## 4. Monte Carlo simulations of a test caseTo investigate properties of the ML estimator we use the Hyades
cluster as a test case. Based on kinematic criteria, including
spectroscopic radial velocities, Perryman et al. (1998) identified
197 stars from the Hipparcos Catalogue as probable members of the
Hyades (stars with `1' in column x of their Table 2). This sample
is subsequently referred to as `Hy0'. The positions and distances of
these stars, as determined by Hipparcos, define the three-dimensional
structure of the cluster assumed in all our simulations. The
astrometric accuracies () are also
taken directly from the Hipparcos Catalogue. In equatorial coordinates
the centroid velocity of the cluster is taken to be
km s ## 4.1. Solutions based on consistent cluster modelsIn this section we examine the performance of the ML estimator in the case when the model assumed in the solution correctly describes the actual kinematics of the cluster. The purpose is to determine the possible biases that are intrinsic to the method, and its behaviour under ideal circumstances. In Sect. 4.2, we then consider how deviations from the assumed model will affect the results. ## 4.1.1. Precision and bias of estimatesResults from 5000 experiments with the basic cluster model applied
to the 197 stars in sample Hy0 are summarised in Table 1
(simulation
In simulation The simulations described above all used the basic cluster model,
in which no systematic velocity field is included. In simulation
## 4.1.2. Kinematically improved parallaxesFig. 2 shows the distribution of parallax errors in simulation
The rms error of 0.43 mas for the kinematically improved
parallaxes indicated in Fig. 2 is in reasonable agreement with
the theoretical Eq. (11) of Paper I. Assuming
mas,
mas yr The standard deviation of the normalised estimation error is 1.28. The formal errors of the estimated parallaxes thus need to be increased by 28 per cent to be consistent with the scatter found in the Monte Carlo experiments. ## 4.1.3. Distribution of the goodness-of-fit valuesFig. 3 shows the cumulative distribution of the
values from simulation
The distribution of the values is of course modified by the rejection procedure described in Sect. 3.7, whereby an upper limit is introduced. As shown by the dashed curves in Fig. 3, this procedure produces a gently truncated exponential distribution of the values, without much affecting the distribution of the small values. ## 4.2. RobustnessRobustness refers to the desirable property of an estimator that the results are relatively insensitive to deviations from the model assumptions. In the context of the basic cluster model two types of deviation have been considered: systematic velocity patterns, and deviations from a Gaussian velocity distribution. The latter may be caused either by individual contaminating field stars or astrometric binaries, or by a more general shape of velocity distribution (e.g. a mixture of different dispersions). The existence of systematic velocity patterns can at least partly
be dealt with by solving components of a linear velocity field as
discussed in Sect. 3.3. However, in practice this solution would
not be accepted unless it gave a significant result for the linear
velocity terms. We should therefore consider the possible biases in
the basic solution produced by velocity fields that are weak enough to
remain undetected. Given the formal errors and scatters in simulations
The use of (primarily) kinematic criteria to identify probable
cluster members, e.g. for the Hyades by Perryman et al. (1998),
precludes that the input sample for the ML estimation contains a large
number of field stars. Also member stars with strongly deviating
proper motion (due to binarity) are rejected a priori. However, a
small number of stars could still have peculiar velocities exceeding
several times the combined standard deviation of observational errors
and . The rejection procedure
described in Sect. 3.7 is intended to eliminate such outliers. To
test the effectiveness of the procedure, we made a simulation in which
the peculiar velocity of an individual star, with probability 0.05,
was multiplied by a factor 10. The centroid astrometric accuracy was
estimated with different rejection limits
. The rms scatter was
0.56 km s In reality we do not expect such a clear-cut distinction between
well-behaved member stars and outliers. A more likely situation is
that of a continuous blend of populations with different kinematic
characteristics. A simple model for this is to assume that the
velocity dispersion itself is a
random variable. Since the dispersion must be positive, a convenient
assumption is that it follows a log-normal distribution with median
value and logarithmic standard
width (thus
is Gaussian with mean value
and standard deviation
). With
km s
A rejection procedure using thus appears to provide excellent protection against outliers and works very well also in more general cases of non-Gaussian velocity dispersions. The accuracy of the resulting centroid velocity is only marginally degraded compared with the nominal case of a Gaussian velocity dispersion. ## 4.3. Unbiased estimation of the velocity dispersionIt was noted in Sect. 4.1.1 that the internal velocity dispersion is strongly underestimated in the ML solutions based on simulated observations. We now turn to investigating this effect more closely, and to finding a remedy for it. The bias in the ML estimate of
apparent in Table 1 is probably related to the circumstance that
we assume an isotropic three-dimensional dispersion of the peculiar
velocities , while in practice only
one component can be measured astrometrically, viz.
perpendicular to the plane
containing the line of sight and the centroid velocity vector. The
radial component of the peculiar
velocities is obviously not determined at all, since that would
require spectroscopic velocities. The remaining tangential component
, parallel to the plane containing
the line of sight and the centroid velocity, is largely absorbed by
the individual distance estimates (provided that
, as is the case for the Hyades
cluster). Thus the measured variance in one direction
() is effectively `spread out' in
all three directions, causing the estimated
to come out much too small. The
effect is compounded by the observational errors in the proper
motions, which are implicitly taken into account in the ML estimation
by reducing even further. This
explains, at least qualitatively, why we obtain
in simulations where the true
dispersion is less than a certain value
( km s It should be remarked here that the existence of this bias does not imply that the present formulation or implementation of the ML method would not be valid. While the ML method is known to perform well in many practical situations, there is no guarantee that it provides unbiased estimates. In the present case the bias seems to be the consequence of an intrinsic anisotropy of the astrometric observations with respect to the mathematical cluster model. The referee has drawn our attention to a practical way in which this difficulty could be avoided, using a variant of the procedure described by Narayanan & Gould (1999a). Instead of forcing a solution with the - physically well motivated - isotropic dispersion, let us assume a triaxial velocity ellipsoid with dispersions , , along the previously defined axes. Applying the ML estimation to this model will of course give much too small a value for and an undetermined . However, may be obtained without bias, and that value could then be adopted as the actual dispersion in all three axes. (Narayanan & Gould use also spectroscopic radial velocities and photometric distances, and so are able to obtain information on all three components; they then impose an isotropic dispersion using the best combined estimate.) We have not adopted that method, mainly because there is a mathematical inconsistency involved in forcing and to equal the independently estimated . However, along a similar line of thought we found another way to deal with the problem. From the proper-motion residuals of the ML solution we compute estimates of the peculiar velocity components and of their observational uncertainties . A posteriori analysis of these data provides an estimate of , and hence of under the hypothesis of isotropic dispersion. Details of the procedure are given in Appendix A.4. Table 2 shows that calculated in this way is a practically unbiased estimate of even for very small dispersions. This value can then be used in Monte Carlo simulations to derive the true uncertainties in all the estimated parameters.
© European Southern Observatory (ESO) 2000 Online publication: April 17, 2000 |