Model | Periods | KS
test^{c} | ||||||

LSSM AR[p] | (s) | (s) | . | |||||

0 | 1 | - | - | 0 | . | 0% | ||

1 | 0.722 | - | 4799 | 93 | . | 5% | ||

2 | 0.701 | - | 26 | . | 1 | 66 | . | 8% |

- | 5011 | |||||||

3 | 0.510 | - | 10 | . | 6 | 88 | . | 2% |

- | 18 | . | 9 | |||||

- | 4798 | |||||||

4 | 0.395 | 236.3 | 71 | . | 1 | 92 | . | 1% |

- | 6 | . | 7 | |||||

- | 4780 | |||||||

. | . | |||||||

^{a} Variance of the observational noise. | ||||||||

^{b} Relaxation time. | ||||||||

^{c} Kolmogorov-Smirnov test for white noise. |

We applied LSSMs with different order AR processes. An LSSM using an AR[0]
process corresponds to a pure white noise process without any temporal
correlation and a flat spectrum. The used Kolmogorov-Smirnov test rejects
this model at any level of significance (see Table 1 (click here)). Without loss
of generality, *Q* is set to unity, the mean and variance are set to 0 and
1, respectively. We see that the X-ray lightcurve of NGC 5506 can be well
modelled with a LSSM AR[1] model, as the residuals between the estimated
AR[1] process and the measured data are consistent with Gaussian white
noise. Figure 3 (click here) shows the distribution and the corresponding
normal quantile plot of the
fit residuals which both display the Gaussian character of the
observational noise. The standard deviation of the distribution is 0.738
which is in good agreement to the estimated observational variance of 0.722
for the LSSM
AR[1] fit (see Table 1 (click here)). Furthermore, the lightcurve of the
estimated AR[1] looks very similar to the temporal behavior of the hidden
process (Fig. 2 (click here)). The corresponding dynamical parameter *a*_{1} of
the LSSM AR[1] fit is 0.9938 which corresponds to a relaxation time of about
4799 s.

**Figure 3:** **a)** Distribution and **b)** normal quantile plot of the
residuals of the LSSM AR[1] fit to the EXOSAT ME NGC 5506 lightcurve (the
dotted lines in a) indicate the mean and rms of the observational noise). A
normal quantile
plot arranges the data in increasing order and plot each data value at a
position that corresponds to its ideal position in a normal distribution.
If the data are normally distributed, all points should lie on a straight
line

The LSSM AR[1] gives a good fit to the EXOSAT NGC 5506 data as the variance of the prediction errors nearly remains constant from model order 1 to 2 and the residuals conforms to white noise. The decrease in the variance for higher model orders might be due to correlations in the modelled noise, generated by the switching of the EXOSAT detectors. Since each detector has its own noise charateristics a regular swapping between background and source detectors would lead to an alternating observational noise level (see Sect. 2). The higher order LSSM AR[p] fits try to model the resulting correlations with additional but negligible relaxators and damped oscillators (, ).

We have used the Durbin-Levinson algorithm (see Sect. 3) to estimate the parameters of a competing simple AR[p] model (see Table 2 (click here)). As expected for time series containing observational noise, the characteristic timescales are underestimated by fitting a simple AR process and the statistical test rejects the AR[p] model. A test for white noise residuals fails, which means that there are still correlations present which cannot be modelled with an AR[p] procces. We have performed AR[p] fits for model orders up to 10 and we never found residuals consitent with white noise, indicating that there is no preferred model order. All occuring relaxators and damped oscillators are insignificant due to their short relaxation timescales compared with the bintime of 30s. As the observational noise is not modelled explicitly in AR models, it is included accidentally in the inherent AR noise term. Thus, any correlation in the observed time series which can be detected in the LSSM fits, is wiped out and the higher order AR fits only reveal fast decaying relaxators and oscillators.

