On the local modeling of count data: multiscale geographically weighted Poisson regression

Abstract

A recent addition to the suite of techniques for local statistical modeling is the implementation of the multiscale geographically weighted regression (MGWR), a multiscale extension to geographically weighted regression (GWR). Using a back-fitting algorithm, MGWR relaxes the restrictive assumption in GWR that all processes being modeled operate at the same spatial scale and allows the estimation of a unique indicator of scale, the bandwidth, for each process. However, the current MGWR framework is limited to use with continuous data making it unsuitable for modeling data that do not typically exhibit a Gaussian distribution. This study expands the application of the MGWR framework to scenarios involving discrete response outcomes (count data following a Poisson’s distribution). Use of this new MGWR Poisson regression (MGWPR) model is demonstrated with a simulated data set and then with COVID-19 case counts within New York City at the zip code level. The results from the simulated data underscore the superiority of the MGWPR model in effectively capturing spatial processes that influence count data patterns, particularly those operating across diverse spatial scales. For empirical data, the results reveal significant spatial variations in relationships between socio-ecological factors and COVID-19 cases – variations often missed by traditional ‘global’ models.

Keywords:

• Local Poisson regression
• spatial process scale
• multiscale geographically weighted Poisson model
• COVID-19
• local scoring algorithm

Data and codes availability statement

The data and code used in the manuscript are openly available on Figshare: https://doi.org/10.6084/m9.figshare.21743021.v1. A local version of the MGWR repository from https://github.com/pysal/mgwr was used as a base code for the experiments. The simulation experiment code files are available in file ‘Simulation_experiment_version-1_IJGIS.ipynb’. The code runs the experiment once – to obtain the results reported in the paper, the code was run 1000 times. The replication of the NYC Covid data study from DiMaggio et al. (2020) uses the data file named ‘nyc_all_data.csv’. The data are compiled from the sources mentioned by DiMaggio et al. (2020) and are enumerated in Table 1 of the manuscript. The code for the NYC replication study is available in the file ‘NYC_replication_code_submission-IJGIS.ipynb’.

Author contributions

Mehak Sachdeva: project administration, conceptualization, software development, graphics production, analysis, writing original draft and editing subsequent drafts. A. Stewart Fotheringham: conceptualization, writing original draft and editing subsequent drafts. Ziqi Li: conceptualization, assistance with writing original draft, software development. Hanchen Yu: analytical development, editing original draft, software development.

Disclosure statement

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

Notes

1 The constant variance assumption for a linear regression model states that the variance of the errors/residuals is assumed to be constant (Poole and O’Farrell 1971).

2 An up-to-date bibliography of all the peer-reviewed journal articles applying the geographically weighted regression framework and its extensions is available here: https://sgsup.asu.edu/sparc/multiscale-gwr

3 We used cores of Intel Xeon Processor E5 v4 Family (E5-2680V4) on the high-performance computing platform at ASU Core research facilities.

4 We used the commonly employed statistical variable selection techniques namely, best subset selection and forward selection (Marhuenda et al. 2014), using the AICc as the diagnostic criterion and both resulted in the same subset of variables as depicted in Equation (21)(21) $$\mathrm{E}\mathrm{ }\left(\mathit{Positive}\mathrm{ }\mathit{cases}\right)\sim \mathit{Poisson}\left[\mathit{Offse}{t}_i\mathrm{ }\right(\mathit{Total}\mathrm{ }\mathit{tests}\left)\text{ exp }\right({\beta }_{0,i,bw0}+\mathrm{ }{\beta }_{1,i,bw1}\mathrm{ }(\mathrm{\%}\mathrm{ }\mathit{Afr}.\mathrm{ }\mathit{American})+{\beta }_{2,i,bw2}\mathrm{ }\left(\mathit{pop}\mathrm{ }\mathit{density}\right){+\beta }_{3,i,bw3}\mathrm{ }\left(\mathrm{\%}\mathrm{ }\mathit{heart}\right)\mathrm{ }+\mathrm{ }{\beta }_{4,i,bw4}\mathrm{ }\left(\mathrm{\%}\mathrm{ }\mathit{Hispanic}\right)+\mathrm{ }{\beta }_{5,i,bw5}\mathrm{ }\left(\mathrm{\%}\mathrm{ }\mathit{pub}\mathrm{ }\mathit{asst}\right)+\mathrm{ }{\beta }_{6,i,bw6}\mathrm{ }(\mathit{schools}\mathrm{ }\mathit{per}\mathrm{ }sq.\mathrm{ }\mathit{mile})\left)\right]$$(21) .

5 We follow da Silva and Fotheringham (2016)’s effective correction criterion to maintain the expected family-wise error rate and to avoid false positives.

6 MGWR desktop software is available for open download at: https://sgsup.asu.edu/sparc/multiscale-gwr; the open-source Python implementation of MGWR is embedded within PySAL: https://github.com/pysal/mgwr

Additional information

Funding

This work is supported by the National Science Foundation (#2117455) awarded to Prof. A. Stewart Fotheringham.

Notes on contributors

Mehak Sachdeva

Mehak Sachdeva is a Faculty Fellow at the Center for Urban Science and Progress within the Tandon School of Engineering at New York University. E-mail: [email protected]. Her research interests include developing and testing spatial analytical methods to model and understand urban processes and phenomena.

A. Stewart Fotheringham

A. Stewart Fotheringham is Regents' Professor of Computational Spatial Science and Director of the Spatial Analysis Research Center in the School of Geographical Sciences and Urban Planning at Arizona State University. E-mail: [email protected]. His research interests include local spatial models, spatial processes, spatial analytics, and spatial interaction modeling.

Ziqi Li

Ziqi Li is an Assistant Professor of Quantitative Geography in the Department of Geography at Florida State University, Tallahassee, FL 32306. E-mail: [email protected]. His research interests include spatial statistical modeling, explainable geospatial artificial intelligence, and their applications in interdisciplinary fields.

Hanchen Yu

Hanchen Yu is a visiting assistance professor in Urban Governance and Design Thrust, Society Hub, The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China. His research interests include spatial analysis, geographic information science, and spatial interaction modeling.