{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:30:13Z","timestamp":1775838613082,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T00:00:00Z","timestamp":1559952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1758786"],"award-info":[{"award-number":["1758786"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IJGI"],"abstract":"<jats:p>Geographically weighted regression (GWR) is a spatial statistical technique that recognizes that traditional \u2018global\u2019 regression models may be limited when spatial processes vary with spatial context. GWR captures process spatial heterogeneity by allowing effects to vary over space. To do this, GWR calibrates an ensemble of local linear models at any number of locations using \u2018borrowed\u2019 nearby data. This provides a surface of location-specific parameter estimates for each relationship in the model that is allowed to vary spatially, as well as a single bandwidth parameter that provides intuition about the geographic scale of the processes. A recent extension to this framework allows each relationship to vary according to a distinct spatial scale parameter, and is therefore known as multiscale (M)GWR. This paper introduces mgwr, a Python-based implementation of MGWR that explicitly focuses on the multiscale analysis of spatial heterogeneity. It provides novel functionality for inference and exploratory analysis of local spatial processes, new diagnostics unique to multi-scale local models, and drastic improvements to efficiency in estimation routines. We provide two case studies using mgwr, in addition to reviewing core concepts of local models. We present this in a literate programming style, providing an overview of the primary software functionality and demonstrations of suggested usage alongside the discussion of primary concepts and demonstration of the improvements made in mgwr.<\/jats:p>","DOI":"10.3390\/ijgi8060269","type":"journal-article","created":{"date-parts":[[2019,6,10]],"date-time":"2019-06-10T03:16:51Z","timestamp":1560136611000},"page":"269","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":663,"title":["mgwr: A Python Implementation of Multiscale Geographically Weighted Regression for Investigating Process Spatial Heterogeneity and Scale"],"prefix":"10.3390","volume":"8","author":[{"given":"Taylor","family":"Oshan","sequence":"first","affiliation":[{"name":"Center for Geospatial Information Science, Department of Geographical Sciences, University of Maryland, College Park, MD 20740, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6345-4347","authenticated-orcid":false,"given":"Ziqi","family":"Li","sequence":"additional","affiliation":[{"name":"Spatial Analysis Research Center, School of Geographical Sciences an Urban Planning, Arizona State University, Tempe, AZ 85281, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1073-7781","authenticated-orcid":false,"given":"Wei","family":"Kang","sequence":"additional","affiliation":[{"name":"Center for Geospatial Sciences, School of Public Policy, University of California, Riverside, CA 92521, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0274-599X","authenticated-orcid":false,"given":"Levi","family":"Wolf","sequence":"additional","affiliation":[{"name":"School of Geographical Sciences, University of Bristol, Bristol BS8 1SS, UK"}]},{"given":"A.","family":"Fotheringham","sequence":"additional","affiliation":[{"name":"Spatial Analysis Research Center, School of Geographical Sciences an Urban Planning, Arizona State University, Tempe, AZ 85281, USA"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"234","DOI":"10.2307\/143141","article-title":"A Computer Movie Simulating Urban Growth in the Detroit Region","volume":"46","author":"Tobler","year":"1970","journal-title":"Econ. Geogr."},{"key":"ref_2","unstructured":"Fotheringham, A.S., Brunsdon, C., and Charlton, M. (2002). Geographically Weighted Regression: The Analysis of Spatially Varying Relationships, John Wiley & Sons."},{"key":"ref_3","first-page":"1247","article-title":"Multi-Scale Geographically Weighted Regression","volume":"107","author":"Fotheringham","year":"2017","journal-title":"Ann. Am. Assoc. Geogr."},{"key":"ref_4","unstructured":"Environmental Systems Research Institute (ESRI) (2018). ArcMap 10.3 Spatial Analyst Toolbox, ESRI."},{"key":"ref_5","unstructured":"Bivand, R., Yu, D., Nakaya, T., and Garcia-Lopez, M.A. (2017). spgwr: Geographically Weighted Regression, R package version 0.6-32."},{"key":"ref_6","unstructured":"Wheeler, D. (2013). gwrr: Fits Geographically Weighted Regression Models with Diagnostic Tools, R package version 0.2-1."