5 Results

AMORE has been tested for a wide range of setups. The results in Table A.1 indicate that AMORE give both acceptable and less acceptable solutions. They are displayed in Figs. 3 and 4.

To better understand what goes on during the genetic evolution we display the results from model A.1-40. Figure 5 displays an example of the evolution of the merit function F for a number of generations. It shows how the initially dispersed individuals gradually find their way, start to cluster together around generation 10, and penetrate the region with acceptable solutions after about 50 generations. After 100 generations the improvements become marginal for this model.

Figure 6 displays for the same model A.1-40 the phenotypical changes of the CMD for several fitnesses during the genetic evolution. The various panels show that the synthetic CMD resembles better and better the "observed'' CMD when the fitness improves. Note that at fitness fem=em0.05 one already gets for the eye appealing solutions.

Figure 7 shows the improvements of the astrophysical parameters as a function of increasing fitness for the models A.1-40 and A.1-51. The panels for distance and extinction show that the distance is systematically underestimated, while the extinction is overestimated. But in general one notices that the astrophysical parameters obtained from model A.1-40 get quite close to the parameters of the CMD to be matched.

Figure 5: Conception diagram of the evolution of the genetic population of model A.1-40 during the optimization process displayed in Fig. 7. Frame a) shows the initial population and the frames b)- l) show the population after several generations up to generation = 400. The outer shaded region indicates solutions for which the difference between the CMDs from the "observed'' and synthetic population is on average between 1-3em$\sigma$. The inner shaded regions marks the region with solutions for which the difference between the "observed'' and synthetic CMDs are less than 1em$\sigma$. Such solutions are close to perfect matches between the "observed'' and synthetic data and are considered to belong to a group of solutions for which one may say "too good to be true''. The solid lines indicate the 10em$\sigma$, 20em$\sigma$, 30em$\sigma$ and 40em$\sigma$ contours.

5.1 Test 1: Parameter values and degeneracy

   
5.1.1 Degeneracy of the parameter space

Table A.1 is displayed in Fig. 3. The clustering in the figure provides an indication that a degeneracy of the parameter space is present near $f\!>\!0.25$ (i.e. $F\!<\!3$, see Eq. (1)). Without a major computational effort it will be difficult to obtain a significant improvement of the parameters once $f\!>\!0.25$. However, $F\!<\!3$ indicates a region in the ( $F_\chi,F_{\rm P}$)-plane for which the systematic offset of the individual parameters from its true value are on average less than $\sqrt{F/n}~\sigma_k\!=\!0.6~\sigma_k$, see Sect. 2.9 for details. In practice it turns out that a strong correlation between three of the eight parameters has the culprit; at least two of them have to change simultaneously in the proper direction in order to improve the fitness (see also Sect. 6.3). They have an average offset of $\sim\!\sqrt{F/3}~\sigma_k\!=\!1~\sigma_k$, while for the remaining parameters this is $\ll$ $1~\sigma_k$.

In addition, Fig. 4 displays the retrieved parameters for all models as a function of fitness. Note, that AMORE systematically underestimates the distance of the test population. On the other hand, the effect of this underestimation is in its turn partially canceled by overestimating the extinction, the upper age limit and the slope of the power-law IMF slightly (see also Fig. 7). Another clue we get from Fig. 4 is that the slope of the SFR $\beta$ is very poorly constrained.

Figure 6: Genetic evolution of the colour-magnitude diagram (CMD) from the first test population. Panel  a) displays the original population to be matched. The physical parameters for this population are described Table 2 and Sect. 4.2. The CMD of the initial trial population is shown in panel b). Panels c)- e) display the resulting CMDs obtained with setup A.1-40 for different fitnesses (see Sect 2.7). The fitnesses f = 0.05, f = 0.19and f = 0.27 are respectively reached after 20, 60 and 80 generations. The dots in the panels b)- e) are used for each matching point, while the red open stars $\bigstar$ in panels c)- j) are the points in the simulation which have no counterpart in the original CMD. Panel f) displays the fitness f = 0.41 as obtained after 341 generations. Note that the CMDs of panels a) and f) as well as c)- e) are visually almost indistinguishable. Panels g)- j) displays the residuals between the simulated and the original CMD ( a; green solid squares $\blacksquare$ ) are those points in the original which have no counterpart in the simulation.

