Fitting FITS Image Data
Sherpa Threads (CIAO 4.15 Sherpa)
This thread models image data of the supernova remnant G21.5-0.9 using a source model expression that involves multiple model components. The Create and Read Fits Image Data section shows how the input file was created.
Last Update: 9 Dec 2022 - updated for CIAO 4.15, no content change.
- Create and Read Fits Image Data
- Filtering Image Data
- Defining a Source Model
- Fitting the model
- Adding an extra component
- Checking the Sherpa session
- Scripting It
- Figure 1: Image of the data
- Figure 2: Filter region defined on the image
- Figure 3: Filtered data
- Figure 4: FITS image file containing filter information
- Figure 5: Residuals of the best-fit model using: g1+bgnd
- Figure 6: Residuals of the best-fit model using: g1+g2+bgnd
- Figure 7: The g2 model contribution to the best-fit model g1+g2+bgnd
Create and Read Fits Image Data
Download the sample data: 1838 (ACIS-S, G21.5-0.9)
unix% download_chandra_obsid 1838
In this example, the data has been reprocessed with chandra_repro. First, an image of the region of interest is created:
unix% punlearn dmcopy unix% dmcopy \ "acisf01838_repro_evt2.fits[sky=box(4059.25,4235.75,521.5,431.5)][bin x=::2,y=::2]" \ image2.fits
Because we plan to model the extended structure, not detailed features in the image, we can use coarser binning to reduce the time taken for the fit—a binning factor of 2 is used. The choice of binning should be determined by the quality of the data and the models to be fit with the data. Increasing the binning factor when creating the image is likely to lose small-scale spatial information.
Figure 1: Image of the data
Setting the coordinate system
The default coordinate system for fitting images is that of logical (also called image), which uses the FITS convention for numbering pixels: the first pixel is centered at (1,1) with the bottom-left corner being at (0.5,0.5) and the top-right corner being at (1.5,1.5).
Although a simple system, it can be difficult when converting the best-fit parameters, such as source location, back to more-meaningful coordinate systems. For this reason, the physical coordinate system is chosen for fitting. This is done with the set_coord function:
Choosing the appropriate statistics
When fitting Chandra or XMM-Newton imaging data, it is almost certain that many pixels will only contain a small number of counts, making the source model fit with Cash statistic, the Poisson log-likelihood, to the Poisson image suitable. Subsequently, subtracting a background data set is not permitted and must either be accounted for with a background model—as shown in this example—or the background signal is ignored.
The CSTAT statistic—equivalent to the XSPEC C statistic—may also be used in fitting image data. The advantage of CSTAT over CASH is that an approximate goodness-of-fit measure may be assigned to a given value of the CSTAT statistic for X-ray spectral fitting.
Setting the fit method
The default fitting method (Levenberg-Marquardt) is not well suited for use with the Cash statistic, so we choose to use the Simplex optimizer instead (the NelderMead method is the same as simplex).
The fit should also be tried with the Monte Carlo (moncar) optimization method to ensure that the global best-fit has been found, rather than a local one.
The current Sherpa status may be reviewed by using the show_all command:
sherpa> show_all() Data Set: 1 Filter: name = image2.fits x0 = Float64 x1 = Float64 y = Float64 shape = (216, 261) staterror = None syserror = None sky = physical crval = [3798.5,4019.5] crpix = [0.5,0.5] cdelt = [2.,2.] eqpos = world crval = [278.386 ,-10.5899] crpix = [4096.5,4096.5] cdelt = [-0.0001, 0.0001] crota = 0 epoch = 2000 equinox = 2000 coord = physical
Filtering Image Data
Here we illustrate several ways of filtering image data within Sherpa.
Define the region to filter
After the data has been displayed, the filter region may be determined from the imager. Alternatively, an existing region file may be used.
