带有相位变差函数型数据的若干主成分分析及应用

函数主成分分析（FPCA）是一种数据驱动的、最常用的有效降维方法，被广泛的应用于生物、经济以及计算机等领域。现有的绝大部分FPCA方法有如下问题：一、没有考虑到相位变差对FPCA的影响。相位变差广泛存在于函数型数据中，通常认为数据注册就可以将其消除。但是由于注册的目标通常和FPCA的目标是不一致的，如果先进行注册就很可能扭曲了函数主成分。二、估计出来的主成分函数缺乏解释性。主要由于估计出来的主成分函数在整个观测区间内几乎处处非零。如果只在局部区域非零，即具有稀疏性局部特征，那么其解释性将大大提升。本项目将首先提出相位变差和FPCA联合分析的框架，避免了两步目标不一致的问题。其次，在该框架下进一步提出稀疏FPCA，提高其解释性。本项目将提出实际可行的优化算法以及证明算法收敛性和估计量的渐近性质。研究结果将应用于分析心电图等数据。

Functional principal component analysis (FPCA) is a data-driven and most commonly used effective dimensionality reduction method, which is widely used in, such as, biology, economy, and computer fields. The most existing FPCA methods have the following problems: 1. The influence of phase variation on FPCA is not fully considered. Phase variation is widely existing in functional data, and it is generally considered that data registration (registration) can eliminate it. However, since the goals of registration are usually inconsistent with the goals of FPCA.If register first, the principal components of the function will be distorted. 2. The estimated principal component function lacks explanatory nature. Mainly because the estimated principal component function is almost non-zero in the entire observation interval. If it is non-zero only in a local area, that is, it has sparse local features, then its explanatory will be greatly improved. This project will first propose a framework for joint analysis of phase variation and FPCA to avoid the problem of inconsistent goals between the two steps. Secondly, sparse FPCA is further proposed in this framework to improve its explanatory nature. This project will propose practical optimization algorithms and prove the convergence of the algorithm and the asymptotic properties of the estimator. The research results will be used to analyze ECG data.