{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:01:17Z","timestamp":1776718877541,"version":"3.51.2"},"reference-count":51,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,1]],"date-time":"2025-07-01T00:00:00Z","timestamp":1751328000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Social Science Foundation of China","award":["24BTJ067"],"award-info":[{"award-number":["24BTJ067"]}]},{"name":"National Social Science Foundation of China","award":["3502Z20231042"],"award-info":[{"award-number":["3502Z20231042"]}]},{"name":"Open Fund of Xiamen Software Supply Chain Security Public Technology Service Platform","award":["24BTJ067"],"award-info":[{"award-number":["24BTJ067"]}]},{"name":"Open Fund of Xiamen Software Supply Chain Security Public Technology Service Platform","award":["3502Z20231042"],"award-info":[{"award-number":["3502Z20231042"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Spatial data not only enables smart cities to visualize, analyze, and interpret data related to location and space, but also helps departments make more informed decisions. We apply a Bayesian quantile regression (BQR) of the partially linear varying coefficient spatial autoregressive (PLVCSAR) model for spatial data to improve the prediction of performance. It can be used to capture the response of covariates to linear and nonlinear effects at different quantile points. Through an approximation of the nonparametric functions with free-knot splines, we develop a Bayesian sampling approach that can be applied by the Markov chain Monte Carlo (MCMC) approach and design an efficient Metropolis\u2013Hastings within the Gibbs sampling algorithm to explore the joint posterior distributions. Computational efficiency is achieved through a modified reversible-jump MCMC algorithm incorporating adaptive movement steps to accelerate chain convergence. The simulation results demonstrate that our estimator exhibits robustness to alternative spatial weight matrices and outperforms both quantile regression (QR) and instrumental variable quantile regression (IVQR) in a finite sample at different quantiles. The effectiveness of the proposed model and estimation method is demonstrated by the use of real data from the Boston median house price.<\/jats:p>","DOI":"10.3390\/e27070715","type":"journal-article","created":{"date-parts":[[2025,7,1]],"date-time":"2025-07-01T09:29:01Z","timestamp":1751362141000},"page":"715","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Modeling Spatial Data with Heteroscedasticity Using PLVCSAR Model: A Bayesian Quantile Regression Approach"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1293-0016","authenticated-orcid":false,"given":"Rongshang","family":"Chen","sequence":"first","affiliation":[{"name":"School of Computer and Information Engineering, Xiamen University of Technology, Xiamen 361024, China"},{"name":"Xiamen Software Supply Chain Security Public Technology Service Platform, Xiamen 361024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9247-5019","authenticated-orcid":false,"given":"Zhiyong","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China"},{"name":"Fujian Provincial Key Laboratory of Statistics and Artificial Intelligence, Fuzhou 350117, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,1]]},"reference":[{"key":"ref_1","unstructured":"Cliff, A.D., and Ord, J.K. (1973). Spatial Autocorrelation, Pion Ltd."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Anselin, L. (1988). Spatial Econometrics: Methods and Models, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-7799-1"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"953","DOI":"10.2307\/2938168","article-title":"Spatial patterns in householed demand","volume":"59","author":"Case","year":"1991","journal-title":"Econometrica"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Cressie, N. (1993). Statistics for Spatial Data, John Wiley and Sons.","DOI":"10.1002\/9781119115151"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1177\/016001769702000107","article-title":"Bayesian estimation of spatial autoregressive models","volume":"20","author":"LeSage","year":"1997","journal-title":"Int. Reg. Sci. Rev."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Kazar, B.M., and Celik, M. (2012). Spatial Autoregressive Model, Springer Press.","DOI":"10.1007\/978-1-4614-1842-9"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1111\/j.1435-5957.2008.00175.x","article-title":"Regional economic growth in Europe: A semiparametric spatial dependence approach","volume":"87","author":"Basile","year":"2008","journal-title":"Pap. Reg. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.jeconom.2009.10.033","article-title":"Profile quasi-maximum likelihood estimation of partially linear spatial autoregressive models","volume":"157","author":"Su","year":"2010","journal-title":"J. Econom."