Derivative observations in Gaussian process models of dynamic systems (original) (raw)

Dynamic systems identification with Gaussian processes

2005

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.

Comprising Prior Knowledge in Dynamic Gaussian Process Models

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. It is shown how uncertainty can be propagated through such models when predicting ahead in time in an iterative manner with Markov Chain Monte Carlo approach. The approach is illustrated with a simple numerical example.

Gaussian process models for systems identification

Different models can be used for nonlinear dy-namic systems identification and the Gaussian process model is a relatively new option with several interesting features: model predictions contain the measure of confidence, the model has a small number of training parameters and facilitated structure determination, and different possibilities of including prior knowledge exist. In this paper the framework for the identification of a dynamic system model based on the Gaussian processes is presented and a short survey with a comprehensive bibliography of published works on application of Gaussian processes for modelling of dynamic systems is given.

EXPLICIT ESTIMATION OF DERIVATIVES FROM DATA AND DIFFERENTIAL EQUATIONS BY GAUSSIAN PROCESS REGRESSION

In this work, we employ the Bayesian inference framework to robustly estimate the derivatives of a function from noisy observations of only the function values at given location points, under the assumption of a physical model in the form of a differential equation governing the function and its derivatives. To overcome the instability of numerical differentiation of the fitted function solely from the data or the prohibitive costs of solving the differential equation on the whole domain, we use the Gaussian processes to jointly model the solution, the derivatives, and the differential equation, by utilizing the fact that differentiation is a linear operator. By regarding the linear differential equation as a linear constraint, we develop the Gaussian process regression with the constraint method (GPRC) at the Bayesian perspective to improve the prediction accuracy of derivatives. For nonlinear equations, we propose a Picard-iteration approximation of linearization around the Gaussian process obtained only from data to iteratively apply our GPRC. Besides, a product of experts method is applied if the initial or boundary condition is also available. We present several numerical results to illustrate the advantages of our new method and show the new estimation of the derivatives from GPRC improves the parameter identification with fewer data samples.

Implementation of Gaussian process models for nonlinear system identification

'Proceedings 5th MATHMOD, Vienna, February 2006', Editors I. Troch, F. Breitenecker, 2006

Abstract This thesis is concerned with investigating the use of Gaussian Process (GP) models for the identification of nonlinear dynamic systems. The Gaussian Process model is a nonparametric approach to system identification where the model of the underlying system is to be identified through the application of Bayesian analysis to empirical data. The GP modelling approach has been proposed as an alternative to more conventional methods of system identification due to a number of attractive features.

Gaussian Process Dynamical Models

2005

This paper introduces Gaussian Process Dynamical Models (GPDM) for nonlinear time series analysis. A GPDM comprises a low-dimensional latent space with associated dynamics, and a map from the latent space to an observation space. We marginalize out the model parameters in closed-form, which amounts to using Gaussian Process (GP) priors for both the dynamics and the observation mappings. This results in a nonparametric model for dynamical systems that accounts for uncertainty in the model. We demonstrate the approach on human motion capture data in which each pose is 62-dimensional. Despite the use of small data sets, the GPDM learns an effective representation of the nonlinear dynamics in these spaces.

Gaussian Processes for Regression and Optimisation

Gaussian processes have proved to be useful and powerful constructs for the purposes of regression. The classical method proceeds by parameterising a covariance function, and then infers the parameters given the training data. In this thesis, the classical approach is augmented by interpreting Gaussian processes as the outputs of linear filters excited by white noise. This enables a straightforward definition of dependent Gaussian processes as the outputs of a multiple output linear filter excited by multiple noise sources. We show how dependent Gaussian processes defined in this way can also be used for the purposes of system identification.

Identification of continuous-time nonlinear systems by using a gaussian process model

IEEJ Transactions on Electrical and Electronic Engineering, 2008

This paper deals with a nonparametric identification of continuous-time nonlinear systems by using a Gaussian process model. Genetic algorithm is applied to train the Gaussian process prior model by minimizing the negative log marginal likelihood of the identification data. The nonlinear term of the objective system is estimated as the predictive mean function of the Gaussian process, and the confidence measure of the estimated nonlinear function is given by the predictive covariance function of the Gaussian process.

An example of Gaussian process model identification

The paper describes the identification of nonlinear dynamic systems with a Gaussian process prior model. This approach is an example of a probabilistic, non-parametric modelling. Gaussian process model can be considered as the special case of radial basis function network and as such an alternative to neural networks or fuzzy black box models. An attractive feature of Gaussian process model is that the variance, associated with the model response, is readily obtained. Variance can be seen as uncertainty of the model and can be used to obtain more accurate multi-step ahead prediction. The method is demonstrated on laboratory pilot plant identification.

Incorporating knowledge about model structure in the identification of Gaussian-process models

2013

Dynamic system identification with Gaussian-process prior model is a probabilistic, nonparametric modelling method for identification. Gaussian-process models provide, besides the prediction, also the information about prediction uncertainty based on the availability or uncertainty of the data used for the modelling. An advantage of this kind of model is a small number of training parameters, a facilitated structure determination and the possibility to include various sorts of prior knowledge into the model. One of possibilities is to include blockstructure knowledge like Hammerstein model structure. The identification procedure of Gaussian-process model with Hammerstein model structure will be presented and illustrated with an example. Key–Words: System identification, Gaussian process models, dynamic systems, Hammerstein model.