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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 ??Z-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 ??H-SVR is compared and contrasted with the least-square SVR (LS-SVR), the Clifford SVR (C-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 ??H-SVR performs significantly better than the C-SVR, the LS-SVR, and the M-SVR in terms of mean-squared error, outlier sensitivity, and support vector sparsity.