Functional variables are often used as predictors in regression problems. A commonly used parametric approach, called scalar-on-function regression, uses the L2 inner product to map functional predictors into scalar responses. This method can perform poorly when predictor functions contain undesired phase variability, causing phases to have disproportionately large influence on the response variable. One past solution has been to perform phase–amplitude separation (as a pre-processing step) and then use only the amplitudes in the regression model. Here we propose a more integrated approach, termed elastic functional regression model (EFRM), where phase-separation is performed inside the regression model, rather than as a pre-processing step. This approach generalizes the notion of phase in functional data, and is based on the norm-preserving time warping of predictors. Due to its invariance properties, this representation provides robustness to predictor phase variability and results in improved predictions of the response variable over traditional models. We demonstrate this framework using a number of datasets involving gait signals, NMR data, and stock market prices.

中文翻译：

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使用函数形状作为预测变量的回归模型

函数变量通常用作回归问题中的预测变量。一种常用的参数方法称为函数标量回归，它使用 L2 内积将函数预测变量映射为标量响应。当预测函数包含不需要的相位可变性时，此方法的性能可能会很差，从而导致相位对响应变量产生不成比例的大影响。过去的一种解决方案是执行相位-幅度分离（作为预处理步骤），然后仅使用回归模型中的幅度。在这里，我们提出了一种更集成的方法，称为弹性函数回归模型 (EFRM)，其中相分离在回归模型内部执行，而不是作为预处理步骤。这种方法概括了功能数据中的相位概念，并且基于预测变量的规范保留时间扭曲。由于其不变性，这种表示提供了对预测器相位可变性的鲁棒性，并导致对响应变量的预测优于传统模型。我们使用许多涉及步态信号、核磁共振数据和股票市场价格的数据集来演示这个框架。