Fuzzy Logic Approach - Computational Intelligence in Time Series Forecasting - page 181

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4.7.3.1.1 Parameters of Gustafson-Kessel Algorithm
The same parameters must be specified beforehand in the Gustafson-Kessel (GK)
clustering algorithm as for the fuzzy c -means algorithm (except for the norm-
inducing matrix A , which is automatically adapted): the number of clusters c , the
fuzziness exponent m , and the termination tolerance parameters. Additional
parameters are the cluster volumes U . Without any prior knowledge, the cluster
volumes is simply fixed at 1 for each cluster. Due to this constraint, the Gustafson-
Kessel algorithm can only find clusters of approximately equal volumes. This is a
drawback of this setting.

4.7.3.1.2 Interpretation of Cluster Covariance Matrix
The cluster covariance matrix provides important information about the shape and
orientation of the cluster. The ratio of the lengths of the cluster's hyperellipsoids
axes is given by the ratio of the square roots of the eigenvalues of the covariance
matrix. The directions of the axes are given by the eigenvectors of covariance
matrix. The Gustafson-Kessel algorithm can be used to detect clusters along linear
subspaces of the data space. These clusters are represented by flat hyperellipsoids,
which can be regarded as hyperplanes. The eigenvector corresponding to the
smallest eigenvalue determines the normal to the hyperplane, and can be used to
compute optimal local linear models from the covariance matrix.

4.7.4 Identification of Antecedent Parameters by Fuzzy Clustering

Using the given data, the identification of antecedent parameters of the Takagi-
Sugeno model is usually done in two steps. In the first step, the antecedent fuzzy
sets of the rules are determined. This can be done manually, from knowledge of the
process, by interviewing the human experts, or by some data-driven technique,
such as a neuro-fuzzy technique, or by the fuzzy clustering method described
earlier, which produces a partitioning of the antecedent (input) space. Once the
fuzzy antecedent parameters are determined, the LSE estimate described earlier is
then applied in order to determine the consequent parameters of the Takagi-Sugeno