{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T08:58:54Z","timestamp":1758704334143,"version":"3.41.2"},"reference-count":24,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03n04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Adv. Data Sci. Adapt. Data Anal."],"published-print":{"date-parts":[[2022,7]]},"abstract":"<jats:p>Singular spectrum analysis (SSA) is a powerful method that is frequently used in dynamical systems theory and time series analysis. However, the algorithm itself is only partially understood. In this paper, we tackle the problem of a thorough interpretation of the complete basic SSA algorithm. We point out the relationship between SSA and Fourier analysis (FA), linear filtering (LF), principal component analysis (PCA), and power spectrum density estimation (PSDE). The main contribution is a reformulation and comprehensive interpretation of SSA by means of a novel class of data-adaptive circulant frames. Based on this class of frames, we introduce a formalism for time series decomposition and reconstruction. This mathematical framework generalizes principle component analysis and SSA. We show that SSA, PCA, FA, LF, and PSDE can be found from special cases of the proposed frame structures. This way we are able to provide a formal description of SSA and its relationship to the aforementioned methods. Whilst previous approaches to the interpretation of SSA relied on approximations, we provide exact mathematical descriptions of the relation between the above-mentioned techniques.<\/jats:p>","DOI":"10.1142\/s2424922x22500085","type":"journal-article","created":{"date-parts":[[2022,6,14]],"date-time":"2022-06-14T07:31:21Z","timestamp":1655191881000},"source":"Crossref","is-referenced-by-count":5,"title":["Singular Spectrum Analysis and Circulant Maximum Variance Frames"],"prefix":"10.1142","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3186-3450","authenticated-orcid":false,"given":"Christopher","family":"Bonenberger","sequence":"first","affiliation":[{"name":"Institute of Neural Information Processing, University of Ulm, James-Franck-Ring, 89081 Ulm, Germany"},{"name":"Institute for Artificial Intelligence, Ravensburg-Weingarten University of Applied Sciences, Weingarten, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wolfgang","family":"Ertel","sequence":"additional","affiliation":[{"name":"Institute for Artificial Intelligence, Ravensburg-Weingarten University of Applied Sciences, Weingarten, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Friedhelm","family":"Schwenker","sequence":"additional","affiliation":[{"name":"Institute of Neural Information Processing, University of Ulm, James-Franck-Ring, 89081 Ulm, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Markus","family":"Schneider","sequence":"additional","affiliation":[{"name":"Institute for Artificial Intelligence, Ravensburg-Weingarten University of Applied Sciences, Weingarten, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2022,7,15]]},"reference":[{"key":"S2424922X22500085BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-87626-5_2"},{"key":"S2424922X22500085BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.05.028"},{"key":"S2424922X22500085BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(86)90031-X"},{"volume-title":"Finite Frames: Theory and Applications","year":"2012","author":"Casazza P. G.","key":"S2424922X22500085BIB004"},{"key":"S2424922X22500085BIB005","doi-asserted-by":"publisher","DOI":"10.1201\/9780203744925"},{"key":"S2424922X22500085BIB006","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2007.900760"},{"key":"S2424922X22500085BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2514-8"},{"key":"S2424922X22500085BIB008","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-8373-3_10"},{"key":"S2424922X22500085BIB009","doi-asserted-by":"publisher","DOI":"10.1016\/0031-3203(81)90082-0"},{"key":"S2424922X22500085BIB010","doi-asserted-by":"publisher","DOI":"10.1002\/wics.1487"},{"key":"S2424922X22500085BIB011","doi-asserted-by":"publisher","DOI":"10.1201\/9781420035841"},{"key":"S2424922X22500085BIB012","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-62436-4"},{"key":"S2424922X22500085BIB014","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2010.07.005"},{"key":"S2424922X22500085BIB015","doi-asserted-by":"publisher","DOI":"10.6339\/JDS.2007.05(2).396"},{"volume-title":"Principal Component Analysis","year":"2002","author":"Jolliffe I.","key":"S2424922X22500085BIB016"},{"volume-title":"Nonlinear Time Series Analysis","year":"2004","author":"Kantz H.","key":"S2424922X22500085BIB017"},{"key":"S2424922X22500085BIB018","doi-asserted-by":"publisher","DOI":"10.1016\/0169-7439(95)00076-3"},{"key":"S2424922X22500085BIB019","doi-asserted-by":"publisher","DOI":"10.1142\/S1793536912500239"},{"key":"S2424922X22500085BIB021","doi-asserted-by":"publisher","DOI":"10.1049\/ip-f-1.1984.0100"},{"volume-title":"Handbook of Time Series Analysis, Signal Processing, and Dynamics","year":"1999","author":"Pollock D. S. G.","key":"S2424922X22500085BIB022"},{"key":"S2424922X22500085BIB024","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0020217"},{"key":"S2424922X22500085BIB025","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195094237.003.0002","volume-title":"Wavelets and Filter Banks","author":"Strang G.","year":"1996"},{"key":"S2424922X22500085BIB026","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(89)90077-8"},{"key":"S2424922X22500085BIB027","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139839099"}],"container-title":["Advances in Data Science and Adaptive Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S2424922X22500085","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,23]],"date-time":"2023-11-23T03:20:27Z","timestamp":1700709627000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S2424922X22500085"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7]]},"references-count":24,"journal-issue":{"issue":"03n04","published-print":{"date-parts":[[2022,7]]}},"alternative-id":["10.1142\/S2424922X22500085"],"URL":"https:\/\/doi.org\/10.1142\/s2424922x22500085","relation":{},"ISSN":["2424-922X","2424-9238"],"issn-type":[{"type":"print","value":"2424-922X"},{"type":"electronic","value":"2424-9238"}],"subject":[],"published":{"date-parts":[[2022,7]]},"article-number":"2250008"}}