Fuzzy logic approaches to structure preserving dimensionality reduction

Sammon's (1969) nonlinear projection method is computationally prohibitive for large data sets, and it cannot project new data points. We propose a low-cost fuzzy rule-based implementation of Sammon's method for structure preserving dimensionality reduction. This method uses a sample and applies Sammon's method to project it. The input data points are then augmented by the corresponding projected (output) data points. The augmented data set thus obtained is clustered with the fuzzy c-means (FCM) clustering algorithm. Each cluster is then translated into a fuzzy rule to approximate the Sammon's nonlinear projection scheme. We consider both Mamdani-Assilian and Takagi-Sugeno models for this. Different schemes of parameter estimation are considered. The proposed schemes are applied on several data sets and are found to be quite effective to project new points, i.e., such systems have good predictability.