},{"key":"ref_7","unstructured":"Yu, H., Fotheringham, A.S., Li, Z., Oshan, T., Kang, W., and Wolf, L.J. (2019). Inference in multiscale geographically weighted regression. Geogr. Anal."},{"key":"ref_8","unstructured":"Lu, B., Harris, P., Charlton, M., Brundson, C., Nayaka, T., and Gollini, I. (2018). GWmodel: Geographically-Weighted Models, R package version 2.0-5."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"982","DOI":"10.1080\/13658816.2016.1263731","article-title":"Geographically weighted regression with parameter-specific distance metrics","volume":"31","author":"Lu","year":"2017","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_10","unstructured":"Li, Z., Fotheringham, A.S., Li, W., and Oshan, T. (2018). Fast Geographically Weighted Regression (FastGWR): A Scalable Algorithm to Investigate Spatial Process Heterogeneity in Millions of Observations. Int. J. Geogr. Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2751","DOI":"10.1068\/a38218","article-title":"Spatial-filtering-based contributions to a critique of geographically weighted regression (GWR)","volume":"40","author":"Griffith","year":"2008","journal-title":"Environ. Plan. A"},{"key":"ref_12","unstructured":"Da Silva, A.R., and Fotheringham, A.S. (2015). The Multiple Testing Issue in Geographically Weighted Regression: The Multiple Testing Issue in GWR. Geogr. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/s10109-005-0155-6","article-title":"Multicollinearity and correlation among local regression coefficients in geographically weighted regression","volume":"7","author":"Wheeler","year":"2005","journal-title":"J. Geogr. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Belsey, D.A., Kuh, E., and Welsch, R.E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity, Wiley.","DOI":"10.1002\/0471725153"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1007\/s11135-006-9018-6","article-title":"A Caution Regarding Rules of Thumb for Variance Inflation Factors","volume":"41","year":"2007","journal-title":"Qual. Quant."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2464","DOI":"10.1068\/a38325","article-title":"Diagnostic Tools and a Remedial Method for Collinearity in Geographically Weighted Regression","volume":"39","author":"Wheeler","year":"2007","journal-title":"Environ. Plan. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s10109-016-0239-5","article-title":"Geographically weighted regression and multicollinearity: Dispelling the myth","volume":"18","author":"Fotheringham","year":"2016","journal-title":"J. Geogr. Syst."},{"key":"ref_18","unstructured":"Oshan, T.M., and Fotheringham, A.S. (2017). A Comparison of Spatially Varying Regression Coefficient Estimates Using Geographically Weighted and Spatial-Filter-Based Techniques: A Comparison of Spatially Varying Regression. Geogr. Anal."},{"key":"ref_19","unstructured":"Murakami, D., Lu, B., Harris, P., Brunsdon, C., Charlton, M., Nakaya, T., and Griffith, D.A. (2017). The importance of scale in spatially varying coefficient modeling. arXivt."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1007\/s11004-010-9284-7","article-title":"The Use of Geographically Weighted Regression for Spatial Prediction: An Evaluation of Models Using Simulated Data Sets","volume":"42","author":"Harris","year":"2010","journal-title":"Math. Geosci."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Lu, B., Yang, W., Ge, Y., and Harris, P. (2018). Improvements to the calibration of a geographically weighted regression with parameter-specific distance metrics and bandwidths. Comput. Environ. Urban Syst.","DOI":"10.1016\/j.compenvurbsys.2018.03.012"},{"key":"ref_22","unstructured":"Comber, A., Chi, K., Quang Huy, M., Nguyen, Q., Lu, B., Huu Phe, H., and Harris, P. (2018). Distance metric choice can both reduce and induce collinearity in geographically weighted regression. Environ. Plan. B Urban Anal. City Sci."}],"container-title":["ISPRS International Journal of Geo-Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2220-9964\/8\/6\/269\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:57:02Z","timestamp":1760187422000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2220-9964\/8\/6\/269"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,6,8]]},"references-count":22,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2019,6]]}},"alternative-id":["ijgi8060269"],"URL":"https:\/\/doi.org\/10.3390\/ijgi8060269","relation":{},"ISSN":["2220-9964"],"issn-type":[{"value":"2220-9964","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,6,8]]}}}