Widely Linear Estimation with Geometric Algebra

Widely Linear Estimation with Geometric Algebra

Tohru Nitta (National Institute of Advanced Industrial Science and Technology (AIST), Japan)
Copyright: © 2013 |Pages: 16
DOI: 10.4018/978-1-4666-3942-3.ch014

Abstract

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.
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Introduction

Suppose a problem to determine the relationship between two kinds of samples as

978-1-4666-3942-3.ch014.m05
and 978-1-4666-3942-3.ch014.m06, where 978-1-4666-3942-3.ch014.m07 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

978-1-4666-3942-3.ch014.m09
by any means. Here, 978-1-4666-3942-3.ch014.m10 can be estimated from 978-1-4666-3942-3.ch014.m11 using parameter 978-1-4666-3942-3.ch014.m12 thus obtained.

Samples 978-1-4666-3942-3.ch014.m13, 978-1-4666-3942-3.ch014.m14, and parameter 978-1-4666-3942-3.ch014.m15 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.

978-1-4666-3942-3.ch014.m16 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 978-1-4666-3942-3.ch014.m17, the complex conjugate number of 978-1-4666-3942-3.ch014.m18, 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).

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Background

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 978-1-4666-3942-3.ch014.m19 is estimated from an observed value denoted by a real-valued random vector 978-1-4666-3942-3.ch014.m20, where978-1-4666-3942-3.ch014.m21 is the set of real numbers and 978-1-4666-3942-3.ch014.m22 is a natural number. Then consider an estimated value978-1-4666-3942-3.ch014.m23expressed as follows:

978-1-4666-3942-3.ch014.m24
(1) where 978-1-4666-3942-3.ch014.m25.The problem is to find a parameter 978-1-4666-3942-3.ch014.m26 that minimizes mean square error 978-1-4666-3942-3.ch014.m27.Such a framework is designated as (real-valued) linear mean square estimation.

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