This chapter is comprised of a brief description of the basics of spectral fitting, their application in XSPEC, and some helpful hints on how to approach particular problems.
Although we use a spectrometer to try to find out the spectrum of a source, what the spectrometer obtains is not the actual spectrum, but rather photon counts (C) within specific instrument channels, (I). This observed spectrum is related to the actual spectrum of the source (f(E)), such that:
The usual alternative is to try to choose a model spectrum, f(E), that can be described in terms of a few parameters (i.e., f(E,p1,p2,...)), and match, or ``fit" it to the data obtained by the spectrometer. For each f(E), a predicted count spectrum (Cp(I)) is calculated and compared to the observed data (C(I)). Then a ``fit statistic'' is computed from the comparison, which enables one to judge whether the model spectrum ``fits'' the data obtained by the spectrometer.
The model parameters then are varied to find the parameter values that give the most desirable fit statistic. These values are referred to as the best-fit parameters. The model spectrum, fb(E), made up of the best-fit parameters is considered to be the best-fit model.
The most common fit statistic in use for determining the ``best-fit" model is , defined as follows:
where is the error for channel I (e.g., if C(I) are counts then is usually estimated by ; see e.g. Wheaton et.al. 1995 for other possibilities).
Once a ``best-fit" model is obtained, one must ask two questions:
The statistic provides a well-known goodness-of-fit criterion for a given number of degrees of freedom (, which is calculated as the number of channels minus the number of model parameters) and for a given confidence level. If exceeds a critical value (tabulated in many statistics texts) one can conclude that fb(E) is not an adequate model for C(I). As a general rule, one wants the ``reduced " (/) to be approximately equal to one ( ). A reduced that is much greater than one indicates a poor fit, while a reduced that is much less than one indicates that the errors on the data have been over-estimated. Even if the best-fit model (fb(E)) does pass the ``goodness-of-fit" test, one still cannot say that fb(E) is the only acceptable model. For example, if the data used in the fit are not particularly good, one may be able to find many different models for which adequate fits can be found. In such a case, the choice of the correct model to fit is a matter of scientific judgement.
The confidence interval for a given parameter is computed by varying the parameter value until the increases by a particular amount above the minimum, or ``best-fit" value.
The amount that the is allowed to increase (also referred to as the critical ) depends on the confidence level one requires, and on the number of parameters whose confidence space is being calculated. The critical for common cases are given in the following table (from Avni, 1976):
There is a good discussion of confidence ranges in Press et al., (1992) for readers who want more details2.1