Model | Periods | KS
test^{c} | ||||

AR[p] | (s) | (s) | ||||

0 | 1 | - | - | 0.0% | ||

1 | 0.9235 | - | 23 | . | 3 | 0.5% |

2 | 0.8814 | - | 55 | . | 6 | 0.3% |

- | 29 | . | 8 | |||

3 | 0.8566 | - | 97 | . | 0 | 0.4% |

197.4 | 40 | . | 6 | |||

4 | 0.8362 | - | 153 | . | 2 | 0.4% |

- | 51 | . | 2 | |||

127.7 | 55 | . | 1 | |||

. | ||||||

^{a} Variance of inherent AR noise. | ||||||

^{b} Relaxation time. | ||||||

^{c} Kolmogorov Smirnov test for white noise. |

One might expect that the resulting best fit LSSM light curve
(Fig. 2 (click here)b) might also be produced by just smoothing the original
lightcurve. This assumption is wrong as a smoothing filter would pass long
timescales and suppress all short time variability patterns. Thus all
information about the variations on short timescales would be lost
(Brockwell & Davis 1989). The Kalman filter concedes not only
the time series values *x*(*t*) but also its prediction errors. These errors
are much smaller than the errors of the observed lightcurve *y*(*t*). In the
case of the NGC 5506 observation (Fig. 2 (click here)) the estimation errors are
about 0.18 counts/s and the errors of *y*(*t*) are about 1.3 counts/s,
respectively. Both lightcurves in Fig. 2 (click here) are shown without error
bars due to reasons of clarity.

We have used Monte Carlo Simulations to determine the error of the
dynamical parameter *a*_{1}. Using the distribution of the estimated
parameters of 1000 simulated AR[1] time series with the best fit
results, we found . As the dynamical
parameter is close to unity the corresponding relaxation time error is
high, with s. To prove the quality of the
LSSM results we have fitted a LSSM AR[1] spectrum to the periodogram
data. This fit yields the dynamical parameter which is consistent with the LSSM AR[1] fit in the time
domain, but the corresponding error is much higher due to the lower
statistical significance of frequency domain fits (see Sect. 2).

The autocovariance function of the AR[1] process is given by:

which is an exponentially decaying function for stationary (|*a*_{1}| <
1) time series, very similar to the temporal behavior of the
autocorrelation function of a shot noise model (Papoulis
1991):

The variable denotes the density and is the lifetime of the shots. This similarity means that an AR[1] process can also be modelled by a superposition of Poisson distributed decaying shots (Papoulis 1991). The shot noise model, which has been used as an alternative to the model, appears to give a good fit to the power spectrum of NGC 5506 (Papadakis & Lawrence 1995; Belloni & Hasinger 1990 and references therein). But instead of all the shots having the same lifetime, Papadakis & Lawrence (1995) used a distribution varying as between and . They fixed arbitrarily at 12000s and found that is around 300s for NGC 5506, much lower than the relaxation time of about 4800s found with the LSSM fit. A possible explanation for this difference could be the distribution of lifetimes. Since the power law slope of the shot noise model is constantly -2 at medium and high frequencies, this distribution is necessary to modify the slope and to maintain a good fit to the spectrum. The advantage of a LSSM is a variable slope at medium frequencies which depends on the dynamical parameter (see Fig. 4 (click here)).

**Figure 4:** Periodogram of the EXOSAT ME X-ray lightcurve of NGC 5506
(dots) and the spectrum of the best fit LSSM AR[1] model in the time domain
(line) (see Fig. 2 (click here)a). The spectra of the higher order LSSM
AR fits differ less than from the LSSM AR[1] spectrum. The dashed
lines display the - spectra of the corresponding frequency
domain fit. The time domain fit yields errors which are more than
3 times smaller (see text for details)

The shot noise model can be regarded as an approximation of an AR[1]
model for values *a*_{1} near unity. The mean density of the Poisson
events then corresponds to the variance *Q* of the dynamical
noise in the LSSM system Eq. (6 (click here)). Thus *Q* could be
used to quantify and compare the rate of the accretion shots occuring
in AGNs.

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