Erratum to “Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes”

The covariance structure of spatial Gaussian predictors (aka Kriging predictors) is generally modeled by parameterized covariance functions; the associated hyperparameters in turn are estimated via the method of maximum likelihood. In this work, the asymptotic behavior of the maximum likelihood of spatial Gaussian predictor models as a function of its hyperparameters is investigated theoretically. Asymptotic sandwich bounds for the maximum likelihood function in terms of the condition number of the associated covariance matrix are established. As a consequence, the main result is obtained:optimally trained nondegenerate spatial Gaussian processes cannot feature arbitrary ill-conditioned correlation matrices. The implication of this theorem on Kriging hyperparameter optimization is exposed. A nonartificial example is presented, where maximum likelihood-based Kriging model training is necessarily bound to fail.

Download Full-text

Bayesian Trigonometric Support Vector Classifier

Neural Computation ◽

10.1162/089976603322297368 ◽

2003 ◽

Vol 15 (9) ◽

pp. 2227-2254 ◽

Cited By ~ 20

Author(s):

Wei Chu ◽

S. Sathiya Keerthi ◽

Chong Jin Ong

Keyword(s):

Loss Function ◽

Gaussian Processes ◽

Likelihood Function ◽

Support Vector ◽

Data Sets ◽

Model Adaptation ◽

Bayesian Techniques ◽

Benchmark Data ◽

Support Vector Classifier ◽

Set Up

This letter describes Bayesian techniques for support vector classification. In particular, we propose a novel differentiable loss function, called the trigonometric loss function, which has the desirable characteristic of natural normalization in the likelihood function, and then follow standard gaussian processes techniques to set up a Bayesian framework. In this framework, Bayesian inference is used to implement model adaptation, while keeping the merits of support vector classifier, such as sparseness and convex programming. This differs from standard gaussian processes for classification. Moreover, we put forward class probability in making predictions. Experimental results on benchmark data sets indicate the usefulness of this approach.

Download Full-text

Truncated covariance matrices and Toeplitz methods in gaussian processes

9th International Conference on Artificial Neural Networks: ICANN '99 ◽

10.1049/cp:19991084 ◽

1999 ◽

Cited By ~ 3

Author(s):

A.J. Storkey

Keyword(s):

Gaussian Processes ◽

Covariance Matrices

Download Full-text

Generalized Inferences about the Mean Vector of Several Multivariate Gaussian Processes

Journal of Probability and Statistics ◽

10.1155/2015/479762 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9

Author(s):

Pilar Ibarrola ◽

Ricardo Vélez

Keyword(s):

Gaussian Processes ◽

Confidence Region ◽

Confidence Regions ◽

Covariance Matrices ◽

Nuisance Parameters ◽

Vector Parameter ◽

Multivariate Gaussian ◽

The Mean ◽

Unknown Vector ◽

Mean Vector

We consider in this paper the problem of comparing the means of several multivariate Gaussian processes. It is assumed that the means depend linearly on an unknown vector parameterθand that nuisance parameters appear in the covariance matrices. More precisely, we deal with the problem of testing hypotheses, as well as obtaining confidence regions forθ. Both methods will be based on the concepts of generalizedpvalue and generalized confidence region adapted to our context.

Download Full-text

A DYNAMICAL APPROACH TO FRACTIONAL BROWNIAN MOTION

Fractals ◽

10.1142/s0218348x94000077 ◽

1994 ◽

Vol 02 (01) ◽

pp. 81-94 ◽

Cited By ~ 25

Author(s):

RICCARDO MANNELLA ◽

PAOLO GRIGOLINI ◽

BRUCE J. WEST

Keyword(s):

Dynamical System ◽

Asymptotic Behavior ◽

Brownian Motion ◽

Correlation Function ◽

Fractional Brownian Motion ◽

Gaussian Processes ◽

The Other ◽

Hurst Coefficient ◽

And Diffusion ◽

Clear Relation

Herein we develop a dynamical foundation for fractional Brownian motion. A clear relation is established between the asymptotic behavior of the correlation function and diffusion in a dynamical system. Then, assuming that scaling is applicable, we establish a connection between diffusion (either standard or anomalous) and the dynamical indicator known as the Hurst coefficient. We argue on the basis of numerical simulations that although we have been able to prove scaling only for "Gaussian" processes, our conclusions may well apply to a wider class of systems. On the other hand, systems exist for which scaling might not hold, so we speculate on the possible consequences of the various relations derived in the paper on such systems.

Download Full-text