Figure 7: Panels a)- h) display the convergence curves for the parameters of models A.1-40 (dotted line; fitness f=0.41) and A.1-51 (long dashed line; fitness f=0.25). The solid line in the frames a)- f) refers to the value adopted for the original population. The long, dot dashed line in frames i) and j) shows the threshold values to be crossed for acceptable solutions, i.e. F < 2 and ${1\over3}\!<\!f\!<\!1$. The short dashed area in frame j) marks the region where degeneracy of the parameter space becomes noticeable (see Sects. 6.3 for details).

   
5.1.2 Determining values for pcross, rcross, rbrood, pcreep, and pcorr

Looking at Table A.1, a result that is immediately obvious is that pcorr = 0.0 has a strong stabilizing effect on the simulation. Its effects overwhelm the effects of the pcreep parameter and lock $f_{\rm A}$ at a certain value, which may be quite good (e.g. models 10, 11 and 12) or quite poor (e.g. models 28, 29 and 30). Setting ${\tt pcorr} > 0$ is thus preferred to avoid getting locked in a low value for $f_{\rm A}$.

In order to compensate for this strong stabilizing effect, we also evaluate in Table 3 the average fitness of the models when we exclude all models which have pcorr = 0.0.

As expected, the rbrood parameter has a strong influence on the amount of computational time needed. Although the models with high values of rbrood are somewhat better than models with low values, this effect is only marginal. Considering that a high value of rbrood lessens the genetic variation in the gene pool while increasing the computational time needed for a run with several factors, it is desirable to have a low value of rbrood.

The different parameters are not independent, as can be seen from Table A.1 and Table 3. Simply taking the best options in Table 3 yields model 134 for the case in which pcorr = 0.0 has not been corrected for, a reasonable, but not an exceptionally good model.

   
5.2 Test 2: Rounding

Table 4 shows the effect of the accuracy of the retrieved values of the parameters on the evaluation of the fitness. If one applies a rounding to one or two significant digits it is not possible to reproduce the expected fitness, i.e. the expected fitness drops from $f_{\rm trial} = 0.44$ to $f_{\rm trial}=0.28$. A better agreement can be obtained by reporting the values of the parameters with the addition of one or more (apparently insignificant) digit(s). A renewed search (Table 4; round-v3) circumvents the local optimum at $f_{\rm trial}=0.28$ and results in a near global fitness of $f_{\rm trial} = 0.42$, which is close enough to the value expected.

The true $\sigma$ line in the table shows that both the $[Z]_{\rm high}$ and the $\beta$ parameter are the weak links in the overall parameter estimation (see also Fig 4).

 

 

Table 5: Fitness statistics when fixing one parameter at its correct value. Averaged fitness values $\overline {f_{\rm A}}$and their associated standard deviation $\sigma _{n-1}$are obtained from simulations with the setup parameters from models 9, 14, 22, 34, 40 and 52. See Table A.2 for additional details.

parameter	$\overline{f_A}$	$\sigma _{n-1}$
log d	0.299	0.081
$A_{\rm V}$	0.274	0.081
$\log t_{\rm low}$	0.332	0.021
$\log t_{\rm high}$	0.304	0.056
$[Z]_{\rm low}$	0.269	0.036
$[Z]_{\rm high}$	0.361	0.036
$\alpha$	0.305	0.066
$\beta$	0.312	0.076

   
5.3 Test 3: Fixing parameters at the correct value

The results of fixing parameters at the correct value are listed in Table A.2 and an example of the diagnostics is listed in Table 6. Details of the individual setups for these tests are given below. In general, the results of the tests for which one of the parameters was set to the correct value were slightly better than the results for the models for which all parameters are set free, see Tables 4 and 5 for additional details. This behaviour is due to the fact that by forcing one parameter to a fixed value the evolutionary path changes. The models were selected from the results with low and intermediate fitness given in Table A.1.

