** Up:** Stellar population synthesis diagnostics

**Subsections**

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 error ellipse.
In this range one can
assume that the errors follow a Normal distribution. The is a measure for the goodness of the fit
for *N* observed points in a CMD and is defined as:
| |
(1) |

| (2) |

where the subscripts (*O*,*S*) refer respectively to the observed and synthetic
CMD, (*c*_{i},*m*_{i}) respectively to the colour and magnitude for each
point *i* in the CMD,
is the error ellipse around each point *i*,
and is the
(colour,emmagnitude) difference between the observed and the
synthetic star. In addition, the reduced merit function
is defined as:

| |
(3) |

where refers to the number of points found within
of the error ellipse for each point between
the observed and the synthetic CMD. Depending on the selection criteria
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 , 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.
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
and ), should in the ideal case
be smaller than the Poisson uncertainty
for the total number of *N*_{O} observed and *N*_{S} synthetic points:

| |
(4) |

or written as the Poisson merit function

| |
(5) |

All the residual points can be placed in a CMD.
This diagram contains indications
about parameters that need to be optimized.
In practice 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.
Both and 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 and are about 1 or smaller.
The best fit can therefore be
obtained by minimizing the global merit function *F*,
which is defined as

| |
(6) |

Both and are in units of , say
and respectively.
The difference between the observed and synthetic
points is therefore on average about ,where 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 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.

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
, each with its own average
error ellipse. Equation (1) can then be re-written to
| |
(7) |

| (8) |

where *M* is the resolution of the binned CMD, i.e. the number of
colour-magnitude bins,
and *k*_{j} 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.
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:

| |
(9) |

Acceptable solutions yield or .
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 for each parameter *k* is defined in such a way
that the global merit *F* changes with 1 when
this parameter is varied:

| |
(10) |

where *F*_{k} is the merit for parameter *k* and
is the merit obtained for the fittest population.
This procedure corresponds in Fig. 2 with a jump
in the ()-plane from the contour of the optimum solution
to a contour displaced by exactly 1.

Note that the contour of the fittest population has the value
for ,while the contour for the estimated uncertainty has the value
for .

** Up:** Stellar population synthesis diagnostics

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