Statistische Methoden für Longitudinale Funktionale Daten
Final Report Abstract
The Emmy-Noether research group Statistical Methods for Longitudinal Functional Data developed statistical methods for complex and dependent functional data such as are collected in modern longitudinal studies. We developed a general framework for regression with functional responses and/or covariates that covers densely or sparsely observed functions, as well as “generalized” non-Gaussian functional data. Different features of the conditional response distribution such as the pointwise mean, median or a quantile can be modeled, or several features simultaneously in generalized additive models for location, scale and shape (GAMLSS) for functional data. The models can contain a large number of flexible linear, smooth and interaction effects of scalar and/or functional covariates, including functional random effects to model the dependence structure of longitudinal or otherwise dependent functional data. Model terms are expanded in spline or functional principal component bases and we derived estimators for functional principal component expansions of functional random effects for a wide range of dependence structures between functions. Models can then be estimated as (generalized) functional additive mixed models, which allows for mixed model based inference, or based on a componentwise gradient boosting approach, which can handle a very large number of model terms and automatically selects or deselects model terms during estimation. We investigated identifiability of regression models with functional responses and/or covariates and developed suitable constraints and diagnostic measures. We derived a Karhunen-Loève theorem for multivariate functional data that is observed on different (dimensional) domains, e.g. functions and images. Establishing a relationship between univariate and multivariate functional principal components for the case of finite or truncated Karhunen-Loève expansions allowed us to estimate the multivariate functional principal components and scores based on their univariate counterparts, and we showed consistency of the resulting estimators. Developed methods were implemented in five packages for the open source R Project for Statistical Computing. We applied our methods to a wide range of data sets from scientific collaborations ranging from medicine and psychology through phonetics to engineering. Scientific results were honored by several awards including the David P. Byar Award of the American Statistical Association in 2016, the Gustav Adolf Lienert Award of the German Region of the International Biometric Society in 2013 and the Wolfgang Wetzel Award of the German Statistical Society in 2012. Further Information: http://www.biostat.statistik.unimuenchen.de/forschung/emmy_noether_pup/index.html
Publications
- (2010). Longitudinal functional principal component analysis. Electronic Journal of Statistics, 4, 1022–1054
Greven, S., Crainiceanu, C. M., Caffo, B. S., and Reich, D.
- (2013). Corrected confidence bands for functional data using principal components. Biometrics, 69(1), 41–51
Goldsmith, J., Greven, S., and Crainiceanu, C. M.
(See online at https://doi.org/10.1111/j.1541-0420.2012.01808.x) - (2013). Longitudinal scalar-on-functions regression with application to tractography data. Biostatistics, 14(3), 447–461
Gertheiss, J., Goldsmith, J., Crainiceanu, C., and Greven, S.
(See online at https://doi.org/10.1093/biostatistics/kxs051) - (2014). Longitudinal high-dimensional principal components analysis with application to diffusion tensor imaging of multiple sclerosis. The Annals of Applied Statistics, 8(4), 2175–2202
Zipunnikov, V., Greven, S., Shou, H., Caffo, B., Reich, D. S., and Crainiceanu, C.
(See online at https://doi.org/10.1214/14-AOAS748) - (2015). Functional additive mixed models. Journal of Computational and Graphical Statistics, 24(2), 477–501
Scheipl, F., Staicu, A.-M., and Greven, S.
(See online at https://doi.org/10.1080/10618600.2014.901914) - (2015). The functional linear array model. Statistical Modelling, 15(3), 279–300
Brockhaus, S., Scheipl, F., Hothorn, T., and Greven, S.
(See online at https://doi.org/10.1177/1471082X14566913) - (2016). Boosting flexible functional regression models with a high number of functional historical effects. Statistics and Computing
Brockhaus, S., Melcher, M., Leisch, F., and Greven, S.
(See online at https://doi.org/10.1007/s11222-016-9662-1) - (2016). Functional linear mixed models for irregularly or sparsely sampled data. Statistical Modelling, 16(1), 67–88
Cederbaum, J., Pouplier, M., Hoole, P., and Greven, S.
(See online at https://doi.org/10.1177/1471082X15617594) - (2016). Generalized functional additive mixed models. Electronic Journal of Statistics, 10(1), 1455–1492
Scheipl, F., Gertheiss, J., and Greven, S.
(See online at https://doi.org/10.1214/16-EJS1145) - (2017). A General Framework for Functional Regression Modelling. Statistical Modelling, Volume: 17 issue: 1-2, page(s): 1-35
Greven, S. and Scheipl, F.
(See online at https://doi.org/10.1177/1471082X16681317)