   
5.3.1 Fixed distance and extinction

Distance and extinction are negatively correlated. When both parameters are set free, a certain degree of degeneracy is to be expected. Fixing one of the parameters at its correct value can break this degeneracy. The result depends strongly on the evolutionary path of the other parameters.

The lower value for the average fitness $\overline{f_{\rm A}}=0.299\hskip 0.15 em\pm\hskip 0.15 em0.081$, when fixing the distance at the correct value, is caused by the presence of one outlier (see Table A.2), which is caused by the age-metallicity degeneracy. Excluding this value results in an average fitness of $\overline{f_{\rm A}}=0.328\hskip 0.15 em\pm\hskip 0.15 em0.039$. In general: the extinction can be reliably retrieved when fixing the distance.

When considering a fixed extinction, the results show a strong variation in both age and metallicity. It should also be noted that the average $^{10}\log$em(distance (pc)) retrieved is only 3.8953em$\pm$em0.0066. This is more than one sigma away from the optimum value for the distance (see Table 4). This is an indication that retrieval of the distance by fixing the extinction is hampered by the age-metallicity degeneracy. Therefore, the distance cannot be reliably retrieved when fixing the extinction to its correct value.

   
5.3.2 Fixed age and metallicity

Fixing one of the age limits results in values for both the age and metallicity which are close to the input values. This is due to the (partial) breaking of the age-metallicity degeneracy. The distance-extinction degeneracy remains. The results also suggest that the age-metallicity degeneracy has a stronger impact on the fitness than the distance-extinction degeneracy.

Fixing the upper metallicity limit to its correct value shows that the values for age and metallicity come closer to their original, input values. This is quite in contrast with the results obtained from fixing the lower metallicity to its correct value. Table A.2 shows that both the high metallicity limit and the slope of the exponential SFR are not well constrained. This behaviour can be accounted to the implicit shape of the linear age-metallicity relation adopted in the HRD-GST. The number of high metallicity stars is smaller than the number of low metallicity stars due to the adopted, exponentially decreasing ($\beta=1$), star formation rate. The consequence is that the high metallicity limit can be better determined when both the low and high limits are determined in union.

 

 

Table 6: Description of the diagnostic statistics for model A.1-40 when fixing one parameter at its correct value, see Sects. 5.3, 2.7 and 6.2.1 for additional details.

model	$N_{\rm O,not}$	$N_{\rm S,not}$	$N_{\rm match}$	$F_\chi$	$F_{\rm P}$	F	$f_{\rm A}$
ideal	0	0	5000	0.000	0.000	0.000	1.000
free	60	60	4940	0.847	0.849	1.438	0.410
fixed: log d (pc)	155	155	4845	0.903	2.192	5.621	0.151
fixed: $A_{\rm V}$	103	103	4897	0.840	1.457	2.827	0.261
fixed: $\log t_{\rm low}$ (yr)	76	76	4924	0.817	1.075	1.823	0.354
fixed: $\log t_{\rm high}$ (yr)	104	104	4896	0.902	1.471	2.978	0.251
fixed: $[Z]_{\rm low}$	94	94	4906	0.836	1.329	2.466	0.289
fixed: $[Z]_{\rm high}$	76	76	4924	0.815	1.075	1.820	0.355
fixed: $\alpha$	74	74	4926	0.834	1.047	1.799	0.357
fixed: $\beta$	80	80	4920	0.727	1.131	1.808	0.356

 

 

Table 7: Fitness statistics when fixing one parameter $\pm \hskip 0.15 em1\sigma$ from its original value. Averaged fitness values $\overline {f_{\rm A}}$ and their associated standard deviation $\sigma _{n-1}$are obtained from simulations with the setup parameters from models 9, 14, 22, 34, 40 and 52. See Table A.2 for additional details.

