Information Consistency-Based Measures for Spatial Stratified Heterogeneity

Abstract

As a typical form of spatial heterogeneity, spatial stratified heterogeneity is widely observed in geographical phenomena. Although the q statistic provides a measure of spatial stratified heterogeneity using variance differences, it is not suitable for nominal target variables and neglects information differences between strata at higher order moments. Based on the mutual information and relative entropy between variables, two spatial stratified heterogeneity measures are proposed for nominal and continuous target variables, respectively. Permutation tests are then used to determine their statistical significance. The proposed measures are suitable for either nominal or continuous target variables. They make no assumptions regarding the distribution of target variables, and return a value of zero only when the distribution of the target variable is independent of the explanatory variable. Experiments on five illustrative data sets and three publicly accessible data sets show that the proposed measures are consistent with the q statistic and can detect the existence of spatial stratified heterogeneity when the q statistic fails, so long as there are significant differences between the distributions in different strata.

作为一种典型的空间异质性, 空间分层异质性普遍存在于地理现象中。q统计采用方差差异来度量空间分层异质性, 但q统计不适于名义目标变量, 并且忽略了高阶层次之间的信息差异。本文基于变量之间的互信息和相对熵, 针对名义和连续目标变量分别提出了空间分层异质性度量, 并利用排列检验来确定它们的统计显著性。该度量适用于名义或连续目标变量, 不需要目标变量分布假设, 只有在目标变量分布与解释变量无关时才返回零值。对五个示意性数据集和三个公开数据集的实验表明, 当不同层次的分布存在显著差异时, 该度量与q统计一致；当q统计失败时, 该度量也能检测到空间分层异质性。

Como forma atípica de heterogeneidad espacial, la heterogeneidad espacial estratificada se observa con amplitud en los fenómenos geográficos. Aunque la estadística q provee una medida de heterogeneidad espacial estratificada usando diferencias de varianza, no es apropiada para variables nominales de objetivo y descuida las diferencias de información entre los estratos a momentos de orden más altos. Con base en la información mutua y la entropía relativa entre variables, se proponen dos medidas de heterogeneidad espacial estratificada para las variables nominal y de objetivo continuo, respectivamente. Después, se utilizan pruebas de permutación con las cuales determinar su significancia estadística. Las medidas propuestas son adecuadas tanto para variables nominales como de objetivo continuo. No formulan ningún supuesto sobre la distribución de las variables de objetivo y solamente devuelven un valor de cero cuando la distribución de la variable objetivo es independiente de la variable explicativa. Los experimentos conducidos con cinco conjuntos de datos ilustrativos y tres conjuntos de datos de acceso público muestran que las medidas propuestas son consistentes con la estadística q y pueden detectar la existencia de heterogeneidad espacial estratificada cuando falla la estadística q, siempre que ocurran diferencias significativas entre las distribuciones en diferentes estratos.

Key Words:

• mutual information
• relative entropy
• spatial heterogeneity
• spatial stratified heterogeneity

关键词:

• 互信息
• 相对熵
• 空间异质性
• 空间分层异质性

Palabras clave:

• entropía relativa
• heterogeneidad espacial
• heterogeneidad espacial estratificada
• información mutua

Notes

1 See https://geodacenter.github.io/data-and-lab/baltim/.

2 See https://geodacenter.github.io/data-and-lab/walnut_hills/.

3 See https://cran.r-project.org/web/packages/geodetector/index.html.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (Grant Nos. 41871286, 42230110, 62072294, 41725006).

Notes on contributors

Hexiang Bai

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

Hui Wang

HUI WANG is a Postgraduate Student in the School of Computer and Information Technology, Shanxi University, Taiyuan, Shanxi, China, 030006. E-mail: [email protected]. His research interests include knowledge-based recommender systems.

Deyu Li

DEYU LI is a Professor in the School of Computer and Information Technology, Shanxi University, Taiyuan, Shanxi, China, 030006. E-mail: [email protected]. His research interests include data mining and knowledge discovery.

Yong Ge

YONG GE [co-corresponding author] is a Full Professor at the Institute of Geographical Science and Natural Resources Research, Chinese Academy of Science, Beijing, China, 100045. 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.