[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83804-en":3,"doc-seo-83804-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83804,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Time Series Decomposition using the Fréchet Distance","Time Series Decomposition using the Fréchet Distance presents a new data analysis problem for decomposing a set of univariate time series into k base curves with bounded length. The objective minimizes the sum of Fréchet distances from each input series to a “Fréchet combination,” formed by individually scaling base curves under a k-dimensional traversal. Two variants are studied: arbitrary base curves or bases restricted to a finite candidate set. A (1+ε)-approximation is provided for the single-base case, plus an exact algorithm for the projection-distance problem, enabling exact results for general k within a fixed candidate set.","arXiv :2607 .04397v 1 [ cs .DS] 5 Jul 2026  \nTime Series Decomposition using the Fr´echet Distance Anne Driemel∗ Jan H¨ockendorff† Ioannis Psarros ‡ Christian Sohler §  \nAbstract  \nIn this paper, we introduce a new data analysis problem that aims to decompose a set of univariate time series into a small set of k base curves of length at most l such that the sum of Fr´echet distances of the time series to a “Fr´echet combination” of the base curves is minimized. Here, a Fr´echet combination allows to combine individually scaled base curves using a k-dimensional traversal. We call the problem of finding a set of optimal base curves the Fr´echet decomposition problem and we consider two variants: (a) the base curves can be arbitrary curves of bounded length and (b) the curves come from a given finite set of candidate curves. We think of the Fr´echet decomposition problem as a Fr´echet variant of principal component analysis. For the case of a single base curve we develop a (1 + ε)-approximation algorithm for the Fr´echet decomposition problem. Additionally we give an exact algorithm for the projection distance problem that asks to compute the distance of one given time series to a given set of k base curves. This allows us to design an exact algorithm for the Fr´echet decomposition problem for general k when curves come from a fixed candidate set.  \n1 Introduction  \nA fundamental problem in exploratory data analysis and forecasting is to decompose a given signal into its (most important) components. Several methods for this problem are known, including principal component analysis (PCA; also known as Karhunen-Loewe transform or empirical orthogonal functions [30, 23 , 42]), complex principal component analysis [25], independent component analysis [27], functional principal component analysis etc. One drawback of principal component analysis is that it cannot deal effectively with individual temporal distortion that may happen for a relevant pattern across the input time series. One reason why this may happen is that the signal is not sampled uniformly, but one could also think of physical effects that could lead to such a behaviour. For example, in the area of oceanography, one of the main factors in sea surface height analysis [38] is a change that corresponds to summer/winter, and so depending on the location on the planet, the underlying signal may be shifted. Some variants of PCA like the complex principal component analysis, are able to find temporal shifts of patterns in the input [25] . However, this comes at the expense of an intrinsic indeterminacy that complicates the interpretation of the results. Kernel PCA [31] can be viewed as addressing similar needs. There are kernels for variants of the  \n∗ University of Bonn, Germany.  \n†University of Cologne, Germany. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– Project Number 459420781 .  \n‡Archimedes, Athena Research Center, Greece. This work has been partially supported by project MIS 5154714 of the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGenerationEU Program.  \n§ University of Cologne, Germany.  \nFr´echet distance [37] and DTW [17, 16] . However, a fundamental problem with this approach is that the decomposition exists only in lifting to a high-dimensional feature space, and there is no reverse mapping from the feature space back to the input space. This is also called the preimage problem [24] . Otherwise classical techniques like spectral analysis [34], wavelet analysis [36] are used to determine patterns with respect to a single time series, which makes them unsuitable for our setting.  \nIn the statistics literature, so-called function registration aims to find suitable warping functions for the input time series to optimize the fit to a template or reference function as a preprocessing step to a (functional) principal component analysis [41, 35] . This is usually done for each input ","cbCaihp0SLf4538G","https://ap.wps.com/l/cbCaihp0SLf4538G","pdf",569822,6,1,31,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What problem does the paper introduce for time series decomposition?\",\"answer\":\"It defines a Fréchet decomposition problem that represents each univariate time series as being close to a Fréchet combination of k base curves, minimizing the sum of Fréchet distances across all series.\"},{\"question\":\"How is a “Fréchet combination” of base curves constructed?\",\"answer\":\"A Fréchet combination allows individually scaled base curves to be combined using a k-dimensional traversal that accounts for temporal alignment under the Fréchet distance.\"},{\"question\":\"What are the two variants of the decomposition problem considered?\",\"answer\":\"The base curves can either be arbitrary curves with bounded length, or they can be chosen from a given finite set of candidate curves.\"}]",1784190523,78,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"time-series-decomposition-using-the-frechet-distance","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/time-series-decomposition-using-the-frechet-distance/83804/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does the paper introduce for time series decomposition?","Question",{"text":76,"@type":77},"It defines a Fréchet decomposition problem that represents each univariate time series as being close to a Fréchet combination of k base curves, minimizing the sum of Fréchet distances across all series.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is a “Fréchet combination” of base curves constructed?",{"text":81,"@type":77},"A Fréchet combination allows individually scaled base curves to be combined using a k-dimensional traversal that accounts for temporal alignment under the Fréchet distance.",{"name":83,"@type":74,"acceptedAnswer":84},"What are the two variants of the decomposition problem considered?",{"text":85,"@type":77},"The base curves can either be arbitrary curves with bounded length, or they can be chosen from a given finite set of candidate curves.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]