Classification of Compositional Data Using Mixture Models:
a Case Study Using Granulometric Data
C. Barcelo, V. Pawlowsky and G. Bohling
1
Introduction
Classification of samples is a fundamental problem in the geosciences. Usually,
an initial unsupervised classification is used to define natural groups of samples.
Then, discriminant analysis is applied to obtain posterior probabilities that a
given sample belongs to each group. Those probabilities may be mapped onto
geographic space, resulting in a regionalized classification. Almost always this
procedure implicitly assumes that the observations forming the different groups
come from multivariate normal distributions. Although the majority of classification procedures based on minimum Mahalanobis distance are robust with respect to deviations from the hypothesis of normality, the above-mentioned associated posterior probabilities, calculated to draw regionalized classification
maps, are more sensitive to departures from the assumed distribution model.
For those cases where a hypothesis of normality can be assumed, McLachlan
and Basford (1988) suggest mixture models as a clustering technique. With a
normal mixture model, a clustering is performed assigning the data to groups
on the basis of estimated posterior probabilities, but it requires maximum-likelihood estimates for the parameters of the model. The EM (expectation/maximization) algorithm provides a convenient way for the iterative computations of
solutions of the likelihood equations. Initial values for the mixture parameters
needed to start the EM algorithm are calculated using memberships derived
from a standard clustering method.
In the case of compositional data, the hypothesis of normality has to be directly rejected because the natural sample space for this kind of data is the simplex. Thus, strictly speaking, the data should be transformed prior to application of the previously discussed methodology. Two approaches have been considered: transforming the data using the log-ratio approach (Aitchison 1986) or
applying multivariate Box-Cox transformations (Barcelo et al. 1996). Assuming
transformed data to come from a mixture of multivariate distributions, the clustering procedure proposed by McLachlan and Basford (1988) can be applied. It
was expected that the resulting groups, the posterior probabilities of group
membership and the associated regionalized classification maps obtained by either of these procedures were, from a statistical point of view, more consistent
a Case Study Using Granulometric Data
C. Barcelo, V. Pawlowsky and G. Bohling
1
Introduction
Classification of samples is a fundamental problem in the geosciences. Usually,
an initial unsupervised classification is used to define natural groups of samples.
Then, discriminant analysis is applied to obtain posterior probabilities that a
given sample belongs to each group. Those probabilities may be mapped onto
geographic space, resulting in a regionalized classification. Almost always this
procedure implicitly assumes that the observations forming the different groups
come from multivariate normal distributions. Although the majority of classification procedures based on minimum Mahalanobis distance are robust with respect to deviations from the hypothesis of normality, the above-mentioned associated posterior probabilities, calculated to draw regionalized classification
maps, are more sensitive to departures from the assumed distribution model.
For those cases where a hypothesis of normality can be assumed, McLachlan
and Basford (1988) suggest mixture models as a clustering technique. With a
normal mixture model, a clustering is performed assigning the data to groups
on the basis of estimated posterior probabilities, but it requires maximum-likelihood estimates for the parameters of the model. The EM (expectation/maximization) algorithm provides a convenient way for the iterative computations of
solutions of the likelihood equations. Initial values for the mixture parameters
needed to start the EM algorithm are calculated using memberships derived
from a standard clustering method.
In the case of compositional data, the hypothesis of normality has to be directly rejected because the natural sample space for this kind of data is the simplex. Thus, strictly speaking, the data should be transformed prior to application of the previously discussed methodology. Two approaches have been considered: transforming the data using the log-ratio approach (Aitchison 1986) or
applying multivariate Box-Cox transformations (Barcelo et al. 1996). Assuming
transformed data to come from a mixture of multivariate distributions, the clustering procedure proposed by McLachlan and Basford (1988) can be applied. It
was expected that the resulting groups, the posterior probabilities of group
membership and the associated regionalized classification maps obtained by either of these procedures were, from a statistical point of view, more consistent