![]() (2015) for the argument of the response function with an eigenbasis is a key step to creating a computationally efficient algorithm. The replacement of the spline basis used by Scheipl et al. First, we combine B-spline bases for the covariate function, X( s), and for its argument, s, ( Marx and Eilers, 2005 Wood, 2006 McLean et al., 2014) with a functional principal component basis for the argument, t, of the response function see (2) below. There are three major contributions in this paper. The methodology is applicable for realistic scenarios involving densely and/or sparsely observed functional responses and predictors, as well as various residual dependence structures. We develop a novel estimation procedure that is an order of magnitude faster than the existing algorithm and discuss inference for the predicted response curves. (2015), but the present paper is the first to investigate them fully. (2013).Īdditive function-on-function regression models where the current mean response depends on the time point itself as well as the full covariate trajectory were introduced by Scheipl et al. Additive and generalized additive models for a scalar response and functional predictors were introduced by McLean et al. Additive models allow nonparametric modeling of the relationship between the response and the predictors while avoiding the so-called curse of dimensionality and being easily interpreted. Their model replaces the linear model Y i = β 0 + β 1 X i ,1 + ⋯ + β p X i, p + ε i, where each of Y i, X i ,1, …, X i, p, i = 1, …, n, is scalar by, Y i = β 0 + f 1( X i ,1) + ⋯ f p( X i, p) + ε i. This paper considers flexible nonlinear regression models that can capture complex relationships between the response and the full covariate trajectory more directly.Īdditive models have enjoyed great popularity since they were introduced by Friedman and Stuetzle (1981) for a scalar response and scalar predictors. A limitation of this approach is that the estimated effects are not easily interpretable. ![]() The linearity assumption was extended to the functional additive model (FAM) of Müller and Yao (2008), which models the effect of the covariate by a sum of smooth functions of the functional principal component scores of the covariate. The functional linear model ( Ramsay and Silverman, 2005 Yao et al., 2005b Wu et al., 2010) assumes that the relationship is linear: the effect of the full covariate trajectory is modeled through a weighted integral using an unknown bivariate coefficient function as the weights. We consider functional regression models that relate the current response to the full trajectory of the covariate. When the current response depends on the past values of the covariate/s, the historical functional linear model ( Malfait and Ramsay, 2003) is more appropriate. One of the commonly known models is the functional concurrent model where the current response relates to the current values of the covariate/s see for example, Ramsay and Silverman (2005) Sentürk and Nguyen (2011) Kim et al. These models are often called function-on-function regression. Regression models where both the response and the covariate are curves have become common in many scientific fields such as medicine, finance, and agriculture.
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