Quaternion vector autoregression (VAR) modeling is a natural extension of real and complex VAR. We demonstrate how a quaternion VAR can be treated as a special case of structured real VAR. We show that generalized least squares and (under Gaussianity) maximum likelihood estimation of the parameters reduces to simple least squares estimation if the innovations are quaternion proper. I. Introduction While real-valued scalar and vector autoregressive processes are ubiquitous in science and

more »

... rics, extensions to complex-valued and quaternion-valued processes are readily found and increasing. Complex-valued AR processes [15] have been applied to temperature forecasting [8], character recognition [14] and shape recognition and extraction [18], [19]. The fading of telecommunication signals is simulated using complex-valued vector autoregressive (VAR) processes in [1]. AR modelling has been extended naturally to the quaternion domain, following other standard complex signal processing tools, such as the polar and singular value decompositions [3], [21], partial least squares and multivariate linear regression [20]. An example quaternion-valued AR process was used in [6], and an adaptive quaternion AR filter is applied to short-term wind forecasting in [4]. An equivalence between the complex-valued or quaternion forms and structured real-valued forms provides these models with a convenient theoretical grounding. In this paper we consider the most complicated of these number fields, the quaternions, and the VAR process model, to show that generalized least squares (GLS) estimation of the model parameters reduces to least squares (LS) estimation if the process is a proper quaternion VAR process; moreover, given Gaussianity, these two estimators are also identical to the maximum likelihood estimator. If ✏ t is any n-length random quaternion-valued innovations vector of the VAR process, the process is a proper quaternion VAR process if the associated 4n ⇥ 4n real-valued covariance matrix has quaternion structure. The less di cult complex case may be treated similarly and details are provided as appropriate. II. Preliminaries A. Quaternion-Valued Matrices A quaternion matrix Q 2 H m⇥n , takes the form Q = A + Bi + Cj + Dk, (A, B, C, D 2 R m⇥n ), where 1, i, j, k are the 4 basis elements which satisfy i 2 = j 2 = k 2 = ijk = 1, ij = ji = k,