6 Discussion

6.1 Relative contribution of the parameters to the fitness

Figure 4 and the Tables 4, 5, 7, A.2 and A.3 provide the hint that not all parameters have an equal contribution to the fitness. It appears that the $\beta$ and $[Z]_{\rm high}$ parameter can vary considerably and still yield a decent fitness. Moreover, Tables A.2 and 5 indicate that knowledge of the value of the $[Z]_{\rm high}$ parameter results in acceptable values for the other parameters.

The origin of this behaviour lies in the implicit definition of the exponential star formation rate (for $\beta\!=\!1$ one has a decreasing star formation towards a younger age) attached to a linear age-metallicity relation. The latter relation will give less metal-richer stars. The small number of stars with higher metallicity induces a larger variation of the $[Z]_{\rm high}$ parameter without affecting significantly the overall fitness.

   
6.2 Convergence

The fine-tuning of the genetic algorithm is a tedious task. It is not straightforward to find the optimum setting for the problem to be solved. One has to balance the exploring quality through crossovers against the variation of the parameters through (creep) mutations.

We did not want to deal with a mutation dominated search, because it tends to move farther away from an optimum parameter setting in the majority of the cases. We used therefore a relatively high crossover probability (pcross) and we set the mutations at a fixed rate, such that on average only 2.8 mutations occur in the gene pool of each individual.

At a certain stage however one requires the variation of other correlated parameters to obtain an improvement. This becomes particularly necessary when approaching the optimum setting of the parameters. A favourable crossover and mutation might do the trick, but it might take a while before this occurs. We introduced in Sect. 2.6.5 the possibility that two parameters might be more sensitive to mutations than others. This approach gave better results for the majority of the trial cases (see Table 3), but it failed to obtain improvements when changes of one parameter were neutralized through the variation of one or more parameters. The distance-extinction and the age-metallicity degeneracies slow down the convergence of AMORE for f>0.3, see Fig. 4.

One of the modifications to consider for future implementation is a two-chromosome approach. In that case acceptable values for the parameters do not shift out of the population if the overall fitness is less, but still reside in the gene pool as a recessive quality. This however, will require a major extension to PIKAIA and a significant amount of genetic research to be done about dominant and recessive qualities in the AMORE gene pool.
Another modification to consider in order to improve the accuracy and to speed up convergence, is to replace the finite resolution of the digital encoding scheme with a genetic coding based on floating point, i.e. each gene on the chromosome is represented by one floating point number. According to Michalewicz (1996) a real encoding scheme can be superior and improve convergence. Such an encoding scheme is indeed to be included in the next release of PIKAIA 2.0 (Charbonneau; in preparation).

   
6.2.1 Unstable solutions

In one test (fixing $A_{\rm V}$ at one sigma above the original value for model 9) no convergence was achieved and the run was aborted. Because AMORE is quite sensitive to rounding these effects can be circumvented by slightly altering the input parameters. We decided in this case against such an action, because that would make the sample inhomogeneous.

   
6.3 Degeneracy

Isochrones for a particular age and metallicity can be mimicked with another set of isochrones of different age and metallicity (Worthey 1994; Charlot et al. 1996, and references cited therein). This degeneracy of the parameter space increases if one considers the distance and the extinction towards a stellar aggregate. There is no straight forward method to circumvent partial degeneracy of the parameters to be explored. One might consider to apply AMORE for the analysis of colour-colour diagrams in order to rule out the distance, to determine the extinction and a number of other parameters, and finally to determine the distance to the stellar population from one of the CMDs. The combined analysis of colour-magnitude and colour-colour diagrams is expected to improve the results obtained by AMORE so far. However, this requires that at least one additional colour should be available for each star considered above a certain detection and completeness threshold. Moreover, as was mentioned in Sect. 5.3.1, knowledge of the extinction does not automatically imply that the distance can be retrieved accurately.

As demonstrated in Sect. 5.1.1 the degeneracy among parameters becomes noticeable for f > 0.25 or F < 3. This corresponds to systematic offset for each parameter of on average $\sim$ $0.6\sigma_k$ and at maximum $\sim$$1\sigma_k$. The Poisson uncertainty of the original population results in a fitness of $f\!=\!0.43$ ( $F\!\simeq\!1.33$). However, solutions with a comparable fitness do exist due to the degeneracy of the parameter space. A direct consequence is that there is an intrinsic offset present among the parameters amounting to on average $\sim$ $0.4\sigma_k$ and at maximum $\sim$ $0.7\sigma_k$. This intrinsic offset is present in the solutions obtained with AMORE and actually is responsible for slowing down the convergence in the fitness range 0.30<f<0.43. It will therefore be nearly impossible to recover in one pass the original input values. However, some improvements might be obtained by averaging the parameter values obtained from AMORE runs with different initial conditions.