Computational Intelligence - May 2013 - 78

The book reviewed here provides an excellent overview
of the current trends in the research of CVNNs for
students and researchers interested in computational
intelligence as well as offers up-to-date theories and
applications of CVNNs for experts and practitioners.
problem of averaging the parameters
of a pool of cooperative CVNNs,
multichannel blind deconvolution of
signals in telecommunications, and
blind source separation of complexvalued signal sources. It is beneficial
that pseudocodes of the learning procedures for all of these problems are
listed in this chapter.
Chapter 3 focuses on an N-dimensional vector neuron, which is a natural
extension of a complex-valued neuron
in two-dimensional space to its
N-dimensional version. First, relevant
neuron models with high-dimensional
parameters are briefly reviewed to locate
the N-dimensional vector neuron. Next,
the author defines the N-dimensional
vector neuron which can represent
N signals as one cluster and reveals its
decision boundary to consist of N
hyperplanes which intersect orthogonally each other. The generalization ability of a single N-dimensional vector
neuron is demonstrated for N-bit parity
problem. Finally, the presented method is
compared with other layered neural networks in terms of the number of neurons, the number of parameters, and the
number of layers.
In Chapter 4, learning algorithms
with feedforward and recur rent
CVNNs are systematically described by
using Wirtinger calculus. The Wirtinger
calculus, which generalizes the concept
of derivatives in complex domain,
enables to perform all the computations of well-known learning algorithms with CVNNs directly in the
complex domain. For feedforward layered CVNNs, the complex gradient
descent algorithm and the complex
Levenberg-Marquardt algorithm are
derived with the complex gradient. For
recurrent type CVNNs, the complex
real-time recurrent learning algorithm
and the complex extended Kalman

filter algorithm are obtained utilizing
the Wirtinger calculus. Computer simulation results are given to verify the
above four algorithms.
Chapter 5 presents associative memory models with Hopfield-type recurrent neural networks based on quaternion, which is a four-dimensional
hypercomplex number. In the introduction to quaternion algebra, the definition of quaternion is given and its analyticity in the quaternionic domain is
described. Then, stability analysis is performed by means of energy functions
for several different types of quaternionvalued neural networks. The different
types of recurrent networks are constructed with bipolar state neurons, continuous state neurons, and multistate
neurons. All of these quaternion-valued
networks are shown to work well as
associative memory models by implementing typical learning rules including
the Hebbian rule, the projection rule,
and the local iterative learning rule.
Chapter 6 concentrates on recurrent-type Clifford neural networks. This
chapter starts with the definition of
Clifford algebra and the basic properties
of the operators in hypercomplex number systems. Subsequently, a Hopfieldtype recurrent Clifford neural network
is proposed as an extension of the classical real-valued Hopfield neural network,
with an appropriate definition of an
energy function for the Clifford neural
network. Finally, under several assumptions on the weight coefficients and the
activation functions, the existence of the
energy function is proved for two specific types of Clifford neural networks.
Chapter 7 provides a meta-cognitive
learning algorithm for a single hidden
layer CVNN, called Meta-cognitive Fully
Complex-valued Relaxation Network
(McFCRN), consisting of a cognitive
component and a meta-cognitive one.

IEEE ComputatIonal IntEllIgEnCE magazInE | may 2013

First, it is explained that the learning
strategy of the neural network (cognitive
part) is controlled by a self-regulatory
learning mechanism (meta-cognitive
part) through sample deletion, sample
learning, and sample reserve. After the
drawbacks of the conventional
meta-cognitive CVNNs such as Metacognitive Fully Complex-valued Radial
Basis Function Network (McFCRBF)
and the Complex-valued Self-regulatory
Resource Allocation Network (CSRAN)
are pointed out, the learning algorithm of
McFCRN is presented with a pseudocode. The performance of McFCRN is
evaluated in a synthetic complex-valued
function approximation problem and
benchmarks of real-valued classification
problems, in comparison with the other
existing methods.
In Chapter 8, a multilayer feedforward
neural network with multi-valued neurons (MLMVNs), found in the monograph [4], is applied to brain-computer
interfacing (BCI) aiming at extracting
relevant information from the human
brain wave activity. Following a general
introduction to the concept of BCI using
EEG recordings, a particular type of BCI
based on Steady-State Visual Evoked
Potential (SSVEP) is focused, in which
the EEG signals are obtained as responses
to the target stimulus, flickering at a
certain frequency. Subsequently, the
MLMVN is presented to decode the
phase-coded SSVEP-based BCI.
The performance of the MLMVN is
demonstrated to show a better result
compared with other methods in terms
of decoding accuracy.
In Chapter 9, complex-valued
B-spline neural networks are developed
to identify a complex-valued Wiener
system, which compr ises a linear
dynamical model followed by a nonlinear static transformation. A CVNN
based on B-spline curves consisting of
many polynomial pieces is presented to
estimate the complex-valued nonlinear
function in the complex-valued Wiener
model. For identification of the system,
an algorithm to estimate the parameters
is given based on Gauss-Newton
method with the aid of De Boor algorithm. An algorithm to compute the

Table of Contents for the Digital Edition of Computational Intelligence - May 2013