Local and global spatio-temporal entropy indices based on distance- ratios and co-occurrences distributions

When it comes to characterize the distribution of ‘things’ observed spatially and identified by their geometries and attributes, the Shannon entropy has been widely used in different domains such as ecology, regional sciences, epidemiology and image analysis. In particular, recent research has taken into account the spatial patterns derived from topological and metric properties in order to propose extensions to the measure of entropy. Based on two different approaches using either distance-ratios or co-occurrences of observed classes, the research developed in this paper introduces several new indices and explores their extensions to the spatio-temporal domains which are derived whilst investigating further their application as global and local indices. Using a multiplicative space-time integration approach either at a macro or micro-level, the approach leads to a series of spatio-temporal entropy indices including from combining co-occurrence and distances-ratios approaches. The framework developed is complementary to the spatio-temporal clustering problem, introducing a more spatial and spatio-temporal structuring perspective using several indices characterizing the distribution of several class instances in space and time. The whole approach is first illustrated on simulated data evolutions of three classes over seven time stamps. Preliminary results are discussed for a study of conflicting maritime activities in the Bay of Brest where the objective is to explore the spatio-temporal patterns exhibited by a categorical variable with six classes, each representing a conflict between two maritime activities

SpatEntropy: Spatial Entropy Measures in R

This article illustrates how to measure the heterogeneity of spatial data presenting a finite number of categories via computation of spatial entropy. The R package SpatEntropy contains functions for the computation of entropy and spatial entropy measures. The extension to spatial entropy measures is a unique feature of SpatEntropy. In addition to the traditional version of Shannon's entropy, the package includes Batty's spatial entropy, O'Neill's entropy, Li and Reynolds' contagion index, Karlstrom and Ceccato's entropy, Leibovici's entropy, Parresol and Edwards' entropy and Altieri's entropy. The package is able to work with both areal and point data. This paper is a general description of SpatEntropy, as well as its necessary theoretical background, and an introduction for new users.Comment: 24 pages, 6 figure

A local scale-sensitive indicator of spatial autocorrelation for assessing high- and low-value clusters in multiscale datasets

Georeferenced user-generated datasets like those extracted from Twitter are increasingly gaining the interest of spatial analysts. Such datasets oftentimes reflect a wide array of real-world phenomena. However, each of these phenomena takes place at a certain spatial scale. Therefore, user-generated datasets are of multiscale nature. Such datasets cannot be properly dealt with using the most common analysis methods, because these are typically designed for single-scale datasets where all observations are expected to reflect one single phenomenon (e.g., crime incidents). In this paper, we focus on the popular local G statistics. We propose a modified scale-sensitive version of a local G statistic. Furthermore, our approach comprises an alternative neighbourhood definition that enables to extract certain scales of interest. We compared our method with the original one on a real-world Twitter dataset. Our experiments show that our approach is able to better detect spatial autocorrelation at specific scales, as opposed to the original method. Based on the findings of our research, we identified a number of scale-related issues that our approach is able to overcome. Thus, we demonstrate the multiscale suitability of the proposed solution

Measuring heterogeneity in urban expansion via spatial entropy

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, i.e. chaotic urban expansion; such idea of chaos is here linked to the concept of entropy. Entropy, firstly introduced in information theory, rapidly became a standard tool in ecology, biology and geography to measure the degree of heterogeneity among observations; in these 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 alternative 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 an absolute measure of urban sprawl and for comparison over space and time.Comment: 23 pages, 7 figure

An Empirical Bayesian Approach to Quantify Multi-Scale Spatial Structural Diversity in Remote Sensing Data

Landscape structure is as much a driver as a product of environmental and biological interactions and it manifests as scale-specific, but also as multi-scale patterns. Multi-scale structure affects processes on smaller and larger scales and its detection requires information from different scales to be combined. Herein, we propose a novel method to quantify multi-scale spatial structural diversity in continuous remote sensing data. We combined information from different extents with an empirical Bayesian model and we applied a new entropy metric and a value co-occurrence approach to capture heterogeneity. We tested this method on Normalized Difference Vegetation Index data in northern Eurasia and on simulated data and we also tested the effect of coarser pixel resolution. We find that multi-scale structural diversity can reveal itself as patches and linear landscape features, which persist or become apparent across spatial scales. Multi-scale line features reveal the transition zones between spatial regimes and multi-scale patches reveal those areas within transition zones where values are most different from each other. Additionally, spatial regimes themselves can be distinguished. We also find the choice of scale need not be informed by typical length-scales, which makes the method easy to implement. The proposed multi-scale approach can be applied to other contexts, following the roadmap we pave out in this study and using the tools available in the accompanying R package StrucDiv

Geological entropy and solute transport in heterogeneous porous media

We propose a novel approach to link solute transport behavior to the physical heterogeneity of the aquifer, which we fully characterize with two measurable parameters: the variance of the log K values ( math formula), and a new indicator (HR) that integrates multiple properties of the K field into a global measure of spatial disorder or geological entropy. From the results of a detailed numerical experiment considering solute transport in K fields representing realistic distributions of hydrofacies in alluvial aquifers, we identify empirical relationship between the two parameters and the first three central moments of the distributions of arrival times of solute particles at a selected control plane. The analysis of experimental data indicates that the mean and the variance of the solutes arrival times tend to increase with spatial disorder (i.e., HR increasing), while highly skewed distributions are observed in more orderly structures (i.e., HR decreasing) or at higher math formula. We found that simple closed-form empirical expressions of the bivariate dependency of skewness on HR and math formula can be used to predict the emergence of non-Fickian transport in K fields considering a range of structures and heterogeneity levels, some of which based on documented real aquifers. The accuracy of these predictions and in general the results from this study indicate that a description of the global variability and structure of the K field in terms of variance and geological entropy offers a valid and broadly applicable approach for the interpretation and prediction of transport in heterogeneous porous media