Functional principal component regression and functional partial least squares
Functional principal component regression and functional partial least squares
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DOI：
10.1198/016214507000000527
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发表时间：
2007-09-01
影响因子：
3.7
通讯作者：
Ogden, R. Todd
中科院分区：
文献类型：
--
作者：
Reiss, Philip T.;Ogden, R. Todd
当信号的维度远远超过其数量时，如化学样品的近红外（NIR）光谱，信号预测器的标量响应的回归呈现出重大挑战。这个问题的大多数解决方案都通过对分量进行回归来降低预测因子的维数[例如，主成分回归（PCR）和偏最小二乘（PLS）或通过平滑方法，其将系数函数限制到样条基的跨度。本文介绍了PCR和PLS的功能版本，它们结合了前面两种降维方法的联合收割机。开发了两个版本的功能PCR，都使用B样条和粗糙度处罚。正则化分量版本将这样的惩罚应用于主分量的构造（即，它使用函数主成分），而正则化回归版本在回归中包含惩罚。对于后一种形式的功能PCR，惩罚参数可以通过广义交叉验证、限制最大似然（REML）或最小均方积分误差准则来选择。同样，我们开发了两个版本的功能PLS。证明了正则化回归函数PCR的渐近收敛性质。模拟研究和分裂样本验证与几个近红外光谱数据集表明，功能PCR和功能PLS，特别是正则化回归版本与REML，提供了优于现有方法的系数函数的估计和预测未来的观测。
Regression of a scalar response on signal predictors, such as near-infrared (NIR) spectra of chemical samples, presents a major challenge when, as is typically the case, the dimension of the signals far exceeds their number. Most solutions to this problem reduce the dimension of the predictors either by regressing on components [e.g., principal component regression (PCR) and partial least squares (PLS)l or by smoothing methods, which restrict the coefficient function to the span of a spline basis. This article introduces functional versions of PCR and PLS, which combine both of the foregoing dimension-reduction approaches. Two versions of functional PCR are developed, both using B-splines and roughness penalties. The regularized-components version applies such a penalty to the construction of the principal components (i.e., it uses functional principal components), whereas the regularized-regression version incorporates a penalty in the regression. For the latter form of functional PCR, the penalty parameter may be selected by generalized cross-validation, restricted maximum likelihood (REML), or a minimum mean integrated squared error criterion. Proceeding similarly, we develop two versions of functional PLS. Asymptotic convergence properties of regularized-regression functional PCR are demonstrated. A simulation study and split-sample validation with several NIR spectroscopy data sets indicate that functional PCR and functional PLS, especially the regularized-regression versions with REML, offer advantages over existing methods in terms of both estimation of the coefficient function and prediction of future observations.