This paper presents a new Gaussian process (GP) surrogate modeling for predicting the outcome of a physical experiment where some experimental inputs are controlled by other manipulating factors. Particularly, we are interested in the case where the control precision is not very high, so the input factor values vary significantly even under the same setting of the corresponding manipulating factors. The case is observed in our main application to carbon nanotube growth experiments, where one experimental input among many is manipulated by another manipulating factors, and the relation between the input and the manipulating factors significantly varies in the dates and times of operations. Due to this variation, the standard GP surrogate that directly relates the manipulating factors to the experimental outcome does not provide a great predictive power on the outcome. At the same time, the GP model relating the main factors to the outcome directly is not appropriate for the prediction purpose because the main factors cannot be accurately set as planned for a future experiment. Motivated by the carbon nanotube example, we propose a two-tiered GP model, where the bottom tier relates the manipulating factors to the corresponding main factors with potential biases and variation independent of the manipulating factors, and the top tier relates the main factors to the experimental outcome. Our two-tier model explicitly models the propagation of the control uncertainty to the experimental outcome through the two GP modeling tiers. We present the inference and hyper-parameter estimation of the proposed model. The proposed approach is illustrated with the motivating example of a closed-loop autonomous research system for carbon nanotube growth experiments, and the test results are reported with the comparison to a benchmark method, i.e. a standard GP model.
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