{"paper":{"title":"Functional linear regression analysis for longitudinal data","arxiv_id":"math/0603132","primary_cat":"math.ST","submitted_at":"2006-03-06T08:09:42+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":true,"role":"contribution","text":"We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.91,"main_claim_p":0.14,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":1,"p":0.93,"id":"s1","own":false,"role":"background","text":"Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.21,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.2,"role_probabilities":{"gap":0.0,"method":0.07,"result":0.0,"background":0.93,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":false,"role":"background","text":"In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit.","hedge":{"p":0.57,"label":"hedged_possibility"},"own_p":0.1,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.41,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.9,"id":"s3","own":true,"role":"method","text":"We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.9,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.1,"interpretation":0.0}},{"i":4,"p":0.36,"id":"s4","own":true,"role":"interpretation","text":"This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.77,"main_claim_p":0.14,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.12,"result":0.23,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.29,"interpretation":0.36}},{"i":5,"p":0.67,"id":"s5","own":true,"role":"contribution","text":"The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.72,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.25,"result":0.03,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.67,"interpretation":0.05}},{"i":6,"p":0.85,"id":"s6","own":true,"role":"result","text":"Asymptotic properties for a sample of $n$ subjects are investigated under mild conditions, as $n\\to \\infty$, and we obtain consistent estimation for the regression function.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.26,"quantitative":false,"quantitative_p":0.23,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.85,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.13,"interpretation":0.0}},{"i":7,"p":0.72,"id":"s7","own":true,"role":"contribution","text":"Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.13,"quantitative":false,"quantitative_p":0.26,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.27,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.72,"interpretation":0.0}},{"i":8,"p":0.97,"id":"s8","own":true,"role":"contribution","text":"A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard $R^2$ to the functional 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