Classification of Compositional Data Using Mixture Models: a Case Study Using Granulometric Data

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1 Classification of Compositional Data Using Mixture Models 1 Classification of Compositional Data Using Mixture Models: a Case Study Using Granulometric Data C. Barceló 1, V. Pawlowsky 2 and G. Bohling 3 1 Escola Politècnica Superior, Departament d'informàtica i Matemàtica Aplicada, Universitat de Girona, Spain 2 E.T.S.E.C.C.P., Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Spain 3 Mathematical Geology Section, Kansas Geological Survey, The University of Kansas, Lawrence, Kansas, USA 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, 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 hypothesis of normality can be assumed, McLachlan & 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

2 2 C. Barceló et al. previously discussed methodology. Two approaches have been considered: transforming the data using the log-ratio approach (Aitchison, 1986) or applying multivariate Box-Cox transformations (Barceló, Pawlowsky & Grunsky, 1996). Now, assuming transformed data to come from a mixture of multivariate distributions, the clustering procedure proposed by McLachlan & Basford (1988) can be applied. It is 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 with the assumed model and thus in principle more reliable than those obtained applying traditional clustering procedures to untransformed compositional data. 2. Case Study We have applied the clustering procedure based on a mixture model using both types of transformations to granulometric data from the Darss Sill area of the Baltic Sea (Fig. 1). As stated in Davis et al. (1995) and in Bohling et al. (1996), the samples of Darss Sill data set represent the weight percentage of each sample within each of eight grain size classes: gravel (>2000 mm), very coarse sand ( mm), coarse sand ( mm), medium sand ( mm), medium fine sand ( mm), fine sand ( mm), very find sand ( mm) and silt (<63 mm). The Darss Sill area represents the primary bottleneck in the exchange of water between the North Sea and the Baltic Sea. In the study area the predominant current flow and the sediment transport direction is from the southwest across the Darss Sill and northeast and east into the Baltic. There is evidence that current patterns in eastern portion of the study area follow a counterclockwise circulating pattern. The Darss Sill acts as a barrier between the Mecklenburgian Bight in the southwest and the Arkona Basin in the east. It consists of glacial tills and is covered in areas with lag sediments. The Kadet Channel was incised in the glacial tills by postglacial drainage. The Subsections 2.1, 2.2 and 2.3 show the results obtained applying different clustering procedures to Darss Sill data set. Results obtained applying the clustering procedure based on mixture models to data transformed by the additive logratio transformation are shown in Subsection 2.2, while the case for data transformed by a multivariate Box-Cox transformation is presented in Subsection 2.3. These results can be compared with those derived from a traditional clustering procedure (Ward's method) applied to untransformed data, which are displayed in Subsection 2.1.

3 Classification of Compositional Data Using Mixture Models 3 Fig. 1. Darss Sill and surrounding area of the eastern part of the Baltic Sea. After Davis et al. (1995) In each Subsection, Tables 1,3 and 5 and the histograms of Figs 2, 5 and 8 summarize the typical mean weight percents (centroids) in the size classes of the groups. Figures 3, 6 and 9 show the ternary diagrams of data amalgamated into three grain size divisions: gravel through medium fine sand (grv-mfs), fine sand (fs), and very fine sand and silt (vfs-slt). Diferents colors are used to represent different groups. Figures 4, 7 and 10 show classification maps obtained interpolating Mahalanobis distances in a regular grid with a 300-meter spacing in each direction using two-step scaled inverse distance squared interpolation scheme that incorporates local estimates of slope. All these maps have been plotted with Surface III (Sampson, 1988). 2.1 Hierarchical Clustering applied to Raw Granulometric Data Davis et al. (1995) and Bohling et al. (1996) apply a hierarchical clustering analysis using Ward's method (Everitt, 1993) to group untransformed samples into classes that are as homogeneous as possible in a minimum variance sense. Because the minimum within-groups sums of squares criterion employed in Ward's method corresponds to the assumption of a spherical, homoscedastic (equal-covariance) normal model for the groups, this method tends to create hyperspherical clusters (Statistical Sciences, 1995). Inspection of the results shows - as stated by above mentioned authors - that the grain size distributions corresponding to the centroids of seven groups (see Table

4 4 C. Barceló et al. 1 and Fig. 2) reflect the well to moderately sorted nature of groups 1 through 6 and the poorly sorted nature of group 7. Table 1. Grain size distribution of centroids of each group (hierarchical clustering) Gr N gravel vcsand csand msand mfsand fsand vfsand silt % % % % % % % weight% grv vcs cs ms mfs fs vfs silt groups Fig. 2. Grain size distribution of centroids of each group (hierarchical clustering)

