{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T09:03:21Z","timestamp":1785402201339,"version":"3.56.0"},"reference-count":45,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T00:00:00Z","timestamp":1778198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research Plan Project of Shaanxi Provincial  Department of Education","award":["20JK0649"],"award-info":[{"award-number":["20JK0649"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12271420"],"award-info":[{"award-number":["12271420"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"award":["12271420"],"award-info":[{"award-number":["12271420"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IJGI"],"abstract":"<jats:p>Currently, two kinds of local spatial deviation indicators, namely local spatial heteroscedastic statistics and local spatial variance, with their associated tests have been proposed for estimating and inferring the characteristics of a spatial process at the second-order moment level, which is of wide potential application in spatial data analysis. Nonetheless, the performance of the indicators with their associated tests remains to be systematically investigated. Due to their mixed application orientations and the difficulty in theoretically comparing their performance, we design proper simulation experiments to assess their performance in estimating the variance function of a spatial process and detecting the local spatial heteroscedasticity and the boundaries of spatial homogeneous clusters. Some worthwhile findings are obtained and their performance from different application orientations is clarified. Based on the findings, two real-life spatial datasets are analyzed to demonstrate the applications of the indicators with their associated tests.<\/jats:p>","DOI":"10.3390\/ijgi15050205","type":"journal-article","created":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T19:35:43Z","timestamp":1778528143000},"page":"205","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Comparison of Local Spatial Deviation Indicators with Their Associated Tests: Evidence from Simulations and Applied Cases"],"prefix":"10.3390","volume":"15","author":[{"given":"Ruochen","family":"Mei","sequence":"first","affiliation":[{"name":"Department of Environmental Science and Engineering, School of Environmental and Chemical Engineering, Xi\u2019an Polytechnic University, Xi\u2019an 710048, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhi","family":"Zhang","sequence":"additional","affiliation":[{"name":"Fundamentals Department, Air Force Engineering University, Xi\u2019an 710043, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiuxia","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Finance and Statistics, School of Science, Xi\u2019an Polytechnic University, Xi\u2019an 710048, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,5,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1111\/j.1538-4632.1981.tb00731.x","article-title":"Generalized procedures for evaluating spatial autocorrelation","volume":"13","author":"Hubert","year":"1981","journal-title":"Geogr. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1111\/j.1538-4632.2008.00727.x","article-title":"A history of the concept of spatial autocorrelation: A geographer\u2019s perspective","volume":"40","author":"Getis","year":"2008","journal-title":"Geogr. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1093\/biomet\/37.1-2.17","article-title":"Notes on continuous stochastic phenomena","volume":"37","author":"Moran","year":"1950","journal-title":"Biometrika"},{"key":"ref_4","first-page":"115","article-title":"The contiguity ratio and statistical mapping","volume":"5","author":"Geary","year":"1954","journal-title":"Inc. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1111\/j.1538-4632.1992.tb00261.x","article-title":"The analysis of spatial association by use of distance statistics","volume":"24","author":"Getis","year":"1992","journal-title":"Geogr. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1191\/030913297676693207","article-title":"Trends in quantitative methods I: Stressing the local","volume":"21","author":"Fotheringham","year":"1997","journal-title":"Prog. Hum. Geogr."},{"key":"ref_7","first-page":"415","article-title":"Exploratory spatial data analysis with local statistics","volume":"47","author":"Unwin","year":"1998","journal-title":"J. R. Stat. Soc. D"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1111\/j.1538-4632.1995.tb00338.x","article-title":"Local indicators of spatial association\u2014LISA","volume":"27","author":"Anselin","year":"1995","journal-title":"Geogr. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1111\/j.1538-4632.1995.tb00912.x","article-title":"Local spatial autocorrelation statistics: Distribution issues and an application","volume":"27","author":"Ord","year":"1995","journal-title":"Geogr. