Extropy: Complementary Dual of Entropy

This article provides a completion to theories of information based on entropy, resolving a longstanding question in its axiomatization as proposed by Shannon and pursued by Jaynes. We show that Shannon's entropy function has a complementary dual function which we call "extropy." The entropy and the extropy of a binary distribution are identical. However, the measure bifurcates into a pair of distinct measures for any quantity that is not merely an event indicator. As with entropy, the maximum extropy distribution is also the uniform distribution, and both measures are invariant with respect to permutations of their mass functions. However, they behave quite differently in their assessments of the refinement of a distribution, the axiom which concerned Shannon and Jaynes. Their duality is specified via the relationship among the entropies and extropies of course and fine partitions. We also analyze the extropy function for densities, showing that relative extropy constitutes a dual to the Kullback-Leibler divergence, widely recognized as the continuous entropy measure. These results are unified within the general structure of Bregman divergences. In this context they identify half the $L_2$ metric as the extropic dual to the entropic directed distance. We describe a statistical application to the scoring of sequential forecast distributions which provoked the discovery.Comment: Published at http://dx.doi.org/10.1214/14-STS430 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Family Business in the Fishing District of Mazara del Vallo. Critical issues and Dynamics

This work-in-progress paper presents the early results of a survey aiming to understand the relationship between family firms and regional development in contemporary economic, by examining the negative effects of “family inertia” on the acquisition of dynamic capabilities necessary to enhance competitiveness and establish inter-firm cooperation strategies in family businesses in the context of an industrial district. Family firms are acknowledged for playing an important role in the creation, organization, and allocation of regional resources and in regional processes. Industrial districts are recognized as fertile fields for the development of inter-firm interrelations that may produce positive effects on both the economic development of the region and on the performance of the individual businesses in the area. On the other hand, in small family business, inertia can prevent the creation of those dynamic capabilities necessary to establish cooperation strategies to enhance competitiveness, frustrating the potential advantages offered by the participation in a district. The family businesses in the District Fisheries of Mazara del Vallo (Sicily, Italy), have been analyzed, to detect their structural and cultural characteristics and their implications on the longevity of firms and on the survival of the district system. At the present stage, the paper describes the characteristics of the area, presents the method followed for the empirical survey and reports initial comments to the findings

Il fenomeno della commoditization nel mercato delle conserve ittiche a base di tonno

Nel complesso scenario competitivo dell’odierno mercato globalizzato nasce la minaccia della commoditization: un processo in cui un prodotto (o un servizio) perde la differenziazione significativa rispetto ai concorrenti. Il prezzo risulta così l’unico parametro di scelta per il consumatore e il prodotto si trasforma in una commodity, diviene cioè “banale”. In una prospettiva empirica, si vuole qui studiare la commoditization nell’industria ittico conserviera del tonno descrivendone le dinamiche e le caratteristiche strutturali che presentano chiari segnali della “banalizzazione”. In particolare, attraverso lo strumento della mappa prezzo-benefici saranno indagati il maturo mercato del tonno in scatola sott’olio e il più giovane mercato dei filetti di tonno sott’olio in vaso di vetro.In the actual complex scenario of the global industrial market, the menace of the commoditization is growing: a process in which a product (or a service) loses its significant differentiation respect to the concurrent. The price results the only parameter for the consumer’s choice and the product became a commodity. In an empirical view, here the goal is to study the commoditization in the industrial production of the tuna-fish, describing the dynamics and the structural features which show clear signals of commoditization. In particular, the well-established market of the olive-oil tuna-fish cans will be investigated using the tool of the price-benefit map in comparison with the younger market of the olive-oil tuna-fish filets sold in glass jar

The Duality of Entropy/Extropy, and Completion of the Kullback Information Complex

The refinement axiom for entropy has been provocative in providing foundations of information theory, recognised as thoughtworthy in the writings of both Shannon and Jaynes. A resolution to their concerns has been provided recently by the discovery that the entropy measure of a probability distribution has a dual measure, a complementary companion designated as “extropy”. We report here the main results that identify this fact, specifying the dual equations and exhibiting some of their structure. The duality extends beyond a simple assessment of entropy, to the formulation of relative entropy and the Kullback symmetric distance between two forecasting distributions. This is defined by the sum of a pair of directed divergences. Examining the defining equation, we notice that this symmetric measure can be generated by two other explicable pairs of functions as well, neither of which is a Bregman divergence. The Kullback information complex is constituted by the symmetric measure of entropy/extropy along with one of each of these three function pairs. It is intimately related to the total logarithmic score of two distinct forecasting distributions for a quantity under consideration, this being a complete proper score. The information complex is isomorphic to the expectations that the two forecasting distributions assess for their achieved scores, each for its own score and for the score achieved by the other. Analysis of the scoring problem exposes a Pareto optimal exchange of the forecasters’ scores that both are willing to engage. Both would support its evaluation for assessing the relative quality of the information they provide regarding the observation of an unknown quantity of interest. We present our results without proofs, as these appear in source articles that are referenced. The focus here is on their content, unhindered. The mathematical syntax of probability we employ relies upon the operational subjective constructions of Bruno de Finetti

Influence of a Specific Aquatic Program on Social and Gross Motor Skills in Adolescents with Autism Spectrum Disorders: Three Case Reports

Swimming pool activities revealed to be efficacious to train psychomotor skills and increase adaptive behaviors in children with Autism Spectrum Disorders (ASD). Therefore, the purpose of this study was to investigate the efficacy of a specific multi-systemic aquatic therapy (CI-MAT) on gross motor and social skills in three adolescents with Autism Spectrum Disorders (ASD). Methods: three adolescents with ASD of which two boys (M1 with a chronological age of 10.3 years and a mental age of 4.7 years; M2 with a chronological age of 14.6 and a mental age inferior to 4 years) and one girl (chronological age of 14.0 and a mental age inferior to 4 years). The study was divided into three phases: baseline, 12-week CI-MAT program and Post-Test. Participants were administered a battery of tests incorporating anthropometric measurements, gross motor development test and a social skills questionnaire before and after a 12-week MAT-CI program. Results: Subjects improved locomotors and object control skills following the CI-MAT program in a different way. Concerning social behaviors, the higher proportion of gains was observed in the sensitivity of other’s presence and eye contact, for the contact domain, and in the comply turn for the relationship domain. Conclusions: The results of this study showed that the CI-MAT program was effective for the development of gross-motor skills and social behaviors in subjects with ASD. Moreover there is an urge to carry out a whole psychological assessment targeting both motor and adaptive development suitable to provide educational and vocational plans of exercises for people with ASD