## 2. Data reduction## 2.1. Source detectionWe investigated the ## 2.2. Variability testingAs mentioned in the introduction, for flare event detection we have used a method developed by Scargle (1998) based on Bayesian statistics. The method is applicable to data that are known to be from a nearly ideal Poisson process, i.e. a class of independent, identically distributed processes, having zero lengths of dead time. The data gathered in Nevertheless, the arrival times of photons registered by
Scargle's (1998) method is designed for application to photon
counting data and uses it for decomposition into a piecewise constant
Poisson process. For example, let us assume that during a continuous
observational interval of length By taking as a background information ( where is the (marginal) probability of the data given , and is the prior probability of model . The term in the denominator is a normalization constant, and we may eliminate it by calculating the ratio of the posterior probabilities instead of probabilities directly. Indeed, the extent to which the data support model over is measured by the ratio of their posterior probabilities and is called posterior odds ratio The first factor on the right-hand side of Eq. (2) is the ratio of
the where The second factor on the right-hand side of Eq. (2) is the prior odds ratio, which will often be equal to 1 (see below), representing the absence of a priori preference for either model. It follows that the Bayes factor is equal to the posterior odds when the prior odds is equal to 1. When , the data favor over , and when the data favor . If we have calculated the odds ratio , in favor of model over , we can find probabilities for model by inverting Eq. (2), giving Applying this approach iteratively to the observational data set, the Scargle (1998) method returns an array of rates, , and a set of so called "change points" , giving the times when an abrupt change in the rate is determined, i.e. a significant variation. This is the most probable partitioning of the observational interval into segments during which the photon arrival rate was discernibly constant, i.e. had no statistically significant variations. Unlike most, this method does not stipulate or predetermine time bins - instead the data themselves determine an effective, non-uniform binning in time. Therefore this data analysis procedure does not itself impose a lower limit to the time scale on which variability can be detected. There are two free parameters in the method that are used to halt the segmentation process: The first is the minimum number of events that are allowed in a block (we have chosen two) and the second is a prior odds ratio that may be applied to disfavor segmentation. The prior odds ratio (second factor in the right-hand side of Eq. (2)) represents the relative likelihood assigned to the two models before the data is considered. Although this would appear to warrant a value of unity, in practice a larger value is used to prevent the method from making an incorrect decision to segment when two models are almost equally likely. To have strong evidence in favor of segmenting, Scargle (1998)
suggests to use as a prior odds ratio a quantity which is equal to the
ratio of the length of the observational interval and the desired time
resolution of the data. For interpreting As is well known, owing to the spacecraft orbits, A number of X-ray sources, having maximum likelihood of existence , were chosen for variability testing. For this purpose, we have extracted from the original observational data the part corresponding to a given source as follows: Around the center of each source a set of photon events are chosen using a circle with radius 2.5 times the Full Width at Half Maximum (FWHM) of a detected source, available from source detection algorithms and, for the corresponding background, an annulus with the same inner radius and an outer radius equal to times of the inner one (it covers the same area in the sky as the source). It should be noted that the used radius for the source extracts the overwhelming majority of photon events corresponding to the source. If in the area used for the background, there is any source with maximum likelihood of existence greater than 0 , the background is taken from the opposite direction of that source sector with angle with inner radius equal to the and outer radius times of the inner one. Finally, as a background for a given source, a sector is used not including any source with and covering equal area in the sky as overwhelming majority of photon events coming from the source. © European Southern Observatory (ESO) 1999 Online publication: April 12, 1999 |