Learning a Gaussian Process Model with Uncertain Inputs

Learning with uncertain inputs is well-known to be a difficult task. In order to achieve this analytically using a Gaussian Process prior model, we expand the original process around the input mean (Delta method), assuming the random input is normally distributed. We thus derive a new process whose covariance function accounts for the randomness of the input. We illustrate the effectiveness of the proposed model on a simple static simulation example and on the modelling of a nonlinear noisy time-series. 1 Background Solving the learning task with uncertain or missing inputs has been the scope of much research and the level of difficulty obviously depends on the type of model used. One can distinguish between different situations, depending on the nature of a particular application. Figure 1 summarizes the main different cases: (a) corresponds for instance to the modelling of a noisy timeseries.1 Case (b) is commonly encountered when the system of interest senses inputs imperfectly and (c) corresponds to clean inputs to the system, but corruption during sensing of the inputs for data collection. We can also imagine a blend of these, with both noisy channels from to system, as in (a) & (b), and independent noise on observations of , as in (c). Technical Report TR-2003-144, Department of Computing Science, University of Glasgow, June, 2003. 1When a state-space representation is used, in which the state is formed of delayed observed values. 1