3.1.11. Error: Calculate the errors of the fitted parameters

3.1.11.1. Overview

This command calculates the error on a certain parameter or parameter range, if that parameter is free (thawn). Standard the 1\sigma error is calculated, which is equivalent to a 68 % confidence level. So \Delta\chi^2 is equal to 1 for a single parameter error. The \Delta\chi^2 value can be set, such that for instance 90 % errors are determined (see the table at the bottom of this page for \Delta\chi^2 values with their corresponding confidence levels).

SPEX determines the error bounds by iteratively modifying the parameter of interest and calculating \chi^2 as a function of the parameter. During this process the other free parameters of the model may vary. The iteration stops when \chi^2 = \chi_{\min}^2 + \Delta \chi^2, where \Delta \chi^2 is a parameter that can be set separately. The iteration steps are displayed. It is advised to check them, because sometimes the fit at a trial parameter converges to a different solution branch, therefore creating a discontinuous jump in \chi^2. In those situations it is better to find the error bounds by varying the search parameter by hand.

Note that SPEX remembers the parameter range for which you did your last error search. This saves you often typing in sector numbers or component numbers if you keep the same spectral component for your next error search.

If the error search reveals a new minimum in \chi^2 space that was not found in the fit, SPEX will save the parameters of this new minimum in the file spex_lower_chi.com. After the error search these parameters can be set through the command log exe spex_lower_chi, in order to direct the model to the new minimum. Note that in the file spex_lower_chi.com the parameter for which the error was calculated is “frozen”.

Warning

A parameter must have the status “thawn” in order to be able to determine its errors.

Warning

The first time that you use the error command, you need to provide the sector number before it is remembered by SPEX. For example, error 1 1 norm.

3.1.11.2. Syntax

The following syntax rules apply:

error [[[#i1:] #i2:] #a:] : Determine the error bars for the parameters specified by the sector range #i1: (optional), component range #i2 (optional) and parameter range #a: (optional). If not specified, the range for the last call will be used. On startup, this is the first parameter of the first component of the first sector.
error dchi #r : This command changes the \Delta\chi^2, to the value #r. Default at startup and recommended value to use is 1, for other confidence levels see the Table below.
error start #r : This command gives an initial guess of the error bar, from where to start searching the relevant error. This can be helpful for determining the errors on normalization parameters, as otherwise SPEX may from a rather small value. To return to the initial situation, put #r=0 (automatic error search).

3.1.11.3. Examples

error norm : Find the error for the normalization of the current component
error 2:3 norm:gamm : determines the error on all free parameters between “norm” and “gamm” for components 2:3
error start 0.01 : Start calculating the error beginning with an initial guess of 0.01
error dchi 2.71 : Calculate from now onwards the 90 % error, for 1 degree of freedom. (Not recommended, use standard 68 % errors instead!)
Table of \Delta\chi^2 as a function of confidence level, P, and degrees of freedom, \nu.

P

1

2

3

4

5

6

68.3%

1.00

2.30

3.53

4.72

5.89

7.04

90%

2.71

4.61

6.25

7.78

9.24

10.6

95.4%

4.00

6.17

8.02

9.70

11.3

12.8

99%

6.63

9.21

11.3

13.3

15.1

16.8

99.99%

15.1

18.4

21.1

13.5

25.7

27.8