The USC Andrew and Erna Viterbi School of Engineering USC Signal and Image Processing Institute USC Ming Hsieh Department of Electrical Engineering University of Southern California

Technical Report USC-SIPI-300

“NonParametric Modeling of Nonlinear Processes Using a Fuzzy Set Theoretic Approach”

by George Mouzouris

August 1996

Effective representation of uncertainty is undoubtedly an important problem in the design of intelligent computational systems. Probability theory and Bayesianism have been the conventional approaches for dealing with uncertainty, but they are by no means complete or uniquely optimal for managing uncertainty and solving problems. The restrictive assumptions usually necessary in the construction of a parametric model can be circumvented by the use of a model-free, nonparametric technique such as a fuzzy logic system (FLS). Fuzzy set theory is a mathematical paradigm that can utilize both numerical and linguistic information in a unified manner, and translate abstract concepts into computable entities. It provides a rigorous algebraic framework where incomplete knowledge, statistical information and lexical imprecision can be combined to construct approximate or interpolative models of complex systems. In this work, we show how this can be achieved; we begin by presenting a formal derivation of nonsingleton fuzzy logic systems (NSFLSs), a generalization of singleton FLSs capable of processing uncertain, set-valued inputs. We show how they can be efficiently computed, investigate their properties, and compare them analytically with singleton FLSs. We prove that NSFLSs can uniformly approximate any real continuous function on a compact set to an arbitrary degree of accuracy, and show with an example in nonlinear adaptive equalization that they outperform singleton systems and come very close to the optimal equalizer. We also demonstrate the superiority of NSFLSs over singleton FLSs by constructing predictive models of several discrete- and continuous-time chaotic systems corrupted by additive noise. We derive a backpropagation (BP) algorithm that allows us to train all NSFLS design parameters, including one that is proportional to input uncertainty. The BP-trained NSFLS is shown to produce useful forecasts of financial markets, and significantly outperform linear regression. Since in most real life problems the pieces of information used to construct a model are correlated, we present a Singular-Value-QR decomposition method that allows us to eliminate redundancy in the fuzzy rulebase, by providing an estimate of the number of necessary rules, and selecting a subset of independent fuzzy basis functions which are sufficient to adequately represent a desired system. We use the method to design nonlinear dynamic identifiers, and to analyze the contribution of linguistic information with varying amounts of numerical information. Using a quantitative measure of uncertainty, we find defuzzification methods that produce minimally ambiguous crisp values, and obtain a constrained learning algorithm that minimizes both mean-squared error and output system uncertainty. We extend static NSFLSs to Dynamic NSFLSs, derive a dynamic backpropagation type of learning algorithm, and use it to model nonlinear dynamic processes of unknown order. Then, we present the concept of a complex-value FLS, which is expressedas a summation of complex exponentials. We then show, by an example in nonlinear modeling, how it can be used in conjunction with an iterative learning algorithm on the complex plane, as a post-processing subsystem of a real-valued FLS that can offer significant performance advantages. Also, by exploiting the properties of complex exponentials, we derive an upper bound for the recall error. Finally, we present our conclusions and directions for future work.

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