A New Approach for the Analysis of Complex Multi-Dimensional Data from Instruments which Detect Individual Photons

J. Garrett Jernigan, Michael Vezie

Space Sciences Laboratory, U. C. Berkeley, Berkeley, CA 94720

Abstract:

We present a full realization of an entirely new approach for the analysis of multi-dimensional data in the form of an finite event list of individual photons, typical of many applications in X-ray and astronomy. The new approach is an inverse Monte Carlo method which starts with a list of raw photons of imperfect information and ends with a finite list of model photons of perfect information. The most significant feature of the new method is that all mathematical steps are carried out without the creation of any spectral or spatial bins of photons. The practical utility of this approach required the invention of a new version of a multi-dimensional Kolmogorov-Smirnov (KS) test.

1. Description of the New Analysis Method and A One Dimensional Example

The essential feature of the new approach is that computation is restricted to constructs only in the form of lists of events (photons) thereby avoiding the use of bins or histograms. One could describe this approach as an inverse Monte Carlo method. The natural figure of merit for comparison of two lists of events is the two sample Kolmogorov-Smirnov (KS) test. The usual KS approach is defined in one dimension as the comparison of the integral of a list of events.

As a simple one dimensional example consider an X-ray spectrum of a hot low density plasma with a solar abundance of elements at a temperature of sixty million degrees Kelvin as detected by a Si sensor with a nearly constant resolution of 170 keV (FWHM) in the energy band from 6 to 8 keV. One expects two Fe X-ray lines at 6.63 keV and 6.93 keV and background. Figure 1(a) is a display of an intermediate step in the method which shows the KS curves for an intermediate version of the model along with the data. Since these two curves do not overlap it is clear that the model at this step in the method is not a good match to the data. In Figure 1(b), the difference between the two KS curves is shown. In later displays only the difference KS curve will be plotted. Note that the maximum absolute difference or the KS parameter is about 0.16 which for a 1000 photon example is not a good fit according to the expected KS distribution. Figure 2 is a histogram of a 5000 photon Monte Carlo realization of the Gaussian spectral response function corresponding to a FWHM of 170 keV.

Figure 3 indicates the final results from the new method plotted in familiar histogram form. Figure 3(a) is a histogram of a Monte Carlo realization of the final model result. Note the presence of the two X-ray lines and a uniform background. Figure 3(b) is the Monte Carlo version of the model photons in Figure 3(a) with the added instrumental response which is plotted Figure 2. Figure 3(c) is the simulated raw data that is not varied during the computation of the method. Figure 3(d) is the difference histogram of model (Figure 3(b)) and raw data (Figure 3(c)). Note that the residuals are as one would expect for a final solution with normal Poisson noise. In Figure 4 (a) through (c) the iterative steps of the method are shown. It is a simple example that fully implements the new method. The left column are the histograms of the Monte Carlo versions of the best model at a particular step in the method. The middle panel at each step in the sequence (a) through (c) is the Monte Carlo realization of the corresponding left panel with the added effect of the instrumental spectral response function shown in Figure 2. The initial model is a uniform distribution of 1000 photons without lines. For each photon of the model (left panel) a random photon is selected from the instrumental spectral response photons. Its deviation from the calibration line at 7 keV is computed and is added to the energy of the photon in the perfect model to represent the detected photon in the middle panel. One iterates the procedure at each step by comparing the middle panel photons with the raw photons (see Figure 3(c)) utilizing the KS difference curve shown in the right panel. The method of iteration used for this simple example is to randomly select a new photon from the current best model in turn for each of the 1000 current photons. On a trial basis one replaces each photon and recomputes the KS parameter. If it is smaller than or equal to the previous KS parameter, the new trial photon replaces the old model photon. Otherwise the trial photon is discarded and the older model photon is retained. This procedure is repeated until the method converges.

2. A Two Dimensional Example

A component of the new method required the invention of a new version of a multi-dimensional Kolmogorov-Smirnov (KS) test. The traditional weakness of the KS test is that it is only normally defined in one dimension and therefore is of limited utility. It has been extended to two dimensions by Peacock (1983, MNRAS, 202, 615) and further refined to higher dimensions by Fasano and Franceschini (1987, MNRAS, 225, 155). The brief length of this paper precludes a detailed explanation of the newly invented extension of the KS test to higher dimensions.

Figure 5 illustrates an example of the analysis of a ASCA image of the Cas A supernova remnant taken in a single long exposure with the GIS instrument. Cas A is not centered in the field of view and therefore the PSF displayed in Figure 5(a) obtained from an independent image of an off-axis point source is both complex and non-axisymmetric. Figure 5(b) is the raw GIS image of Cas A. Figure 5(c) is the final deconvolved image or model derived by the two dimensional version of the new method. Note that this image shows the expected shell like structure of the Cas A supernova remnant. Figure 5(d) is the final model image after convolution with the PSF in Figure 5(a). The new method converges by a two dimensional KS comparison of the raw image, Figure 5(b), and the smeared model in Figure 5(d).