{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T13:22:01Z","timestamp":1771248121348,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T00:00:00Z","timestamp":1771200000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Basic Research Fund in Shenzhen Natural Science Foundation","award":["JCYJ20240813104924033"],"award-info":[{"award-number":["JCYJ20240813104924033"]}]},{"name":"Humanities and Social Sciences Foundation of Ministry of Education of China","award":["25YJA910002"],"award-info":[{"award-number":["25YJA910002"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["72173104"],"award-info":[{"award-number":["72173104"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In signal processing and information analysis, the detection and identification of anomalies present in signals constitute a critical research focus. Accurately discerning these deviations using probabilistic, statistical, and information-theoretic methods is essential for ensuring data integrity and supporting reliable downstream analysis. Outlier detection in functional data aims to identify curves or trajectories that deviate significantly from the dominant pattern\u2014a process vital for data cleaning and the discovery of anomalous events. This task is challenging due to the intrinsic infinite dimensionality of functional data, where outliers often appear as subtle shape deformations that are difficult to detect. Moving beyond conventional approaches that discretize curves into multivariate vectors, we introduce a novel framework that projects functional data into a low-dimensional space of meaningful features. This is achieved via a tailored weighting scheme designed to preserve essential curve variations. We then incorporate the Mahalanobis distance to detect directional outlyingness under non-Gaussian assumptions through a robustified bootstrap resampling method with data-driven threshold determination. Simulation studies validated its superior performance, demonstrating higher true positive and lower false positive rates across diverse anomaly types, including magnitude, shape-isolated, shape-persistent, and mixed outliers. The practical utility of our approach was further confirmed through applications in environmental monitoring using seawater spectral data, character trajectory analysis, and population data underscoring its cross-domain versatility.<\/jats:p>","DOI":"10.3390\/e28020233","type":"journal-article","created":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T12:37:28Z","timestamp":1771245448000},"page":"233","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Outlier Detection in Functional Data Using Adjusted Outlyingness"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4230-3053","authenticated-orcid":false,"given":"Zhenghui","family":"Feng","sequence":"first","affiliation":[{"name":"School of Science, Harbin Institute of Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaodan","family":"Hong","sequence":"additional","affiliation":[{"name":"School of Economics, Xiamen University, Xiamen 361005, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yingxing","family":"Li","sequence":"additional","affiliation":[{"name":"School of Business, Sun Yat-Sen University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaofei","family":"Song","sequence":"additional","affiliation":[{"name":"School of Science, Harbin Institute of Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-0305-9530","authenticated-orcid":false,"given":"Ketao","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Science, Harbin Institute of Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/BF02293704","article-title":"When the data are functions","volume":"47","author":"Ramsay","year":"1982","journal-title":"Psychometrika"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s41237-018-0066-8","article-title":"Multivariate functional clustering and its application to typhoon data","volume":"46","author":"Misumi","year":"2019","journal-title":"Behaviormetrika"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"104420","DOI":"10.1016\/j.chemolab.2021.104420","article-title":"Homogeneity tests for functional data based on depth-depth plots with chemical applications","volume":"219","author":"Laniado","year":"2021","journal-title":"Chemom. 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