Spatial Association from the Perspective of Mutual Information

Abstract

Measures of spatial association are important to reveal the spatial structures and patterns in geographical phenomena. They have utility for spatial interpolation, stochastic simulation, and causal inference, among others. Such measures are abundantly available for continuous spatial variables, whereas for categorical spatial variables they are less well developed. In this research, we developed a measure of spatial association for categorical spatial variables coined the entropogram, quantifying its spatial association using mutual information. Mutual information concerns information shared by pairs of random variables at different locations as revealed by their observed joint frequency distribution and marginal frequency distributions. The developed new measure is modeled as a function of lag in analogy to the variogram. Whereas existing measures focus mainly on interstate relationships, the entropogram models the spatial correlation in categorical spatial variables holistically. In this way, the entropogram imparts multiple advantages, for example, simplifying the representation of spatial structure for categorical variables and facilitating communication. The entropogram also reflects variation in the spatial correlation between different states. We first explored the properties of the entropogram in a simulation study. Then, we applied the entropogram to analyze the spatial association of land cover types in Qinxian, Shanxi, China. We conclude that the entropogram provides a suitable addition to existing measures of spatial association for applications in a wide range of disciplines where the categorical spatial variable is of interest.

在揭示地理现象的空间结构和空间模式方面, 空间关联算法具有非常重要的作用, 可用于空间插值、随机模拟和因果推理等。尽管有很多针对连续空间变量的算法, 但缺乏针对分类空间变量的算法。我们开发了一个针对分类空间变量的空间关联算法—熵图, 通过互信息来量化其空间关联。根据联合频率分布和边际频率分布, 互信息考虑了处于不同位置的随机变量的信息共享。本文的新算法是一个类似于变差函数的滞后函数。尽管现有算法主要侧重状态之间的关系, 熵图对分类空间变量的空间相关性进行了一体化建模。因此, 熵图拥有多种优势: 简化了分类变量的空间结构表达、促进了交流。熵图还反映了不同状态之间在空间相关性上的变化。我们首先通过模拟探索了熵图的性质。然后, 利用熵图分析了中国山西省沁县土地覆盖类型的空间关联。我们的结论是, 对于涉及分类空间变量的众多学科应用, 熵图是一个适宜的空间关联算法。

Las medidas de asociación espacial son importantes para revelar las estructuras y patrones espaciales de los fenómenos geográficos. Son útiles en la interpolación espacial, en la simulación estocástica y en la inferencia causal, entre otras. Tales medidas están disponibles en abundancia para las variables espaciales continuas, en tanto que para las variables espaciales categóricas están menos desarrolladas. En esta investigación, desarrollamos una medida de asociación espacial para las variables espaciales categóricas, acuñando el entropograma y cuantificando su asociación espacial por medio de información mutua. La información mutua concierne a la información compartida con pares de variables aleatorias en diferentes localidades, según se revela por la observación de sus distribuciones de frecuencias conjuntas y sus distribuciones de frecuencias marginales. Las nueva medida desarrollada se modela como una función de desfase por analogía en el variograma. Mientras las medidas existentes se enfocan principalmente en las relaciones interestatales, los modelos del entropograma modelan de manera holística la correlación espacial en variables espaciales categóricas. Así, el entropograma conlleva múltiples ventajas, como simplificar la representación de la estructura espacial para las variables categóricas y facilitando la comunicación. El entropograma también refleja variación en la correlación espacial entre diferentes estados. Primero exploramos las propiedades del entropograma en un estudio de simulación. Luego lo aplicamos para analizar la asociación espacial de los tipos de cobertura del suelo en Qinxian, Shanxi, China. Concluimos que el entropograma es una adición apropiada a las medidas existentes de asociación espacial para las aplicaciones en un amplio rango de disciplinas donde la variable espacial categórica es de interés.

Key Words:

• categorical data
• entropogram
• multicategorical random function
• mutual information
• spatial association

关键词:

• 分类数据
• 熵图
• 多分类随机函数
• 互信息
• 空间关联。

Palabras clave:

• asociación espacial
• datos categóricos
• entropogram
• función aleatoria multicategórica
• información mutua

Acknowledgments

The authors would like to thank Professor Gerard Heuvelink for his valuable comments and Managing Editor Jennifer Cassidento for her kind help. Wen-Bin Zhang conceived and designed the study, built the model, collected data, finalized the analysis, interpreted the findings, and wrote the original article. Yong Ge designed the study and interpreted the findings. Hexiang Bai and Yan Jin collected data and reviewed the article. Alfred Stein reviewed and edited the article. Peter M. Atkinson conceived and designed the study, interpreted the findings, and reviewed and edited the article. All authors read and approved the final article. The data sets analyzed during this study are available in the Figshare repository at the following link: https://doi.org/10.6084/m9.figshare.21687905.v2.

Additional information

Funding

This study was supported by the National Natural Science Foundation of China (No. 41725006, 42230110, 41871286).

Notes on contributors

Wen-Bin Zhang

WEN-BIN ZHANG is a PhD Candidate in the University of Chinese Academy of Science and a research assistant in the Institute of Geographical Science and Natural Resources Research, Chinese Academy of Science, Beijing 100101, China. E-mail: [email protected]. His research interests include GIScience and complexity. Applications concern health geography, population dynamics, and discovering the facts of the world.

Yong Ge

YONG GE (corresponding author) is a Full Professor in the Institute of Geographical Science and Natural Resources Research, Chinese Academy of Science, Beijing 100101, China. E-mail: [email protected]. Her research interests include spatial statistics and spatial data science including machine learning. Applications concern poverty, land use–land cover change detection, and scaling Earth science data.

Hexiang Bai

HEXIANG BAI is a Professor in the School of Computer and Information Technology at Shanxi University, Taiyuan, Shanxi, China, 030006. E-mail: [email protected]. His research interests include spatial statistics and rough sets theory-based spatial data mining.

Yan Jin

YAN JIN is a Lecturer in the School of Geographic and Biologic Information, Nanjing University of Posts and Telecommunications, Nanjing 210023, China. E-mail: [email protected]. Her current research interests include spatial statistics, data fusion, and remote sensing applications.

Alfred Stein

ALFRED STEIN is a Full Professor in the Department of Earth Observation and Geo-information science at the University of Twente, Enschede 7522 NH, The Netherlands. E-mail: [email protected]. His research interests include spatial statistics and image analysis with a specific focus on satellite image analysis. Applications concern geohealth, climate, environment, agriculture, natural vegetation, and urban development.

Peter M. Atkinson

PETER M. ATKINSON is a Distinguished Professor in the Lancaster Environment Centre at Lancaster University, Lancaster LA1 4YQ, UK. E-mail: [email protected]. His research interests are in spatial statistics and spatial data science including machine learning and artificial intelligence, with specific attention on issues relating to spatial scale and the change-of-support problem. Applications include land cover–land use change, climate-induced vegetation phenology change, and disease transmission systems.