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From the view points of functional data analysis, we assume that we have p-dimensional functions corresponding to n objects. We denote the functions for n objects depending on a variable t as Z{t) = {zi{t)}{i = 1, 2 , . . , n). It is realistic that values Z{tj){j = 1, 2 , . . , m) are given. We restrict ourselves Graphical Representation of Functional Clusters and MDS 33 to two dimensional functional data and to the one dimensional domain. The number of clusters k is prespecified as a user parameter.

2 Graphical Representations of Functional Clustering The purpose of cluster analysis is to find relatively homogeneous clusters of objects based on measured characteristics. Sometimes, we divide methods of cluster analysis into two groups: hierarchical clustering methods and nonhierarchical clustering methods. Hierarchical clustering refers to the formation of a recursive clustering of the objects data: a partition into two clusters, each of which is itself hierarchically clustered. , n}, where Sij is the dissimilarity between objects i and j , and n is the size of the objects.

2 Selected Results According to the J [/MP-criterion the optimal number of clusters for the d a t a at hand is 37. The G K M algorithm also results in a 37-cluster solution, where sets of 174 to 394 individual consumer profiles are assigned to 15 meaningful clusters with centroids ^ 1 , . . ,^15. Moreover, 22 outliers are assigned to single-object clusters. Applying the SGNN algorithm with auax = 81 ( = L) results in a neural network t h a t comprises 14 prototypes ^ 1 , . . , ^14.