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2 Method

2.1 The chi-squared merit function

The method is based on minimizing a chi-squared merit function between all the observed points in a CMD which have a corresponding point in the synthetic CMD within a $3-5\sigma(c_i,m_i)$ error ellipse. In this range one can assume that the errors follow a Normal distribution. The $\chi^2$is a measure for the goodness of the fit for N observed points in a CMD and is defined as:
\quad \chi^2(O,S) & = & \sum_{i=1}^N 
 ...{{\delta m(c_i,m_i)}\over{\sigma_i(c_{i,O},m_{i,O})}}\right)^2 \;,\end{eqnarray} (1)
where the subscripts (O,S) refer respectively to the observed and synthetic CMD, (ci,mi) respectively to the colour and magnitude for each point i in the CMD, $\sigma_i(c_{i,O},m_{i,O})$ is the error ellipse around each point i, and $\delta m(c_i,m_i)$ is the (colour,emmagnitude) difference between the observed and the synthetic star. In addition, the reduced merit function $F_\chi$ is defined as:
\quad F_\chi = \overline{\chi^2} = \sqrt{\chi^2(O,S)/N_{\rm match}} \;,\end{displaymath} (3)
where $N_{\rm match}$ refers to the number of points found within $3-5\sigma(c_i,m_i)$ of the error ellipse for each point between the observed and the synthetic CMD. Depending on the selection criteria $\overline{\chi^2}$ is defined to be smaller than or equal to 5. In general this is one of the functions that should be minimized. Acceptable models are those with $F_\chi\!\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
 ...\offinterlineskip\halign{\hfil$\scriptscriptstyle ... , i.e. models for which the difference in the (colour, magnitude) of the matched points between the observed and synthetic CMDs is on average less than 1$\sigma$.

2.2 The Poisson merit function

There are some points which do not have counterparts in the observed or the synthetic CMD, due to the limits imposed in the comparison. For a good fit the number of unmatched points, observed and simulated (respectively $N_{O,\rm not}$ and $N_{S,\rm not}$), should in the ideal case be smaller than the Poisson uncertainty for the total number of NO observed and NS synthetic points:
\quad N_{O,\rm not} + N_{S,\rm not} \mathrel{\mathchoice {\v...
 ...riptstyle ... (4)
or written as the Poisson merit function $F\rm _P$
\quad F_{\rm P} = {{N_{O,\rm not} +
N_{S,\rm not}}\over{\sqr...
 ...halign{\hfil$\scriptscriptstyle ... (5)
All the residual points can be placed in a CMD. This diagram contains indications about parameters that need to be optimized. In practice $F\rm _P$ will not be smaller than 1, due to simplifications adopted for the model CMD, to a not optimum representation of some evolutionary phases or even to limitations in the transformation from the theoretical to the observational plane. In Sect. 3 the diagnostics derived from the CMD filled with the residuals are explained in more detail. The CMDs filled with the residuals are hereafter also referred to as the diagnostic diagrams.

2.3 The global merit function

Both $F_\chi$ and $F\rm _P$ span a two-dimensional plane, see for example Fig. 2, which displays the merit for the various models (see Sect. 3). An acceptable model is obtained when both $F_\chi$ and $F\rm _P$ are about 1 or smaller. The best fit can therefore be obtained by minimizing the global merit function F, which is defined as
\quad F = F_\chi^2 + F\rm _P^2 \;.\end{displaymath} (6)
Both $F\rm _P$ and $F_\chi$ are in units of $\sigma$, say $\sigma\rm _P$ and $\sigma_\chi$respectively. The difference between the observed and synthetic points is therefore on average about $\sqrt{F}\sigma$,where $\sigma\!=\!\sigma(F_{\rm P},F_\chi)$ is a function of the average chi-squared difference of the matched points combined with the Poisson uncertainty of the unmatched points. In general the minimization of the global merit function is mainly due to minimization of the Poisson merit function, i.e. the reduction of the number of unmatched points. An acceptable fit of the data is obtained when $F\!\simeq\!2$ or smaller.

Note that this procedure is not limited to finding the best fit for a single-colour, magnitude diagram. It can easily be adjusted to fit multi-colour, magnitude diagrams.

2.4 Speeding up

A comparison between the observed and the synthetic CMD on a point by (nearest) point basis can slow down the fitting procedure considerably, especially when a CMD consists of many data points. To speed up the whole procedure one can make a concession in accuracy by binning the N data points in M average, colour-magnitude boxes $(\overline{c}_j,\overline{m}_j)$, each with its own average error ellipse. Equation (1) can then be re-written to
\chi^2(O,S) & = & \sum_{j=1}^M 
{(\overline{c}_{j,O}-...{\sigma}_j(\overline{c}_{j,O},\overline{m}_{j,O})}}\right)^2 \;,\end{eqnarray} (7)
where M is the resolution of the binned CMD, i.e. the number of colour-magnitude bins, and kj is the number of points in each bin. This method has not been applied here, because the gain is small for the low number of objects used for the examples in Sect. 3.

2.5 Optimizing with a genetic algorithm

This paper does not deal with the actual implementation of an automated search program, which will be the subject of a forthcoming paper (Ng et al., in preparation). The following part has been included for completeness as an example for a possible approach.

Genetic algorithms are a class of heuristic search techniques, capable of finding in a robust way the optimum setting for a problem (Charbonneau 1995; Charbonneauem&emKnapp 1996 and references cited therein). The optimum setting is searched with a so-called fitness parameter f, which ranges from zero (worst) to one (best). The fitness parameter f can be expressed as follows in terms of the global merit function:
\quad f = {1\over{1+F}} \;.\end{displaymath} (9)
Acceptable solutions yield $F\!\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
 ...\offinterlineskip\halign{\hfil$\scriptscriptstyle ... or $f\!\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
 ...ineskip\halign{\hfil$\scriptscriptstyle ... . An estimate of the uncertainty of the input parameters can be obtained by doing multiple, time consuming simulations. However, another way to estimate the uncertainty is to vary for the fittest solution one parameter at a time. The fitness $f_{\sigma,k}$for each parameter k is defined in such a way that the global merit F changes with 1$\sigma(F_{\rm P},F_\chi)$ when this parameter is varied:
\quad f_{\sigma,k} = {1\over{1+\vert\sqrt{F_k}-\sqrt{F_{\rm min}}-1\vert}} \;,\end{displaymath} (10)
where Fk is the merit for parameter k and $F_{\rm min}$ is the merit obtained for the fittest population. This procedure corresponds in Fig. 2 with a jump in the ($F_{\rm P},F_\chi$)-plane from the contour of the optimum solution to a contour displaced by exactly 1$\sigma(F_{\rm P},F_\chi)$.

Note that the contour of the fittest population has the value $f_{\sigma,k}\!=\!{1\over2}$ for $F_k\!=\!F_{\rm min}$,while the contour for the estimated uncertainty has the value $f_{\sigma,k}\!=\!1$ for $\sqrt{F_k}\!=\!\sqrt{F_{\rm min}}+1$.

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