11 Conclusions

The ampligram may be considered as an analogy to signal decomposition into Fourier components. In that case different components correspond to different frequencies. In the present case different components correspond to different wavelet coefficient magnitudes, being equivalent to spectral densities. A useful property of the ampligram is that when integrated along the coefficient magnitude axis, the original signal should be obtained. However, if the original signal spans over a large range of amplitudes, the energy leakage into the ampligram band will occur and the integral of the ampligram will show values larger than those of the original signal.

The time scale spectrum of the ampligram tells us more than the original wavelet spectrum does. The time scale spectrum reveals individual signal components and indicates the statistical properties of each component: deterministic or stochastic. The ampligram and its time scale spectrum seems to be a useful tool to study processes resulting in a mixture of stochastic and deterministic components. The most important point of the method is that it may be used to remove the influence of the Poisson statistics in the photon data and in this way to extract real deterministic luminosity variations. As it is shown by simulations in Sects. 9 and 10 of this work, the method is capable to extract weak, deterministic variations embedded in a totally Poisson-like series of events. The entropy of amplitude distributions at the individual levels of wavelet coefficient magnitude was found to be a useful measure of the occurrence of deterministic components in an ampligram. In the case of X-ray luminosity variations in the AGN it is expected that the described data analysis technique will provide a conclusive proof of the existence of building blocks. The efficient decomposition of the luminosity variation data may be used to study the deterministic, quasi-periodic phenomena, like tones and chirps.

Acknowledgements

The authors are indebted to A.W. Wernik of Space Research Center, Warsaw, Poland, for support in the area of non-linear filtering using wavelets and to A.G. Pacholczyk and W.R. Stoeger of Steward Observatory, University of Arizona for continuing discussions in the course of this work and for support in matters concerning the AGNs. The software used in this work has been developed by Pär-Ola Nilsson, Jan Karlsson and Fredrik Rutqvist of the Umeå Division of the Swedish Institute of Space Physics. The data used to demonstrate the present analysis technique are ROSAT PSPC photon event history files obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center.