On the notion of ‘bandwidth’ in geographically weighted regression models of spatially varying processes

Abstract

Models designed to capture spatially varying processes are now employed extensively in the social and environmental sciences. The main strength of such models is their ability to represent relationships that vary across locations through locally varying parameter estimates. However, local models of spatial processes also provide information on the nature of these spatially varying relationships through the estimation of a ‘bandwidth’ parameter. This paper examines bandwidth at a conceptual, operational and empirical level within the framework of geographically weighted regression, one of the more frequently employed local spatial models. We outline how bandwidth relates to three characteristics of spatial processes: variation, dependence and strength.

Keywords:

• Bandwidth
• spatial scale
• MGWR
• spatially varying coefficients
• spatial processes

Data and codes availability statement

The data and code that support the findings of this study are openly available in Figshare at https://www.doi.org/10.6084/m9.figshare.14340368.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In Bayesian spatially varying coefficients models the parameter is referred to as a ‘decay’ (Finley et al. 2007) or a ‘range’ (Banerjee et al. 2014) parameter.

2 While this requires us to assume that covariance kernels have been specified in a particular way, this assumption is not onerous. When specified as a ‘decay’ kernel, large ‘decay’ parameter values indicate locality. Defining the bandwidth as inversely proportional to a decay parameter is sufficient. Either form has been used in Bayesian local models, but the bandwidth form is solely used in the GWR literature.

3 Fundamentally, this is a partitioning of the covariance matrix of β_i into its diagonal and off-diagonal elements.

4 Here, we use binary first-order rook contiguity.

5 While some spatial processes related to physical phenomena, such as soils, appear to be ‘random’, this is often an issue of data and scale, rather than the true underlying process itself (Webster 2000).

Additional information

Funding

This work was supported by the National Science Foundation with Award Number [#2117455].

Notes on contributors

A. Stewart Fotheringham

A. Stewart Fotheringham is Regents’ Professor of Computational Spatial Science and is a member of the National Academy of Sciences, Academia Europaea and the Academy of Social Sciences. His research interests are in spatial analysis, particularly local spatial modeling. He identified the research problem, led the research design and discussion of the results and led the writing of the paper.

Hanchen Yu

Hanchen Yu is a Postdoctoral Researcher in Center for Geographic Analysis at Harvard University. His research interests include spatial analysis, spatial econometrics and spatial statistics. He contributed towards the research design, formal data analysis, discussion of the results and the writing of the paper.

Levi John Wolf

Levi John Wolf is a Senior Lecturer (Assistant Professor) at the University of Bristol and a Fellow at the Alan Turing Institute. He works on novel methods, concepts and computation in spatial statistics with a focus on understanding segregation, sorting, inequality and redistricting. He contributed towards the research design, formal data analysis, discussion of the results and the writing of the paper.

Taylor M. Oshan

Taylor M. Oshan is an Assistant Professor in the Center for Geospatial Information Science within the Department of Geographical Sciences at the University of Maryland, College Park. His research interests are centered on developing methods to analyze spatial and temporal processes and applying them in the context of urban health and transportation, as well as building open-source tools. In particular, his work has focused on spatial interaction models and local multivariate statistical models. He contributed towards the research design, discussion of the results and the writing of the paper.

Ziqi Li

Ziqi Li is a Lecturer in GIScience at the University of Glasgow. His research interests broadly include spatial analysis and modeling, spatial statistical learning, interpretable machine learning and their applications in multidisciplinary fields. He contributed towards the research design, discussion of the results and the writing of the paper.