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1080\/13658810601034267","article-title":"Local statistical spatial analysis: Inventory and prospect","volume":"21","author":"Boots","year":"2007","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1111\/j.1538-4632.1997.tb00960.x","article-title":"A note on the extremities of local Moran\u2019s Iis and their impact on global Moran\u2019s I","volume":"29","author":"Tiefelsdorf","year":"1997","journal-title":"Geogr. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1111\/j.1435-5597.1998.tb00710.x","article-title":"Some practical applications of Moran\u2019s I\u2019s exact conditional distribution","volume":"77","author":"Tiefelsdorf","year":"1998","journal-title":"Pap. Reg. Sci."},{"key":"ref_13","first-page":"187","article-title":"The saddlepoint approximation of Moran\u2019s I\u2019s and local Moran\u2019s Ii\u2019s reference distribution and their numerical evaluation","volume":"34","author":"Tiefelsdorf","year":"2002","journal-title":"Geogr. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/PL00011461","article-title":"Global and local spatial autocorrelation in bounded regular tessellations","volume":"2","author":"Boots","year":"2000","journal-title":"J. Geogr. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1068\/a3550","article-title":"Statistical test for local patterns of spatial association","volume":"35","author":"Leung","year":"2003","journal-title":"Environ. Plan. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1111\/gean.12164","article-title":"A local indicators of multivariate spatial association: Extending Geary\u2019s c","volume":"51","author":"Anselin","year":"2019","journal-title":"Geogr. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10109-019-00299-x","article-title":"Operational local joint count statistics for cluster detection","volume":"21","author":"Anselin","year":"2019","journal-title":"J. Geogr. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1111\/gean.12266","article-title":"Partial and semi-partial statistics of spatial associations for multivariate areal data","volume":"53","author":"Eckardt","year":"2021","journal-title":"Geogr. Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1784","DOI":"10.1080\/13658816.2022.2072850","article-title":"BiFlowAMOEBA for the identification of arbitrarily shaped clusters in bivariate flow data","volume":"36","author":"Liu","year":"2022","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_20","first-page":"1216","article-title":"Confounded local inference: Extending local Moran statistics to handle confounding","volume":"114","author":"Wolf","year":"2024","journal-title":"Ann. Am. Assoc. Geogr."},{"key":"ref_21","first-page":"1185","article-title":"A reciprocal statistic for detecting the full range of local patterns of bivariate spatial association","volume":"115","author":"Tao","year":"2025","journal-title":"Ann. Am. Assoc. Geogr."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1186\/1476-072X-8-11","article-title":"Spatio-temporal cluster analysis of the incidence of Campylobacter cases and patients with general diarrhea in a Danish county, 1995\u20132004","volume":"8","author":"Jepsen","year":"2009","journal-title":"Int. J. Health Geogr."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1515","DOI":"10.1080\/13658816.2010.511717","article-title":"Analysing spatio-temporal autocorrelation with LISTA-Viz","volume":"24","author":"Hardisty","year":"2010","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1111\/j.1435-5957.2011.00402.x","article-title":"A spatio-temporal measure of spatial dependence: An example using real estate data","volume":"92","author":"Legros","year":"2013","journal-title":"Pap. Reg. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1068\/a130063p","article-title":"A unified bootstrap test for local patterns of spatiotemporal association","volume":"47","author":"Yan","year":"2015","journal-title":"Environ. Plan. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"e2463","DOI":"10.1002\/env.2463","article-title":"Testing for local structure in spatiotemporal point pattern data","volume":"29","author":"Siino","year":"2018","journal-title":"Environmetrics"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1615","DOI":"10.1080\/13658816.2023.2204345","article-title":"Spatiotemporal flow L-function: A new method for identifying spatiotemporal clusters in geographical flow data","volume":"37","author":"Yan","year":"2023","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1111\/gean.12323","article-title":"Applying local indicators of spatial association to analyze longitudinal data: The absolute perspective","volume":"55","author":"Tao","year":"2023","journal-title":"Geogr. Anal."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1007\/s00168-011-0492-y","article-title":"Local spatial heteroscedasticity (LOSH)","volume":"48","author":"Ord","year":"2012","journal-title":"Ann. Reg. