Data-Driven Time-Frequency Analysis via Nonlinear Optimization

California Institute Of Technology, Pasadena CA

Investigators

Abstract

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.