{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T02:53:01Z","timestamp":1780714381118,"version":"3.54.1"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T00:00:00Z","timestamp":1752624000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T00:00:00Z","timestamp":1752624000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education","award":["NRF-2018R1D1A1B07045707"],"award-info":[{"award-number":["NRF-2018R1D1A1B07045707"]}]},{"name":"Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education","award":["NRF-2021R1F1A1048309"],"award-info":[{"award-number":["NRF-2021R1F1A1048309"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Stat"],"published-print":{"date-parts":[[2025,12]]},"DOI":"10.1007\/s00180-025-01653-y","type":"journal-article","created":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T12:10:17Z","timestamp":1752667817000},"page":"5315-5352","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Non-parametric entropy tests for spatial dependence"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8739-6631","authenticated-orcid":false,"given":"Christian H.","family":"Wei\u00df","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hee-Young","family":"Kim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,7,16]]},"reference":[{"issue":"1","key":"1653_CR1","doi-asserted-by":"publisher","first-page":"222","DOI":"10.1080\/03610926.2017.1301476","volume":"47","author":"G Arbia","year":"2018","unstructured":"Arbia G, Bee M, Espa G, Santi F (2018) Fitting spatial regressions to large datasets using unilateral approximations. Commun Stat Theory Methods 47(1):222\u2013238","journal-title":"Commun Stat Theory Methods"},{"issue":"6","key":"1653_CR2","doi-asserted-by":"publisher","first-page":"613","DOI":"10.3390\/e21060613","volume":"21","author":"C Bandt","year":"2019","unstructured":"Bandt C (2019) Small order patterns in big time series: a practical guide. Entropy 21(6):613","journal-title":"Entropy"},{"issue":"3","key":"1653_CR3","doi-asserted-by":"publisher","DOI":"10.1063\/5.0132602","volume":"33","author":"C Bandt","year":"2023","unstructured":"Bandt C (2023) Statistics and contrasts of order patterns in univariate time series. Chaos Interdiscip J Nonlinear Sci 33(3):033124","journal-title":"Chaos Interdiscip J Nonlinear Sci"},{"issue":"17","key":"1653_CR4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.88.174102","volume":"88","author":"C Bandt","year":"2002","unstructured":"Bandt C, Pompe B (2002) Permutation entropy: a natural complexity measure for time series. Phys Rev Lett 88(17):174102","journal-title":"Phys Rev Lett"},{"issue":"4","key":"1653_CR5","doi-asserted-by":"publisher","DOI":"10.1063\/5.0136912","volume":"33","author":"C Bandt","year":"2023","unstructured":"Bandt C, Wittfeld K (2023) Two new parameters for the ordinal analysis of images. Chaos Interdiscip J Nonlinear Sci 33(4):043124","journal-title":"Chaos Interdiscip J Nonlinear Sci"},{"issue":"3","key":"1653_CR6","doi-asserted-by":"publisher","first-page":"631","DOI":"10.2307\/1427527","volume":"25","author":"S Basu","year":"1993","unstructured":"Basu S, Reinsel GC (1993) Properties of the spatial unilateral first-order ARMA model. Adv Appl Probab 25(3):631\u2013648","journal-title":"Adv Appl Probab"},{"issue":"2","key":"1653_CR7","doi-asserted-by":"publisher","first-page":"192","DOI":"10.1111\/j.2517-6161.1974.tb00999.x","volume":"36","author":"JE Besag","year":"1974","unstructured":"Besag JE (1974) Spatial interaction and the statistical analysis of lattice systems. J R Stat Soc Ser B 36(2):192\u2013225","journal-title":"J R Stat Soc Ser B"},{"key":"1653_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2021.104798","volume":"186","author":"A Betken","year":"2021","unstructured":"Betken A, Dehling H, N\u00fc\u00dfgen I, Schnurr A (2021) Ordinal pattern dependence as a multivariate dependence measure. J Multivar Anal 186:104798","journal-title":"J Multivar Anal"},{"issue":"5","key":"1653_CR9","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1080\/07474938.2017.1365431","volume":"38","author":"MV Caballero-Pintado","year":"2019","unstructured":"Caballero-Pintado MV, Matilla-Garc\u00eda M, Ruiz