函数型线性判别分析

Linear Discriminant Analysis for Functional Data

摘要: 本文针对输入为函数型数据的分类问题提出了一个函数型线性判别分析方法。通过引入函数范数来度量类内距离和类间距离，从而构造了函数型线性判别分析的优化模型。进一步，通过利用基函数方法将无穷维函数空间优化模型转化为有限维优化模型，从而使模型易于求解。由于数据被函数化后，可对函数求一阶导数或二阶导数。利用求导数后的数据可进一步提高分类效果。最后，数值实验部分展示了函数型线性判别分析方法的可行性和有效性。

Abstract: In this paper, functional linear discriminant analysis method is proposed for the classification problem of input as functional data. By introducing the functional norm to measure the distance within-class and between-class, an optimization model of functional linear discriminant analysis is constructed. Furthermore, by using the basis function method to transform the infinite dimensional function space into a finite dimensional optimization model, then this model is easy to solve. Since the data is functional, the first derivative or the second derivative of the function can be found. The classification result can be further improved by using the data after the derivative. Finally, the numerical experiments show the feasibility and effectiveness of the functional linear discriminant analysis method.