A problem present in the three-gaussian model fitted to the data is the large number of free parameters, 22 in total per triple image. It is clear that some of these parameters must be correlated as the three exposures were intended to be as similar as possible. Therefore a model with smaller number of degrees of freedom would be interesting to consider because:
a) The fitting process would be more robust with a smaller number of free parameters.
b) The fitting algorithm would be faster.
Principal Components Analysis (PCA) (Brosche 1973; Whitney 1983, for example) is a method which first obtains a new set of uncorrelated variables by determining the set of eigenvectors of the original data correlation matrix. The corresponding eigenvalues give the portion of total variation in the original data for which each eigenvector accounts for. Then the eigenvalues are sorted and split into two groups: The ones with larger values putatively represent the signal, and the remaining ones represent the noise. The programme used (Lentes 1985) contains a modification of the correlation matrix in order to take care for error covariance.
We found that it is possible to reduce the number of free parameters to not less than 7. This minimum is defined by the number of eigenvalues with larger values representing the actual data.
A question can be posed now: Is there any set of less than 22 parameters which
seems suitable a priori to represent the data? Some characteristics
in our problem
can help us to find a possible candidate. For example, the three exposures
should in
principle be identical, as they were supposed to be given the same exposure
time in each
individual plate and were taken under approximately the same
atmospheric conditions, as there was only about half an hour between
consecutive
exposures. More explicitly, we have found that, in average, in the current
data the values of the parameters in an individual exposure are compatible with the case
of being equal to the analogue ones in the other two exposures for each single star.
This means that it is basically correct to use only one peak density value A for the
three exposures, and the same holds for 
, 
, s and t. The only
parameter left, the background, was found to be varying in a not at all regular way over
the plate (Fig. 12 (click here)). However, the scale of this variation is much larger than
just the 
mm windows containing the triple images and no errors are
introduced when taking it to be constant. Thus, it was considered to be the same inside
the whole frame containing the triple image
The reasonings above, thus, lead in total to a set of 12 free parameters, that is, the 6 mentioned just before and 6 position coordinates for the centres of the three exposures. These have been the degrees of freedom of the new model applied in Sect. 7 (click here).
The question remains whether it is possible to only have 7 degrees of freedom
without loosing an important fraction of the original information. First, one
would expect a strong correlation between and (and
it is so indeed) hence one of them can be computed as a function of the
other one. One has to pay attention to the fact that because of the presence
of field curvature this function will depend on the distance to the plate
centre.
Moreover, could it be possible to know the position of each
exposure only knowing the triangle central coordinates? For this to be
correct the triangle geometry should be constant through the
plate. The angles which the position vectors of each exposure in a triplet
form with an x-axis parallel to the x plate scan axis and as origin the
triangles geometric centre have been calculated, together with their rms. Its
dependence on the position in the plate and on the distance to the plate
centre was studied and it was found that exposure 1 suffers a displacement when
approaching one of the plate corners. It also happens to exposure 3, but
to a smaller degree. The total dispersion of the angle values is equivalent
to an image displacement along an arc of length of 
,
and 
for exposures 1, 2 and 3,
respectively. One can compare this
with the residuals when performing the comparison with an external catalogue
(Table 3 (click here)). If the total error in the image position is taken to
be 
, we obtain
, which is larger than . Thus, angle
dispersion is not the main source of error in the position in this case. We
have also checked that the ratios among the triangle sides are not dependent
on the magnitude.
A model for the three exposure positions as a function only of the geometrical
triangle centre position in the plate is then possible and this
would lead to a set of 7 free parameters, as predicted by the PCA: A,
(or ), s, t, 
, 
and the
background (B).