An Adaptive Approach for Nonlinear and Nonstationary Data Analysis

Analyzing data from real world is a challenge; we have to face the limitations imposed by reality: nonstationarity, nonlinearity. The traditional methods cannot fully accommodate these restrictions. To alleviate this difficulty, various assumptions and approximations have often been invoked to process and analyze data. Unfortunately, methods based on the most common assumptions, linearity and stationarity, are not as effective as is expected in revealing useful information and the key information of data remains concealed most of the time. In this chapter, we will focus on adaptive methods that can overcome many limitations of traditional methods. As will be illustrated, many of the difficulties could be traced back to the lack of logical definition of frequency, a critical physical quantity. In fact, once the frequency is properly defined and extracted from the data, many difficult tasks, such as quantification of degree of nonlinearity and nonstationarity and determination of the trend, can be carried out with ease. It is also showed that the accurate determination of the frequency has to depend on the adaptive method and the Hilbert-Huang Transform (HHT) can serve this purpose HHT, consists of Empirical Mode Decomposition (EMD) and the Hilbert Spectral Analysis (HSA), is a new adaptive method that can provide time-frequency-energy expression of data without using a priori basis, such as in wavelet. Related most recent developments, such as the Nonlinear Matching Pursuit method, Ensemble Empirical Mode Decomposition (EEMD), Instantaneous Frequency computations, Trend determination, Time-dependent Intrinsic Correlation (TIDC), density representation of Hilbert Spectrum, and the extension of the time series analysis method to multi-dimensional data, will also be discussed. These latter advances have made the HHT method much more robust and mature, and many applications of HHT are progress and had produced viable results.