Kolmogorovs Spline Complex Network and Adaptive Dynamic Modeling of Data

Introduction

This chapter describes specific neural network architecture with complex weights and potentially complex inputs in the context of adaptive dynamic modeling of time-varying multidimensional data. The technological and scientific developments in many areas of human activity have reached a level, requiring adequate changes in traditional methods of data modeling. Consider just two examples. One is related to coal-operating power stations. Coal might provide a fuel for world power industry for hundreds of years, making a good alternative to rapidly decreasing oil. But coal combustion produces harmful pollutions. Mitigation of this problem by controlling combustion requires modeling of power data, which are time-variant, highly multidimensional, nonlinear, non-stationary, and influenced by complicated, interacting chemical, electro-magnetic, and mechanical processes. There is no way to do such modeling by traditional methods of control theory (Ljung, 2000; Astrom & Wittenmark, 1995). The second example is related to defense, in particular to problems of detection, identification, and tracking targets in the clutter environment, utilizing sensors, such as radar, sonar, infrared (Hovanessian, 1984; Scolnik, 1990), and others. A possibility of having multiple moving and interacting targets, clutters, and sensors makes these problems extremely difficult for solution in real applications. The methods for solution of these problems, basically Bayesian ones (Antony, 1995; Congdon, 2006; Stone, Barlow, & Corvin, 1999), are founded on the theory developed 30-50 years ago, and inadequate to currently existing reality.

There is a long history of signal and noise representation, utilizing complex numbers in signal processing (Oppenheim & Schafer, 1975; Rihaczek & Hershkowitz, 1996; Haykin, 2001). Relatively recently it was recognized that complex representation of inputs and adaptively adjusted weights may be helpful in neural network (net) modeling, especially for pattern recognition (Kim & Guest, 1990; Leung & Haykin, 1991; Georgiu & Koutsougeras, 1992; Nitta, 1997, 2003; Arena, Fortuna, Muscato, & Xibilia, 1998; Aizenberg, Aizenberg, & Vandewalle, 2000; Igelnik, Tabib-Azar, & Leclair, 2001a; Hirose, 2003, 2006).

This chapter has the following objectives: 1) introducing basic principles, ideas, and algorithms of adaptive neural net modeling of time-varying, highly multidimensional data; 2) introducing a specific complex-valued neural network, the Kolmogorov’s Spline Complex Network (KSCN), which might be advantageous especially in various tasks of pattern recognition.