This paper proposes a linear regression model for scalar-valued responses and two-way functional (bivariate) predictors. Our motivation stems from the quality evaluation of products based on optical emission spectroscopy data from virtual metrology of semiconductor manufacturing. We focus on multivariate cases where the smoothness and shapes of the data vary significantly across variables. We propose a two-step solution to this problem, consisting of decomposition and prediction. First, we decompose the two-way functional data into pairs of component functions using functional singular value decomposition. Next, we build functional linear models for the decomposed functional variables and obtain the final predictor by averaging the models. Results from numerical studies, including simulation studies and real data analysis, demonstrate the promising empirical properties of the proposed approach, especially when the number of predictors is large.

中文翻译：

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半导体制造中双向函数数据的模型平均预测

本文提出了标量值响应和双向函数（双变量）预测器的线性回归模型。我们的动机源于基于半导体制造虚拟计量的发射光谱数据对产品进行质量评估。我们关注多变量情况，其中数据的平滑度和形状因变量而异。我们提出了这个问题的两步解决方案，包括分解和预测。首先，我们使用函数奇异值分解将双向函数数据分解为分量函数对。接下来，我们为分解的函数变量建立函数线性模型，并通过对模型进行平均来获得最终的预测变量。数值研究（包括模拟研究和真实数据分析）的结果证明了所提出方法的有希望的经验特性，特别是当预测变量数量很大时。