Continuous-time Gaussian Processes Research Articles

Exact Inference for Continuous-Time Gaussian Process Dynamics

Many physical systems can be described as a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. While this scheme is mathematically tempting, it can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. We tackle this task by leveraging higher-order numerical integrators. These integrators provide the necessary tools to discretize dynamical systems with arbitrary accuracy. However, most higher-order integrators require dynamics evaluations at intermediate time steps, making exact GP inference intractable. In previous work, this problem is often addressed by approximate inference techniques. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to enable direct inference, we propose to leverage multistep and Taylor integrators. We demonstrate how exact inference schemes can be derived for these types of integrators. Further, we derive tailored sampling schemes that allow one to draw consistent dynamics functions from the posterior. The learned model can thus be integrated with arbitrary integrators, just like a standard dynamical system. We show empirically and theoretically that our approach yields an accurate representation of the continuous-time system.

Event-Based Stereo Visual Odometry With Native Temporal Resolution via Continuous-Time Gaussian Process Regression

Event-Based Stereo Visual Odometry With Native Temporal Resolution via Continuous-Time Gaussian Process Regression

Mixtures of Gaussian Processes for Robot Motion Planning Using Stochastic Trajectory Optimization

Robot motion planning methods based on trajectory optimization can efficiently generate feasible and optimal trajectories by minimizing a suitable cost function, even in high-dimensional spaces. However, the main drawback of these methods lies in their proneness to infeasible local minima, especially in complex environments. To mitigate this issue, we propose a novel motion planning method that represents trajectories as samples from a mixture of continuous-time Gaussian processes (MGP) and employs stochastic optimization in order to update the MGP parameters in a cost-minimizing manner. The contributions of the proposed trajectory optimization method arise from the introduced mixture representation and stochastic gradient estimation, dominantly enabling better exploration of the trajectory space and including nondifferentiable optimizing costs. We evaluated the proposed method in multiple simulation benchmarks featuring 7 degree-of-freedom (DOF) robot arms and a 10DOF mobile manipulator. We also conducted a real-world experiment with a 14DOF dual-arm robot. The experimental results demonstrated that the proposed method achieves higher success rate than several state-of-the-art methods, while the advantages stemming from MGPs and stochastic optimization, like trajectory smoothness, support of nondifferentiable cost functions, multiple trajectory solutions, and the ability to tackle high-dimensional planning problems, are inherently kept.

Anisotropic GPMP2: A Fast Continuous-Time Gaussian Processes Based Motion Planner for Unmanned Surface Vehicles in Environments With Ocean Currents

In the past decade, there is an increasing interest in the deployment of unmanned surface vehicles (USVs) for undertaking ocean missions in dynamic, complex maritime environments. The success of these missions largely relies on motion planning algorithms that can generate optimal navigational trajectories to guide a USV. Apart from minimising the distance of a path, when deployed a USVs’ motion planning algorithms also need to consider other constraints such as energy consumption, the affected of ocean currents as well as the fast collision avoidance capability. In this paper, we propose a new algorithm named anisotropic GPMP2 to revolutionise motion planning for USVs based upon the fundamentals of GP (Gaussian process) motion planning (GPMP, or its updated version GPMP2). Firstly, we integrated the anisotropy into GPMP2 to make the generated trajectories follow ocean currents where necessary to reduce energy consumption on resisting ocean currents. Secondly, to further improve the computational speed and trajectory quality, a dynamic fast GP interpolation is integrated in the algorithm. Finally, the new algorithm has been validated on a WAM-V 20 USV in a ROS environment to show the practicability of anisotropic GPMP2. Note to Practitioners—The work reported in this article will be significant for USVs to conduct missions in complex, dynamic maritime environments where various obstacles and time-varying ocean currents exit. We develop this novel motion planning algorithm based on Gaussian process and optimise the trajectory using probabilistic inferences. The new algorithm can generate collision free trajectories that also minimise the influences caused by adverse ocean currents in a highly efficient way. In addition, the planning has been undertaken in a continuous-time domain making the generated trajectory have a guaranteed smoothness and readily feasible for autopilots to track. We use a coastal area with time-varying vortexes to present a challenging practical maritime environment. The presented algorithm integrates the available information about a fluid field regarding energy consumption and hazard level, along with the density of obstacles to plan a navigational route efficiently. To increase the practical performance of the proposed method, diverse models for generating ocean currents need to be developed in the future to tackle unpredictable situations.

