Gaussian process

About: Gaussian process is a research topic. Over the lifetime, 18944 publications have been published within this topic receiving 486645 citations. The topic is also known as: Gaussian stochastic process.

Optimal Steering of a Linear Stochastic System to a Final Probability Distribution—Part III

Abstract: We address the problem of steering the state of a linear stochastic system to a prescribed distribution over a finite horizon with minimum energy, and the problem to maintain the state at a stationary distribution over an infinite horizon with minimum power. For both problems the control and Gaussian noise channels are allowed to be distinct, thereby, placing the results of this paper outside of the scope of previous work both in probability and in control. The special case where the disturbance and control enter through the same channels has been addressed in the first part of this work that was presented as Part I. Herein, we present sufficient conditions for optimality in terms of a system of dynamically coupled Riccati equations in the finite horizon case and in terms of algebraic conditions for the stationary case. We then address the question of feasibility for both problems. For the finite-horizon case, provided the system is controllable, we prove that without any restriction on the directionality of the stochastic disturbance it is always possible to steer the state to any arbitrary Gaussian distribution over any specified finite time-interval. For the stationary infinite horizon case, it is not always possible to maintain the state at an arbitrary Gaussian distribution through constant state-feedback. It is shown that covariances of admissible stationary Gaussian distributions are characterized by a certain Lyapunov-like equation and, in fact, they coincide with the class of stationary state covariances that can be attained by a suitable stationary colored noise as input. We finally address the question of how to compute suitable controls numerically. We present an alternative to solving the system of coupled Riccati equations, by expressing the optimal controls in the form of solutions to (convex) semi-definite programs for both cases. We conclude with an example to steer the state covariance of the distribution of inertial particles to an admissible stationary Gaussian distribution over a finite interval, to be maintained at that stationary distribution thereafter by constant-gain state-feedback control.

Building Efficient Response Surfaces of Aerodynamic Functions with Kriging and Cokriging

Abstract: In this paper, we compare the global accuracy of different strategies to build response surfaces by varying sampling methods and modeling techniques. The aerodynamic test functions are obtained by deforming the shape of a transonic airfoil. For comparisons, a robust strategy for model fit using a new efficient initialization technique followed by a gradient optimization was applied. First, a study of different sampling methods proves that including a posteriori information on the function to sample distribution can improve accuracy over classical space-filling methods such as Latin hypercube sampling. Second, comparing kriging and gradient-enhanced kriging on two- to six-dimensional test cases shows that interpolating gradient vectors drastically improves response-surface accuracy. Although direct and indirect cokriging have equivalent formulations, the indirect cokriging outperforms the direct approach. The slow linear phase of error convergence when increasing sample size is not avoided by cokriging. Thus, the number of samples needed to have a globally accurate surface stays generally out of reach for problems considering more than four design variables.

Variational Bayesian multinomial probit regression with Gaussian process priors

Abstract: It is well known in the statistics literature that augmenting binary and polychotomous response models with gaussian latent variables enables exact Bayesian analysis via Gibbs sampling from the parameter posterior. By adopting such a data augmentation strategy, dispensing with priors over regression coefficients in favor of gaussian process (GP) priors over functions, and employing variational approximations to the full posterior, we obtain efficient computational methods for GP classification in the multiclass setting. The model augmentation with additional latent variables ensures full a posteriori class coupling while retaining the simple a priori independent GP covariance structure from which sparse approximations, such as multiclass informative vector machines (IVM), emerge in a natural and straightforward manner. This is the first time that a fully variational Bayesian treatment for multiclass GP classification has been developed without having to resort to additional explicit approximations to the nongaussian likelihood term. Empirical comparisons with exact analysis use Markov Chain Monte Carlo (MCMC) and Laplace approximations illustrate the utility of the variational approximation as a computationally economic alternative to full MCMC and it is shown to be more accurate than the Laplace approximation.

MIMO Gaussian Channels With Arbitrary Inputs: Optimal Precoding and Power Allocation

Abstract: In this paper, we investigate the linear precoding and power allocation policies that maximize the mutual information for general multiple-input-multiple-output (MIMO) Gaussian channels with arbitrary input distributions, by capitalizing on the relationship between mutual information and minimum mean-square error (MMSE). The optimal linear precoder satisfies a fixed-point equation as a function of the channel and the input constellation. For non-Gaussian inputs, a nondiagonal precoding matrix in general increases the information transmission rate, even for parallel noninteracting channels. Whenever precoding is precluded, the optimal power allocation policy also satisfies a fixed-point equation; we put forth a generalization of the mercury/waterfilling algorithm, previously proposed for parallel noninterfering channels, in which the mercury level accounts not only for the non-Gaussian input distributions, but also for the interference among inputs.

Hypothesis Testing in Time Series via the Empirical Characteristic Function: A Generalized Spectral Density Approach

Abstract: The standardized spectral density completely describes serial dependence of a Gaussian process. For non-Gaussian processes, however, it may become an inappropriate analytic tool, because it misses the nonlinear processes with zero autocorrelation. By generalizing the concept of the standardized spectral density, I propose a new spectral tool suitable for both linear and nonlinear time series analysis. The generalized spectral density is indexed by frequency and a pair of auxiliary parameters. It is well defined for both continuous and discrete random variables, and requires no moment condition. Introduction of the auxiliary parameters renders the spectrum able to capture all pairwise dependencies, including those with zero autocorrelation. The standardized spectral density can be derived by properly differentiating the generalized spectral density with respect to the auxiliary parameters at the origin. The consistency of a class of Parzen's kernel-type estimators for the generalized spectral dens...