of a term over the permutation of its levels provides a connection with the variance of the level counts. Moreover, using mean aberrations for symmetric $s^m$ designs with $s$ prime, we derive a new formula for computing the GWLP of ${\mathcal F}$. It is computationally easy, does not use complex numbers and also provides a clear way to interpret the GWLP. As case studies, we consider non-isomorphic orthogonal arrays that have the same GWLP. The different distributions of the mean aberrations suggest that they could be used as a further tool to discriminate between fractions.

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arXiv2015

Generalized Minimum Aberration mixed-level orthogonal arrays A general approach based on sequential integer quadratically constrained quadratic programming

Orthogonal Fractional Factorial Designs and in particular Orthogonal Arrays are frequently used in many fields of application, including medicine, engineering and agriculture. In this paper we present a methodology and an algorithm to find an orthogonal array, of given size and strength, that satisfies the generalized minimum aberration criterion. The methodology is based on the joint use of polynomial counting functions, complex coding of levels and algorithms for quadratic optimization and puts no restriction on the number of levels of each factor.

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arXiv2014

$D$-optimal saturated designs: a simulation study

In this work we focus on saturated $D$-optimal designs. Using recent results, we identify $D$-optimal designs with the solutions of an optimization problem with linear constraints. We introduce new objective functions based on the geometric structure of the design and we compare them with the classical $D$-efficiency criterion. We perform a simulation study. In all the test cases we observe that designs with high values of $D$-efficiency have also high values of the new objective functions.

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arXiv2013

Random Latin squares and Sudoku designs generation

Uniform random generation of Latin squares is a classical problem. In this paper we prove that both Latin squares and Sudoku designs are maximum cliques of properly defined graphs. We have developed a simple algorithm for uniform random sampling of Latin squares and Sudoku designs. It makes use of recent tools for graph analysis. The corresponding SAS code is annexed.

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arXiv2013

A Characterization of Saturated Designs for Factorial Experiments

In this paper we study saturated fractions of factorial designs under the perspective of Algebraic Statistics. We define a criterion to check whether a fraction is saturated or not with respect to a given model. The proposed criterion is based purely on combinatorial objects. Our technique is particularly useful when several fractions are needed. We also show how to generate random saturated fractions with given projections, by applying the theory of Markov bases for contingency tables.

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arXiv2013

Random generation of optimal saturated designs

Efficient algorithms for searching for optimal saturated designs are widely available. They maximize a given efficiency measure (such as D-optimality) and provide an optimum design. Nevertheless, they do not guarantee a \emph{global} optimal design. Indeed, they start from an initial random design and find a local optimal design. If the initial design is changed the optimum found will, in general, be different. A natural question arises. Should we stop at the design found or should we run the algorithm again in search of a better design? This paper uses very recent methods and software for discovery probability to support the decision to continue or stop the sampling. A software tool written in SAS has been developed.

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arXiv2012

Saturated fractions of two-factor designs

In this paper we study saturated fractions of a two-factor design under the simple effect model. In particular, we define a criterion to check whether a given fraction is saturated or not, and we compute the number of saturated fractions. All proofs are constructive and can be used as actual methods to build saturated fractions. Moreover, we show how the theory of Markov bases for contingency tables can be applied to two-factor designs for moving between the designs with given margins.

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arXiv2009

Generation of Fractional Factorial Designs

The joint use of counting functions, Hilbert basis and Markov basis allows to define a procedure to generate all the fractions that satisfy a given set of constraints in terms of orthogonality. The general case of mixed level designs, without restrictions on the number of levels of each factor (like primes or power of primes) is studied. This new methodology has been experimented on some significant classes of fractional factorial designs, including mixed level orthogonal arrays.

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arXiv2007

2-level fractional factorial designs which are the union of non trivial regular designs

Every fraction is a union of points, which are trivial regular fractions. To characterize non trivial decomposition, we derive a condition for the inclusion of a regular fraction as follows. Let $F = \sum_αb_αX^α$ be the indicator polynomial of a generic fraction, see Fontana et al, JSPI 2000, 149-172. Regular fractions are characterized by $R = \frac 1l \sum_{α\in \mathcal L} e_αX^α$, where $α\mapsto e_α$ is an group homeomorphism from $\mathcal L \subset \mathbb Z_2^d$ into $\{-1,+1\}$. The regular $R$ is a subset of the fraction $F$ if $FR = R$, which in turn is equivalent to $\sum_t F(t)R(t) = \sum_t R(t)$. If $\mathcal H = \{α_1 >... α_k\}$ is a generating set of $\mathcal L$, and $R = \frac1{2^k}(1 + e_1X^{α_1}) ... (1 + e_kX^{α_k})$, $e_j = \pm 1$, $j=1 ... k$, the inclusion condition in term of the $b_α$'s is % \begin{equation}b_0 + e_1 b_{α_1} + >... + e_1 ... e_k b_{α_1 + ... + α_k} = 1. \tag{*}\end{equation} % The last part of the paper will discuss some examples to investigate the practical applicability of the previous condition (*). This paper is an offspring of the Alcotra 158 EU research contract on the planning of sequential designs for sample surveys in tourism statistics.

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