Use the Region → Shape menu in DS9 to choose a region shape, then left-click on the display to draw the desired shape. For further instructions on how to create regions in DS9, see the Using CIAO Region Files thread.
Figure 2: Filter region defined on the image
After the desired region size is set, as shown in Figure 2, use the Region → List Regions menu to get the region definition, first setting the format to CIAO and the coordinate system to Physical in the menu that pops up:
# Region file format: CIAO version 1.0 circle(4072.46,4249.34,108)
The region may also be saved to a file; see the DS9 thread for instructions.
Define the filter on the command line
sherpa> image_getregion(coord="physical") 'circle(4072.46,4249.34,108);'
The image_getregion command be default returns region strings in logical/image coordinates, so the coordinates must be explicitly set to "physical".
The filter is now set on the command-line with the region definition:
sherpa> notice2d("circle(4072.46,4249.34,108)") dataset 1: Field() -> Circle(4072.46,4249.34,108)
or by formatting the string returned by image_getregion with the Python replace string method:
One may also specify a FITS or ASCII file which contains the region definition:
Figure 3: Filtered data
Load filter information from file
2-D data sets may be filtered with filtering information read from FITS image files using the load_filter function. The filter image input to load_filter should contain 1s and 0s to indicate which pixels should be noticed/ignored, and should match the source image in shape and number of pixels. When the new ignore argument of load_filter is set to False (default), the pixels in the data image which correspond to those marked by 1s in the filter image will be noticed, and the reverse when ignore is True.
The filter information contained in a FITS image filter file would be applied to the source image data set using the load_filter function as follows:
sherpa> load_filter("load_filter_image.fits", ignore=False)
An example of a FITS image file which can be used to filter the source image might look like that shown in Figure 4, where the pixels to be noticed are those which have a value of 1 (because ignore=False), and those which should be excluded from the analysis are the pixels with a value of 0.
Figure 4: FITS image file containing filter information
Defining a Source Model
We wish to fit the image using a source model expression that involves multiple model components. By the end of the thread, we will use two two-dimensional Gaussian functions together with a constant background component, but for now we begin with only one Gaussian.
Since this is an example, these choices are not physically motivated but just an empirical choice to describe the emission.
It may be necessary to modify the initial parameter values and boundaries before performing the fit. For complex models, physical knowledge of the problem can guide the initial settings of parameters and their ranges, and improve the final convergence of the fit. For simple, single-component models the Sherpa guess function may be used to automatically set sensible starting values and ranges for the model parameters. To have Sherpa automatically query for the initial parameter values when a model is established, set 'paramprompt(True)' (the default value is False).
Note that the ellipticity and its angle (theta) are already frozen, so they do not vary during the fit.
sherpa> g1.ampl = 20 sherpa> g1.fwhm = 20 sherpa> g1.xpos = 4065.5 sherpa> g1.ypos = 4250.5
The upper-limits of the fwhm, xpos, and ypos parameters are constrained to roughly the size of the image in physical coordinates (300). ampl is restricted to the range 1:1000.
sherpa> g1.fwhm.max = 4300 sherpa> g1.xpos.max = 4300 sherpa> g1.ypos.max = 4300 sherpa> g1.ampl.min = 1 sherpa> g1.ampl.max = 1000
The current source model definition may be displayed:
sherpa> show_model() Model: 1 gauss2d.g1 Param Type Value Min Max Units ----- ---- ----- --- --- ----- g1.fwhm thawed 20 1.17549e-38 4300 g1.xpos thawed 4065.5 -3.40282e+38 4300 g1.ypos thawed 4250.5 -3.40282e+38 4300 g1.ellip frozen 0 0 0.999 g1.theta frozen 0 -6.28319 6.28319 radians g1.ampl thawed 20 1 1000
The image_fit command displays a montage of the data, current model, and residuals that can be used to see how well the model describes the data.
