On HR calculus, quaternion valued stochastic gradient, and adaptive three dimensional wind forecasting

Short term forecasting of wind field in the quaternion domain is addressed. This is achieved by casting the three components of wind speed (two horizontal and a vertical) into a pure quaternion and adding air temperature as a scalar component, to form the full quaternion. First, HR calculus is introduced in order to provide a unifying framework for the calculation of the derivatives of both analytic quaternion valued functions and real functions of quaternion variables, such as the standard cost function (error power). The analysis shows that the maximum change in the gradient is in the direction of the conjugate of the weight vector, conforming with the gradient calculation in the complex domain. For rigour, we also illustrate that the widely linear model is required in order to capture full second order information within three- and four-dimensional quaternion valued signals. The so established framework is used to illustrate a convenient way to derive the recently introduced quaternion least mean square (QLMS) and the widely linear QLMS (WL-QLMS). Simulations on short term prediction of real world wind signals support the approach.