Studies on the complexity reduction with orthogonal transformation

Studies on complexity reduction for simplifying fuzzy rule-based models using the orthogonal transformation method are described, where the output space as well as the input space are considered. Generally, the conventional clustering algorithms are applied only to data set of input space. In some applications, data set of the output space is integrated into the input space and clustering is performed jointly. However, in this case the essential role of the output in the model is overlooked. This situation can be improved by using clustering methods where both spaces are used in the clustering process. One such a model is radial basis function network (RBFN) where clusters can be determined by the orthogonal least squares (OLS) algorithm. The OLS algorithm is an orthogonal transformation method using the Gram-Schmidt orthogonalisation process. The present work describes the implementation of OLS algorithm along the principal components of the RBFN-matrix and shows that the method itself is equivalent to complexity reduction where basis functions are in effect orthogonal. The main implication of this is the essential enhancement on the effectiveness of clustering. Furthermore, an important computational gain is achieved by the simplification of the complexity reduction process at the moderate expense of computing principal components.