Statistical inference for stationary linear models with tapered data

In this paper, we survey some recent results on statistical inference (parametric and nonparametric statistical estimation, hypotheses testing) about the spectrum of stationary models with tapered data, as well as, a question concerning robustness of inferences, carried out on a linear stationary process contaminated by a small trend. We also discuss some question concerning tapered Toeplitz matrices and operators, central limit theorems for tapered Toeplitz type quadratic functionals, and tapered Fejer-type kernels and singular integrals. These are the main tools for obtaining the corresponding results, and also are of interest in themselves. The processes considered will be discrete-time and continuous-time Gaussian, linear or Levy-driven linear processes with memory.

Cross-entropy based stochastic optimization of robot trajectories using heteroscedastic continuous-time Gaussian processes

High dimensional robot motion planning has recently been approached with trajectory optimization methods that efficiently minimize a suitable objective function in order to generate robot trajectories that are both optimal and feasible. However, finding a globally optimal solution is often an insurmountable problem in practice and state-of-the-art trajectory optimization methods are thus prone to local minima, mainly in cluttered environments. In this paper, we propose a novel trajectory planning algorithm that employs stochastic optimization in order to find a collision-free trajectory generated from a continuous-time Gaussian process (GP). The contributions of the proposed motion planning method stem from introducing the heteroscedasticity of the GP, together with exploited sparsity for efficient covariance estimation, and a cross-entropy based stochastic optimization for importance sampling based trajectory optimization. We evaluate the proposed method on three simulated scenarios: a maze benchmark, a 7DOF robot arm planning benchmark and a 10DOF mobile manipulator trajectory planning example and compare it to a state-of-the-art GP trajectory optimization method, namely the Gaussian process motion planner 2 algorithm (GPMP2). Our results demonstrate the following: (i) the proposed method yields a more thorough exploration of the solution space in complex environments than GPMP2, while having comparable execution time, (ii) the introduced heteroscedasticity generates GP priors better suited for collision avoidance and (iii) the proposed method has the ability to efficiently tackle high-dimensional trajectory planning problems.

Continuous-Time Look-Ahead Optimization of Energy Storage in Real-Time Balancing and Regulation Markets

This article proposes a novel continuous-time look-ahead optimization model for scheduling balancing energy and regulation capacity provided by energy storage (ES) devices and generating units in real-time markets. The proposed model implements the receding horizon control approach, where at each time, the operation of ES devices and generating units are co-optimized to minimize the total real-time system operation cost for providing the services over the control horizon in future. A continuous-time Gaussian process is developed to characterize the uncertainty and variability of load, which is used to update the load forecast over each control horizon. The proposed continuous-time problem is solved in a finite-dimensional function space spanned by Bernstein polynomials, which converts the problem into a solvable mixed-integer linear programming problem. The proposed model is implemented on the IEEE reliability test system. Numerical results show the optimal participation of ES in balancing and regulation markets considerably reduces the total real-time operation cost of the system.

Power of the MOSUM test for online detection of a transient change in mean

In this article we discuss an online moving sum (MOSUM) test for detection of a transient change in the mean of a sequence of independent and identically distributed (i.i.d.) normal random variables. By using a well-developed theory for continuous time Gaussian processes and subsequently correcting the results for discrete time, we provide accurate approximations for the average run length (ARL) and power of the test. We check theoretical results against simulations, compare the power of the MOSUM test with that of the cumulative sum (CUSUM), and briefly consider the cases of nonnormal random variables and weighted sums.

