Abstract
We propose a general framework for regression models with functional response containing a potentially large number of flexible effects of functional and scalar covariates. Special emphasis is put on historical functional effects, where functional response and functional covariate are observed over the same interval and the response is only influenced by covariate values up to the current grid point. Historical functional effects are mostly used when functional response and covariate are observed on a common time interval, as they account for chronology. Our formulation allows for flexible integration limits including, e.g., lead or lag times. The functional responses can be observed on irregular curve-specific grids. Additionally, we introduce different parameterizations for historical effects and discuss identifiability issues.The models are estimated by a component-wise gradient boosting algorithm which is suitable for models with a potentially high number of covariate effects, even more than observations, and inherently does model selection. By minimizing corresponding loss functions, different features of the conditional response distribution can be modeled, including generalized and quantile regression models as special cases. The methods are implemented in the open-source R package FDboost. The methodological developments are motivated by biotechnological data on Escherichia coli fermentations, but cover a much broader model class.
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Acknowledgments
Special thanks to Markus Luchner and Gerald Striedner for providing us with the fermentation data. We thank Fabian Scheipl for useful discussions. The work of Sarah Brockhaus and Sonja Greven was supported by the German Research Foundation (DFG) through Emmy Noether grant GR 3793/1-1. The work of Michael Melcher and Friedrich Leisch was supported by the Federal Ministry of Traffic, Innovation and Technology (bmvit), the Federal Ministry of Economy, Family and Youth (BMWFJ), the Styrian Business Promotion Agency (SFG), the Standortagentur Tirol and the ZIT Technology Agency of the City of Vienna through the COMET-Funding Program managed by the Austrian Research Promotion Agency (FFG). The computational results presented have been in part achieved using the Vienna Scientific Cluster (VSC). We thank the reviewers and the associate editor for their useful comments.
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Web Appendices, Tables, and Figures referenced in Sects. 2, 3, 5, and 6 as well as reproducible R code for the simulation study are available online with this paper. (ZIP 1533 KB)
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Brockhaus, S., Melcher, M., Leisch, F. et al. Boosting flexible functional regression models with a high number of functional historical effects. Stat Comput 27, 913–926 (2017). https://doi.org/10.1007/s11222-016-9662-1
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DOI: https://doi.org/10.1007/s11222-016-9662-1