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Another point is worth mentioning in comparing EDMA with the
MLE: even for the simple case of three landmarks in two-dimensions,
we found the likelihood very difficult to maximize numerically using
the Dryden and Mardia (1991) distribution. Despite estimating the
variance-covariance matrix using a Cholesky decomposition to guar-
antee positive semi-definiteness, many data sets required new starting
values and/or a preliminary sizing step for the variance-covariance
parameters in order to achieve convergence. Despite repeated
attempts there were still 2 data sets (out of 100) in the first parameter
configuration for which we could not obtain convergence.
The practical utility of the maximum likelihood method of estima-
tion based on Mardia-Dryden distribution is therefore somewhat
suspect, especially when one is dealing with a large number of land-
marks for three-dimensional objects and a medium sample size. Such
a situation is not at all uncommon and should be clear from the typi-
cal data analyses presented in this topic as well as various papers cited
herein.
As a practical alternative to the maximum likelihood estimation,
Goodall and Bose (1987) and Goodall (1991) discuss Procrustes super-
imposition based estimators of the mean form, mean shape, and the
variance-covariance matrix. See also Bookstein (1991) for similar esti-
mators based on Bookstein shape coordinates. Kent (1994) and Kent
and Mardia (1997) show that the Procrustes estimators can also be
viewed as estimators based on tangent space approximation to the
Kendall shape space. These estimators are computationally easier
than the maximum likelihood estimators. Unfortunately, these estima-
tors, under the Gaussian perturbation model with general covariance
structure, are inconsistent for mean form as well as mean shape (Lele,
1993, Kent and Mardia, 1997).
Moreover, simulations reported in Lele (1993) and the discussion in
Part 1 of this chapter show that the Procrustes estimator of the vari-
ability does not correspond to the variance-covariance matrix used in
the Gaussian perturbation model. Given these results, we find the
Procrustes estimators inadequate. They are not discussed here in
detail. See Dryden and Mardia (1998) for a detailed discussion of the
Procrustes and related approaches.
3.14 Computational algorithms
3.14.1 Algorithms for estimation of mean and variance parameters
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