Asymptotic near-efficiency of the “Gibbs-energy (GE) and empirical-variance” estimating functions for fitting Matérn models - II: Accounting for measurement errors via “conditional GE mean”

Consider one realization of a continuous-time Gaussian process Z which belongs to the Matérn family with known “regularity” index ν>0. For estimating the autocorrelation-range and the variance of Z from n observations on a fine grid, we studied in Girard (2016) the GE–EV method which simply retains the empirical variance (EV) and equates it to a candidate “Gibbs energy (GE)”, i.e. the quadratic form zTR−1z∕n where z is the vector of observations and R is the autocorrelation matrix for z associated with a candidate range. The present study considers the case where the observation is z plus a Gaussian white noise whose variance is known. We propose to simply bias-correct EV and to replace GE by its conditional mean given the observation. We show that the ratio of the large-n mean squared error of the resulting CGEM–EV estimate of the range-parameter to the one of its maximum likelihood estimate, and the analog ratio for the variance-parameter, have the same behavior than in the no-noise case: they both converge, when the grid-step tends to 0, toward a constant, only function of ν, surprisingly close to 1 provided ν is not too large. We also obtain, for all ν, convergence to 1 of the analog ratio for the microergodic-parameter. Furthermore, we discuss possible non-normality of the noise, and the impact of a “not small enough” grid-step.

Continuous-time Gaussian process motion planning via probabilistic inference

We introduce a novel formulation of motion planning, for continuous-time trajectories, as probabilistic inference. We first show how smooth continuous-time trajectories can be represented by a small number of states using sparse Gaussian process (GP) models. We next develop an efficient gradient-based optimization algorithm that exploits this sparsity and GP interpolation. We call this algorithm the Gaussian Process Motion Planner (GPMP). We then detail how motion planning problems can be formulated as probabilistic inference on a factor graph. This forms the basis for GPMP2, a very efficient algorithm that combines GP representations of trajectories with fast, structure-exploiting inference via numerical optimization. Finally, we extend GPMP2 to an incremental algorithm, iGPMP2, that can efficiently replan when conditions change. We benchmark our algorithms against several sampling-based and trajectory optimization-based motion planning algorithms on planning problems in multiple environments. Our evaluation reveals that GPMP2 is several times faster than previous algorithms while retaining robustness. We also benchmark iGPMP2 on replanning problems, and show that it can find successful solutions in a fraction of the time required by GPMP2 to replan from scratch.

Multi-agent Gaussian Process Motion Planning via Probabilistic Inference

Abstract This paper deals with motion planning for multiple agents by representing the problem as a simultaneous optimization of every agent’s trajectory. Each trajectory is considered as a sample from a one-dimensional continuous-time Gaussian process (GP) generated by a linear time-varying stochastic differential equation driven by white noise. By formulating the planning problem as probabilistic inference on a factor graph, the structure of the pertaining GP can be exploited to find the solution efficiently using numerical optimization. In contrast to planning each agent’s trajectory individually, where only the current poses of other agents are taken into account, we propose simultaneous planning of multiple trajectories that works in a predictive manner. It takes into account the information about each agent’s whereabouts at every future time instant, since full trajectories of each agent are found jointly during a single optimization procedure. We compare the proposed method to an individual trajectory planning approach, demonstrating significant improvement in both success rate and computational efficiency.

Asymptotic near-efficiency of the “Gibbs-energy and empirical-variance” estimating functions for fitting Matérn models — I: Densely sampled processes

Consider one realization of a continuous-time Gaussian process Z which belongs to the Matérn family with known regularity index ν>0. For estimating the autocorrelation-range and the variance of Z from n observations on a fine grid, we propose two simple estimating functions based on the “candidate Gibbs energy” (GE) and the empirical variance (EV). Here a candidate GE designates the quadratic form zTR−1z/n where z is the vector of observations and R is the autocorrelation matrix for z associated with a candidate range. We show that the ratio of the large-n mean squared error of the resulting GE–EV estimate of the range-parameter to the one of its maximum likelihood estimate, and the analog ratio for the variance-parameter, both converge, when the grid-step tends to 0, toward a constant, only function of ν, surprisingly close to 1 provided ν is not too large. This latter condition on ν has not to be imposed to obtain the convergence to 1 of the analog ratio for the microergodic-parameter. Possible extensions of this approach, which could be rather easily implemented, are briefly discussed.

Maxima and sum for discrete and continuous time Gaussian processes

We study the asymptotic relation among the maximum of continuous weakly and strongly dependent stationary Gaussian process, the maximum of this process sampled at discrete time points, and the partial sum of this process. It is shown that these two extreme values and the sum are asymptotically independent if the grid of the discrete time points is sufficiently sparse and the Gaussian process is weakly dependent, and asymptotically dependent if the grid points are Pickands grids or dense grids.

