Two approaches to extraction of information from time series.
Time series analysis (signal processing) is a large applied discipline which includes many techniques
and approaches. They can be systematized in different ways: linear and nonlinear, parametric and
non-parametric, based on construction of a mathematical model and "direct" (dispensing with modeling).
As for the latter systematization, one can consider the following methods as "direct":
1) Traditional statistical analysis which includes calculations of mean values, variances,
correlation function, probability distribution function, etc;
2) Fourier-analysis and wavelet analysis belong to a wide family of the so-called "linear"
methods since they give complete characterization of linear processes;
3) Nonlinear methods include reconstruction of phase orbit from time series, estimation
of fractal dimensions, entropies, Lyapunov exponents, etc.
Methods based on model construction historically and logically can be divided into to big groups:
1) Construction of linear stochastic models (the most popular among them are autoregressive - moving average models).
This field has been called "system identification".
2) Construction of nonlinear dynamical models (as a rule, maps or ordinary differential equations). It relies on
ideas and techniques of nonlinear dynamics. Therefore it is called "reconstruction of dynamical systems".
- Box G., Jenkins G. Time series analysis. Forecast and control. Golden-Day, San-Francisco, 1970.
- Ljung L. System identification. Theory for the user. Moscow, 1991. 432 p.
- Kendall J, Stewart A. Statistical inferences and relationships. Moscow, 1973. Multidimensional statistical analysis and time series. Мoscow, 1976.
- Jenkins G., Watts D. Spectral analysis and its applications. Moscow, 1978. 316 p.
- Rabiner L.R., Gold B. Theory and applications of digital signal processing. Moscow, 1978. 495 p.
- Astafieva N.M. Wavelet-analysis: basics of theory and examples of applications // Sov. Uspekhi fizicheskih nauk. 1996. V. 166, No. 11. P. 1145-1170.
- Koronovskii A.A., Hramov A.Ye. Continuous wavelet-analysis in applications to nonlinear dynamics problems. Saratov, "College", 2003. 216 p.
- Kantz H., Schreiber T. Nonlinear time series analysis. Cambridge University Press, Cambridge, 1997.
- Chaos and Its Reconstructions // Eds. G. Gouesbet, S. Meunier-Guttin-Cluzel, O. Menard. Nova Science Publishers, New York, 2003.
- Malinetskiy G.G. Chaos. Structures. Numerical experiment (introduction to nonlinear dynamics). Moscow, Editorial URSS, 2000. 256 p.
- Anishchenko V.S., Astakhov V.V., Vadivasova T.Ye., et al. Nonlinear effects in chaotic and stochastic systems. Moscow-Izhevsk, ICR, 2003.
- Bezruchko B.P., Smirnov D.A. Mathematical modeling and chaotic time series. Saratov, "College", 2005. 320 p.