Cultural Heritage Applications: Archaeological Ceramics and Building Restoration - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 131

6.1.3 Motivation for an ICAMM Application
Let us consider a probabilistic classification context where some selected features
are organized as elements of vectors belonging to an observation space to be
divided into K classes C fg k ¼ 1...K. Given an observed feature vector X ; we
want to determine the most probable class. More formally, we want to determine
the class C k that maximizes the conditional probabilityp ð C k = x Þ . Since classes are
not directly observed, Bayes theorem is used to express p ð C k = x Þ in terms of
the class-conditioned observation probability density p ð x = C k Þ in the form
p ð C k = x Þ¼ p ð x = C k Þ p ð C k Þ= p ð x Þ: Note that p ð x Þ is a scaling factor that is irrelevant
to the maximization of p ð C k = x Þ , and that a priori probability p ð C k Þ is assumed to
be known (or equal to 1 = K for all classes). Hence, the key problem focuses on
estimation of p ð x = C k Þ .
A non-parametric classifier tries to estimate p ð x = C k Þ from a training set of
observation vectors, but this becomes progressively intractable as the dimension
of the observation space (number of features) increases, because the required size
of the training set becomes prohibitive. On the other hand, a parametric classifier
assumes a given form for p ð C k = x Þ and, thus, tries to estimate the required
parameters from the training observation set [ 9 ]. Most of the classifiers from
parametric approaches consider Gaussian densities to simplify the problem in the
absence of other information that could lead to better choices. Moreover, both
parametric and non-parametric classifiers are very much complicated in semi-
supervised scenarios, i.e., when part of the observed vectors belonging to the
training set have unknown classes [ 10 ].
Therefore, procedures that would be of interest in the general area of classifi-
cation should combine the following characteristics: the versatility of the non-
parametric approach (from the point of view of the assumed form of p ð x = C k Þ ); the
simplicity of the parametric methods (in the sense that most effort will concentrate
on the estimation of a finite set of parameters); and operate in semi-supervised
scenarios. This is especially remarkable in the area of non-destructive classifica-
tion of materials. On one hand, the prediction of the joint-density of some selected
features is almost impossible (Gaussianity is an assumption that is too restrictive in
many cases). On the other hand, there are some applications where the available
set of specimens used to obtain the training set can hardly be classified. This
happens, for example, when the specimen cannot be destroyed to find the true
inner state or when the definition of the K classes is not clearly known a priori.
The classification problem considered in this application has the conditions
necessary for verifying the usefulness of a versatile classifier that is capable of
working with semi-supervised training. Ceramic composition is assumed to be
different in different historic and proto-historic periods, so there should be
opportunities to classify the pieces from features derived from ultrasonic analysis.
Nevertheless, exact modelling of the propagation of ultrasound in ceramic and
statistical characterization of the features is a complex matter. Hence, it is
advisable not to assume particular parametric distributions (like normal density) in