Mar\u00edn M (2019) Symbolic correlation integral. Econom Rev 38(5):533\u2013556","journal-title":"Econom Rev"},{"key":"1653_CR10","volume-title":"Statistics for spatio-temporal data","author":"N Cressie","year":"2011","unstructured":"Cressie N, Wikle CK (2011) Statistics for spatio-temporal data. Wiley, Hoboken"},{"key":"1653_CR11","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.spasta.2018.09.003","volume":"29","author":"S De Iaco","year":"2019","unstructured":"De Iaco S, Posa D, Cappello C, Maggio S (2019) Isotropy, symmetry, separability and strict positive definiteness for covariance functions: a critical review. Spat Stat 29:89\u2013108","journal-title":"Spat Stat"},{"issue":"4","key":"1653_CR12","doi-asserted-by":"publisher","first-page":"858","DOI":"10.1016\/j.csda.2009.11.025","volume":"54","author":"P Duchesne","year":"2010","unstructured":"Duchesne P, de Micheaux PL (2010) Computing the distribution of quadratic forms: further comparisons between the Liu-Tang-Zhang approximation and exact methods. Comput Stat Data Anal 54(4):858\u2013862","journal-title":"Comput Stat Data Anal"},{"issue":"5","key":"1653_CR13","doi-asserted-by":"publisher","first-page":"838","DOI":"10.1016\/j.regsciurbeco.2013.04.007","volume":"43","author":"H Elsinger","year":"2013","unstructured":"Elsinger H (2013) Comment on: a non-parametric spatial independence test using symbolic entropy. Reg Sci Urban Econ 43(5):838\u2013840","journal-title":"Reg Sci Urban Econ"},{"issue":"4","key":"1653_CR14","doi-asserted-by":"publisher","first-page":"1235","DOI":"10.1007\/s00477-023-02626-7","volume":"38","author":"S Fischer","year":"2024","unstructured":"Fischer S, Oesting M, Schnurr A (2024) Multivariate motion patterns and applications to rainfall radar data. Stoch Environ Res Risk Assess 38(4):1235\u20131249","journal-title":"Stoch Environ Res Risk Assess"},{"issue":"sup1","key":"1653_CR15","doi-asserted-by":"publisher","first-page":"439","DOI":"10.2307\/143157","volume":"46","author":"P Gould","year":"1970","unstructured":"Gould P (1970) Is Statistix inferens the geographical name for a wild goose? Econ Geogr 46(sup1):439\u2013448","journal-title":"Econ Geogr"},{"issue":"1","key":"1653_CR16","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1111\/stan.12298","volume":"78","author":"D Karlis","year":"2024","unstructured":"Karlis D, Chutoo A, Khan NM, Jowahee V (2024) The multilateral spatial integer-valued process of order 1. Stat Neerl 78(1):4\u201324","journal-title":"Stat Neerl"},{"issue":"1","key":"1653_CR17","first-page":"59","volume":"30","author":"RA Kempton","year":"1981","unstructured":"Kempton RA, Howes CW (1981) The use of neighbouring plot values in the analysis of variety trials. J R Stat Soc Ser C 30(1):59\u201370","journal-title":"J R Stat Soc Ser C"},{"issue":"2\u20133","key":"1653_CR18","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1016\/j.regsciurbeco.2009.11.003","volume":"40","author":"F L\u00f3pez","year":"2010","unstructured":"L\u00f3pez F, Matilla-Garc\u00eda M, Mur J, Ruiz Mar\u00edn M (2010) A non-parametric spatial independence test using symbolic entropy. Reg Sci Urban Econ 40(2\u20133):106\u2013115","journal-title":"Reg Sci Urban Econ"},{"key":"1653_CR19","unstructured":"Mohr M, Wilhelm F, Hartwig M, M\u00f6ller R, Keller K (2020) New approaches in ordinal pattern representations for multivariate time series. Proceedings of Thirty-Third International Flairs Conference (FLAIRS 33), Association for the Advancement of Artificial Intelligence (AAAI), Washington, USA. pp 124\u2013129"},{"issue":"3","key":"1653_CR20","doi-asserted-by":"publisher","first-page":"655","DOI":"10.2307\/1426425","volume":"12","author":"DK Pickard","year":"1980","unstructured":"Pickard DK (1980) Unilateral Markov fields. Adv Appl Probab 12(3):655\u2013671","journal-title":"Adv Appl Probab"},{"key":"1653_CR21","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/j.spasta.2018.04.005","volume":"28","author":"T Rajala","year":"2018","unstructured":"Rajala T, Redenbach C, S\u00e4rkk\u00e4 A, Sormani M (2018) A review on anisotropy analysis of spatial point patterns. Spat Stat 28:141\u2013168","journal-title":"Spat