Alfredo Rizzi

Representation, synthesis, variability and data preprocessing of a three-way data set

Abstract In the first section of this paper we describe the structures (vectors and matrices) on which a three-way data set X can be organized, and the information we can point out when using these structures. Many three-way analyses are based on pooled representations of X , that are systematically studied. The information given by a three-way data set can be synthesized according to the structures utilized to represent X . In the second section we define one-way, two-way and three-way syntheses of X . Also the variability of a three-way data set is evaluated, in section three, according three different levels: one-way or fiber variability, two-way or slab variability and three-way variability. The syntheses and the variability indices of X can be used for data preprocessing of X , which is here discussed in section four. Furthermore we discuss, in Section 5, the Principal Matrices Analysis on the base of three-way variability indices.

Sharp bounds for the maximum of the chi-square index in a class of contingency tables with given marginals

Our research is motivated by the problem of finding a joint frequency distribution with given marginals which is ‘as far as possible’ from the independent distribution with the same marginals. If the distance between these two distributions is measured by Pearsons chi-square index, the problem can be formulated as maximizing a quadratic convex separable function subject to transportation constraints. We present three heuristics for this problem: two of them are greedy heuristics; the third one amounts to solve a linear relaxation of the problem, and yields also an upper bound of the optimum; thus one can estimate the relative error E of any given heuristic. The lower bounds given by the three heuristics may be improved via Frank-Wolfes algorithm. Numerical experiments on 600 randomly generated test problems with up to 50 rows and 100 columns show that the above heuristics provide sharp bounds on the optimum (often one has E < 0.01). Even more interestingly, these bounds become sharper and sharper as the problem size increases.

Assessment of risk factors for pancreatic resection for cancer.

Abstract. A series of 101 consecutive patients undergoing pancreatic resection for cancer was retrospectively analyzed to define factors that may affect the immediate postoperative outcome. Overall morbidity and mortality were 28.7% and 10.9%, respectively, although these figures were greatly reduced during the last years; the complication rate dropped from 55.6% (1981–1987) to 20.0% (1993–1995) and the mortality from 16.7% to 6.7%. At univariate statistical analysis the patient characteristics (sex, age, American Society of Anesthesiologists [ASA] class, nutritional status, jaundice), tumor characteristics (site, size, TNM stage, and grading), and type of surgery were found not to affect postoperative morbidity and mortality. In contrast, a significantly lower rate of complications was observed in patients not undergoing gastric resection, in those who received 3 units or less of blood intraoperatively, and in subjects operated more recently (after 1990). At multivariate analysis the period when the operation was performed was the only independent variable that affected the immediate postoperative outcome. Among the examined factors, only the experience acquired over time regarding the intra- and perioperative treatment of these patients seems able to lower the rate of postoperative complications.

Metrics in Symbolic Data Analysis

The Authors consider the general problem of similarity and dissimilarity measures in Symbolic Data Analysis. First they examine the classical definitions of elementary event, assertion object, hierarchical dependences and logical dependences, then they consider some well-known measures resemblance measures between two objects (Sokal-Michener, Roger-Tanimoto, Sokal-Sneath, Dice-Czekanowski-Sorenson, Russel-Rao). For resemblance measures based on aggregation functions, the authors consider the proposals of Gowda-Diday, De Baets et al., Malerba et al., Vladutu et al., and Ichino-Yaguchi.

Statistical Methods for Cryptography

In this note, after recalling certain results regarding prime numbers, we will present the following theorem of interest to cryptography: Let two discrete s.v.’s (statistical variable) X, Y assume the value: 0, 1, 2, …, m − 1. Let X be uniformly distributed, that is, it assumes the value \(i(i = 0,1,\ldots ,m - 1)\) with probability 1 ∕ m and let the second s.v. Y assume the value i with probability \(({p}_{i}\,:\,\sum\limits_{i=1}^{m-1}{p}_{i} =\ 1,{p}_{i}\,\geq \,0)\). If the s.v. \(Z = X + Y\) (mod m) is uniformly distributed and m is a prime number, at least one of the two s. v. X and Y is uniformly distributed.

A New Method for Ranking n Statistical Units

In many research problems it is useful to summarize some indices or indicators to express a synthetic, indirect measure of a concept which is revealed by p variables observed in each statistical unit. This is because the p variables are considered to be indirect measures of a complex (perhaps indefinable) concept. Within this context and for ranking the n statistical units the author suggests the index:

Statistical Analysis of Papal Encyclicals

R_i = (\operatorname{sgn} {\mathbf{ }}c_i 1)(\sum\limits_r {c_{ir}^2 } )^{1/2}

Advances in data science and classification

where the c ir (i = 1, 2,...,n; r = 1, 2,..., p) represent the values of the p principal components connected with the i-th statistical unit. This index is applied for ranking the 20 Italian Regions for quality of life for the years 2000–2002. The results are compared with those that are furnished by the single source method.