A Theoretical Foundation for the Widely Linear Processing of Quaternion-Valued Data ()
1. Introduction
The widely linear (WL) estimation method has been proved mathematically to be effective for estimation problems using complex-valued data. Estimation using WL uses complex conjugate parameters in addition to complex-valued parameters [1]. It has been applied to communications [2,3] and adaptive filters [4], together with so-called augmented complex statistics, a concept introduced by Picinbono et al. Moreover, an extension of the WL method to quaternion-valued case has been presented [5], which fully exploits available statistical information. A quaternion, a four-dimensional number invented by W. R. Hamilton in 1843, is an extension of a complex number. Let 
be a four-dimensional vector space over 
with an ordered basis, denoted by 
and 
where 
is a set of real numbers. Any quarternion 
is expressed as 
where
. The three basis elements 
satisfy the relations
(1)
(2)
Quaternion algebra has been used in fields such as robotics, computer vision, neural networks, signal processing, and communications (e.g. [6-8]).
In this paper, we present a theoretical foundation for quaternion-valued WL estimation: it is proved that the estimation error obtained using the quaternion-valued WL estimation method is smaller than that obtained using the usual quaternion-valued estimation method [9].
2. The WL Model
In this section, the complex-valued WL model and the quaternion-valued WL model are reviewed.
2.1. The Complex-Valued WL Model
Let 
be a scalar complex-valued random variable to be estimated in terms of an observation that is a complex-valued random vector 
where 
is a set of complex numbers, and 
is a natural number. That is, 
is a true value and 
is an observed value. In complex-valued linear mean square estimation (LMSE), the problem is to find an estimate written as
(3)
where
, and 
represents the complex conjugation and transposition. Then, the objective of the problem is to find the parameter 
that minimizes the estimation error
.
In the meantime, the complex-valued widely linear mean square estimation (WLMSE) problem can be stated as follows: Consider the scalar 
defined as
(4)
where
, and 
is the complex conjugate of
. Then, the objective of the problem is to find parameters 
that minimize the estimation error 
.
Picinbono et al. gave a theoretical foundation for the complex-valued WLMSE described above: it was proved that the estimation error obtained using the complexvalued WLMSE method is smaller than that obtained using the usual complex-valued LMSE method: 
where the equality holds only in exceptional cases [1].
Figures 1 and 2 demonstrate the effectiveness of complex-valued widely linear estimation. Samples of complex-valued random variable x and y are axisymmetric about the broken line: the white circle, the black circle, and the christcross in Figure 1 correspond to the white circle, the black circle, and the christcross in Figure 2, respectively, and the three marks in Figure 1 and the three marks in Figure 2 are all axisymmetric about the broken line. It is therefore difficult to estimate the three samples in Figure 1 by performing rotation, reduction, or expansion of the three samples in Figure 2 about the origin. Nevertheless, inversion of the samples about the real axis, i.e., assuming it as the samples of conjugate complex-valued random variable, allows to be estimated by rotation, reduction, and expansion about the origin. This is an intuitive explanation that the complex-valued widely linear estimation is superior to the conventional linear estimation method.
Figure 1. Three samples of a complex-valued random variable x.
Figure 2. Three samples of a complex-valued random variable y.
2.2. The Quaternion-Valued WL Model
The quaternion-valued WL model is a natural extension of the complex-valued WL model described in Section 2.1. Let 
be a scalar quaternion-valued random variable to be estimated in terms of an observation that is a quaternion-valued random vector
. That is, y is a true value and 
is an observed value.
In quaternion-valued linear mean square estimation (LMSE), the problem is to find an estimate written as
(5)
where
, N is a natural number, and H represents the quaternionic conjugation and transposition.
The quaternion-valued widely linear mean square estimation (WLMSE) problem can be stated as follows: Consider the scalar 
defined as
(6)
where
, N is a natural number, H represents the quaternionic conjugation and transposition, and 
is the quaternionic conjugate of
. Then, the objective of the problem is to find parameters 
that minimize 
.
Took and Mandic derived an augmented quaternion least mean squares (AQLMS) algorithm for quaternion valued adaptive filters based on the quaternion-valued WL model, and confirmed its effectiveness via computer simulations [5]. Actually, the experimental results on the Lorenz attractor, real-world wind forecasting, and data fusion via quaternion spaces support the approach. Consequently, computer simulations proved that the quarternion-valued WL estimation method is superior to the usual quaternion-valued linear estimation method.
Moreover, the three perpendicular quaternion involutions can be introduced into the quaternion-valued WLMSE, which are given by
(7)
(8)
(9)
where 
[10-12]. The quarternion-valued WLMSE using (6) is an initial insight. However, in order to achieve the complete description of the second order statistics in
, we need to consider the involutions ((7)-(9)). Actually, the quaternion-valued widely linear mean square estimation (WLMSE) problem using the involutions has been formulated as follows [12]: Consider the scalar 
defined as
(10)
where
. Then, the objective of the problem is to find parameters 
that minimize
.