We decide to include a background component before fitting, and so add in a constant model to the source expression. The value of the background model (0.2) was estimated from a radial profile. The upper limit is set to 100 as a reasonable constraint.
sherpa> set_source(g1+const2d.bgnd) sherpa> bgnd.c0 = 0.2 sherpa> bgnd.c0.max = 100
The model is now:
sherpa> show_model(); Model: 1 (gauss2d.g1 + const2d.bgnd) Param Type Value Min Max Units ----- ---- ----- --- --- ----- g1.fwhm thawed 20 1.17549e-38 4300 g1.xpos thawed 4065.5 -3.40282e+38 4300 g1.ypos thawed 4250.5 -3.40282e+38 4300 g1.ellip frozen 0 0 0.999 g1.theta frozen 0 -6.28319 6.28319 radians g1.ampl thawed 20 1 1000 bgnd.c0 thawed 0.2 -3.40282e+38 100
Fitting the model
We are now ready to fit the data:
sherpa> fit() Dataset = 1 Method = neldermead Statistic = cash Initial fit statistic = 28202 Final fit statistic = -49863.4 at function evaluation 581 Data points = 9171 Degrees of freedom = 9166 Change in statistic = 78065.4 g1.fwhm 57.7311 g1.xpos 4061.99 g1.ypos 4241.4 g1.ampl 23.6888 bgnd.c0 0.274221
The best-fit values for the Gaussian's FWHM and amplitude, and the background level are close to the values we guessed previously. Since we are using the PHYSICAL coordinate system the xpos, ypos, and fwhm parameters are given in the physical system. For this ACIS image this is the SKY coordinate system, which means the FWHM corresponds to 57.8 * 0.492 = 28.5 arcseconds. This can be calculated at the Sherpa prompt since we can easily access parameter values:
sherpa> g1.fwhm.val * 0.492
If g1.fwhm * 0.492 is just used, then the desired answer is not returned; the .val suffix for the parameter name is required to return the value at the Sherpa prompt.
The fit can be visualised using image_fit, which displays the data, model, and residual image. Here we are interested in just the residuals:
Figure 5: Residuals of the best-fit model using: g1+bgnd
The resulting image is shown in Figure 5. Although the Gaussian does a reasonable job at describing the radial profile of the large-scale emission, the residuals do appear to be spatially correlated. This suggests that the model used here is insufficient to describe all the structure in the source; however, it will suffice for this example.
sherpa> save_model("model.fits") sherpa> save_resid("resid.fits")
Here, save_model saves the model array associated with 2-D data set id=1 to the FITS image file model.fits, and the save_resid function saves the simple data-minus-model counts residuals array to the FITS image file resid.fits.
Calculating errors on the fit parameters
Note that the statistic value is negative because we are using the Cash statistic—which is a maximum-likelihood statistic—rather than a χ2 statistic. This means that we cannot get an "absolute" measure of the fit—i.e. how well the model actually fits the data. However, we can still use the conf command to estimate errors on these parameters, since this relies on the change in the statistic value rather than the absolute value:
sherpa> conf() g1.ypos lower bound: -0.177778 g1.fwhm lower bound: -0.281862 g1.ypos upper bound: 0.177778 g1.fwhm upper bound: 0.281862 g1.xpos lower bound: -0.179686 g1.xpos upper bound: 0.179686 g1.ampl lower bound: -0.24621 g1.ampl upper bound: 0.248712 bgnd.c0 lower bound: -0.00937002 bgnd.c0 upper bound: 0.00952193 Dataset = 1 Confidence Method = confidence Iterative Fit Method = None Fitting Method = neldermead Statistic = cash confidence 1-sigma (68.2689%) bounds: Param Best-Fit Lower Bound Upper Bound ----- -------- ----------- ----------- g1.fwhm 57.7311 -0.281862 0.281862 g1.xpos 4061.99 -0.179686 0.179686 g1.ypos 4241.4 -0.177778 0.177778 g1.ampl 23.6888 -0.24621 0.248712 bgnd.c0 0.274221 -0.00937002 0.00952193
Adding an extra component
The residual image and profile of the fit show excess emission at the center of the source. We will add another Gaussian component to try and account for this emission. The upper parameter limits used for g1 are also used for g2.
