The field of nonlinear signal characterization and nonlinear signal processing has attracted a growing number of researchers in the past three decades. This comes from the fact that linear techniques have some limitations in certain areas of signal processing. Numerous nonlinear techniques have been introduced to complement the classical linear methods and as an alternative when the assumption of linearity is inappropriate. Two of these techniques are higher order statistics (HOS) and nonlinear dynamics theory (chaos). They have been widely applied to time series characterization and analysis in several fields, especially in biomedical signals. Both HOS and chaos techniques have had a similar evolution. They were first studied around 1900: the method of moments (related to HOS) was developed by Pearson and in 1890 Henri Poincaré found sensitive dependence on initial conditions (a symptom of chaos) in a particular case of the three-body problem. Both approaches were replaced by linear techniques until around 1960, when Lorenz rediscovered by coincidence a chaotic system while he was studying the behaviour of air masses. Meanwhile, a group of statisticians at the University of California began to explore the use of HOS techniques again. However, these techniques were ignored until 1980 when Mendel (Mendel, 1991) developed system identification techniques based on HOS and Ruelle (Ruelle, 1979), Packard (Packard, 1980), Takens (Takens, 1981) and Casdagli (Casdagli, 1989) set the methods to model nonlinear time series through chaos theory. But it is only recently that the application of HOS and chaos in time series has been feasible thanks to higher computation capacity of computers and Digital Signal Processing (DSP) technology. The present article presents the state of the art of two nonlinear techniques applied to time series analysis: higher order statistics and chaos theory. Some measurements based on HOS and chaos techniques will be described and the way in which these measurements characterize different behaviours of a signal will be analized. The application of nonlinear measurements permits more realistic characterization of signals and therefore it is an advance in automatic systems development.
In digital signal processing, estimators are used in order to characterize signals and systems. These estimators are usually obtained using linear techniques. Their mathematical simplicity and the existence of a unifying linear systems theory made their computation easy. Furthermore, linear processing techniques offer satisfactory performance for a variety of applications.
However, linear models and techniques cannot solve issues such as nonlinearities due to noise, to the production system of the signal, system nonlinearities in digital signal acquisition, transmission and perception, nonlinearities introduced by the processing method and nonlinear dynamics behaviour. Therefore, the application of linear processing techniques leads to less realistic characterization of certain systems and signals. As a result of the shortcomings of linear techniques, analysis procedures are being revised and nonlinear techniques are being applied in computing estimators and models and in signal characterization to increase the possibilities of digital signal processing.
HOS is a field of statistical signal processing which has become very popular in the last 25 years. To date almost all digital signal processing have been based on second order statistics (autocorrelation function, power spectrum). HOS use extra information which can be used to get better estimates of noisy situation and nonlinearities.
Chaos theory (nonlinear dynamical theory) is a long-term unpredictable behaviour in a nonlinear dynamic system caused by sensitive on initial conditions. Therefore, irregularities in a signal can be produced not only by random external input but also by chaotic behaviour.
Both nonlinear techniques have been used in signals characterization and numerous automatic classification systems have been developed using HOS and chaos features in many fields. Texture classification (Coroyer, Declercq, Duvaut, 1997), seismic event prediction (Van Zyl, 2001), fault diagnosis in machine condition monitoring through vibration signals (Samanta, Al.Balushi. & Al-Araimi, 2006), (Wang & Lin, 2003) and economy (Hommes & Manzan, 2006) are some examples.
Key Terms in this Chapter
Polyespectra: The Fourier transform of cumulants. The second order polyspectra is the power spectrum. Most HOS work on polyspectra focusses attention on the bispectrum and the trispectrum.
Lyapunov Exponents: Quantity that characterizes the rate of separation of infinitesimally close trajectories in a dynamical system. The maximal Lyapunov exponent (MLE) determines the predictability of a dynamical system. A positive MLE means a chaotic system.
Cumulants: The kth order cumulant is a function of the moments of orders up to and including k.
Reconstructed Phase Space: Phase space obtained from a time series through embedding techniques such as principal component analysis or the method of delays.
Kolmogorov-Sinai Entropy: Measurement of information loss per unit of time in phase space.
Bicoherence: It is a normalised version of the bispectrum. The bicoherence takes values bounded between 0 and 1, which make it a convenient measure for quantifying the phase coupling in a signal.
Attractor: A region in the phase space to which all trajectories converge after a transition time. It is the long term behaviour of a dynamical system.
HOS: Higher order statistics is a field of statistical signal processing that uses more information than autocorrelation functions and spectrum. It uses moments, cumulants and polyspectra. They can be used to get better estimates of parameters in noisy situations, or to detect nonlinearities in the signals.
Chaos: Long-term unpredictable behaviour caused by sensitive dependence on initial conditions.