## The Basics of Spectral Fitting
Although we use a spectrometer to measure the spectrum of a source,
what the spectrometer obtains is not the actual spectrum, but rather
photon counts () within specific instrument channels, (). This
observed spectrum is related to the actual spectrum of the source
() by:
Where is the instrumental response and is proportional to the probability that an incoming photon of energy will be detected in channel . Ideally, then, we would like to determine the actual spectrum of a source, , by inverting this equation, thus deriving for a given set of . Regrettably, this is not possible in general, as such inversions tend to be non-unique and unstable to small changes in . (For examples of attempts to circumvent these problems see Blissett & Cruise (1979); Kahn & Blissett (1980); Loredo & Epstein (1989)). The usual alternative is to choose a model spectrum, , that can be described in terms of a few parameters (i.e., ), and match, or “fit” it to the data obtained by the spectrometer. For each , a predicted count spectrum () is calculated and compared to the observed data (). Then a “fit statistic” is computed from the comparison and used 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, , 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 (generally unknown) error for channel (e.g., if 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: - How confident can one be that the observed can have been
produced by the best-fit model ? The answer to this
question is known as the “goodness-of-fit” of the model. 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 is
*not*an adequate model for . As a general rule, one wants the “reduced ” ( ) to be approximately equal to one (i.e. ). 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 () does pass the “goodness-of-fit” test, one still cannot say that 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 judgment. - For a given best-fit parameter (), what is the range of
values within which one can be confident the true value of the
parameter lies? The answer to this question is the “confidence
interval” for the parameter. 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):
Confidence Parameters 1 2 3 0.68 1.00 2.30 3.50 0.90 2.71 4.61 6.25 0.99 6.63 9.21 11.30
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