XMM-Newton Science Analysis System


eimsim (eimsim-2.4) [xmmsas_20190531_1155-18.0.0]

Attempt to match detected and simulated sources

This function may be performed alone by calling the script with entrystage and finalstage=`compare'. The actual processing is done by a task called srccompare.

In order to asses how well the source detection machinery performs, we need some way to (i) match every detection with a unique member of the list of simulated sources which is the most likely identification, and (ii) measure the probablity that the match arose by chance. The obvious answer to the first requirement seems to be to find that simulated source which is `nearest' in both position and flux to the detected source. This intuition can be quantified by imagining that both simulated and detected sources are represented by points in an abstract 3-dimensional space in which the first two axes record the source position, and the third records the source flux. Let us define a quantity $R$ in this space by the equation

$$R^2 = \left( \frac{x_\mathrm{sim}-x_\mathrm{det}}{\sigma_x} ... ...eft( \frac{S_\mathrm{sim}-S_\mathrm{det}}{\sigma_S} \right)^2,$$

where $x$, $y$ and $S$ represent position and flux respectively. The $\sigma$ quantities represent the uncertainties which were determined by the source-detection procedure. For each detected source, we define its `matching simulated source' as the one which minimizes $R$ for that detection. Let us denote this minimum value of $R$ by $R_\mathrm{match}$. The probability can then be obtained as follows. First, consider the ellipsoidal surface defined by

$$R^2_\mathrm{match} = \left( \frac{x-x_\mathrm{det}}{\sigma_x... ...right)^2 + \left( \frac{S-S_\mathrm{det}}{\sigma_S} \right)^2.$$

From the definition of $R_\mathrm{match}$, this ellipsoid has the following properties:

• It is centred on the `position' in this abstract 3-dimensional space of the detected source.
• The principle axes of the ellipse preserve the ratios between the uncertainties. Indeed one can visualize the process of searching for a match as `inflating' the ellipsoid as one inflates a balloon, until its edge intersects a simulated source.
• The ellipsoid just touches the matching simulated source.
• No other simulated source is found inside it.

Intuition suggests that the larger the ellipse, or the larger the value of $R_\mathrm{match}$, the less likely it is that the detection is `genuine'. Again we quantify this intuition by integrating the probability density distribution of simulated sources in position and flux over the ellipsoidal volume to give $\eta$, the expectation value for the number of simulated sources which would fall inside the ellipsoid by chance. Ok, we said above that there are zero sim sources within the ellipsoid - but that was in a single, particular case. What we want to test now is the null hypothesis, ie to ask how many simulated sources, on average, we would expect to land inside our ellipsoid if we threw the chips at random.

Having calculated $\eta$, it is fairly easy to see that the probability $P_\mathrm{null}$ of the null hypothesis is given by

$$P_\mathrm{null} = 1 - exp(-\eta).$$ (3)

There is a slight issue here, in that the simulated sources are not evenly distributed in $S$: the number of sources per flux interval increases greatly at low flux. This leads to a bias towards matching with fainter sources. In previous versions of eimsim I assumed that this was a bad thing, and took steps to transform the flux coordinate to correct for this. This is the point of the FLUXRAND business described in section 4.3.1. Now I am no longer sure that this is the case. In real life, we expect the gradient of number density with flux to bias the detected flux - this is called Eddington bias. Maintaining this bias during the matching stage ought to help correct for this. What concerns me more now is that the + and - flux uncertainties ought not to be the same in a simple flux scale: one would expect that the + one ought to be larger. Perhaps then the correct way to transform the flux scale before matching is to take its square root, which should even up the uncertainties. What I have done is provide the facility in eimsim to do any one of three things, namely (i) leave the flux alone; (ii) transform it to the FLUXRAND scale, in which the simulated sources are evenly distributed; (iii) transform the flux scale by taking square roots of flux. Comparison of empirical results ought to show which is the best procedure.

The following additional columns are written to the list of detected sources:

Name	Data type	Units	Comment
X	4-byte real	arcsec	$X$-coordinate of det source.
Y	4-byte real	arcsec	$Y$-coordinate of det source.
X_ERR	4-byte real	arcsec	$X$-coordinate error of det source.
Y_ERR	4-byte real	arcsec	$Y$-coordinate error of det source.
SIM_X	4-byte real	arcsec	$X$-coordinate of matching sim source.
SIM_Y	4-byte real	arcsec	$Y$-coordinate of matching sim source.
SIM_FLUX	4-byte real	erg cm$^{-2}$ s$^{-1}$	Flux of matching sim source.
SIM_INDX	4-byte int		From simlist column INDEX.
SIM_INV_SENSY	4-byte real		From simlist column INV_SENSY.
R_SIGMAS	4-byte real		$R_\mathrm{match}$.
MATCH_PNULL	4-byte real		$P_{\rm {null}}$ from equation 3.
SIM_LINF	4-byte real		From simlist column FLUXRAND.
FLAG	4-byte int

If the user chooses to take the square root of the flux coordinate then the following additional columns are written:

ROOTF	4-byte real		Square root of det source FLUX.
ROOTF_ERR	4-byte real		The appropriate error in L.
SIM_ROOTF	4-byte real		Square root of sim source FLUX.

The FLAG column is hardly used at present, but may be found useful in further analysis. Only bit 0 is set by task srccompare. If the same simulated source is `claimed' by more than one detected source, bit 0 of the flag column is set for all the claimants except that with the smallest value of MATCH_PNULL.

This section also writes a keyword COMPARED=`T' to the table header.