Properties of three dimensional vector autoregressive model

The invariance and covariance of extracted features from an object under certain transformation play quite important roles in the fields of pattern recognition and image understanding. For instance, in order to recognize a three dimentional (3D) object, we need specific features extracted from a given object. These features should be independent of the pose and the location of an object. To extract such feature, One of the authors has presented the 3D vector autoregressive (VAR) model. This 3D VAR model is constructed on the quaternion, which is the basis of SU(2) (the rotation group in two dimensional complex space). Then the 3D VAR model is defined by the external products of 3D sequential data and the autoregressive (AR) coefficients, unlike the conventinaol An models. Therefore the 3D VAR model has some prominent features. For example, The AR coefficients of the 3D VAR model behave like vectors under any three dimensional rotation. In this paper, we present the recursive computation of 2D VAR coefficients and 3D VAR coefficients. This method reduce the cost of computation of of VER coefficients. We also define the partial correlation (PARCOR) vectors for the 2D VER model and 3D VER model from the point of view of data compression and pattern recognition.