1 Introduction

There is some evidence that X-ray photons from astronomical sources can not be fully described by a Poisson process. There seems be a deterministic modulation of the photon series which is reflected in observed wavelet spectra of photon counts. The following experiment may be performed using the photon event data in order to test the above statement: Let us select 5 consecutive photon events observed at instants: $t_{\rm {1}}$, $t_{\rm {2}}$, $t_{\rm {3}}$, $t_{\rm {4}}$ and $t_{\rm {5}}$. Let assume that the occurrence of a photon at $t_{\rm {3}}$ is conditioned by occurrences at preceding instants $t_{\rm {1}}$, $t_{\rm {2}}$ and following instants $t_{\rm {4}}$ and $t_{\rm {5}}$. Using a large population of sets of 5 consecutive photons it is possible to create a statistical model of the photon train using, for example, the neural network technique. That technique, being a kind of a non-linear interpolation, has been used to reconstruct uniformly sampled data, even when many data points were missing in an observed chaotic process [Liszka1996].

If the photon data would follow a true Poisson process, it would be impossible to create such a model. However, it has been found, that even a simplest model consisting of a single back-propagation network, could be trained with rms errors usually less than 30%. This means that the conditional probability of a photon event at $t_{\rm {3}}$, conditioned by photon events at $t_{\rm {1}}$, $t_{\rm {2}}$, $t_{\rm {4}}$ and $t_{\rm {5}}$:

$$P\left(t_{\mathrm{3}} \vert t_{\mathrm{1}}, t_{\mathrm{2}}, t_{\mathrm{4}}, t_{\mathrm{5}} \right) > 0.$$ (1)

The above fact is an indication that the photon events do not follow a Poisson process. In a true Poisson process it would be impossible to predict the occurrence of an event at $t_{\rm {3}}$. Two examples of prediction of photon events at $t_{\rm {3}}$ for photon data from NGC 4051 observed by the ROSAT satellite are shown in Fig. 1. A single back-propagation neural network with 9 processing elements in the hidden layer has been used to construct a model of the photon series. Using a more complex model of hybrid type [Liszka1996] it would be possible to obtain even better prediction accuracy (cf. Fig. 1).

Figure 1: Two examples of prediction of photon events at t3 for photon data from NGC 4051 observed by the ROSAT satellite. t3 is expressed in seconds counted from t1

Another proof for deterministic variations of the photon series is the fact that if the image is divided into two or four equal parts, variations of photon counts in individual parts are correlated. An example of photon-count variations in NGC 5548 recorded by the ROSAT satellite (ROR 701242) in two halves of the image is shown in Fig. 2.

Figure 2: An example of photon-count variations in NGC 5548 (ROR 701242) in two halves of the image. Photon energies < 0.5 keV, 2 second sampling bins

However, it would also be the case if variations would be imposed by the measuring instrument. In such case the short time variations of different sources in the same image would be correlated. This type of correlation has not been found, see an example in Fig. 3.

Figure 3: An example of photon-count variations in NGC 5548 (ROR 701242) (upper graph) together with simultaneous variations of another source in the same image

The conclusion of the above experiments is that the short-term variations of photon counts recorded by the satellite contain deterministic information, most likely corresponding to intensity variations of the source itself.

In the case of X-rays from AGN there is probably a physical source of deterministic variations of the photon flux. It seems that there are individual physical luminosity producing events in the AGN source itself which together constitute the X-ray light curve. There are two proposals for the nature of these elementary events. Pacholczyk & Stoeger Pacholczyk94 propose "building blocks'' in the X-ray photon flux from active galactic nuclei resulting from ballistic events due to smaller black holes passing through the accretion disk of the largest black hole in the cluster. Another proposal about the nature of these events is that they may be magnetohydrodynamic flares in the accretion disk around a single supermassive black hole [De Vries & Kuijpers1992].

In the present study we discuss methods to extract the deterministic component from the photon event histories. These methods are described in the remainder of the paper. It will be shown that wavelet spectra seem to be a useful tool to study the short-term temporal variations in the photon counting rate, even in cases of very low counting rates. Here we shall employ wavelet transform methods together with principal component analysis and nonlinear filtering to extract the deterministic components in AGN X-ray variability. The methods may be useful for understanding the dynamics of X-ray photon-count fluctuation. In the present paper the data analysis will be illustrated using a photon event history file from NGC 5548 (ROR 701246) and from the X-ray pulsar 1E2259+586 (ROR 400314), both for photon energies > 0.5 keV. The wavelet scalogram for NGC 5548 (ROR 701246) is shown in Fig. 4.

Figure 4: The wavelet scalogram for NGC 5548 (ROR 701246), photon energies > 0.5 keV. The vertical axis shows logarithm of time scales in s