We present a novel approach for explaining Gaussian processes (GPs) that can utilize the full analytical covariance structure present in GPs.

Applying Physics-Informed Gaussian Process Regression to the eigenvalue problem $(\mathcal{L}-λ)u = 0$ poses a fundamental challenge, where the null source term results in a trivial predictive mean and a degenerate marginal likelihood. Drawing inspiration from system identification, we construct a transfer function-type indicator for the unknown eigenvalue/eigenfunction using the physics-informed Gaussian Process posterior. We demonstrate that the posterior covariance is only non-trivial when $λ$ corresponds to an eigenvalue of the partial differential operator $\mathcal{L}$, reflecting the existence of a non-trivial eigenspace, and any sample from the posterior lies in the eigenspace of the linear operator. We demonstrate the effectiveness of the proposed approach through several numerical examples with both linear and non-linear eigenvalue problems.

Understanding driver interactions is critical to designing autonomous vehicles to interoperate safely with human-driven cars. We consider the impact of these interactions on the policies drivers employ when navigating unsigned intersections in a driving simulator. The simulator allows the collection of naturalistic decision-making and behavior data in a controlled environment. Using these data, we model the human driver responses as state-based feedback controllers learned via Gaussian Process regression methods. We compute the feedback gain of the controller using a weighted combination of linear and nonlinear priors. We then analyze how the individual gains are reflected in driver behavior. We also assess differences in these controllers across populations of drivers. Our work in data-driven analyses of how drivers determine their policies can facilitate future work in the design of socially responsive autonomy for vehicles.

Motivated by recent work involving the analysis of leveraging spatial correlations in sparsified mean estimation, we present a novel procedure for constructing covariance estimator. The proposed Random-knots (Random-knots-Spatial) and B-spline (Bspline-Spatial) estimators of the covariance function are computationally efficient. Asymptotic pointwise of the covariance are obtained for sparsified individual trajectories under some regularity conditions. Our proposed nonparametric method well perform the functional principal components analysis for the case of sparsified data, where the number of repeated measurements available per subject is small. In contrast, classical functional data analysis requires a large number of regularly spaced measurements per subject. Model selection techniques, such as the Akaike information criterion, are used to choose the model dimension corresponding to the number of eigenfunctions in the model. Theoretical results are illustrated with Monte Carlo simulation experiments. Finally, we cluster multi-domain data by replacing the covariance function with our proposed covariance estimator during PCA.

Early approaches to multiple-output Gaussian processes (MOGPs) relied on linear combinations of independent, latent, single-output Gaussian processes (GPs).

Large-scale Gaussian process inference has long faced practical challenges due to time and space complexity that is superlinear in dataset size.

Here we propose to model this causal interaction using integro-differential equations and causal kernels that allow for a rich analysis of effective connectivity.

We consider a modification of the covariance function in Gaussian processes to correctly account for known linear operator constraints.

We consider the problem of estimating the latent state of a spatiotemporally evolving continuous function using very few sensor measurements.

Gaussian processes are rich distributions over functions, with generalization properties determined by a kernel function. When used for long-range extrapolation, predictions are particularly sensitive to the choice of kernel parameters. It is therefore critical to account for kernel uncertainty in our predictive distributions. We propose a distribution over kernels formed by modelling a spectral mixture density with a L evy process. The resulting distribution has support for all stationary covariances--including the popular RBF, periodic, and Mat ern kernels-- combined with inductive biases which enable automatic and data efficient learning, long-range extrapolation, and state of the art predictive performance. The proposed model also presents an approach to spectral regularization, as the L evy process introduces a sparsity-inducing prior over mixture components, allowing automatic selection over model order and pruning of extraneous components. We exploit the algebraic structure of the proposed process for O (n) training and O (1) predictions. We perform extrapolations having reasonable uncertainty estimates on several benchmarks, show that the proposed model can recover flexible ground truth covariances and that it is robust to errors in initialization.