sherpa> freeze(g1) sherpa> set_source(gauss2d.g2+g1+bgnd) sherpa> g2.fwhm = 10 sherpa> g2.ampl = 100 sherpa> g2.xpos = 4065.5 sherpa> g2.ypos = 4250.5 sherpa> g2.fwhm.max=4300 sherpa> g2.xpos.max=4300 sherpa> g2.ypos.max=4300 sherpa> g2.ampl.min=1 sherpa> g2.ampl.max=1000 sherpa> fit() Dataset = 1 Method = neldermead Statistic = cash Initial fit statistic = -47880.5 Final fit statistic = -55165.8 at function evaluation 622 Data points = 9171 Degrees of freedom = 9166 Change in statistic = 7285.34 g2.fwhm 5.90544 g2.xpos 4062.33 g2.ypos 4239.66 g2.ampl 247.756 bgnd.c0 0.268177
Since the g1 model was frozen, those values did not change in the fit:
sherpa> show_model() Model: 1 ((gauss2d.g2 + gauss2d.g1) + const2d.bgnd) Param Type Value Min Max Units ----- ---- ----- --- --- ----- g2.fwhm thawed 5.90544 1.17549e-38 4300 g2.xpos thawed 4062.33 -3.40282e+38 4300 g2.ypos thawed 4239.66 -3.40282e+38 4300 g2.ellip frozen 0 0 0.999 g2.theta frozen 0 -6.28319 6.28319 radians g2.ampl thawed 247.756 1 1000 g1.fwhm frozen 57.7311 1.17549e-38 4300 g1.xpos frozen 4061.99 -3.40282e+38 4300 g1.ypos frozen 4241.4 -3.40282e+38 4300 g1.ellip frozen 0 0 0.999 g1.theta frozen 0 -6.28319 6.28319 radians g1.ampl frozen 23.6888 1 1000 bgnd.c0 thawed 0.268177 -3.40282e+38 100
To see if the central component is elliptical—as suggested by the earlier residual image (Figure 5)—we "thaw" the ellipticity and position-angle parameters of the second Gaussian. They are set to non-zero values to help the fit.
sherpa> thaw(g2.ellip) sherpa> thaw(g2.theta) sherpa> g2.ellip = 0.1 sherpa> g2.theta = 1 sherpa> fit() Dataset = 1 Method = neldermead Statistic = cash Initial fit statistic = -55208.1 Final fit statistic = -55311.9 at function evaluation 658 Data points = 9171 Degrees of freedom = 9164 Change in statistic = 103.816 g2.fwhm 7.16588 g2.xpos 4062.28 g2.ypos 4239.67 g2.ellip 0.314245 g2.theta 0.836301 g2.ampl 250.614 bgnd.c0 0.268045
The residuals of this fit are shown in Figure 6.
Figure 6: Residuals of the best-fit model using: g1+g2+bgnd
To see how reliable the ellipticity and theta values are, we calculate the 1σ errors using the conf() method. In this case we explicitly list the parameters we are interested in (g2.ellip and g2.theta) to avoid calculating the limits for the other parameters.
sherpa> conf(g2.ellip, g2.theta) g2.theta lower bound: -0.0402018 g2.ellip lower bound: -0.0216664 g2.ellip upper bound: 0.0210256 g2.theta upper bound: 0.0405143 Dataset = 1 Confidence Method = confidence Iterative Fit Method = None Fitting Method = neldermead Statistic = cash confidence 1-sigma (68.2689%) bounds: Param Best-Fit Lower Bound Upper Bound ----- -------- ----------- ----------- g2.ellip 0.314245 -0.0216664 0.0210256 g2.theta 0.836301 -0.0402018 0.0405143
We can use the image_source_component command to image one, or a combination of, individual unconvolved source model components in order to visualize the contribution to the full model being used to fit the image data.
