A Division Algebraic Framework for Multidimensional Support Vector Regression

Alistair Shilton, Daniel T. H. Lai, Marimuthu Palaniswami

Abstract: In this paper, division algebras are proposed as an elegant basis upon which to extend support vector regression (SVR) to multidimensional targets. Using this framework, a multitarget SVR called ¿ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z</sub> -SVR is proposed based on an ¿-insensitive loss function that is independent of the coordinate system or basis used. This is developed to dual form in a manner that is analogous to the standard ¿-SVR. The ¿ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> -SVR is compared and contrasted with the least-square SVR (LS-SVR), the Clifford SVR ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> -SVR), and the multidimensional SVR (M-SVR). Three practical applications are considered: namely, 1) approximation of a complex-valued function; 2) chaotic time-series prediction in 3-D; and 3) communication channel equalization. Results show that the ¿ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> -SVR performs significantly better than the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> -SVR, the LS-SVR, and the M-SVR in terms of mean-squared error, outlier sensitivity, and support vector sparsity.