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IMA Hot Topics Workshop
Instantaneous Frequencies and Trends for Nonstationary Nonlinear Data
September 7-9, 2011

Ingrid DaubechiesDuke University
Jianqing FanPrinceton University
Thomas HouCalifornia Institute of Technology
Norden HuangNational Central University

Data are one of the most important links that we have with the physical world. Developing effective data analysis methods is an important path through which we can understand the underlying processes of natural phenomena. In real-world experimental and theoretical studies, we often deal with signals that have ever-changing frequency. Chirp signal is one class of the signals used by bats as well as in radar. The frequency content in speech, though not exactly a chirp, is also ever changing, and many of the consonants are produced through highly nonlinear mechanisms such as explosion or friction. Furthermore, for any nonlinear system, the frequency is modulating not only among different oscillation periods, but also within one period such as in the nonlinear pendulum problem. The traditional concept of frequency based on Fourier analysis is not applicable here. It is essential that we develop efficient adaptive data analysis methods that can extract hidden physical information in these data and define the Instantaneous Frequency in a rigorous way. "Trend" and "detrending" are frequently encountered in data analysis. In many applications, such as climatic data analysis, the trend is one of the most critical quantities sought. In other applications, such as in computing the correlation function and in spectral analysis, it is necessary to remove the trend from the data, known as detrending, lest the result be overwhelmed by the non-zero mean and large tilting term; therefore, detrending is often a necessary step before meaningful spectral results can be obtained. As a result, trend and detrending are both of great interest and importance in data analysis. So far, there has been a lack of proper definition for the trend in nonlinear non-stationary data. A definitive and quantitative study on trend and detrending is clearly needed. In this workshop, we will explore the issues involved with trend determination and instantaneous frequency. There has been some recent exciting progress in developing new mathematical theory and effective computational algorithms to define trend and instantaneous frequency. These efforts involve a number of mathematical tools, including nonlinear variational methods, optimization, sparse representation of data, compressed sensing, TVD-based denoising methods, multiscale analysis, randomized algorithms, and statistical methods. This workshop will bring together experts from these areas to exchange ideas and identify new research opportunities for this emerging research area. One of the main objectives of the workshop is to promote research that leads to the discovery and understanding of the underlying processes in order to provide a base for building predictive models. An extension of the trend study is the problem of regression, which is also of great interests to a broad research community, including the econometrics/finance community.