A multiscale measure of spatial dependence based on a discrete Fourier transform

ABSTRACT

The measurement of spatial dependence within a set of observations or the residuals from a regression is one of the most common operations within spatial analysis. However, there appears to be a lack of appreciation for the fact that these measurements are generally based on an a priori definition of a spatial weights matrix and hence are limited to detecting spatial dependence at a single spatial scale. This paper highlights the scale-dependence problem with current measures of spatial dependence and defines a new, multi-scale approach to defining a spatial weights matrix based on a discrete Fourier transform. This approach is shown to be able to detect statistically significant spatial dependence which other multi-scale approaches to measuring spatial dependence cannot. The paper thus serves as a warning not to rely on traditional measures of spatial dependence and offers a more comprehensive, and scale-free, approach to measuring such dependence.

KEYWORDS:

• Multiscale
• spatial dependence
• Moran’s I
• Fourier transform
• spatial weights matrix

Acknowledgements

The first author would like to acknowledge the financial support of the award of grants from the US National Science Foundation (1841403, 1758786, and 2117455), the China Data Institute, and the Future Data Lab. The second author would like to acknowledge the financial support of grant 2117455 from the National Science Foundation.

Disclosure statement

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

Data and codes availability statement

The data and codes that support the findings of this study are available with the identifier(s) at the link https://www.doi.org/10.6084/m9.figshare.13574468.

Notes

1. Note that although here we use Moran’s I as our measure of spatial dependence, our findings and comments would apply equally to other measures of spatial dependence such as Geory’s C which also depends on an a priori defined spatial weighting matrix.

2. Here, in accordance with the literature, we use ‘autocorrelation’ as a measure of spatial dependence.

Additional information

Funding

This work was supported by the NSF [1758786,1841403,2117455]; China Data Institute [NA]; Future Data Lab [NA].

Notes on contributors

Hanchen Yu

Hanchen Yu is a postdoctoral researcher in Center for Geographic Analysis at Harvard. His research interests include spatial analysis, spatial econometrics and spatial statistics. He contributed to the idea, study design and methodology, and manuscript writing of this paper.

A. Stewart Fotheringham

A. Stewart Fotheringham is Regents’ Professor of Computational Spatial Science in the School of Geographical Sciences and Urban Planning at Arizona State University. His research interests are in the analysis of spatial data using statistical, mathematical and computational methods. He contributed to the study design, interpretation of the results, and the writing of this paper.