多尺度数据分析中的若干优化问题

In this project we plan to discuss optimization methods that help us to approximately find the sparse representation of multiscale data. The discussion is based on the empirical mode decomposition algorithm, which has been widely used in many scientific fields. The main problems we want to solve are: further dicussion on definition of intrinsic mode functions, including the analysis on properties of instantaneous frequency and envelope function; further discussion on and improvement of matching pursuit algorithm, which is frequently used in signal processing; uniqueness and stability of mode decomposition of signals; how to determine and construct a shape function, and how to decompose signals including several modes with different shape functions. We aim to discover new methods of representation and analysis in multiscale data. The results in this field will quite valuable in both theories and applications, and are expected to be wildly used in many important areas such as space industy, national defence, telecommunication, medical science, meteorology, and physical geography, etc.

本项目拟在数据分析领域中采用最优化方法探讨多尺度数据的稀疏表示的近似算法，讨论以当前广泛应用的经验模式分解算法为基础展开系列理论分析与算法研究。主要内容包括：固有模态函数内涵的深入剖析、其瞬时频率和振幅函数性质的分析；对信号处理中常用的匹配追踪法的进一步探讨与改进；信号分解为模态函数形式的唯一性与稳定性；波形函数的新的判别与构造算法及多个不同波形函数模式的信号的分解等。旨在多尺度数据分析中开拓新的表示和分析方法。该课题的研究成果具有重要的理论与应用价值，在航天、国防、电讯、医学、气象、地球物理等国民经济部门有广泛的应用前景。

本项目围绕多尺度数据的稀疏表示进行研究，取得以下结果：.1. 通过振幅函数和瞬时频率对模态函数的尺度分离性质进行了定义，并得到模态函数的一些基本性质；.2. 证明了在尺度分离及各模态函数的频率存在间隙条件下，多尺度信号的稀疏表示在误差允许的范围内是唯一的；.3. 证明了匹配追踪算法提取模态函数的有效性，有效性结论的条件及不满足该条件时出现模态混淆的例子也显示了双层算法的必要性；.4. 在匹配追踪算法的基础上提出了多尺度信号的双层算法以避免模态混淆现象、增加算法稳定性，并开发出该算法的完整matlab软件包；.5. 提出了减弱末端效应的延拓-优化算法。