Stat"},{"issue":"8","key":"1653_CR22","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0040689","volume":"7","author":"HV Ribeiro","year":"2012","unstructured":"Ribeiro HV, Zunino L, Lenzi EK, Santoro PA, Mendes RS (2012) Complexity-entropy causality plane as a complexity measure for two-dimensional patterns. PLoS One 7(8):e40689","journal-title":"PLoS One"},{"issue":"10","key":"1653_CR23","doi-asserted-by":"publisher","first-page":"1149","DOI":"10.1080\/13658810500105887","volume":"19","author":"AK Saha","year":"2005","unstructured":"Saha AK, Arora MK, Gupta RP, Virdi ML, Csaploivics E (2005) GIS\u2014based route planning in landslide\u2014prone areas. Int J Geogr Inf Sci 19(10):1149\u20131175","journal-title":"Int J Geogr Inf Sci"},{"key":"1653_CR24","doi-asserted-by":"publisher","first-page":"234","DOI":"10.2307\/143141","volume":"46","author":"W Tobler","year":"1970","unstructured":"Tobler W (1970) A computer movie simulating urban growth in the Detroit region. Econ Geogr 46:234\u2013240","journal-title":"Econ Geogr"},{"issue":"9","key":"1653_CR25","doi-asserted-by":"publisher","DOI":"10.1063\/5.0094943","volume":"32","author":"CH Wei\u00df","year":"2022","unstructured":"Wei\u00df CH (2022) Non-parametric tests for serial dependence in time series based on asymptotic implementations of ordinal-pattern statistics. Chaos Interdiscip J Nonlinear Sci 32(9):093107","journal-title":"Chaos Interdiscip J Nonlinear Sci"},{"key":"1653_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.spasta.2023.100800","volume":"59","author":"CH Wei\u00df","year":"2024","unstructured":"Wei\u00df CH, Kim H-Y (2024) Using spatial ordinal patterns for non-parametric testing of spatial dependence. Spat Stat 59:100800","journal-title":"Spat Stat"},{"issue":"3","key":"1653_CR27","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1080\/10485252.2023.2231565","volume":"36","author":"CH Wei\u00df","year":"2024","unstructured":"Wei\u00df CH, Schnurr A (2024) Generalized ordinal patterns in discrete-valued time series: nonparametric testing for serial dependence. J Nonparametr Stat 36(3):573\u2013599","journal-title":"J Nonparametr Stat"},{"issue":"3","key":"1653_CR28","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1080\/00401706.2022.2157883","volume":"65","author":"CH Wei\u00df","year":"2023","unstructured":"Wei\u00df CH, Testik MC (2023) Non-parametric control charts for monitoring serial dependence based on ordinal patterns. Technometrics 65(3):340\u2013350","journal-title":"Technometrics"},{"issue":"3\/4","key":"1653_CR29","doi-asserted-by":"publisher","first-page":"434","DOI":"10.2307\/2332724","volume":"41","author":"P Whittle","year":"1954","unstructured":"Whittle P (1954) On stationary processes in the plane. Biometrika 41(3\/4):434\u2013449","journal-title":"Biometrika"}],"updated-by":[{"DOI":"10.1007\/s00180-025-01670-x","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T00:00:00Z","timestamp":1760745600000}}],"container-title":["Computational Statistics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-025-01653-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00180-025-01653-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-025-01653-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T08:40:43Z","timestamp":1763368843000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00180-025-01653-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,16]]},"references-count":29,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["1653"],"URL":"https:\/\/doi.org\/10.1007\/s00180-025-01653-y","relation":{},"ISSN":["0943-4062","1613-9658"],"issn-type":[{"value":"0943-4062","type":"print"},{"value":"1613-9658","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,16]]},"assertion":[{"value":"2 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 June 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 July 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 October 2025","order":5,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":6,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s00180-025-01670-x","URL":"https:\/\/doi.org\/10.1007\/s00180-025-01670-x","order":8,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}