ABSTRACT

Functional linear regression is a widely used approach to model functional responses with respect to functional inputs. However, classical functional linear regression models can be severely affected by outliers. We therefore introduce a Fisher-consistent robust functional linear regression model that is able to effectively fit data in the presence of outliers. The model is built using robust functional principal component and least squares regression estimators. The performance of the functional linear regression model depends on the number of principal components used. We therefore introduce a consistent robust model selection procedure to choose the number of principal components. Our robust functional linear regression model can be used alongside an outlier detection procedure to effectively identify abnormal functional responses. A simulation study shows our method is able to effectively capture the regression behavior in the presence of outliers, and is able to find the outliers with high accuracy. We demonstrate the usefulness of our method on jet engine sensor data. We identify outliers that would not be found if the functional responses were modeled independently of the functional input, or using nonrobust methods.

摘要

函数线性回归是一种广泛使用的方法，用于对与函数输入相关的函数响应进行建模。然而，经典的函数线性回归模型可能会受到异常值的严重影响。因此，我们引入了一个 Fisher 一致的稳健函数线性回归模型，该模型能够在存在异常值的情况下有效地拟合数据。该模型是使用稳健的函数主成分和最小二乘回归估计量构建的。函数线性回归模型的性能取决于所使用的主成分的数量。因此，我们引入了一致的稳健模型选择程序来选择主成分的数量。我们强大的功能线性回归模型可以与异常值检测程序一起使用，以有效识别异常功能响应。模拟研究表明，我们的方法能够有效地捕捉存在异常值的回归行为，并且能够以高精度找到异常值。我们证明了我们的方法对喷气发动机传感器数据的有用性。如果功能响应是独立于功能输入建模的，或者使用非鲁棒性方法，我们会识别出不会被发现的异常值。