Variable selection in multivariate linear models for functional data via sparse regularization
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Variable selection in multivariate linear models for functional data via sparse regularization Hidetoshi Matsui1 · Yuta Umezu2 Received: 2 October 2018 / Accepted: 8 July 2019 © Japanese Federation of Statistical Science Associations 2019
Abstract We consider the use of sparse regularization for the problem of variable selection in multivariate linear models where the predictors are given as functions and the responses are scalars. Observations corresponding to the predictors are assumed to be measured repeatedly at discrete time points, and, therefore, they are treated as smooth functional data. Parameters included in the functional multivariate linear model are estimated by the penalized least squares method with an 𝓁1 ∕𝓁2-type penalty. We construct a blockwise descent algorithm for deriving the estimates of the functional multivariate linear model. We also provide a model selection criterion for evaluating the model. To investigate the effectiveness of the proposed method, we apply it to the analysis of simulated data and real data. Keywords Functional data analysis · Lasso · Multivariate regression · Variable selection
1 Introduction Functional data analysis (FDA) has received considerable attention in different fields of application, including bioscience, system engineering, and meteorology, and a number of applications have been reported (see, e.g., Ramsay and Silverman 2005; Kokoszka and Reimherr 2017). The basic idea behind functional data analysis is to express data observed longitudinally as a smooth function, and then draw information from the collection of functional data. Functional regression analysis, which models the relationship between predictors and responses, either This work was supported by JSPS KAKENHI Grand No. 16K16020. * Hidetoshi Matsui [email protected]‑u.ac.jp 1
Faculty of Data Science, Shiga University, 1‑1‑1 Banba, Hikone, Shiga, Japan
2
Department of Computer Science, Nagoya Institute of Technology, Gokiso‑cho, Showa‑ku, Nagoya, Aichi, Japan
13
Vol.:(0123456789)
Japanese Journal of Statistics and Data Science
or both are given as functions, has been widely studied. In many cases, this has been applied to models with functional predictors and scalar responses (James 2002; Cardot et al. 2003; Rossi et al. 2005; Müller and Yao 2008 and references therein); on the other hand, models with functional predictors and functional responses were considered in Malfait and Ramsay (2003), Harezlak et al. (2007), Scheipl et al. (2015) and Matsui (2019). In this paper, we consider the use of sparse regularization in the construction of a functional regression model with functional predictors and multiple scalar responses. Recently, sparse regularization techniques have been introduced to the construction of functional regression models. Sparse regularization can provide estimates in which some of the values are exactly zero, and this can simplify variable selection problems. Details of sparse regularization are presented in Bühlmann and van de Geer (2011) and Hastie et al. (201
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