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Astron. Astrophys. 334, 201-209 (1998)

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3. The Linear State Space Model (LSSM)

The LSSM analysis is a recently developed tool to model stochastic time series (Gantert 1993). It has e.g. been used to describe medical time series (hand-tremor data, Timmer 1995). König & Timmer 1997 were the first to apply the new method to astronomical data by successfully fitting lightcurves of active galactic nuclei with first order LSSMs (see also König et al. 1997). The mathematical background of the LSSMs and the associated fitting-procedure have been discussed in detail by König & Timmer 1997 and in the references therein. Here, we shall concentrate on first order LSSMs, for we found them to be appropriate to describe the Cyg X-1 lightcurves (see Sect. 4).

3.1. The autoregressive process

One possibility to model irregular variability (i.e. temporal variations that cannot be forecasted exactly) is to assume that it is caused by a chaotic process (Voges et al. 1987, Unno et al. 1990). The LSSMs, on the other hand, are based on the alternative assumption, that the variability of the observed system is produced by a stochastic process. The LSSMs use the rather general autoregressive (AR) formulation for the stochastic system-lightcurve [FORMULA].

The AR processes were first introduced by Yule 1927to model the variability of Wolf's sunspot numbers and have been well studied since then (e.g. by  Scargle 1981). An AR process of order p (AR[p]) is defined by:

[EQUATION]

The time series [FORMULA] is sampled at discrete times [FORMULA] with time resolution [FORMULA]. The purely stochastic component [FORMULA], is a Gaussian random variable with mean 0 and variance [FORMULA]. Since Eq. (5) is of the same structure as a regression equation for the variables [FORMULA] and [FORMULA] the name autoregressive process has been assigned to it.

In an AR lightcurve, for each [FORMULA] the value of the random variable [FORMULA] is correlated with the values of the process at earlier times. This correlation is decreasing with increasing time differences between two lightcurve values (expressed by an exponentially decaying autocorrelation function). The stochastic component [FORMULA] is the reason that the process does not simply come to rest. [FORMULA] is an intrinsic property of the system-variability: the system-noise.

The temporal correlations in the AR lightcurve are characterized by the dynamical parameters [FORMULA]. The p dynamical parameters are related to p temporal parameters, describing the temporal structures in the lightcurves: depending on the process, the [FORMULA] represent stochastic relaxators with relaxation times [FORMULA], or damped stochastic oscillators with relaxation times [FORMULA] and periods P, or both (Honerkamp 1994). In case of an first order AR process there is only one dynamical parameter [FORMULA]. For stationary processes [FORMULA] must be [FORMULA] 1 and only positive values of a lead to plausible physical models. The corresponding temporal parameter, [FORMULA], is the relaxation time of a stochastic relaxator, one representation of which is an exponentially decaying shot as described by Eq. (3).

3.2. The observation-noise

For the determination of the temporal parameters of the system-process the noise which is caused by the measurement (i.e. photon statistic, particle background) has to be considered because it disturbs the temporal structures of the system-lightcurve. If the observation of a system-lightcurve is modeled with a plain AR process, the so called observation-noise will lead to an underestimation of the true temporal system-parameters (Robinson & Nather 1979, König & Timmer 1997). In order to solve this general problem, a LSSM consists of two equations: the system-equation for the intrinsic system-lightcurve and the observation-equation for the measured lightcurve. The latter describes the observed lightcurve [FORMULA] by explicitly considering the influence of the observation-noise [FORMULA] on the system-lightcurve.

The important model for the analysis of the short-term variability of Cyg X-1 is the LSSM of first order :

[EQUATION]

Like the system-noise [FORMULA], the observation-noise [FORMULA] is modeled by a Gaussian noise component; c is a constant normalization factor. The equations for higher order LSSMs are given by König & Timmer 1997 S(the temporal parameters of an LSSM of order p (LSSM[p]) correspond to the p dynamical parameters of Eq. 5).