5 Classification of Compositional Data Using Mixture Models 5 A genetic interpretation can be assigned to the results of sediment classification. The poorly sorted group 1, with an unusually high proportion of gravel and coarse sand, represents the remnants from submarine erosion of a glacial till. Group 2 also is erosional in origin, but shows a transitional character. Group 3 belongs to a depositional facies but its relatively poor sorting suggests that it is a transitional sediment. The more typical depositional facies include group 4 which has transitional characteristics, group 5 which is typical of depocenters, and group 6, which represents a sediment that has bypassed the depocenter and has been deposited in a distal position. Group 7 represents a channel environment where erosion, transportation, and deposition change very rapidly. The ternary diagram of Fig. 3 also reflects the sequence from group 1 to group 6 and the erratic location in the diagram of samples belonging to group 7. Note the difficulty to distinguish samples from groups 1 and 2 in this ternary diagram. Fig. 3. Ternary diagram for three-part amalgamation (hierarchical clustering) In spite of the fact that group distributions deviate from the assumed model of multivariate normality with a common covariance matrix, Bohling et al. (1997) contend that the classification map shown in Fig. 4 makes geologic sense: the groups form geographic sequences corresponding to the source area to depocenter sequence represented by groups 1 to 6.

6 6 C. Barceló et al. Fig. 4. Regionalized classification based on hierarchical clustering procedure applied to raw granulometric data Table 2. Resubstitution results for linear discriminant analysis (hierarchical clustering) Assigned True group Group Total N N wrong Error rate 6.8% 12.1% 12.8% 10.8% 0.5% 1.8% 6.4% Overall error rate: 7.3% The overall error rate (see Table 2) of the linear discriminant rules associated to this classification under the assumption of equal prior probabilities is equal to 7.3% and can be considered reasonably low. Samples of group 5 are the best characterized, whereas there is a significant confusion between samples of groups

7 Classification of Compositional Data Using Mixture Models 7 1 and 2. Although the confusion between samples belonging to adjacent groups seems to be natural, it is surprising that almost 8% of samples from group 3 were wrongly classified in group Mixture Clustering Model over the alr-transformed Data Since granulometric data are compositional, attention should be paid more to ratios of components than to their absolute values. For this reason, Aitchison (1986) suggests applying the additive logratio transformation (alr) to compositional data. This transformation breaks the constant sum constraint which distinguishes this kind of data. Since one cannot compute the logarithm of zero, zero data values must be adjusted to some small positive value before the alr transformation is performed (Aitchison, 1986, Chap. 11). This adjustment has been done using a value of δ=0.05 as the estimated maximum rounding error (Bohling et al., 1997). Bohling et al. (1997) demonstrate that the computed results do not vary greatly when delta is varied over a reasonable range, despite the large number of zeros in the data set. Then, assuming that alr-transformed data come frome a mixture of seven homoscedastic normal multivariate distributions, the EM-algorithm is applied as a clustering technique. The sample is considered to be a completely unclassified, although the initial estimations of the parameters of the mixture, needed to start the EM algorithm, are calculated from the seven groups which are obtained when Ward's method is applied to raw granulometric data (see Subsection 2.1). This procedure provides a new classification of data in seven groups (see Table 3 and Fig. 5). These new groups are, from a statistical point of view, very well delimited, because the overall error rate of the associated linear discrimination rule applied to transformed data is equal to 1.2%. However, the ternary diagram of Fig. 6 doesn't reflect the separation of the groups. The new groups are quite different from the initial hierarchic cluster groups of Subsection 2.1. The new clustering tends to move samples from initial hierarchic groups to new central groups 3 and 4 (see Table 4). Tests of normality applied to marginal distributions in each of the seven groups show that the hypotesis of multivariate normality cannot be supported a posteriori. In spite of that, the classification map of Fig. 7 has a globally similar pattern as the map of Fig. 4, although less differentiated in the eastern part, while more chaotic in the Kadet channel. A geologic interpretation of obtained results would be necessary to determine the validity of the applied procedure.

8 8 C. Barceló et al. Table 3. Grain size distribution of centroids of each group (alr-clustering) Gr N gravel vcsand csand msand mfsand fsand vfsand silt % % % % % % % weight% grv vcs cs ms mfs fs vfs silt groups Fig. 5. Grain size distribution of centroids of each group (alr-clustering)

9 Classification of Compositional Data Using Mixture Models 9 Fig. 6. Ternary diagram for three-part amalgamation (alr-clustering) Fig. 7. Regionalized classification based on a mixture model as a clustering procedure applied to data transformed by the alr-transformation

10 10 C. Barceló et al. Table 4. Hierarchical clustering groups versus alr-groups Hierarchic Put into new alr-groups (initial) groups Total Total Mixture Clustering Model over the BC-transformed Data In some cases, if the logratio transformation cannot adjust compositional data to a mixture of multivariate normal distributions, a Box-Cox (BC) transformation can be a good alternative in order to obtain a better adjustment. In such cases it is necessary to choose a denominator to form ratios and to estimate the vector λ of the transformation (Barceló, Pawlowsky & Grunsky, 1996). For the Darss Sill data set, the variable "fine sand %" was used as the denominator of the ratios and the vector λ of the BC-transformation was estimated as λ=(-0.76,-0.33,0.61,0.08,0.13,0.02,-0.02) t. The seven groups obtained from hierarchical clustering analysis were used to estimate λ with the additional hypothesis of homoscedasticity of transformed groups. After the BC-transformation was applied to the granulometric data, the EM-algorithm was executed in the same fashion as in the alr case (Subsection 2.2). In Table 5 and Fig. 8 it can be seen that the new groups 1 to 6 follow again a genetic sequence, from coarser to finer granulometry, whereas group 7 resembles a residual clustering group. From a statistical point of view, new groups seem to be well defined because a small overall error rate of 2.5% is associated to the corresponding linear discriminant rule when applied to transformed data. However, neither the genetic sequence nor the separation of groups are visible in the ternary diagram of Fig. 9. Table 6 makes evident that samples from hierarchic group 2 are split and distributed among new groups 1 and 3, samples from hierarchic groups 3 and 5 move to new group 4, while samples of new group 5 come mainly from hierarchic group 6. The new groups 5 and 6 look like groups 6 and 7 of hierarchic cluster. It seems that only six groups would be necessary to obtain a reasonable clustering of available data. Some of this facts are evident when maps of Fig. 4 and Fig. 10 are compared.