On trend estimation under monotone Gaussian subordination with long-memory: application to fossil pollen series

Fossil pollen data from stratigraphic cores are irregularly spaced in time due to non-linear age–depth relations. Moreover, their marginal distributions may vary over time. We address these features in a nonparametric regression model with errors that are monotone transformations of a latent continuous-time Gaussian process Z(T). Although Z(T) is unobserved, due to monotonicity, under suitable regularity conditions, it can be recovered facilitating further computations such as estimation of the long-memory parameter and the Hermite coefficients. The estimation of Z(T) itself involves estimation of the marginal distribution function of the regression errors. These issues are considered in proposing a plug-in algorithm for optimal bandwidth selection and construction of confidence bands for the trend function. Some high-resolution time series of pollen records from Lago di Origlio in Switzerland, which go back ca. 20,000 years are used to illustrate the methods.

Moment bounds and central limit theorems for Gaussian subordinated arrays

A general moment bound for sums of products of Gaussian vector’s functions extending the moment bound in Taqqu (1977, Lemma 4.5) [28] is established. A general central limit theorem for triangular arrays of nonlinear functionals of multidimensional non-stationary Gaussian sequences is proved. This theorem extends the previous results of Breuer and Major (1983) [5], Arcones (1994) [1] and others. A Berry–Esseen-type bound in the above-mentioned central limit theorem is derived following Nourdin et al. (2011) [20]. Two applications of the above results are discussed. The first one refers to the asymptotic behavior of a roughness statistic for continuous-time Gaussian processes and the second one is a central limit theorem satisfied by long memory locally stationary processes.

A Non-Parametric Estimator of the Spectral Density of a Continuous-Time Gaussian Process Observed at Random Times

In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a nonparametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and application to biological data are also provided.

On the limiting spectral distribution of the covariance matrices of time-lagged processes

We consider two continuous-time Gaussian processes, one being partially correlated to a time-lagged version of the other. We first give the limiting spectral distribution for the covariance matrices of the increments of the processes when the span between two observations tends to zero. Then, we derive the limiting distribution of the eigenvalues of the sample covariance matrices. This result is obtained when the number of paths of the processes is asymptotically proportional to the number of observations for each single path. As an application, we use the second moment of this distribution together with auxiliary volatility and correlation estimates to construct an adaptive estimator of the time lag between the two processes. Finally, we provide an asymptotic theory for our estimation procedure.

Nonparametric Identification of Population Models: An MCMC Approach

The paper deals with the nonparametric identification of population models, that is models that explain jointly the behavior of different subjects drawn from a population, e.g., responses of different patients to a drug. The average response of the population and the individual responses are modeled as continuous-time Gaussian processes with unknown hyperparameters. Within a Bayesian paradigm, the posterior expectation and variance of both the average and individual curves are computed by means of a Markov Chain Monte Carlo scheme. The model and the estimation procedure are tested on both simulated and experimental pharmacokinetic data.

Prediction of Fractional Brownian Motion-Type Processes

We introduce a class of continuous-time Gaussian processes with stationary increments via moving-average representation with good MA coefficient. The class includes fractional Brownian motion with Hurst index less than 1/2 as a typical example. It also includes processes which have different indices corresponding to the local and long-time properties, repsectively. We derive some basic properties of the processes, and, using the results, we establish a prediction formula for them. The prediction kernel in the formula is given explicitly in terms of MA and AR coefficients.

A theory of robust long-run variance estimation

Long-run variance estimation can typically be viewed as the problem of estimating the scale of a limiting continuous time Gaussian process on the unit interval. A natural benchmark model is given by a sample that consists of equally spaced observations of this limiting process. The paper analyzes the asymptotic robustness of long-run variance estimators to contaminations of this benchmark model. It is shown that any equivariant long-run variance estimator that is consistent in the benchmark model is highly fragile: there always exists a sequence of contaminated models with the same limiting behavior as the benchmark model for which the estimator converges in probability to an arbitrary positive value. A class of robust inconsistent long-run variance estimators is derived that optimally trades off asymptotic variance in the benchmark model against the largest asymptotic bias in a specific set of contaminated models.