4 Calibration and classification

The assumed number of model populations (groups) is constrained by the limited number of sample stars (357) and by the number of parameters to be fitted (18 per group). Its relevant value may be determined by means of a Wilks test (Soubiran et al. [1990]; Wilks [1963]). This test basically checks the significance of the likelihood increase obtained when the number of free parameters (in particular the number of groups) is increased. Considering that we are dealing with 2 or 3 types of variable stars and 2 or 3 galactic populations (Luri et al. [1996b]; Alvarez et al. [1997]), several computations were carried out with 2, 3 and 4 groups. Wilks test indicated that the four-groups solution was still significant. Computation with five groups was not pursued until convergence because the number of free parameters was obviously too high (89 for 357 stars). To this respect, it is worth remarking that Group 1 was always clearly separated, while the other groups were mixed when less than 4 groups were used.

The fitted parameters of the model distributions corresponding to the 4-groups solution are given in Table 1 and in Sect. 5.1, with $1\sigma$ error bars derived from Monte Carlo simulations. The de-biased model distributions of period are shown in Fig. 3. The tridimensional {P, M_K, (V-K)₀} distributions of the calibrated data and of the model populations are displayed in Fig. 4.

Figure 4: Calibrated period-luminosity-colour distributions: individual data and main symmetry plane (i.e. de-biased PLC relation) of the model distribution of each group

The groups may get the following interpretation in terms of kinematics: Group 1 (121 stars: 102 Miras, 6 SRa, 13 SRb) and Group 2 (96 stars: 54 SRb, 26 Miras, 16 SRa) have very similar kinematics, corresponding to old disk stars. Group 3 (125 stars: 102 SRb, 12 Miras, 11 SRa) has a younger kinematics than the previous ones. Group 4 (14 stars: 13 Miras, 1 SRa) has the kinematics of extended-disk or halo stars. One can immediately see that the SRb stars constitute two populations of different ages. SRa stars are spread over all groups, with no clear "preference''. Most Miras appear in Group 1, with the same kinematics as the old SRb population.