3.3. The power spectrum

The power spectrum [FORMULA] of a first order Linear State Space Model has the following form (König & Timmer 1997):

[EQUATION]

The examples for [FORMULA] displayed in Fig. 3 b show the flat top as well as red noise. The observational white noise floor is also provided by Eq. 8 but has been omitted in Fig. 3 b ([FORMULA]) to allow the direct comparison with the standard shot noise spectra (Fig. 3 a): In contrary to the fixed red noise slope [FORMULA] of the standard shot noise spectra, the slope of [FORMULA] can be modeled by adjusting a, i.e. [FORMULA]. All the features of the hard state power spectrum of Cyg X-1 can thus be reproduced by the power spectrum of the LSSM[1] which is defined by only one temporal parameter [FORMULA]. Furthermore, Eq. 8 does not need to be corrected for binning effects, since the definition of the LSSM already is discrete in t (see Eq. 6 and 7).

Principally, LSSMs can be evaluated by fitting their power spectra to the measured periodogram. However, fits in the frequency domain do not allow the explicit modeling of the observational noise and may be influenced by several other sources of uncertainties, especially by spectral leakage, which contaminates the power spectrum of each finite time series (Deeter & Boynton 1982, Deeter 1984). To avoid these uncertainties our LSSM analysis was performed in the time domain by direct comparison of the model-lightcurves and the measured lightcurves. This method allows fits with a statistical significance higher than that of the frequency domain fits (König & Timmer 1997).

3.4. The fitting procedure

The parameters of a LSSM[p] fit are estimated with the help of the maximum likelihood procedure: the set of parameters is derived, for which the probability of observing the measured lightcurve [FORMULA] is at maximum. We have obtained those parameters using the expectation-maximization (EM) algorithm (Honerkamp 1994). The method is iterative, starting from a set of initial parameter values. In the expectation step, the Kalman filter is applied, which allows to estimate the unobservable system-lightcurve [FORMULA] by using the observed lightcurve [FORMULA] and a given set of LSSM parameters. In the following maximization step the likelihood function is maximized, taking the estimated [FORMULA] into account and providing a new set of parameters. The results of this iterative procedure are best estimates for the LSSM[p] parameters and for the intrinsic system-lightcurve (a detailed example is discussed in Sect. 4.1). Note, that the latter is not produced using a smoothing filter: all variability timescales of the observed lightcurve (see e.g. Fig. 4 a) are still present in the derived system-lightcurve (see e.g. Fig. 4 b).

[FIGURE] Fig. 4. a One of the 900 EXOSAT ME Cyg X-1 lightcurves with a length of 16 s and a bin time of 16 ms, which were modeled by LSSMs. Mean countrate: 633.1 counts/sec/half, standard deviation: 234.5 counts/sec/half. b Best estimate for the autoregressive system-lightcurve, which has been obtained by applying a LSSM of order [FORMULA] 1 to the lightcurve displayed above (see Table 2). Both lightcurves are shown without error bars for reasons of clarity.

The length and the time resolution of lightcurves, which are analyzed by LSSMs, have to be chosen considering the following requirements:

  • The duration of the lightcurve has to be several times the relaxation timescale, which is expected to be a few tenths of a second (shot noise models), or the relaxator cannot be found.
  • The bin time has to be well below the relaxation timescale to allow us to distinguish the relaxator from statistical fluctuations.

Simulations by König 1997 have shown that the duration should be at least 5 times larger and that the bin time should be at least 10 times smaller than [FORMULA]. In order to fulfill these criteria without an unreasonable increase in LSSM computing time, we have fitted lightcurves with a bin time of 16 ms and length of 16 s (e.g. Fig. 4 a). Although this time interval is shorter than the intervals typically studied in the context of shot noise analysis, it is long enough for the determination of the relaxator.

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© European Southern Observatory (ESO) 1998

Online publication: May 12, 1998

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