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Test on 100 Up: THE
ASTEROID IDENTIFICATION PROBLEM Previous: 3.3
Normalization/weighting of the
We have selected a number of algorithms to propose identifications.
They are based on the computation of a number of ``distances'' in the elements
space ,
which depend not only upon the difference
but also upon the uncertainties, as expressed by the normal matrices
and
.
We use:
The only caution to be used is to compute, for a reduced identification
of the subset
of the orbital elements
,
not the restriction
of the normal matrix but the marginal normal matrix
obtained by inverting the corresponding portion of the covariance matrix,
as discussed in Section 2.3.
The algorithm to propose identifications should be as follows. First,
over all the candidate couples, the distance
should be computed and tested against some control value
.
The logic is that the orbit plane is, in most cases, significantly better
determined than the other orbital elements. Thus the corresponding marginal
normal matrices are typically well conditioned, and the problems of numerical
instability and nonlinearity are much less severe.
On the other hand, the use of this criterion is not enough to discriminate
the strong identification candidates. If the control
is kept too low, some real identifications could be missed; if it is too
high, the number of proposed identifications would be huge.
Thus the following step in the algorithm is to compute, for the couples
selected on the basis of the
criterion, the
distance
and to test it against another control
.
If this test is also positive, the
distance is computed and tested
.
The problem is to determine values of the controls
and
,
in such a way that real identifications are not discarded, and the number
of proposed identifications is not too large. These ``distances'' are dimensionless,
and they do not have a straightforward physical interpretation; thus it
is not possible to decide a range of acceptable values of these controls
a priori in such a way that they should be satisfied by all real identifications.
These acceptable ranges would have to accommodate all the possible effects
of the three difficulties discussed in Section 3, namely bad conditioning,
nonlinearity, and problems in residual weighting.
Therefore we have adopted a criterion based upon experimental evidence:
we have computed the values of ,
and
found in a test set of already identified orbits, as discussed in the following
Subsection.