Here we image the extra g2 Gaussian component to quickly visualize its contribution to the fit of the emission.
The g2 model component image is shown in Figure 7.
Figure 7: The g2 model contribution to the best-fit model g1+g2+bgnd
Checking the Sherpa session
The final overall status of this Sherpa session may be viewed as follows:
sherpa> show_all() Data Set: 1 Filter: Circle(4072.46,4249.34,108) name = image2.fits x0 = Float64 x1 = Float64 y = Float64 shape = (216, 261) staterror = None syserror = None sky = physical crval = [ 3798.5, 4019.5] crpix = [ 0.5, 0.5] cdelt = [ 2., 2.] eqpos = world crval = [ 278.386 , -10.5899] crpix = [ 4096.5, 4096.5] cdelt = [-0.0001, 0.0001] crota = 0 epoch = 2000 equinox = 2000 coord = physical Model: 1 ((gauss2d.g2 + gauss2d.g1) + const2d.bgnd) Param Type Value Min Max Units ----- ---- ----- --- --- ----- g2.fwhm thawed 7.16588 1.17549e-38 4300 g2.xpos thawed 4062.28 -3.40282e+38 4300 g2.ypos thawed 4239.67 -3.40282e+38 4300 g2.ellip thawed 0.314245 0 0.999 g2.theta thawed 0.836301 -6.28319 6.28319 radians g2.ampl thawed 250.614 1 1000 g1.fwhm frozen 57.7311 1.17549e-38 4300 g1.xpos frozen 4061.99 -3.40282e+38 4300 g1.ypos frozen 4241.4 -3.40282e+38 4300 g1.ellip frozen 0 0 0.999 g1.theta frozen 0 -6.28319 6.28319 radians g1.ampl frozen 23.6888 1 1000 bgnd.c0 thawed 0.268045 -3.40282e+38 100 Optimization Method: NelderMead name = simplex ftol = 1.1920928955078125e-07 maxfev = None initsimplex = 0 finalsimplex = 9 step = None iquad = 1 verbose = 0 reflect = True Statistic: Cash Poisson Log-likelihood function. Counts are sampled from the Poisson distribution, and so the best way to assess the quality of model fits is to use the product of individual Poisson probabilities computed in each bin i, or the likelihood L: L = (product)_i [ M(i)^D(i)/D(i)! ] * exp[-M(i)] where M(i) = S(i) + B(i) is the sum of source and background model amplitudes, and D(i) is the number of observed counts, in bin i. The Cash statistic _ is derived by (1) taking the logarithm of the likelihood function, (2) changing its sign, (3) dropping the factorial term (which remains constant during fits to the same dataset), and (4) multiplying by two: C = 2 * (sum)_i [ M(i) - D(i) log M(i) ] The factor of two exists so that the change in cash statistic from one model fit to the next, (Delta)C, is distributed approximately as (Delta)chi-square when the number of counts in each bin is high. One can then in principle use (Delta)C instead of (Delta)chi-square in certain model comparison tests. However, unlike chi-square, the cash statistic may be used regardless of the number of counts in each bin. The magnitude of the Cash statistic depends upon the number of bins included in the fit and the values of the data themselves. Hence one cannot analytically assign a goodness-of-fit measure to a given value of the Cash statistic. Such a measure can, in principle, be computed by performing Monte Carlo simulations. One would repeatedly sample new datasets from the best-fit model, fit them, and note where the observed Cash statistic lies within the derived distribution of Cash statistics. Alternatively, the `cstat` statistic can be used. Notes ----- The background should not be subtracted from the data when this statistic is used. It should be modeled simultaneously with the source. The Cash statistic function evaluates the logarithm of each data point. If the number of counts is zero or negative, it's not possible to take the log of that number. The behavior in this case is controlled by the `truncate` and `trunc_value` settings in the .sherpa.rc file: - if `truncate` is `True` (the default value), then `log(trunc_value)` is used whenever the data value is <= 0. The default is `trunc_value=1.0e-25`. - when `truncate` is `False` an error is raised. References ---------- ..  "Parameter estimation in astronomy through application of the likelihood ratio", Cash, W. 1979, ApJ 228, 939 http://adsabs.harvard.edu/abs/1979ApJ...228..939C Fit:Dataset = 1 Method = neldermead Statistic = cash Initial fit statistic = -55208.1 Final fit statistic = -55311.9 at function evaluation 658 Data points = 9171 Degrees of freedom = 9164 Change in statistic = 103.816 g2.fwhm 7.16588 g2.xpos 4062.28 g2.ypos 4239.67 g2.ellip 0.314245 g2.theta 0.836301 g2.ampl 250.614 bgnd.c0 0.268045 Confidence:Dataset = 1 Confidence Method = confidence Iterative Fit Method = None Fitting Method = neldermead Statistic = cash confidence 1-sigma (68.2689%) bounds: Param Best-Fit Lower Bound Upper Bound ----- -------- ----------- ----------- g2.ellip 0.314245 -0.0216664 0.0210256 g2.theta 0.836301 -0.0402018 0.0405143
The commands used to run this thread may be saved in a text file such as fit.py, which can then be executed as a script with exec(open("fit.py").read()). Alternatively, the Sherpa session can be saved to a binary file with the save command (restored with the restore command), or to an editable ASCII file with save_all.
The Sherpa script command may be used to save everything typed on the command line in a Sherpa session:
sherpa> script(filename="sherpa.log", clobber=False)
(Note that restoring a Sherpa session from such a file could be problematic since it may include syntax errors, unwanted fitting trials, et cetera.)
In this thread, we have shown you how you can fit a two-dimensional model to your image data. As with fitting one-dimensional data, care must be taken to avoid reaching a local, rather than global, minimum. It is suggested that you fit your data using different optimization methods—for instance the fit above could be re-done with the Monte Carlo ("moncar") method—and to try different initial parameter values when fitting.
In the example above, the fit was not very complex: possible additions would have been to link the centers of the two Gaussians to be the same; simultaneously fit the same model to more than one dataset (e.g. for multiple observations of a source or when analysing the data from the three imaging cameras in XMM-Newton); include an exposure map in the fit to account for instrumental features such as chip gaps and bad columns; or convolve the model by the PSF during fitting.
|15 May 2008||rewritten for Sherpa Beta|
|16 Jul 2008||minor update to use simplex rather than the neldermead optimisation method as they are the same|
|11 Dec 2008||updated for CIAO 4.1|
|05 Jan 2010||updated for CIAO 4.2|
|07 Jun 2010||updated to include save_model and save_resid functions for saving 2-D fit results|
|12 Jun 2010||updated for CIAO 4.2 Sherpa v2: load_filter now accepts filter files in FITS image format for filtering 2-D data sets. S-Lang version of thread removed|
|15 Dec 2010||updated for CIAO 4.3: the new image_source_component function is available for visualizing the contribution of individual unconvolved model components to a fit|
|30 Jan 2012||reviewed for CIAO 4.4 (no changes)|
|05 Dec 2013||reviewed for CIAO 4.6: no changes|
|09 Dec 2014||reviewed for CIAO 4.7: updated example file and fit results|
|31 Oct 2016||reviewed for CIAO 4.9: updated fit results|
|12 Dec 2018||reviewed for CIAO 4.11, no content change|
|28 Mar 2022||updated for CIAO 4.14, fits updated using data reprocessed with CIAO 4.14 and CalDB 4.9.7.|
|09 Dec 2022||updated for CIAO 4.15, no content change.|