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Bellman, R.E. (1961). Adaptive Control Processes, Princeton University Press.","DOI":"10.1515\/9781400874668"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1111\/j.2517-6161.1993.tb01939.x","article-title":"Varying-coefficient models","volume":"55","author":"Hastie","year":"1993","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_11","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_12","doi-asserted-by":"crossref","first-page":"e2485","DOI":"10.1002\/env.2485","article-title":"Estimation and inference in spatially varying coefficient models","volume":"29","author":"Mu","year":"2018","journal-title":"Environmetrics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1198\/016214503000170","article-title":"Spatial modeling with spatially varying coefficient processes","volume":"98","author":"Gelfand","year":"2003","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Li, S.S., Chen, J.B., and Chen, D.Q. (2024). PQMLE and Generalized F-Test of Random Effects Semiparametric Model with Serially and Spatially Correlated Nonseparable Error. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8070386"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"de Boor, C. (1978). A Practical Guide to Splines, Springer.","DOI":"10.1007\/978-1-4612-6333-3"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s00180-021-01123-1","article-title":"Bayesian analysis of partially linear, single-index, spatial autoregressive models","volume":"37","author":"Chen","year":"2021","journal-title":"Comput. Stat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1214\/ss\/1038425655","article-title":"Flexible smoothing with B-splines and penalties","volume":"11","author":"Eilers","year":"1996","journal-title":"Stat. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1093\/biomet\/82.4.711","article-title":"Reversible jump Markov chain Monte Carlo computation and Bayesian model determination","volume":"82","author":"Green","year":"1995","journal-title":"Biometrika"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1198\/016214503000143","article-title":"Generalized nonlinear modeling with multivariate free-knot regression splines","volume":"98","author":"Holmes","year":"2003","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2617","DOI":"10.1016\/j.csda.2008.12.010","article-title":"Bayesian estimation and variable selection for single index models","volume":"53","author":"Wang","year":"2009","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1214\/13-AOS1201","article-title":"A Semiparametric spatial dynamic model","volume":"42","author":"Sun","year":"2014","journal-title":"Ann. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jeconom.2014.12.005","article-title":"A spatial autoregressive model with a nonlinear transformation of the dependent variable","volume":"186","author":"Xu","year":"2015","journal-title":"J. Econom."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1080\/07350015.2016.1146145","article-title":"Semiparametric spatial autoregressive models with endogenous regressors: With an application to crime data","volume":"36","author":"Hoshino","year":"2017","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Liu, T., Xu, D.K., and Ke, S.Q. (2024). A Semiparametric Bayesian Approach to Heterogeneous Spatial Autoregressive Models. Entropy, 26.","DOI":"10.3390\/e26060498"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"33","DOI":"10.2307\/1913643","article-title":"Regression quantiles","volume":"46","author":"Koenker","year":"1978","journal-title":"Econometrica"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4396","DOI":"10.1080\/03610918.2022.2154365","article-title":"Quantile regression for partially linear varying coefficient spatial autoregressive models","volume":"53","author":"Dai","year":"2016","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1016\/S0167-7152(01)00124-9","article-title":"Bayesian quantile regression","volume":"54","author":"Yu","year":"2001","journal-title":"Stat. Probab. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1111\/j.1541-0420.2009.01269.x","article-title":"Bayesian quantile regression for longitudinal studies with nonignorable missing data","volume":"66","author":"Yuan","year":"2010","journal-title":"Biometrics"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1214\/14-BA911","article-title":"Bayesian tail risk interdependence using quantile regression","volume":"10","author":"Bernardi","year":"2015","journal-title":"Bayesian Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1080\/02664763.2011.620082","article-title":"Variable selection in quantile regression via Gibbs sampling","volume":"39","author":"Alhamzawi","year":"2012","journal-title":"J. Appl. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1102","DOI":"10.1214\/12-AOS1005","article-title":"Bayesian empirical likelihood for quantile regression","volume":"40","author":"Yang","year":"2012","journal-title":"Ann. Stat."