Agathe Girard

Dynamic systems identification with Gaussian processes

This paper describes the identification of nonlinear dynamic systems with a Gaussian process (GP) prior model. This model is an example of the use of a probabilistic non-parametric modelling approach. GPs are flexible models capable of modelling complex nonlinear systems. Also, an attractive feature of this model is that the variance associated with the model response is readily obtained, and it can be used to highlight areas of the input space where prediction quality is poor, owing to the lack of data or complexity (high variance). We illustrate the GP modelling technique on a simulated example of a nonlinear system.

Propagation of uncertainty in Bayesian kernel models - application to multiple-step ahead forecasting

The object of Bayesian modelling is predictive distribution, which, in a forecasting scenario, enables evaluation of forecasted values and their uncertainties. We focus on reliably estimating the predictive mean and variance of forecasted values using Bayesian kernel based models such as the Gaussian process and the relevance vector machine. We derive novel analytic expressions for the predictive mean and variance for Gaussian kernel shapes under the assumption of a Gaussian input distribution in the static case, and of a recursive Gaussian predictive density in iterative forecasting. The capability of the method is demonstrated for forecasting of time-series and compared to approximate methods.

Adaptive, cautious, predictive control with Gaussian process priors

Abstract Nonparametric Gaussian Process models, a Bayesian statistics approach, are used to implement a nonlinear adaptive control law. Predictions, including propagation of the state uncertainty are made over a K-step horizon. The expected value of a quadratic cost function is minimised, over this prediction horizon, without ignoring the variance of the model predictions. The general method and its main features are illustrated on a simulation example.

Gaussian processes: prediction at a noisy input and application to iterative multiple-step ahead forecasting of time-series

With the Gaussian Process model, the predictive distribution of the output corresponding to a new given input is Gaussian. But if this input is uncertain or noisy, the predictive distribution becomes non-Gaussian. We present an analytical approach that consists of computing only the mean and variance of this new distribution (Gaussian approximation). We show how, depending on the form of the covariance function of the process, we can evaluate these moments exactly or approximately (within a Taylor approximation of the covariance function). We apply our results to the iterative multiple-step ahead prediction of non-linear dynamic systems with propagation of the uncertainty as we predict ahead in time. Finally, using numerical examples, we compare the Gaussian approximation to the numerical approximation of the true predictive distribution by simple Monte-Carlo.

INCORPORATING LINEAR LOCAL MODELS IN GAUSSIAN PROCESS MODEL

Abstract Identification of nonlinear dynamic systems from experimental data can be difficult when, as often happens, more data are available around equilibrium points and only sparse data are available far from those points. The probabilistic Gaussian Process model has already proved to model such systems efficiently. The purpose of this paper is to show how one can relatively easily combine measured data and linear local models in this model. Also, using previous results, we show how uncertainty can be propagated through such models when predicting ahead in time in an iterative manner. The approach is illustrated with a simple numerical example.