3 Time-series decomposition using wavelet transform

Many time series observed in physics consist of a deterministic part with a superimposed stochastic component. A powerful technique to separate both components has been proposed by Farge & Philipovitch Farge93 and implemented in a practically usable software by Wernik & Grzesiak Wernik97. In that method, being a kind of non-linear filtering, called also the threshold filtering, a wavelet frequency spectrum of the time series is calculated. The time series is decomposed into two parts in the following way:

• A deterministic "strong'' part is obtained by setting to zero all wavelet coefficients less than a certain threshold level. The inverse wavelet transform is used to calculate the corresponding time series.
• A stochastic "weak'' part is obtained by setting to zero all wavelet coefficients greater than that threshold level. The inverse wavelet transform is also used here to calculate the corresponding time series.
• New wavelet spectra are calculated for each partial time series.

Signal discrimination using the magnitude of wavelet coefficients as a discrimination criterion would correspond to discrimination with respect to the spectral density when using the Fourier transform. The stochastic part must follow a Gaussian probability distribution function. As a measure of departure from a Gaussian distribution the kurtosis is used. If the threshold is properly selected, the integral of the kurtosis of the stochastic part over the entire frequency range reaches a minimum.

In the present problem the method will be applied in the opposite manner. In the case of a photon train, reaching the measuring instrument at a low rate, there will be a dominating Poisson statistics modulated with a weak deterministic component. A low threshold will then be used to separate a weak, deterministic component from a strong, Poisson component.