{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:48:18Z","timestamp":1760147298155,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,26]],"date-time":"2023-01-26T00:00:00Z","timestamp":1674691200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2021YFA1000102"],"award-info":[{"award-number":["2021YFA1000102"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>As spatial correlation and heterogeneity often coincide in the data, we propose a spatial single-index varying-coefficient model. For the model, in this paper, a robust variable selection method based on spline estimation and exponential squared loss is offered to estimate parameters and identify significant variables. We establish the theoretical properties under some regularity conditions. A block coordinate descent (BCD) algorithm with the concave\u2013convex process (CCCP) is composed uniquely for solving algorithms. Simulations show that our methods perform well even though observations are noisy or the estimated spatial mass matrix is inaccurate.<\/jats:p>","DOI":"10.3390\/e25020230","type":"journal-article","created":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T01:58:59Z","timestamp":1674784739000},"page":"230","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Robust Variable Selection with Exponential Squared Loss for the Spatial Single-Index Varying-Coefficient Model"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3995-3637","authenticated-orcid":false,"given":"Yezi","family":"Wang","sequence":"first","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}]},{"given":"Zhijian","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}]},{"given":"Yunquan","family":"Song","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,26]]},"reference":[{"key":"ref_1","unstructured":"Cliff, A.D. 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