Kolmogorovs Spline Complex Network and Adaptive Dynamic Modeling of Data

Kolmogorovs Spline Complex Network and Adaptive Dynamic Modeling of Data

Boris Igelnik (BMI Research Inc., USA)
DOI: 10.4018/978-1-60566-214-5.ch003
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This chapter describes the clustering ensemble method and the Kolmogorovs Spline Complex Network, in the context of adaptive dynamic modeling of time-variant multidimensional data. The purpose of the chapter is to give an introduction in these subjects and to stimulate a participation of both young and experienced researchers in a solution of challenging and important for theory and practice problems related to this area.
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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.

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Editorial Advisory Board
Table of Contents
Sven Buchholz
Tohru Nitta
Chapter 1
Masaki Kobayashi
Information geometry is one of the most effective tools to investigate stochastic learning models. In it, stochastic learning models are regarded as... Sample PDF
Complex-Valued Boltzmann Manifold
Chapter 2
Takehiko Ogawa
Network inversion solves inverse problems to estimate cause from result using a multilayer neural network. The original network inversion has been... Sample PDF
Complex-Valued Neural Network and Inverse Problems
Chapter 3
Boris Igelnik
This chapter describes the clustering ensemble method and the Kolmogorovs Spline Complex Network, in the context of adaptive dynamic modeling of... Sample PDF
Kolmogorovs Spline Complex Network and Adaptive Dynamic Modeling of Data
Chapter 4
V. Srinivasa Chakravarthy
This chapter describes Complex Hopfield Neural Network (CHNN), a complex-variable version of the Hopfield neural network, which can exist in both... Sample PDF
A Complex-Valued Hopfield Neural Network: Dynamics and Applications
Chapter 5
Mitsuo Yoshida, Takehiro Mori
Global stability analysis for complex-valued artificial recurrent neural networks seems to be one of yet-unchallenged topics in information science.... Sample PDF
Global Stability Analysis for Complex-Valued Recurrent Neural Networks and Its Application to Convex Optimization Problems
Chapter 6
Yasuaki Kuroe
This chapter presents models of fully connected complex-valued neural networks which are complex-valued extension of Hopfield-type neural networks... Sample PDF
Models of Complex-Valued Hopfield-Type Neural Networks and Their Dynamics
Chapter 7
Sheng Chen
The complex-valued radial basis function (RBF) network proposed by Chen et al. (1994) has found many applications for processing complex-valued... Sample PDF
Complex-Valued Symmetric Radial Basis Function Network for Beamforming
Chapter 8
Rajoo Pandey
The equalization of digital communication channel is an important task in high speed data transmission techniques. The multipath channels cause the... Sample PDF
Complex-Valued Neural Networks for Equalization of Communication Channels
Chapter 9
Cheolwoo You, Daesik Hong
In this chapter, the complex Backpropagation (BP) algorithm for the complex backpropagation neural networks (BPN) consisting of the suitable node... Sample PDF
Learning Algorithms for Complex-Valued Neural Networks in Communication Signal Processing and Adaptive Equalization as its Application
Chapter 10
Donq-Liang Lee
New design methods for the complex-valued multistate Hopfield associative memories (CVHAMs) are presented. The author of this chapter shows that the... Sample PDF
Image Reconstruction by the Complex-Valued Neural Networks: Design by Using Generalized Projection Rule
Chapter 11
Naoyuki Morita
The author proposes an automatic estimation method for nuclear magnetic resonance (NMR) spectra of the metabolites in the living body by magnetic... Sample PDF
A Method of Estimation for Magnetic Resonance Spectroscopy Using Complex-Valued Neural Networks
Chapter 12
Michele Scarpiniti, Daniele Vigliano, Raffaele Parisi, Aurelio Uncini
This chapter aims at introducing an Independent Component Analysis (ICA) approach to the separation of linear and nonlinear mixtures in complex... Sample PDF
Flexible Blind Signal Separation in the Complex Domain
Chapter 13
Nobuyuki Matsui, Haruhiko Nishimura, Teijiro Isokawa
Recently, quantum neural networks have been explored as one of the candidates for improving the computational efficiency of neural networks. In this... Sample PDF
Qubit Neural Network: Its Performance and Applications
Chapter 14
Shigeo Sato, Mitsunaga Kinjo
The advantage of quantum mechanical dynamics in information processing has attracted much interest, and dedicated studies on quantum computation... Sample PDF
Neuromorphic Adiabatic Quantum Computation
Chapter 15
G.G. Rigatos, S.G. Tzafestas
Neural computation based on principles of quantum mechanics can provide improved models of memory processes and brain functioning and is of primary... Sample PDF
Attractors and Energy Spectrum of Neural Structures Based on the Model of the Quantum Harmonic Oscillator
Chapter 16
Teijiro Isokawa, Nobuyuki Matsui, Haruhiko Nishimura
Quaternions are a class of hypercomplex number systems, a four-dimensional extension of imaginary numbers, which are extensively used in various... Sample PDF
Quaternionic Neural Networks: Fundamental Properties and Applications
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