Data-Driven Time-Frequency Analysis via Nonlinear Optimization

本研究提出一种新的数据驱动时频分析方法来研究非线性和非平稳数据。关键思想是使用非线性优化在尽可能大的字典中寻找信号的最稀疏时频表示。这种方法的动机是物理应用以及从许多科学和工程应用中产生的多尺度数据中提取瞬时频率和趋势的需要。虽然已经引入了几种从多尺度信号中提取瞬时频率的方法，但这些方法受到各种限制，并且没有坚实的数学基础。该研究人员及其同事开发的数据驱动时频分析方法提供了瞬时频率的数学严格定义。该研究者及其同事开发了一种基于l1正则化非线性最小二乘的高效非线性匹配追踪方法来分解信号。该方法可用于提取信号的瞬时频率和趋势等物理意义信息。初步结果表明，该方法可以准确有效地分解大范围的物理信号。将这种方法应用于一些来自地球科学和生物医学应用的真实世界数据，已经产生了一些新的发现。本提案的主要目标之一是对该方法进行严格的收敛研究，并将其应用于解决生物医学和地球科学应用中的一些具有挑战性的现实问题。开发有效的数据分析方法是从海量数据中理解趋势、周期等隐藏模式的重要途径。到目前为止，大多数数据分析方法使用预先确定的基础来处理数据。这些方法大多只能处理线性和平稳数据。为了更好地理解隐藏在数据中的物理机制，人们需要开发有效的方法来处理数据的非平稳性和非线性。这种方法需要使用数据驱动的基础，这种基础可以适应数据，而不是先验地确定。该研究人员及其同事开发的数据驱动时频方法具有坚实的数学基础，并采用了一种新颖的非线性优化技术。将该方法应用于热带海洋9年AMSU数据，发现了一个新的近年度趋势。该方法已应用于分析血压波数据，为心血管疾病患者的诊断提供了一种全新的方法。所提出的方法可以为分析现实世界的数据提供一种全新的方法。这项研究将有助于培养这一新兴研究领域的学生和博士后。在这个项目中开发的知识、技术和工具将通过在开放文献中发表的方式进行传播，并将开发的软件工具作为开源提供。

This investigator proposes to develop a new data-driven time-frequency analysis method to study nonlinear and non-stationary data. The key idea is to look for the sparsest time-frequency representation of a signal over the largest possible dictionary using nonlinear optimization. Such a method is motivated by physical applications and the need to extract instantaneous frequency and trend from multiscale data arising from many scientific and engineering applications. Although several methods have been introduced to extract instantaneous frequency from a multiscale signal, these methods suffer from various limitations and do not have a solid mathematical foundation. The data-driven time-frequency analysis method developed by this investigator and his colleagues provides a mathematically rigorous definition of instantaneous frequency. This investigator and his colleagues have developed an efficient nonlinear matching pursuit method based on L1-regularized nonlinear least squares to decompose the signal. This method can be used to extract physically meaningful information of the signal such as instantaneous frequency and trend. The preliminary results show that this method can decompose a wide range of physical signals accurately and efficiently. Applications of this method to some real world data from geo-science and biomedical applications have led to some new discoveries. One of the main objectives of this proposal is to carry out a rigorous convergence study of this method and apply it to solve some challenging real world problems in biomedical and geo-science applications. Developing effective data analysis methods is an important path to understand some hidden patterns such as trend and cycles from the massive amount of data. So far, most data analysis methods use a predetermined basis to process data. Most of these methods can handle only linear and stationary data. To better understand the physical mechanisms hidden in data, one needs to develop effective methods that can handle the non-stationarity and nonlinearity of the data. Such methods require the use of a data-driven basis that is adaptive to the data instead of being determined a priori. The data-driven time-frequency method developed by this investigator and his colleagues has a solid mathematical foundation and uses a novel nonlinear optimization technique. Application of this method to the 9 year AMSU data over tropical oceans has led to the discovery of a new near-annual trend. This method has been applied to analyze blood pressure wave data, leading to a completely new way of diagnosing patients with cardiovascular diseases. The proposed method could provide a completely new way to analyze real world data. The proposed research will help train students and postdocs in this emerging research area. The knowledge, techniques and tools developed in this project will be disseminated through publishing in the open literature, and making available as open-source the software tools that are developed.