{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:10:26Z","timestamp":1760213426162,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2016,8,20]],"date-time":"2016-08-20T00:00:00Z","timestamp":1471651200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental\u00a0Research Funds\u00a0for\u00a0the\u00a0Central Universities of Central South University","award":["2016zzts011"],"award-info":[{"award-number":["2016zzts011"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper presents a multiple artificial neural networks (MANN) method with interaction noise for estimating the occurrence probabilities of different classes at any site in space. The MANN consists of several independent artificial neural networks, the number of which is determined by the neighbors around the target location. In the proposed algorithm, the conditional or pre-posterior (multi-point) probabilities are viewed as output nodes, which can be estimated by weighted combinations of input nodes: two-point transition probabilities. The occurrence probability of a certain class at a certain location can be easily computed by the product of output probabilities using Bayes\u2019 theorem. Spatial interaction or redundancy information can be measured in the form of interaction noises. Prediction results show that the method of MANN with interaction noise has a higher classification accuracy than the traditional Markov chain random fields (MCRF) model and can successfully preserve small-scale features.<\/jats:p>","DOI":"10.3390\/a9030056","type":"journal-article","created":{"date-parts":[[2016,8,22]],"date-time":"2016-08-22T10:40:33Z","timestamp":1471862433000},"page":"56","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Multiple Artificial Neural Networks with Interaction Noise for Estimation of Spatial Categorical Variables"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2087-5838","authenticated-orcid":false,"given":"Xiang","family":"Huang","sequence":"first","affiliation":[{"name":"Department of Statistics, Central South University, Changsha 410012, China"}]},{"given":"Zhizhong","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Statistics, Central South University, Changsha 410012, China"}]}],"member":"1968","published-online":{"date-parts":[[2016,8,20]]},"reference":[{"key":"ref_1","unstructured":"Richardson, D. (2016). International Encyclopedia of Geography, Wiley-Blackwell. in press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s11004-007-9081-0","article-title":"Markov chain random fields for estimation of categorical variables","volume":"39","author":"Li","year":"2007","journal-title":"Math. Geol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s11771-016-3062-8","article-title":"Theoretical generalization of Markov chain random field from potential function perspective","volume":"23","author":"Huang","year":"2016","journal-title":"J. Cent. South Univ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2071","DOI":"10.1080\/13658816.2011.600253","article-title":"A multinomial logistic mixed model for the prediction of categorical spatial data","volume":"25","author":"Cao","year":"2011","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Huang, X., Li, J., Liang, Y., Wang, Z., Guo, J., and Jiao, P. (2016). Spatial hidden Markov chain models for estimation of petroleum reservoir categorical variables. J. Petrol. Explor. Prod. Technol., 1\u201312.","DOI":"10.1007\/s13202-016-0251-9"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1426","DOI":"10.1080\/13658816.2015.1133819","article-title":"Prediction of categorical spatial data via Bayesian updating","volume":"30","author":"Huang","year":"2016","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"973","DOI":"10.3390\/algor2030973","article-title":"Advances in artificial neural networks-methodological development and application","volume":"2","author":"Huang","year":"2009","journal-title":"Algorithms"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/s101090200090","article-title":"Learning in neural spatial interaction models: a statistical perspective","volume":"4","author":"Fischer","year":"2002","journal-title":"J. Geogr. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Karlsson, C., Andersson, M., and Norman, T. (2015). Handbook of Research Methods and Applications in Economic Geography, Edward Elgar.","DOI":"10.4337\/9780857932679"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1007\/s001680050082","article-title":"A genetic-algorithms based evolutionary computational neural network for modelling spatial interaction data","volume":"32","author":"Fischer","year":"1998","journal-title":"Ann. Reg. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1111\/1467-9787.00288","article-title":"Neural network modeling of constrained spatial interaction flows: Design, estimation, and performance issues","volume":"43","author":"Fischer","year":"2003","journal-title":"J. Reg. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1080\/02693799308901949","article-title":"Artificial neural networks for land-cover classification and mapping","volume":"7","author":"Civco","year":"1993","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1111\/j.1939-7445.2007.tb00215.x","article-title":"Modeling the spatial distribution of mineral deposits using neural networks","volume":"20","author":"Skabar","year":"2007","journal-title":"Nat. Resour. Model."},{"key":"ref_14","unstructured":"Openshaw, S., and Openshaw, C. (1997). Artificial Intelligence in Geography, Wiley."},{"key":"ref_15","unstructured":"Haykin, S. (1994). Neural Networks: A Comprehensive Foundation, Macmillan."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1080\/014311697218700","article-title":"Introduction neural networks in remote sensing","volume":"18","author":"Atkinson","year":"1997","journal-title":"Int. J. Remote Sens."},{"key":"ref_17","unstructured":"Bishop, C.M. (2006). Patter Recognition and Machine Learning, Springer."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1007\/s11004-008-9165-5","article-title":"The Tau model for data redundancy and information combination in earth sciences: Theory and application","volume":"40","author":"Krishnan","year":"2008","journal-title":"Math. Geosci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1007\/s11004-007-9117-5","article-title":"The Nu expression for probabilistic data integration","volume":"39","author":"Polyakova","year":"2007","journal-title":"Math. Geol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1111\/j.1365-2389.2011.01362.x","article-title":"An efficient maximum entropy approach for categorical variable prediction","volume":"62","author":"Allard","year":"2011","journal-title":"Eur. J. Soil Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"297","DOI":"10.2747\/1548-1603.42.4.297","article-title":"Application of transiograms to Markov chain simulation and spatial uncertainty assessment of land-cover classes","volume":"42","author":"Li","year":"2005","journal-title":"GISci. Remote Sens."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1080\/13658810600607816","article-title":"Transiogram: a spatial relationship measure for categorical data","volume":"20","author":"Li","year":"2006","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1080\/13658810903127991","article-title":"Linear interpolation and joint model fitting of experimental transiograms for Markov chain simulation of categorical spatial variables","volume":"24","author":"Li","year":"2010","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation, Oxford University Press.","DOI":"10.1093\/oso\/9780195115383.001.0001"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3082","DOI":"10.1016\/j.csda.2008.09.012","article-title":"CART algorithm for spatial data: Application to environmental and ecological data","volume":"53","author":"Bel","year":"2009","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1023\/A:1014009426274","article-title":"Conditional simulation of complex geological structures using multiple-point statistics","volume":"34","author":"Strebelle","year":"2002","journal-title":"Math. Geol."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/9\/3\/56\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:28:52Z","timestamp":1760210932000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/9\/3\/56"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,8,20]]},"references-count":26,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,9]]}},"alternative-id":["a9030056"],"URL":"https:\/\/doi.org\/10.3390\/a9030056","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2016,8,20]]}}}