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**The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis.**
*(English)*
Zbl 0945.62093

The spectrum resulting from the Fourier spectral analysis will make little physical sense if the system is not linear and the data are not strictly periodic or stationary. A new method for analysing nonlinear and nonstationary data is developed in this paper. The method is based on the empirical model decomposition (EMD) with which any data set can be decomposed into a finite (and usually small) number of intrinsic mode functions (IMF). The decomposition is based on the direct extraction of the energy associated with various intrinsic time scales.

The IMFs have well-behaved Hilbert transforms that enable to calculate instantaneous frequencies so that any event can be localized both on the time and on the frequency axis (one can see the decomposition as the expansion of the data in terms of IMFs which is linear or nonlinear, complete and almost orthogonal). The local energy and the instantaneous frequency derived from the IMFs through the Hilbert transform can provide a full energy-frequency-time distribution which is designated as the Hilbert spectrum.

Numerous examples demonstrate the new method. These examples are from (1) the numerical results of the classical nonlinear equations systems and (2) data concerning natural phenomena as wave, tide, tsunami, altimeter, earthquake and wind data.

The IMFs have well-behaved Hilbert transforms that enable to calculate instantaneous frequencies so that any event can be localized both on the time and on the frequency axis (one can see the decomposition as the expansion of the data in terms of IMFs which is linear or nonlinear, complete and almost orthogonal). The local energy and the instantaneous frequency derived from the IMFs through the Hilbert transform can provide a full energy-frequency-time distribution which is designated as the Hilbert spectrum.

Numerous examples demonstrate the new method. These examples are from (1) the numerical results of the classical nonlinear equations systems and (2) data concerning natural phenomena as wave, tide, tsunami, altimeter, earthquake and wind data.

Reviewer: T.Cipra (Praha)

### MSC:

62M15 | Inference from stochastic processes and spectral analysis |

46N30 | Applications of functional analysis in probability theory and statistics |

62P12 | Applications of statistics to environmental and related topics |

65P99 | Numerical problems in dynamical systems |