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1111\/j.1467-9876.2010.00725.x","article-title":"Bayesian quantile regression for count data with application to environmental epidemiology","volume":"59","author":"Lee","year":"2010","journal-title":"J. R. Stat. Soc. Ser. C"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1080\/02664763.2018.1508557","article-title":"Bayesian spatial quantile regression for areal count data, with application on substitute care placements in Texas","volume":"46","author":"King","year":"2019","journal-title":"J. Appl. Stat."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1214\/12-BA708","article-title":"Spatial quantile multiple regression using the asymmetric Laplace process","volume":"7","author":"Lum","year":"2012","journal-title":"Bayesian Anal."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1198\/jasa.2010.ap09237","article-title":"Bayesian spatial quantile regression","volume":"106","author":"Reich","year":"2012","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_36","first-page":"2305","article-title":"Spatial quantile autoregression for season within year daily maximum temperature data","volume":"17","author":"Gelfand","year":"2023","journal-title":"Ann. Appl. Stat."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1111\/rssb.12467","article-title":"Joint quantile regression for spatial data","volume":"83","author":"Chen","year":"2021","journal-title":"J. R. Stat. Soc. Ser. Stat. Methodol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s11749-023-00895-6","article-title":"Bayesian joint quantile autoregression","volume":"33","author":"Gelfand","year":"2024","journal-title":"Test"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Chen, Z.Y., Chen, M.H., and Ju, F.Y. (2022). Bayesian P-splines quantile regression of partially linear varying coefficient spatial autoregressive models. Symmetry, 14.","DOI":"10.3390\/sym14061175"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1080\/00949655.2010.496117","article-title":"Gibbs sampling methods for bayesian quantile regression","volume":"81","author":"Kozumi","year":"2011","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1080\/03610918908812785","article-title":"An easily implemented generalised inverse Gaussian generator","volume":"18","author":"Dagnapur","year":"1989","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/biomet\/57.1.97","article-title":"Monte Carlo sampling methods using Markov chains and their applications","volume":"57","author":"Hastings","year":"1970","journal-title":"Biometrika"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1063\/1.1699114","article-title":"Equations of state calculations by fast computing machine","volume":"21","author":"Metropolis","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_44","unstructured":"Tanner, M.A. (1993). Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, Springer. [2nd ed.]."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1080\/01621459.1994.10476829","article-title":"The collapsed Gibbs sampler in Bayesian computations with applications to a gene regulation problem","volume":"89","author":"Liu","year":"1994","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.csda.2013.07.018","article-title":"Bayesian analysis of generalized partially linear single-index models","volume":"68","author":"Poon","year":"2013","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/0167-6377(96)00030-2","article-title":"General hit-and-run Monte Carlo sampling for evaluating multidimensional integrals","volume":"19","author":"Chen","year":"1996","journal-title":"Oper. Res. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"LeSage, P.J., and Pace, R.K. (2009). Introduction to Spatial Econometrics, CRC Press.","DOI":"10.1201\/9781420064254"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1214\/ss\/1177011136","article-title":"Inference from iterative simulation using multiple sequences","volume":"7","author":"Gelman","year":"1992","journal-title":"Stat. Sci."},{"key":"ref_50","unstructured":"Biv, R., Nowosad, J., and Lovelace, R. (2025, May 01). spData: Datasets for Spatial Analysis. R Package Version 2.3.4. Available online: https:\/\/CRAN.R-project.org\/package=spData."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/25-BA1507","article-title":"A New Family of Error Distributions for Bayesian Quantile Regression","volume":"1","author":"Yan","year":"2025","journal-title":"Bayesian Anal."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/7\/715\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:02:34Z","timestamp":1760032954000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/7\/715"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,1]]},"references-count":51,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["e27070715"],"URL":"https:\/\/doi.org\/10.3390\/e27070715","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,1]]}}}