函数型数据分析中机器学习辅助正交基选择
Machine Learning Assisted Orthonormal Basis Selection for Functional Data Analysis
摘要：
在函数数据方法的实现中，正交基的初始选择的影响在过去没有得到太多的关注。通常，考虑几种标准基（例如傅立叶、小波、样条等）来变换观察到的函数数据，并且在没有任何正式标准的情况下做出选择，所述正式标准指示哪种基对于将数据初始变换为函数是优选的。为了解决这个问题，我们提出了一个严格的数据驱动的正交基选择方法。该方法使用最近推出的正交样条基地称为splendium获得有效的正交化的B样条。该算法以机器学习方式从数据中学习，以有效地放置节点。最优性标准是基于平均（每个功能数据点）均方误差，并利用在学习算法和比较研究。后者表明效率，这是特别明显的稀疏的功能数据，并在较小程度上在分析复杂的物理系统的响应。
英文摘要：
In implementations of the functional data methods, the effect of the initial choice of an orthonormal basis has not gained much attention in the past. Typically, several standard bases such as Fourier, wavelets, splines, etc. are considered to transform observed functional data and a choice is made without any formal criteria indicating which of the bases is preferable for the initial transformation of the data into functions. In an attempt to address this issue, we propose a strictly data-driven method of orthogonal basis selection. The method uses recently introduced orthogonal spline bases called the splinets obtained by efficient orthogonalization of the B-splines. The algorithm learns from the data in the machine learning style to efficiently place knots. The optimality criterion is based on the average (per functional data point) mean square error and is utilized both in the learning algorithms and in comparison studies. The latter indicates efficiency that is particularly evident for the sparse functional data and to a lesser degree in analyses of responses to complex physical systems.
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