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In this paper, a robust support vector regression (RSVR) method with uncertain input and output data is studied. First, the data uncertainties are investigated under a stochastic framework and two linear robust formulations are derived. Linear formulations robust to ellipsoidal uncertainties are also considered from a geometric perspective. Second, kernelized RSVR formulations are established for nonlinear regression problems. Both linear and nonlinear formulations are converted to second-order cone programming problems, which can be solved efficiently by the interior point method. Simulation demonstrates that the proposed method outperforms existing RSVRs in the presence of both input and output data uncertainties.