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Astron. Astrophys. 363, 311-315 (2000)

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2. Wavelet entropy

Fourier analysis is an adequate tool for detecting and quantifying constant periodic fluctuations in time series. For intermittent and transient multiscale phenomena, the wavelet transform is able to detect time evolutions of the frequency distribution. The continuous wavelet transform represents an optimal localized decomposition of time series, [FORMULA], as a function of both time t and frequency (scale) a, from a convolution integral:

[EQUATION]

where [FORMULA] is called an analysing wavelet if it verifies the following admissibility condition:

[EQUATION]

where:

[EQUATION]

is the related Fourier transform. In the definition, a and [FORMULA] denote the dilation (scale factor) and translation (time shift parameter), respectively. We define the local wavelet spectrum:

[EQUATION]

where [FORMULA] denotes the peak frequency of the analysing wavelet [FORMULA]. From the local wavelet spectrum we can derive a mean or global wavelet spectrum, [FORMULA]:

[EQUATION]

The relationship between the ordinary Fourier spectrum [FORMULA] and the mean wavelet spectrum [FORMULA] is given by:

[EQUATION]

indicating that the mean wavelet spectrum is the average of the Fourier spectrum weighted by the square of the Fourier transform of the analysing wavelet [FORMULA] shifted at frequency k. Here we used the family of complex analyzing wavelets consisting of a plane wave modulated by a Gaussian, called Morlet wavelet, (Torrence & Compo 1998):

[EQUATION]

where [FORMULA] is the non dimensional frequency here taken to be equal to 6 in order to satisfy the admissibility condition, Eq. (2). Following Quian Quiroga et al. (1999), we define a wavelet entropy as a function of time:

[EQUATION]

where:

[EQUATION]

is the energy probability distribution for each scale level. From the definition, follows that an ordered activity corresponds to a narrow frequency distribution of energy, with low wavelet entropy, and a random activity corresponds to a broad frequency distribution, with high wavelet entropy. Of course, higher values for wavelet entropy means higher dynamical complexity, higher irregular behaviour, lower predictability. The application of the wavelet entropy is optimal for non-stationary signals.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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