{"data":{"id":"10.6084/m9.figshare.1623184","type":"dois","attributes":{"doi":"10.6084/m9.figshare.1623184","prefix":"10.6084","suffix":"m9.figshare.1623184","identifiers":[],"alternateIdentifiers":[],"creators":[{"name":"J. O. Ramsay","affiliation":[],"nameIdentifiers":[]}],"titles":[{"title":"A Functional Estimate of Covariation"}],"publisher":"Taylor \u0026 Francis","container":{},"publicationYear":2015,"subjects":[{"subject":"Biochemistry"},{"subject":"Medicine"},{"subject":"Neuroscience"},{"subject":"Biotechnology"},{"subject":"39999 Chemical Sciences not elsewhere classified","subjectScheme":"FOR"},{"subject":"FOS: Chemical sciences","schemeUri":"http://www.oecd.org/science/inno/38235147.pdf","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"FOS: Chemical sciences","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"Immunology"},{"subject":"FOS: Clinical medicine","schemeUri":"http://www.oecd.org/science/inno/38235147.pdf","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"FOS: Clinical medicine","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"19999 Mathematical Sciences not elsewhere classified","subjectScheme":"FOR"},{"subject":"FOS: Mathematics","schemeUri":"http://www.oecd.org/science/inno/38235147.pdf","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"FOS: Mathematics","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"Cancer"},{"subject":"110309 Infectious Diseases","subjectScheme":"FOR"},{"subject":"FOS: Health sciences","schemeUri":"http://www.oecd.org/science/inno/38235147.pdf","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"FOS: Health sciences","subjectScheme":"Fields of Science and Technology (FOS)"},{"subject":"Computational Biology"}],"contributors":[],"dates":[{"date":"2015-12-14","dateType":"Created"},{"date":"2017-02-16","dateType":"Updated"},{"date":"2015","dateType":"Issued"}],"language":null,"types":{"ris":"DATA","bibtex":"misc","citeproc":"dataset","schemaOrg":"Dataset","resourceType":"Dataset","resourceTypeGeneral":"Dataset"},"relatedIdentifiers":[{"relationType":"IsSupplementTo","relatedIdentifier":"10.1080/10618600.2015.1124041","relatedIdentifierType":"DOI"}],"relatedItems":[],"sizes":["119813 Bytes"],"formats":[],"version":null,"rightsList":[{"rights":"Creative Commons Attribution 4.0 International","rightsUri":"https://creativecommons.org/licenses/by/4.0/legalcode","schemeUri":"https://spdx.org/licenses/","rightsIdentifier":"cc-by-4.0","rightsIdentifierScheme":"SPDX"}],"descriptions":[{"description":"The analysis of functional data calls for a bivariate functional covariance function σ(\u003ci\u003es\u003c/i\u003e, \u003ci\u003et\u003c/i\u003e) that may be evaluated at any discrete set of points to define a variance-covariance matrix Σ. This article uses finite element methodology to construct a representation of a functional Choleski factor λ(\u003ci\u003ew\u003c/i\u003e, \u003ci\u003es\u003c/i\u003e) to define σ(\u003ci\u003es\u003c/i\u003e, \u003ci\u003et\u003c/i\u003e) = ∫λ(\u003ci\u003ew\u003c/i\u003e, \u003ci\u003es\u003c/i\u003e)λ(\u003ci\u003ew\u003c/i\u003e, \u003ci\u003et\u003c/i\u003e) \u003ci\u003edw\u003c/i\u003e. An estimate of Σ-1 is especially important for applications and, where the eigenstructure of the covariance permits, this is readily available since the resulting Σ is almost always positive definite. A simulation study compares the performance of estimates of Σ and Σ-1 to those from the classic covariance matrix estimate and an estimate using glasso package in R. The method’s capability of constraining estimates of Σ-1 to be strongly band-structured resulted in superior estimates. The real data application is to the smoothing of the Fels female growth data where σ(\u003ci\u003es\u003c/i\u003e, \u003ci\u003et\u003c/i\u003e) estimates the residual covariance structure in the presence of sampling points varying from one case to another. Supplementary materials are available online.","descriptionType":"Abstract"}],"geoLocations":[],"fundingReferences":[],"xml":"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