11 Classification of Compositional Data Using Mixture Models 11 Table 5. Grain size distribution of centroids of each group (BC-clustering) Gr N gravel vcsand csand msand mfsand fsand vfsand silt % % % % % % % weight% grv vcs cs ms mfs fs vfs silt groups Fig. 8. Grain size distribution of centroids of each group (BC-clustering)

12 12 C. Barceló et al. Fig. 9. Ternary diagram for three-part amalgamation (BC-clustering) Fig. 10. Regionalized classification based on a mixture model as a clustering procedure applied to data transformed by the BC-transformation

13 Classification of Compositional Data Using Mixture Models 13 Table 6. Hierarchical clustering groups versus BC-groups Hierarchic Put into new BC-groups (initial) groups Total Total Comparison of the Three Clustering Models From the graphical and numerical display of the centroids of each group it can be seen that the three approaches admit an ordering of coarser to finer predominant components for the centroids, except for group 7. This gradient can also be observed in the regionalized classification with hierarchical clustering and with the Box-Cox transformation, while with the alr-transformation some spatial "discontinuities" can be observed, specially for data assigned to group 1 surrounded by data belonging either to group 5 or to group 4. On the other hand, resubstitution results for linear discriminant analysis show that the overall error rate is smallest for the alr-approach (1.2%), followed by Box-Cox clustering (2.5%) and by hierarchical clustering coming last (7.3%). If we look at the error rates for each group, it can be seen that in the first two cases (alr and Box-Cox transformations) the most important concentration of misclassifications is concentrated in group 4, while in the hierarchical clustering procedure misclassification appears to be significant by comparison, although acceptable from a statistical point of view, for groups 2, 3 and 4, and less important for group 1. To check if the two approaches using transformed data are just an improvement of hierarchical clustering, crosstabulation of group memberships are useful. Tables 4 and 6 clearly evidence the breakup of the initial clusters, forming new groups in two clearly different manners except for the initial group 6, which is completely merged into group 5 in both cases. Finally, some common features can be observed in the regionalized classification: a smooth variation in the eastern part of the study area, and a certainly erratic behaviour in the Darss Sill. 3. Conclusions

14 14 C. Barceló et al. Interpretion of results from a cluster analysis procedure is never the result of merely looking at some numbers. Groups have to be reasonable in our case from a geological point of view, and thus the opinion of a geologist is lacking at this point. Nevertheless, natural groups that are clearly different will be quite stable if subjected to different clustering methods. This is not the case of Darss Sill data set, where the three different approaches presented clearly lead to different groupings of data. In the ternary diagram of Fig. 3, the data seem to follow a fairly continous distribution. This fact suggests that the data might be better treated with a continuous approach, possibly in an appropiate transformed space. In such a case, available methods have to be carefully checked for underlying hypotheses, particularly concerning assumed models. The necessity of a more profound study of available information is a must in the present case. References Aitchison, J., 1986, The Statistical Analysis of Compositional Data: Chapmann and Hall, London (GB), 416 p. Barceló, C., Pawlowsky, V., and Grunsky, E., 1996, Some aspects of transformations of compositional data and the identification of outliers: Mathematical Geology, vol. 28, no. 4, pp Bohling, G. C., Davis, J. C., Olea, R. A., and Harff, J., Singularity and nonnormality in the classification of compositional data: submitted to Mathematical Geology, Davis, J. C., Harff, J., Lemke, W., Olea, R. M., Tauber, F., Bohling, G.C., and Zhou Di, 1995, Analysis of sedimentary facies by regionalized classification: Kansas Geological Survey Open File Report 95-4, 28 p. Everitt, B.S., 1993, Cluster Analysis: Edward Arnold, New York (USA), 3rd ed., 170 p. Harff, J., and Davis, J.C., 1990, Regionalization in geology by multivariate classification: Mathematical Geology, vol. 22, no. 5, pp McLachlan, G. J., and Basford, K.E., 1988, Mixture models. Inference and applications to clustering: Marcel Dekker, New York (USA), 253 p. Sampson, R. J., 1988, Surface III User's Manual, Kansas Geological Survey. Statistical Sciences, 1995, S-PLUS Guide to Statistical and Mathematical Analysis, Version 3.3, Seattle: StatSci, a division of MathSoft, Inc.