4.1. LSSM fits of various order to one exemplary lightcurve
Table 2 shows the results of LSSM fits of order 0-5 to the lightcurve segment displayed in Fig. 4 a: The relaxation timescale of about 0.2 s, which is derived by the LSSM fit, is also found by the fits of higher order (the LSSM represents uncorrelated white noise and therefore does not yield a temporal parameter). All other temporal correlations are decaying within a few time bins. Since they are due to statistical fluctuations they are negligible (König 1997). This means, that the dynamics of the lightcurve can most likely be described by one relaxation timescale. If this is the case, the LSSM fit should deliver an adequate description of the observed lightcurve and the goodness of fit should not increase significantly for higher order LSSMs.
Table 2. Results of LSSM fits of various order to the lightcurve in Fig. 4 a.
A statistical test on the residuals, i.e. the difference between the derived system-lightcurve and the observed lightcurve, was performed to determine the quality of the LSSM fits: If a given LSSM can describe the observation, the residuals should be an uncorrelated white noise realization (Eq. 7). In this case all the temporal correlations of the observed lightcurve are modeled by the system-lightcurve. We use the Kolmogorov-Smirnov (KS) test to quantify the goodness of the LSSM fits. It compares the power spectrum of the residuals to a flat white noise power spectrum.
In Table 2 we list the KS probabilities for the residual lightcurves to actually be white noise realizations. For the LSSM fit a very high probablity of 93.5 is reached, which means that the observed lightcurve can be well modeled by an LSSM process. This is consistent with the visual impression from Fig. 4: the LSSM estimate (b) tracks the observed lightcurve (a) very closely. Higher order LSSM fits do not improve the fit significantly. The LSSM (white noise) is of course rejected at any level of confidence.
The distribution of the LSSM model residuals is shown in Fig. 5 a. Its mean and variance are 0.7 counts/sec/half and (179.9 counts/sec/half)2, respectively. Thus the variance of the observation-noise is about 82 of the upper limit as given by the mean countrate of the original lightcurve and assuming a pure Poisson process. The lower half of Fig. 5 displays the normal quantile plot of the fit residuals: it arranges the residuals in increasing order and for each data value indicates the position that corresponds to the same probability in a normal distribution. If the data are normally distributed, all points should lie on a straight line. Fig. 5 b shows that the distribution of the residuals can be approximated by a Gaussian distribution, but is compressed for low and stretched for high countrates. This indicates that the fit residuals are Poisson-distributed as expected for the observation-noise (mean countrate of the original lightcurve: 10.1 counts per 16 ms time bin). The LSSM model parameters reflect the Poisson distribution of the observation-noise by underestimating the Gaussian variance given as (143.8 counts/sec/half)2 in this example.
4.2. LSSM fits of first order to a sample of lightcurves
In order to obtain a reliable value for the temporal parameter , we have fitted LSSMs to a sample of 900 lightcurve segments covering a total length of four hours. We have used 1600 s of each of the nine observations listed in Table 1. The 1600 s consist of 100 equally long, uninterrupted segments with a temporal resolution of 16 ms.
For each of the nine observations 100 values for the relaxator were obtained. In Table 3, the mean , the standard deviation , and the most probable value of these distributions are listed. Averaging , , and over the nine observations gives (0.28 0.04) s, (0.13 0.03) s, and (0.19 0.03) s, respectively. All distributions exhibit the same asymmetric form, in the sense that is always lower than but still higher than the 1 deviation. This kind of distribution is consistent with the results that have been obtained from simulations of AR lightcurves based on one temporal parameter (Hamilton & Wu 1987).
Table 3. Results of the LSSM(AR) fits to EXOSAT ME observations of Cyg X-1.
The lightcurves of three of the observations (No. 1, 2, and 7 of Table 3) take the whole Argon energy range (1-20 keV) into account, i.e. their absolute countrates may be compared. Even though the analyzed segments of one of those three observations (No. 7) have a considerably smaller mean countrate than those of the other two, the corresponding distribution of is not significantly different. This means that no correlation between and the luminosity can be seen, which is confirmed within each of the nine observations: the mean countrates and the relaxation times of the 100 segments are not correlated.
Since the observations provide consistent results, all 900 values for can be combined, in order to arrive at a distribution with better statistics (Fig. 6). Note, that this non-Gaussian distribution, resulting from the maximum likelihood estimation of , cannot be described analytically (Hamilton & Wu 1987). In addition, simulations have shown that the distribution of is contaminated by bad fit results at its "high end" due to numerical effects (Hamilton & Wu 1987, König 1997). Therefore, can be expected to overestimate the "true" relaxation timescale of Cyg X-1, whereas the most probable value should be a better estimator.
© European Southern Observatory (ESO) 1998
Online publication: May 12, 1998