Data-Driven Time-Frequency Analysis via Nonlinear Optimization

这位研究者建议发展一种新的数据驱动的时频分析方法来研究非线性和非平稳数据。其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的时频表示。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。虽然已经提出了几种从多尺度信号中提取瞬时频率的方法，但这些方法都受到各种限制，并且没有坚实的数学基础。这位研究人员和他的同事开发的数据驱动的时频分析方法为瞬时频率提供了数学上的严格定义。这位研究人员和他的同事们提出了一种基于L1正则化非线性最小二乘的有效的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、趋势等有物理意义的信息。初步结果表明，该方法能够准确、高效地对大范围的物理信号进行分解。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，导致了一些新的发现。该方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了解趋势、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都是使用预定的基础来处理数据。这些方法中的大多数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，这种基础是适应数据的，而不是事先确定的。这位研究人员和他的同事们开发的数据驱动时频方法具有坚实的数学基础，并使用了一种新的非线性优化技术。将该方法应用于热带海洋AMSU的9年资料，发现了一种新的近年度趋势。该方法已被应用于血压波动数据的分析，为心血管疾病患者的诊断开辟了一条全新的途径。提出的方法可以为分析真实世界的数据提供一种全新的方法。这项拟议的研究将有助于培训这一新兴研究领域的学生和博士后。在该项目中开发的知识、技术和工具将通过在开放文献中出版并将所开发的软件工具作为开放源码提供来传播。

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.