Functional linear regression that's interpretable

By Gareth M. James, Jing Wang, Ji Zhu. 2009

270 citations recorded by the source corpus.

Regression models to relate a scalar $Y$ to a functional predictor $X(t)$ are becoming increasingly common. Work in this area has concentrated on estimating a coefficient function, $\beta(t)$, with $Y$ related to $X(t)$ through $\int\beta(t)X(t) dt$. Regions where $\beta(t)\ne0$ correspond to places where there is a relationship between $X(t)$ and $Y$. Alternatively, points where $\beta(t)=0$ indicate no relationship. Hence, for interpretation purposes, it is desirable for a regression procedure to be capable of producing estimates of $\beta(t)$ that are exactly zero over regions with no apparent relationship and have simple structures over the remaining regions. Unfortunately, most fitting procedures result in an estimate for $\beta(t)$ that is rarely exactly zero and has unnatural wiggles making the curve hard to interpret. In this article we introduce a new approach which uses variable selection ideas, applied to various derivatives of $\beta(t)$, to produce estimates that are both interpretable, flexible and accurate. We call our method "Functional Linear Regression That's Interpretable" (FLiRTI) and demonstrate it on simulated and real-world data sets. In addition, non-asymptotic theoretical bounds on the estimation error are presented. The bounds provide strong theoretical motivation for our approach.

Subfields: Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/660dae476f0136edfa36f227847ff5c46e2804f8

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Citations in the corpus

The functional linear array model (2015, background): “…as the scalar product of the functional predictor and a smooth coefficient function, in the context of linear models (e.g., Reiss and Ogden, 2007; James et al., 2009), generalized linear models (e.g., Marx and Eilers, 1999; Müller and Stadtmüller, 2005; Wood, 2011; Gertheiss et al., 2013) or…”
Locally Sparse Estimator for Functional Linear Regression Models (2017, methodology): “Therefore, the proposed SLoS estimator overcomes the shortcoming of FLiRTI (James, Wang, and Zhu 2009).”
Penalized Functional Regression (2011, unknown): “…Müller and Stadtmüller (2005) derived confidence intervals for a low-dimension approach to functional regression and others (Reiss and Ogden 2007; James, Wang, and Zhu 2009) discussed empirical or bootstrap confidence intervals, we are unaware of existing explicit derivations of confidence…”
Selection of time instants and intervals with Support Vector Regression for multivariate functional data (2020, methodology): “For example, the algorithm proposed in James et al. (2009) focused on the interpretability by selecting time instants.”
Truncated linear models for functional data (2014, methodology): “…(2013), introduced methods based on smoothing splines; Baı́llo (2009) suggested kernel techniques and made comparisons with parametric approaches; James et al. (2009) developed variable selection ideas; Mas and Pumo proposed an alternative formulation of the functional linear model; He et al.…”
Using basis expansions for estimating functional PLS regression Applications with chemometric data (2010, background): “The case of an scalar response variable and a functional predictor has been also studied in [3,18].”
A generalized Dantzig selector with shrinkage tuning (2009, methodology): “More recently James et al. (2008a) developed the DASSO algorithm, a generalization of least angle regression, which can fit the coefficient paths for both the Lasso and the Dantzig selector.”