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1111\/j.1538-4632.1996.tb00936.x","article-title":"Geographically weighted regression: A method for exploring spatial nonstationarity","volume":"28","author":"Brunsdon","year":"1996","journal-title":"Geogr. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1007\/s00168-014-0605-5","article-title":"A note on the null distribution of the local spatial heteroscedasticity (LOSH) statistic","volume":"52","author":"Xu","year":"2014","journal-title":"Ann. Reg. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1080\/13658816.2017.1402914","article-title":"A statistical test on the local effects of spatially structured variance","volume":"32","author":"Westerholt","year":"2018","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.regsciurbeco.2017.10.015","article-title":"A scan test for spatial groupwise heteroscedasticity in cross-sectional models with an application on houses prices in Madrid","volume":"68","author":"Chasco","year":"2018","journal-title":"Reg. Sci. Urban Econ."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3315","DOI":"10.1111\/tgis.13004","article-title":"A new urban change detection method based on the local G and local spatial heteroscedasticity statistics","volume":"26","author":"Chen","year":"2022","journal-title":"Trans. GIS"},{"key":"ref_35","first-page":"77","article-title":"Local spatial autocorrelation and local spatial heterogeneity analysis of subjective poverty: A study on the provinces of Turkey","volume":"55","author":"Tatli","year":"2025","journal-title":"Rev. Reg. Stud."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"e70026","DOI":"10.1111\/tgis.70062","article-title":"A method for evaluating point segregation and its statistical power","volume":"29","author":"Sadahiro","year":"2025","journal-title":"Trans. GIS"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1503","DOI":"10.1080\/13658816.2022.2048388","article-title":"Statistical tests for the local spatial variance","volume":"36","author":"Rogerson","year":"2022","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1080\/01621459.1974.10482955","article-title":"Robust tests for the equality of variances","volume":"69","author":"Brown","year":"1974","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1111\/gean.12304","article-title":"The importance of null hypothesis: Understanding differences in local Moran\u2019s Ii under heteroskedasticity","volume":"54","author":"Sauer","year":"2022","journal-title":"Geogr. Anal."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"152","DOI":"10.2307\/3545210","article-title":"Spatial heterogeneity against heteroscedasticity: An ecological paradigm versus statistical concept","volume":"66","author":"Dutilleul","year":"1993","journal-title":"Oikos"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1111\/j.2517-6161.1995.tb02031.x","article-title":"Controlling the false discovery rate: A practical and powerful approach to multiple testing","volume":"57","author":"Benjamini","year":"1995","journal-title":"J. R. Stat. Soc. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1111\/j.0016-7363.2006.00682.x","article-title":"Controlling the false discovery rate: A new application to account for multiple and dependent tests in local statistics of spatial association","volume":"38","author":"Castro","year":"2006","journal-title":"Geogr. Anal."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v063.i17","article-title":"GWmodel: An R package for exploring spatial heterogeneity using geographically weighted models","volume":"63","author":"Gollini","year":"2015","journal-title":"J. Stat. Softw."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1080\/10095020.2014.917453","article-title":"The GWmodel R package: Further topics for exploring spatial heterogeneity using geographically weighted models","volume":"17","author":"Lu","year":"2014","journal-title":"Geo-Spat. Inf. Sci."},{"key":"ref_45","unstructured":"Fotheringham, A.S., Brunsdon, C., and Charlton, M. (2002). Geographically Weighted Regression: The Analysis of Spatially Varying Relationships, John Wiley and Sons."}],"container-title":["ISPRS International Journal of Geo-Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2220-9964\/15\/5\/205\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T04:23:58Z","timestamp":1779337438000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2220-9964\/15\/5\/205"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,8]]},"references-count":45,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["ijgi15050205"],"URL":"https:\/\/doi.org\/10.3390\/ijgi15050205","relation":{},"ISSN":["2220-9964"],"issn-type":[{"value":"2220-9964","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,8]]}}}