Entropy is widely employed in many applied sciences to measure the heterogeneity of observations. Recently, many attempts have been made to build entropy measures for spatial data, in order to capture the influence of space over the variable outcomes. The main limit of these developments is that all indices are computed conditional on a single 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 reviews some recent developments, based on univariate distributions, and compares them to a new approach which considers the properties of entropy measures linked to bivariate distributions. This perspective introduces substantial innovations. Firstly, Shannon's entropy may be decomposed into two terms: spatial mutual information, accounting for the role of space in determining the variable outcome, and spatial global residual entropy, summarizing the remaining heterogeneity carried by the variable itself. Secondly, these terms both satisfy the additivity property, being sums of partial entropies measuring what happens at different distance classes. The proposed indices are used for measuring the spatial entropy of a marked point pattern on rainforest tree species. The new entropy measures are shown to be more informative and to answer a wider set of questions than the current proposals of the literature.

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A very recent proposal of a set of entropy measures for spatial data, based on building pairs of realizations, allows to split the data heterogeneity that is usually assessed via Shannon's entropy into two components: spatial mutual information, identifying the role of space, and spatial residual entropy, measuring heterogeneity due to other sources. A further decomposition into partial terms deeply investigates the role of space at specific distance ranges. The present work proposes improvements to the method and adds relevant results proving that the new set of spatial entropies satisfies a list of desirable properties. We extend the methodology to sets of realizations greater than pairs. We also show that the approach is more general, better performing and more interpretable than the most popular proposals in the literature, thanks to the property of additivity and a new way of computing entropy that explicitly discards the order within sets. A novel procedure for building the necessary quantities for computations is also provided. A comparative study illustrates the superior performance of the new set of measures over representative spatial configurations. Practical questions are answered by means of a case study on land use data.

The lack of efficiency in urban diffusion is a debated issue, important for biologists, urban specialists, planners and statisticians, both in developed and new developing countries. Many approaches have been considered to measure urban sprawl, roughly identified as chaotic urban expansion; such idea of chaos is here linked to the concept of entropy. Entropy, firstly introduced in information theory, has rapidly become a standard tool in ecology, biology, and geography to measure the degree of heterogeneity among observations; in such contexts, entropy measures should include spatial information. The aim of this paper is to employ a rigorous spatial entropy‐based approach to measure urban sprawl associated to the diffusion of metropolitan cities. In order to assess the performance of the considered measures, a comparative study is run over archetypical urban scenarios; afterwards, measures are used to quantify the degree of disorder in the urban expansion of three cities in Europe. Results are easily interpretable and can be used both as absolute measures of urban sprawl and for comparison over space and time.

a b s t r a c tThis work introduces a Bayesian approach to detecting multiple unknown changepoints over time in the inhomogeneous intensity of a spatio-temporal point process with spatial and temporal dependence within segments. We propose a new method for detecting changes by fitting a spatio-temporal log-Gaussian Cox process model using the computational efficiency and flexibility of integrated nested Laplace approximation, and by studying the posterior distribution of the potential changepoint positions. In this paper, the context of the problem and the research questions are introduced, then the methodology is presented and discussed in detail. A simulation study assesses the validity and properties of the proposed methods. Lastly, questions are addressed concerning potential unknown changepoints in the intensity of radioactive particles found on Sandside beach, Dounreay, Scotland.