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Internal Seminars 2017: "The use of spatial information in entropy measures"

08/05/2017 dalle 13:15 alle 17:30

Dove Dipartimento di Scienze Statistiche, via delle Belle Arti n. 41 -Aula III 2° piano

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Linda Altieri
Research fellow, Dipartimento di Scienze Statistiche - Bologna


The concept of entropy, firstly introduced in information theory, rapidly became popular in many applied sciences via Shannon's formula to measure the degree of heterogeneity among observations. A rather recent research field aims at accounting for space in entropy measures, as a generalization when the spatial location of occurrences ought to be accounted for. The main limit of these developments is that all indices are computed conditional on a single chosen distance and do not cover the whole spatial configuration of the phenomenon under study. Moreover, most of them do not satisfy the desirable additivity property between local and global spatial measures.

This work follows and extends the route for including spatial components in entropy measures. Starting from the probabilistic properties of Shannon's entropy for categorical variables, it investigates the characteristics of the quantities known as residual entropy and mutual information, when space is included as a second dimension. This way, the proposal of entropy measures based on univariate distributions is extended to the consideration of bivariate distributions, in a setting where the probabilistic meaning of all components is well defined. As a direct consequence, substantial innovations are achieved. Firstly, Shannon’s entropy of a variable may be decomposed into one term, spatial mutual information, accounting for the role of space in determining the variable outcome, and another term, spatial global residual entropy, summarizing the remaining information carried by the variable itself. Secondly, the two terms both satisfy the additivity property, being sums of partial entropies measuring what happens at different distance classes.

The superiority of the proposed indices is assessed both via their theoretical properties and via a thorough comparative study. Moreover, they are used for measuring the spatial entropy of a case study consisting on a marked point pattern on rainforest tree species.


Alessandra Luati
Silvia Cagnone