parameter	offset	$\overline{f_A}$	$\sigma _{n-1}$
log d	$-1\sigma$	0.291	0.048
	$+1\sigma$	0.280	0.066
$A_{\rm V}$	$+1\sigma$	0.284	0.020
$\log t_{\rm low}$	$-1\sigma$	0.258	0.044
	$+1\sigma$	0.279	0.012
$\log t_{\rm high}$	$-1\sigma$	0.229	0.022
	$+1\sigma$	0.207	0.014
$[Z]_{\rm low}$	$-1\sigma$	0.254	0.066
	$+1\sigma$	0.315	0.027
$[Z]_{\rm high}$	$-1\sigma$	0.270	0.054
	$+1\sigma$	0.283	0.018
$\alpha$	$-1\sigma$	0.266	0.025
	$+1\sigma$	0.273	0.027
$\beta$	$-1\sigma$	0.232	0.071
	$+1\sigma$	0.276	0.011

5.3.3 Fixed slope for the power-law IMF

Fixing the slope $\alpha$ of the power-law IMF to its correct value ensures stability in the magnitude direction of the CMD. This implies that the degeneracy in the colour dependent parameters, like age, metallicity, and partly the star formation rate, becomes more apparent. Although the overall results are quite good, only model A.2-14 is significantly affected by this degeneracy. If we leave out model A.2-14 from the statistics the average fitness becomes $\overline{f_{\rm A}}=0.331\pm0.020$.

5.3.4 Fixed slope for the exponential SFR

The exponential star formation rate parameter $\beta$is tied to the age and metallicity range, see also Sect. 5.3.2. Fixing the parameter $\beta$ better constrains in particular the upper metallicity limit. However, it does not avoid that the genetic evolution enters into a local age-metallicity gap, see Table A.2. Excluding model A.2-34 improves the average fitness in Table 5 to $\overline{f_{\rm A}}=0.342 \pm 0.029$.

   
5.4 Test 4: Fixing parameters at the wrong value

The results of fixing the parameters at a $\pm \hskip 0.15 em1\sigma$offset from its original value are listed in Table A.3. An example of the diagnostics of these tests are given in Table 8. Details of individual setups are given below.

   
5.4.1 Erroneous distance

A wrong assumption about the distance gives in a vertical shift in the CMD (all other parameters give a diagonal shift) and cannot be masked out through a correlated change of any of the other parameters. This results in quite a wide range in the values of the other parameters, except the value of $\alpha$. The unexpected result is that irrespective if the distance is too short or too far: the power-law IMF slope flattens!

If the distance is overestimated there are more synthetic stars present at fainter magnitudes. To get relatively more synthetic stars at brighter magnitudes one needs to flatten the power-law IMF slope.

If the distance is underestimated then more synthetic stars are present at brighter magnitudes. One expects that a steeper power-law IMF slope is required as compensation. This is not always true, see Table A.3. Stars pop up at the lower end of the main sequence. They are taken away from the stars located at brighter and brighter magnitudes. One therefore requires also in this case a flatter IMF slope.

A flatter slope of the power-law IMF can be a hint that the distance of the stellar aggregate is wrong. Or it might be a hint that the zero point of the adopted synthetic photometric system is different from the actual photometric system used.

Recognizing that the slope is indeed flatter than the majority of the other cases outlined in Table 4 one may start to explore the assumption that the distance is wrong: release the constraint during the next exploration.

The sensitivity to the distance implies that AMORE can be used to determine the distance to a stellar aggregate quite reliably. A bonus is that due to an initially wrongly assumed distance the extinction is in most cases better constrained.

5.4.2 Erroneous extinction, age and metallicity

In (V, V-I) CMDs a strong correlation between extinction, age and metallicity exists (see also Ng & Bertelli 1996 and references cited therein). A higher value of the extinction can be compensated by a younger age and/or a lower metallicity. Indeed, the results in Table A.3 show that this actually occurs for the lower age and metallicity limit. The upper age and metallicity limit, however, drifts away in the opposite direction to compensate for "erroneous corrections'' applied by other parameters.