However, no theoretical proof for the superiority of the quaternion-valued WL estimation method on estimation errors has been given to date, as it has been for the complex-valued WL estimation method. In the complexvalued setting, Picinbono et al. proved that the estimation error obtained with the complex-valued WLMSE is smaller than the error obtained using the complex-valued LMSE [1].
3. A Theoretical Foundation of the Quaternion-Valued WL Model
In this section, we show the superiority of the quarternion-valued WLMSE method. The main result is as follows: the estimation error obtained using the quarternionvalued WL estimation method based on (6) is smaller than that obtained using the usual quarternion-valued linear estimation method, except in rare cases. The result is obtainable in the same manner described by [1]. However, the noncommutativity of the quaternion products must be considered during the analysis (
for any
).
Define
Therein, X is a set of scalar quaternion-valued random variables that constitutes a linear space, and which becomes a Hilbert subspace of the one-dimensional quarternion-valued Hilbert space 
with the scalar product
. Then, for a true value
, an observed value
, and the estimate
, the following equations hold:
(12)
(13)
where 
means that all the components of 
are orthogonal to u with the scalar product 
. From (12) and (13), we obtain
(14)
(15)
Consequently, from (6), (14), (15), the following equations hold (see the appendices for the detail of the derivation):
(16)
(17)
where 
Equations (16) and (17) yield
(18)
(19)
where we assume that 
and 
exist (see the appendices for the detail of the derivation). Then, the estimation error 
is calculated from (6), (16), and (17) as follows:
(20)
The estimation error 
in the quaternion-valued LMSE can be obtained using (5) as shown below.
(21)
Then, from (16) and (18), (20), (21), the quantity 
can be expressed as
(22)
which is the difference between the estimation error of the quaternion-valued LMSE and that of the quarternion-valued WLMSE. Equation (22) is nonnegative because the matrix 
is positive-semidefinite (see the appendices for the proof). Furthermore, (22) is equal to zero only when one of the following conditions holds:
(23)
(24)
Equation (23) is an exceptional case, and (24) means that the true value y can be estimated with probability of one, which is a rare case.
4. Discussions
In the previous section, a theoretical foundation of the quaternion-valued WL model based on (6) is given, which guarantees the superiority of the quaternion-valued WLMSE method. It is difficult to analyze the quaternion-value WLMSE method using (7)-(9) which contains all the necessary second order statistical information because the process of the analysis is very complicated due to the three involutions if we adopt the way of proof used in [1]. To solve the problem, another approach would be needed.
5. Conclusion
We have presented a theoretical foundation for the quaternion-valued WL estimation method. It was proved that the estimation error obtained using the quaternion-valued WL estimation method is smaller than that obtained using the usual quaternion-valued linear estimation method, except in rare cases. In our future studies, we will proceed with analyses of the WL estimation based on the Clifford algebra [13].
6. Acknowledgements
The author extends special thanks to Prof. B. Picinbono, the Laboratoire des Signaux et Systèmes, Supélec, Gifsur Yvette, France for help in resolving several questions, and the anonymous reviewers for valuable comments.
Appendices
Proof of Equations (16) and (17):
(25)
So, we obtain
(26)
Similarly, from Equation (13) we obtain
(27)
Here,
(28)
Note that 
due to the noncommutativity of the quaternion products. Thus, from Equations (26) and (28),
(29)
which means that Equation (16) holds.
It also follows from Equations (27) and (28),
(30)
The left hand side of Equation (30) is given as:
(31)
Equations (30) and (31) complete the proof of Equation (17). ■
Proof of Equations (18) and (19):
From Equation (16), we obtain
(32)
Equations (17) and (34) lead Equation (18). And also, from Equation (17), we obtain
(33)
Equations (16) and (33) lead Equation (19). ■
Proof that the matrix 
is positivesemidefinite:
Consider the following problem. Let 
and 
be two random vectors such that
. The problem is to estimate linearly 
in terms of
, which means to find the N-dimensional quaternion valued matrix M such that 
is the best linear mean square (LMS) estimation of 
in terms of
. The solution is given by the orthogonality principle saying that 
is orthogonal to the observation 
or 
That is, 
where 
and 
. Assuming that 
exists, the solution is obviously
(34)
Here, the matrix 
is positivesemidefinite because for any 
such that
,
. Using the definition 
and Equation (34) yield by simple algebra
(35)
where 
and
.
Suppose now that
. This yields
, 
, 
and
. Thus we obtain from Equation (35)
(36)
As seen above, since 
is positive-semidefinite, the right-hand side of Equation (36) is positive-semidefinite. ■