4. The entropy of amplitude distributions of ampligrams
There is another method for studying differences in dynamical properties between sources and groups of sources. It has been described in Sect. 7 of Paper I. For each level of wavelet coefficient magnitude in the ampligram a normalized amplitude distribution is created. For levels with pure stochastic components, Gaussian amplitude distributions will be obtained. For levels with deterministic components, broader amplitude distributions, often revealing structures, will be found.
It is useful to quantify the character of the distribution by calculating the distribution entropy for each level of wavelet coefficient magnitude. The distribution entropy, , is given by Shannon & Weaver (1949):
Here is the measured probability density, B is number of distribution bins (100 in our case) and M = 20 for a low-20 ampligram. The results of the analysis performed for the 2 groups of objects with lower apparent luminosities, listed in Table 2 are shown in Fig. 4.
It may be seen that the S1 objects show lower minimum values of entropy for low levels of wavelet coefficient magnitudes (14%) where most of deterministic structures is known to occur in the ampligrams. Thus S1 ampligrams show more pronounced deterministic structures (lower entropy) than the ampligrams for QSOs.
It must be remembered, since the entropy is a logarithmic variable, that calculation of averages and standard deviations are really not possible. It may be seen from Fig. 4 that the difference between the entropy for S1 and QSO is undoubtly significant up to 6% of the wavelet coefficient magnitude. Between 6% and 14% of the wavelet coefficient magnitude there might still be significant differences.
For strong sources, like NGC5548 (S1 type source) and 3C 273 (QSO type source) the situation is different (see Fig. 5). The reasons for the difference are given below.
The entropy may be used as a measure of the distribution properties, but it must be remembered that the relation between the entropy and the character of the distribution is not simple. As it has been pointed out in Paper I, the entropy is relatively low for a narrow distribution of pure Gaussian noise, it increases when the distribution broadens due to presence of semi-regular components with different periods and finally it decreases when deterministic components become stronger. See Fig. 6.
It is interesting to study more in detail the shape of the distributions for different groups of sources and how it varies with the apparent luminosity of the source. Since the entropy is a logarithmic quantity a conventional statistics can not be used. For that reason neural network models for both types of sources (S1 and QSO) were constructed, where the entropy of the ampligram distributions was modeled as a function of the percentage of the maximum wavelet coefficient magnitude and of the apparent luminosity of the source (average photon counts during the observation periods). A simple model architecture with a back-propagation neural network having two input processing elements (wavelet coefficient magnitude and average counts), seven processing elements in the hidden layer and one output processing element for the entropy was employed. The number of processing elements in the hidden layer should be related to the number of expected subpopulations in the data. A measure of the model's accuracy may be obtained using a test subset of data. Scatter plots of entropy retrieved from the respective model versus the measured entropy are shown in Fig. 7.
The general dependence of the distribution entropy on the apparent luminosity (average counting rate) for both groups of sources, as retrieved from respective models is shown in graphs of Fig. 8.
Both these figures illustrate properties of the entropy curves for different sources. A vertical section of the graph corresponds to curves shown in Fig. 5. For the QSO objects there is a monotonous increase of entropy towards small wavelet coefficients due to broadening of the distribution. It is valid for all apparent luminosities in the investigated range. For the S1 objects the entropy increases with decreasing wavelet coefficients up to a certain apparent luminosity, after which the deterministic components become so distinct that the entropy starts to decrease again towards small wavelet coefficients. The latter situation corresponds to case (c) in Fig. 10.
The curves of Fig. 5 are in agreement with the above graphs remembering that that the average counting rate for analyzed observation periods is for NGC 5548 is 1.8 photon/sec and for 3C 273 2.3 photons/sec. It may be seen from graphs of Fig. 8 that there would be a larger difference between the curves of Fig. 5 if NGC5548 would have the same apparent luminosity as 3C273.
There is another interesting difference in the structure of entropy variations. If the upper limit of the ampligram will be increased from standard 20% without changing the number of intervals of wavelet coefficient magnitude, the width of a single interval/bandwidth will be increased. A test has been done for NGC 5548 and 3C 273 to see the influence of the upper limit and of the bandwidth on the entropy variations. The curves of Fig. 9 correspond to the upper limit of the ampligram of 20% and correspondingly to a bandwidth of 1%. In the left graph of Fig. 9 the upper limit of the ampligram has been increased to 30% and thus the bandwidth to 1.5%. Respective numbers for the right graph of Fig. 9 are 40% and 2%. It may be seen that there is no significant change in entropy values between Fig. 5 and the left graph of Fig. 9. However, there is a significant change in the right graph. The change is not just a transformation of the horizontal scale due to increase of the upper limit. The decrease of entropy for NGC 5548 must be due to increased bandwidth. It means that the deterministic structures/building blocks in S1 has a finite width in the spectral density domain and that doubled bandwidth is needed to reveal their presence.
© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000