A higher value of the star formation index results in a lower number of stars at the upper age metallicity limit. To get a sufficient number of high metallicity stars one has to stretch the upper metallicity limit to a slightly higher value.

We further notice that a wrong value for the age and metallicity does not affect the extinction significantly. Our findings indirectly supports the method to determine high resolution ( $4\hbox{$^\prime$ }\times4\hbox{$^\prime$ }$) extinction maps towards the Galactic bulge by Schultheis et al. (1999) with the data obtained for the DeNIS project (Epchtein et al. 1997).

5.4.3 Erroneous slope for power-law IMF

The slope of the power-law IMF is very strongly constrained (Ng 1998) for the test population. As a consequence the changes in the values of the remaining parameters are not extremely large. The slightly larger value of the slope $\alpha$pushes slightly more synthetic stars to fainter magnitudes, introducing a relative deficiency of stars at brighter magnitudes. This is compensated through a younger age, a decrease of the lower metallicity limit and an increase of the upper metallicity limit. The higher value for the upper age and metallicity limit compensates in its turn for the overestimation of the exponential star formation index.

A lower value of $\alpha$ is partly compensated for by lowering the upper age limit, lowering the lower metallicity limit and overestimating the upper metallicity limit.

5.4.4 Erroneous index for exponential SFR

A larger index for an exponentially decreasing SFR pushes more stars of the population to the blue edge of the CMD, resulting in a slightly bluer stellar population. AMORE compensates this by mainly increasing the upper metallicity limit, i.e. reddening the synthetic stellar population.

A lower value for the SFR index has the opposite effect. AMORE compensates for the now slightly redder population by lowering the upper metallicity limit, making the population bluer.

 

 

Table 8: Description of the diagnostic statistics for simulations with setup parameters from model A.1-40. One of the parameters is forced to a value $\pm \hskip 0.15 em1\sigma$ from its original value. See Sects. 5.4 and 2.7 for additional details.

model	offset	$N_{\rm O,not}$	$N_{\rm S,not}$	$N_{\rm match}$	$F_\chi$	$F_{\rm P}$	F	$f_{\rm A}$
ideal		0	0	5000	0.000	0.000	0.000	1.000
free		60	60	4940	0.847	0.849	1.438	0.410
fixed: log d (pc)	$-1\sigma$	91	91	4909	0.788	1.287	2.278	0.305
	$+1\sigma$	85	85	4915	0.722	1.202	1.967	0.337
fixed: $A_{\rm V}$	$+1\sigma$	98	98	4902	0.932	1.386	3.790	0.264
fixed: $\log t_{\rm low}$ (yr)	$-1\sigma$	91	91	4901	0.897	1.287	2.460	0.289
	$+1\sigma$	94	94	4906	0.905	1.329	2.587	0.278
fixed: $\log t_{\rm high}$ (yr)	$-1\sigma$	106	106	4894	0.977	1.499	3.202	0.238
	$+1\sigma$	126	126	4874	1.017	1.782	4.211	0.192
fixed: $[Z]_{\rm low}$	$-1\sigma$	78	78	4922	0.907	1.103	2.043	0.329
	$+1\sigma$	80	80	4920	0.835	1.131	1.977	0.336
fixed: $[Z]_{\rm high}$	$-1\sigma$	148	148	4852	0.896	2.094	5.188	0.162
	$+1\sigma$	94	94	4906	0.867	1.329	2.519	0.284
fixed: $\alpha$	$-1\sigma$	95	95	4905	0.886	1.344	2.590	0.279
	$+1\sigma$	103	103	4897	0.894	1.457	2.921	0.255
fixed: $\beta$	$-1\sigma$	145	145	4855	0.930	2.051	5.071	0.165
	$+1\sigma$	103	103	4897	0.834	1.457	2.818	0.262