Laplace Approximation for Divisive Gaussian Processes for Nonstationary Regression

The standard Gaussian Process regression (GP) is usually formulated under stationary hypotheses: The noise power is considered constant throughout the input space and the covariance of the prior distribution is typically modeled as depending only on the difference between input samples. These assumptions can be too restrictive and unrealistic for many real-world problems. Although nonstationarity can be achieved using specific covariance functions, they require a prior knowledge of the kind of nonstationarity, not available for most applications. In this paper we propose to use the Laplace approximation to make inference in a divisive GP model to perform nonstationary regression, including heteroscedastic noise cases. The log-concavity of the likelihood ensures a unimodal posterior and makes that the Laplace approximation converges to a unique maximum. The characteristics of the likelihood also allow to obtain accurate posterior approximations when compared to the Expectation Propagation (EP) approximations and the asymptotically exact posterior provided by a Markov Chain Monte Carlo implementation with Elliptical Slice Sampling (ESS), but at a reduced computational load with respect to both, EP and ESS

Overlapping Mixtures of Gaussian Processes for the data association problem

In this work we introduce a mixture of GPs to address the data association problem, i.e. to label a group of observations according to the sources that generated them. Unlike several previously proposed GP mixtures, the novel mixture has the distinct characteristic of using no gating function to determine the association of samples and mixture components. Instead, all the GPs in the mixture are global and samples are clustered following "trajectories" across input space. We use a non-standard variational Bayesian algorithm to efficiently recover sample labels and learn the hyperparameters. We show how multi-object tracking problems can be disambiguated and also explore the characteristics of the model in traditional regression settings

Pseudospectral Model Predictive Control under Partially Learned Dynamics

Trajectory optimization of a controlled dynamical system is an essential part of autonomy, however many trajectory optimization techniques are limited by the fidelity of the underlying parametric model. In the field of robotics, a lack of model knowledge can be overcome with machine learning techniques, utilizing measurements to build a dynamical model from the data. This paper aims to take the middle ground between these two approaches by introducing a semi-parametric representation of the underlying system dynamics. Our goal is to leverage the considerable information contained in a traditional physics based model and combine it with a data-driven, non-parametric regression technique known as a Gaussian Process. Integrating this semi-parametric model with model predictive pseudospectral control, we demonstrate this technique on both a cart pole and quadrotor simulation with unmodeled damping and parametric error. In order to manage parametric uncertainty, we introduce an algorithm that utilizes Sparse Spectrum Gaussian Processes (SSGP) for online learning after each rollout. We implement this online learning technique on a cart pole and quadrator, then demonstrate the use of online learning and obstacle avoidance for the dubin vehicle dynamics.Comment: Accepted but withdrawn from AIAA Scitech 201

A Specialized Probability Density Function for the Input of Mixture of Gaussian Processes

Part 2: Machine LearningInternational audienceMixture of Gaussian Processes (MGP) is a generative model being powerful and widely used in the fields of machine learning and data mining. However, when we learn this generative model on a given dataset, we should set the probability density function (pdf) of the input in advance. In general, it can be set as a Gaussian distribution. But, for some actual data like time series, this setting or assumption is not reasonable and effective. In this paper, we propose a specialized pdf for the input of MGP model which is a piecewise-defined continuous function with three parts such that the middle part takes the form of a uniform distribution, while the two side parts take the form of Gaussian distribution. This specialized pdf is more consistent with the uniform distribution of the input than the Gaussian pdf. The two tails of the pdf with the form of a Gaussian distribution ensure the effectiveness of the iteration of the hard-cut EM algorithm for MGPs. It demonstrated by the experiments on the simulation and stock datasets that the MGP model with these specialized pdfs can lead to a better result on time series prediction in comparison with the general MGP models as well as the other classical regression methods

Towards population-based structural health monitoring, part I : homogeneous populations and forms

Data-driven models in Structural Health Monitoring (SHM) generally require comprehensive datasets, recorded from systems in operation, which are rarely available. One potential solution to this problem, considers that information might be transferred, in some sense, between similar systems. As a result, a population-based approach to SHM suggests methods to both model and transfer this valuable information, by considering different groups of structures as populations. Specifically, in this work, a method is proposed to model a population of nominally-identical systems, where (complete) datasets are only available from a subset of members. The framework attempts to build a general model, referred to as the population form, which can be used to make predictions across a group of homogeneous systems. First, the form is demonstrated through applications to a simulated population – with a single experimental (test-rig) member; secondly, the form is applied to data recorded from a group of operational wind turbines