Determine asymmetric model parameter errors for a model component

What you will learn

You will learn how to use cterror to derive asymmetric model parameter errors from Fermi-LAT data using a likelihood profile method.

The ctlike tool computes errors for each fitted parameter value based on the covariance matrix of the fit. This assumes a Gaussian approximation of the uncertainties around the best fitting values. Uncertainties may however by asymmetric.

To test the symmetry of the uncertainties you can use the cterror tool. cterror will make use of the likelihood profile to determine the confidence interval for each fitted parameter, allowing thus also for asymmetric errors. You use cterror as follows:

$ cterror
Input event list, counts cube or observation definition XML file [events.fits] obs.xml
Source of interest [Crab] Vela
Input model definiton XML file [$CTOOLS/share/models/crab.xml] vela_results.xml
Output model definiton XML file [results.xml] vela_errors.xml

The confidence intervals can then be extracted from the cterror.log log file that was created by cterror. As you will see, the parameter errors are essentially symmetric and very close to the values determined using the covariance matrix.

2019-04-04T15:04:51: +================================================+
2019-04-04T15:04:51: | Compute error for source "Vela" parameter "RA" |
2019-04-04T15:04:51: +================================================+
2019-04-04T15:04:51:  Confidence level ..........: 68 %
2019-04-04T15:04:51:  Log-likelihood difference .: 0.494473
2019-04-04T15:04:51:  Initial factor range ......: [128.819554251886, 128.851989886469]
2019-04-04T15:05:10:  Lower parameter factor ....: 128.834
2019-04-04T15:05:10:  Upper parameter factor ....: 128.837
2019-04-04T15:05:10:  Error from curvature ......: 0.00162178172918532 deg
2019-04-04T15:05:10:  Error from profile ........: 0.00169067577726878 deg
2019-04-04T15:05:10:  Negative profile error ....: 0.00165900035287336 deg
2019-04-04T15:05:10:  Positive profile error ....: 0.0017223512016642 deg
2019-04-04T15:05:10:
2019-04-04T15:05:10: +=================================================+
2019-04-04T15:05:10: | Compute error for source "Vela" parameter "DEC" |
2019-04-04T15:05:10: +=================================================+
2019-04-04T15:05:10:  Confidence level ..........: 68 %
2019-04-04T15:05:10:  Log-likelihood difference .: 0.494473
2019-04-04T15:05:10:  Initial factor range ......: [-45.1949881925207, -45.1721741960936]
2019-04-04T15:05:31:  Lower parameter factor ....: -45.1847
2019-04-04T15:05:31:  Upper parameter factor ....: -45.1824
2019-04-04T15:05:31:  Error from curvature ......: 0.00114069982135549 deg
2019-04-04T15:05:31:  Error from profile ........: 0.0011654855352532 deg
2019-04-04T15:05:31:  Negative profile error ....: 0.00115852325606625 deg
2019-04-04T15:05:31:  Positive profile error ....: 0.00117244781444015 deg
2019-04-04T15:05:31:
2019-04-04T15:05:31: +=======================================================+
2019-04-04T15:05:31: | Compute error for source "Vela" parameter "Prefactor" |
2019-04-04T15:05:31: +=======================================================+
2019-04-04T15:05:31:  Confidence level ..........: 68 %
2019-04-04T15:05:31:  Log-likelihood difference .: 0.494473
2019-04-04T15:05:31:  Initial factor range ......: [1e-07, 9.09784331943508]
2019-04-04T15:06:15:  Lower parameter factor ....: 3.93674
2019-04-04T15:06:15:  Upper parameter factor ....: 4.89558
2019-04-04T15:06:15:  Error from curvature ......: 4.74564602405264e-10 ph/cm2/s/MeV
2019-04-04T15:06:15:  Error from profile ........: 4.79421031332425e-10 ph/cm2/s/MeV
2019-04-04T15:06:15:  Negative profile error ....: 4.15456324070932e-10 ph/cm2/s/MeV
2019-04-04T15:06:15:  Positive profile error ....: 5.43385738593917e-10 ph/cm2/s/MeV
2019-04-04T15:06:15:
2019-04-04T15:06:15: +====================================================+
2019-04-04T15:06:15: | Compute error for source "Vela" parameter "Index1" |
2019-04-04T15:06:15: +====================================================+
2019-04-04T15:06:15:  Confidence level ..........: 68 %
2019-04-04T15:06:15:  Log-likelihood difference .: 0.494473
2019-04-04T15:06:15:  Initial factor range ......: [1.03239332172832, 1.65461803910225]
2019-04-04T15:07:01:  Lower parameter factor ....: 1.31084
2019-04-04T15:07:01:  Upper parameter factor ....: 1.37275
2019-04-04T15:07:01:  Error from curvature ......: 0.0311112358686962
2019-04-04T15:07:01:  Error from profile ........: 0.0309517300207367
2019-04-04T15:07:01:  Negative profile error ....: 0.0326607212488756
2019-04-04T15:07:01:  Positive profile error ....: 0.0292427387925978
2019-04-04T15:07:01:
2019-04-04T15:07:01: +==========================================================+
2019-04-04T15:07:01: | Compute error for source "Vela" parameter "CutoffEnergy" |
2019-04-04T15:07:01: +==========================================================+
2019-04-04T15:07:01:  Confidence level ..........: 68 %
2019-04-04T15:07:01:  Log-likelihood difference .: 0.494473
2019-04-04T15:07:01:  Initial factor range ......: [0.001, 2.80853448937906]
2019-04-04T15:07:45:  Lower parameter factor ....: 0.81239
2019-04-04T15:07:45:  Upper parameter factor ....: 1.17294
2019-04-04T15:07:45:  Error from curvature ......: 181.899981373502 MeV
2019-04-04T15:07:45:  Error from profile ........: 180.277267455479 MeV
2019-04-04T15:07:45:  Negative profile error ....: 177.144641582696 MeV
2019-04-04T15:07:45:  Positive profile error ....: 183.409893328263 MeV
2019-04-04T15:07:45:
2019-04-04T15:07:45: +====================================================+
2019-04-04T15:07:45: | Compute error for source "Vela" parameter "Index2" |
2019-04-04T15:07:45: +====================================================+
2019-04-04T15:07:45:  Confidence level ..........: 68 %
2019-04-04T15:07:45:  Log-likelihood difference .: 0.494473
2019-04-04T15:07:45:  Initial factor range ......: [0.3080285257564, 0.866644263519715]
2019-04-04T15:08:32:  Lower parameter factor ....: 0.559037
2019-04-04T15:08:32:  Upper parameter factor ....: 0.615056
2019-04-04T15:08:32:  Error from curvature ......: 0.0279307868881658
2019-04-04T15:08:32:  Error from profile ........: 0.0280092058454933
2019-04-04T15:08:32:  Negative profile error ....: 0.028299015035617
2019-04-04T15:08:32:  Positive profile error ....: 0.0277193966553695