{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:25:34Z","timestamp":1760059534980,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T00:00:00Z","timestamp":1750377600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper reinforces the previously proposed moving linear (ML) model approach for time series analysis by introducing theoretically grounded enhancements. The ML model flexibly decomposes a time series into constrained and remaining components, enabling the extraction of trends and fluctuations with minimal structural assumptions. Building on this framework, we present two key improvements. First, we develop a theoretically justified evaluation criterion that facilitates coherent estimation of model parameters, particularly the width of the time interval. Second, we enhance the extended ML (EML) model by introducing a new outlier detection and estimation method that identifies both the number and locations of outliers by maximizing the reduction in AIC. Unlike the earlier version, the reinforced EML model simultaneously estimates outlier effects and improves model fit within a unified, likelihood-based framework. Empirical applications to economic time series illustrate the method\u2019s superior ability to detect meaningful anomalies and produce stable, interpretable decompositions. These contributions offer a generalizable and theoretically supported approach to modeling nonstationary time series with structural disturbances.<\/jats:p>","DOI":"10.3390\/axioms14070479","type":"journal-article","created":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T03:05:27Z","timestamp":1750388727000},"page":"479","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Reinforcing Moving Linear Model Approach: Theoretical Assessment of Parameter Estimation and Outlier Detection"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-7405-6529","authenticated-orcid":false,"given":"Koki","family":"Kyo","sequence":"first","affiliation":[{"name":"Digital Transformation Center, Gifu Shotoku Gakuen University, 1-1 Takakuwanishi, Yanaizu-cho, Gifu 501-6194, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/s41549-023-00089-x","article-title":"A moving linear model approach for extracting cyclical variation from time series data","volume":"19","author":"Kyo","year":"2023","journal-title":"J. Bus. Cycle Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1080\/13873954.2024.2416631","article-title":"An integrated approach for decomposing time series data into trend, cycle and seasonal components","volume":"30","author":"Kyo","year":"2024","journal-title":"Math. Comput. Model. Dyn. Syst."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ren, H., Xu, B., Wang, Y., Yi, C., Huang, C., Kou, X., Xing, T., Yang, M., Tong, J., and Zhang, Q. (2019, January 4\u20138). Time-series anomaly detection service at Microsoft. Proceedings of the KDD \u201919: The 25th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, Anchorage, AK, USA.","DOI":"10.1145\/3292500.3330680"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"3328","DOI":"10.1016\/j.jksus.2020.09.018","article-title":"An algorithm for outlier detection in a time series model using backpropagation neural network","volume":"32","author":"Vishwakarma","year":"2020","journal-title":"J. King Saud Univ.\u2014Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"101672","DOI":"10.1016\/j.ecoinf.2022.101672","article-title":"Detecting outliers in a univariate time series dataset using unsupervised combined statistical methods: A case study on surface water temperature","volume":"69","author":"Jamshidi","year":"2022","journal-title":"Ecol. Inform."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Kyo, K. (2023, January 24\u201327). An approach for the identification and estimation of outliers in a time series with a nonstationary mean. Proceedings of the 2023 World Congress in Computer Science, Computer Engineering, & Applied Computing (CSCE\u201923), Las Vegas, NV, USA.","DOI":"10.1109\/CSCE60160.2023.00244"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s10260-025-00780-6","article-title":"Enhancing business cycle analysis by integrating anomaly detection and components decomposition of time series data","volume":"34","author":"Kyo","year":"2025","journal-title":"Stat. Methods Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Kyo, K., and Noda, H. (2025). Analyzing mechanisms of business fluctuations involving time-varying structure in Japan: Methodological proposition and empirical study. Comput. Econ.","DOI":"10.1007\/s10614-025-10971-8"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Kitagawa, G. (2020). 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Big Data, Data Mining and Data Science: Algorithms, Infrastructures, Management and Security, De Gruyter.","DOI":"10.1515\/9783111344553-001"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/479\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:55:24Z","timestamp":1760032524000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/479"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,20]]},"references-count":14,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070479"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070479","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,6,20]]}}}