This chapter reviews the widely linear estimation for complex numbers, quaternions, and geometric algebras (or Clifford algebras) and their application examples. It was proved effective mathematically to add , the complex conjugate number of , as an explanatory variable in estimation of complex-valued data in 1995. Thereafter, the technique has been extended to higher-dimensional algebras. The widely linear estimation improves the accuracy and the efficiency of estimation, then expands the scalability of the estimation framework, and is applicable and useful for many fields including neural computing with high-dimensional parameters.
TopIntroduction
Suppose a problem to determine the relationship between two kinds of samples as
and
, where
expresses transposition, obtained from a certain system as that for determining a relationship between a long-term interest rate and a gold bullion price. Assume linear relationship between the two samples, i.e.,
In this case, what is necessary is merely to determine a real-valued coefficient vector
by any means. Here,
can be estimated from
using parameter
thus obtained.
Samples 
, 
, and parameter 
are all real values in the above-mentioned problem. However, it is necessary to treat complex-valued data with Fourier transform and wavelet transformation, for example, in fields such as communications, image processing, and speech processing. For cases where numbers other than complex numbers are used, it is also necessary to treat two-dimensional data as one unit.
has been used conventionally as an explanatory variable in such cases. It was pointed out recently that there are cases in which sufficient estimates cannot be acquired. Moreover, it was proved effective mathematically to add 
, the complex conjugate number of 
, as an explanatory variable. This technique causes the improvement of the accuracy and the efficiency of estimation, then expands the scalability of the estimation framework, and is applicable and useful for neuro-computing. This chapter describes this widely linear estimation for complex numbers, quaternions, and Clifford numbers (Dorst, Fontijne & Mann, 2007).
TopBackground
First, a technique to treat real-valued data is described. Assume a case in which a true value denoted by a real-valued random variable 
is estimated from an observed value denoted by a real-valued random vector 
, where
is the set of real numbers and 
is a natural number. Then consider an estimated value
expressed as follows:
(1) where
.The problem is to find a parameter
that minimizes mean square error
.Such a framework is designated as (real-valued) linear mean square estimation.