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 ($C$) within specific instrument channels, ($I$). This observed spectrum is related to the actual spectrum of the source ($f(E)$) by:

$$C(I) = \int f(E) R(I,E) dE$$ (2.1)

Where $R(I,E)$ is the instrumental response and is proportional to the probability that an incoming photon of energy $E$ will be detected in channel $I$. Ideally, then, we would like to determine the actual spectrum of a source, $f(E)$, by inverting this equation, thus deriving $f(E)$ for a given set of $C(I)$. Regrettably, this is not possible in general, as such inversions tend to be non-unique and unstable to small changes in $C(I)$. (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, $f(E)$, 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 ($C_p(I)$) is calculated and compared to the observed data ($C(I)$). 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, $f_b(E)$, 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 $\chi^2$, defined as follows:

$$\chi^2 = \sum {(C(I)-C_p(I))^2\over{(\sigma(I))^2}}$$ (2.2)

where $\sigma(I)$ is the (generally unknown) error for channel $I$ (e.g., if $C(I)$ are counts then $\sigma(I)$ is usually estimated by $\sqrt{C(I)}$; see e.g. Wheaton et al. (1995) for other possibilities).

Once a “best-fit” model is obtained, one must ask two questions:

1. How confident can one be that the observed $C(I)$ can have been produced by the best-fit model $f_b(E)$? The answer to this question is known as the “goodness-of-fit” of the model. The $\chi^2$ statistic provides a well-known-goodness-of-fit criterion for a given number of degrees of freedom ($\nu$, which is calculated as the number of channels minus the number of model parameters) and for a given confidence level. If $\chi^2$ exceeds a critical value (tabulated in many statistics texts) one can conclude that $f_b(E)$ is not an adequate model for $C(I)$. As a general rule, one wants the “reduced $\chi^2$” ( $\sim\chi^2/\nu$) to be approximately equal to one (i.e. $\chi^2 \sim \nu$). A reduced $\chi^2$ that is much greater than one indicates a poor fit, while a reduced $\chi^2$ that is much less than one indicates that the errors on the data have been over-estimated. Even if the best-fit model ($f_b(E)$) does pass the “goodness-of-fit” test, one still cannot say that $f_b(E)$ 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 judgment.

2. For a given best-fit parameter ($p1$), 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 $\chi^2$ increases by a particular amount above the minimum, or “best-fit” value. The amount that the $\chi^2$ is allowed to increase (also referred to as the critical $\Delta\chi^2$) 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