On the validity of parametric block correlation matrices with constant within and between group correlations

We consider the set Bp of parametric block correlation matrices with p blocks of various (and possibly different) sizes, whose diagonal blocks are compound symmetry (CS) correlation matrices and off-diagonal blocks are constant matrices. Such matrices appear in probabilistic models on categorical data, when the levels are partitioned in p groups, assuming a constant correlation within a group and a constant correlation for each pair of groups. We obtain two necessary and sufficient conditions for positive definiteness of elements of Bp. Firstly we consider the block average map $\phi$, consisting in replacing a block by its mean value. We prove that for any A $\in$ Bp , A is positive definite if and only if $\phi$(A) is positive definite. Hence it is equivalent to check the validity of the covariance matrix of group means, which only depends on the number of groups and not on their sizes. This theorem can be extended to a wider set of block matrices. Secondly, we consider the subset of Bp for which the between group correlation is the same for all pairs of groups. Positive definiteness then comes down to find the positive definite interval of a matrix pencil on Sp. We obtain a simple characterization by localizing the roots of the determinant with within group correlation values

A warped kernel improving robustness in Bayesian optimization via random embeddings

This works extends the Random Embedding Bayesian Optimization approach by integrating a warping of the high dimensional subspace within the covariance kernel. The proposed warping, that relies on elementary geometric considerations, allows mitigating the drawbacks of the high extrinsic dimensionality while avoiding the algorithm to evaluate points giving redundant information. It also alleviates constraints on bound selection for the embedded domain, thus improving the robustness, as illustrated with a test case with 25 variables and intrinsic dimension 6

ANOVA kernels and RKHS of zero mean functions for model-based sensitivity analysis

International audienceGiven a reproducing kernel Hilbert space H of real-valued functions and a suitable measure mu over the source space D (subset of R), we decompose H as the sum of a subspace of centered functions for mu and its orthogonal in H. This decomposition leads to a special case of ANOVA kernels, for which the functional ANOVA representation of the best predictor can be elegantly derived, either in an interpolation or regularization framework. The proposed kernels appear to be particularly convenient for analyzing the e ffect of each (group of) variable(s) and computing sensitivity indices without recursivity

Derivative based global sensitivity measures

The method of derivative based global sensitivity measures (DGSM) has recently become popular among practitioners. It has a strong link with the Morris screening method and Sobol' sensitivity indices and has several advantages over them. DGSM are very easy to implement and evaluate numerically. The computational time required for numerical evaluation of DGSM is generally much lower than that for estimation of Sobol' sensitivity indices. This paper presents a survey of recent advances in DGSM concerning lower and upper bounds on the values of Sobol' total sensitivity indices $S\_{i}^{tot}$. Using these bounds it is possible in most cases to get a good practical estimation of the values of $S\_{i}^{tot}$. Several examples are used to illustrate an application of DGSM

Derivative based global sensitivity measures

International audienceThe method of derivative based global sensitivity measures (DGSM) has recently become popular among practitioners. It has a strong link with the Morris screening method and Sobol' sensitivity indices and has several advantages over them. DGSM are very easy to implement and evaluate numerically. The computational time required for numerical evaluation of DGSM is generally much lower than that for estimation of Sobol' sensitivity indices. This paper presents a survey of recent advances in DGSM concerning lower and upper bounds on the values of Sobol' total sensitivity indices $S_{i}^{tot}$. Using these bounds it is possible in most cases to get a good practical estimation of the values of $S_{i}^{tot}$. Several examples are used to illustrate an application of DGSM

Cokriging for multivariate Hilbert space valued random fields: application to multi-fidelity computer code emulation

In this paper we propose Universal trace co-kriging, a novel methodology for interpolation of multivariate Hilbert space valued functional data. Such data commonly arises in multi-fidelity numerical modeling of the subsurface and it is a part of many modern uncertainty quantification studies. Besides theoretical developments we also present methodological evaluation and comparisons with the recently published projection based approach by Bohorquez et al. (Stoch Environ Res Risk Assess 31(1):53–70, 2016. https://doi.org/10.1007/s00477-016-1266-y). Our evaluations and analyses were performed on synthetic (oil reservoir) and real field (uranium contamination) subsurface uncertainty quantification case studies. Monte Carlo analyses were conducted to draw important conclusions and to provide practical guidelines for all future practitioners

Polar Gaussian Processes and Experimental Designs in Circular Domains

Predicting on circular domains is a central issue that can be addressed by Gaus- sian process (GP) regression. However, usual GP models do not take into account the geometry of the disk in their covariance structure (or kernel), which may be a drawback at least for industrial processes involving a rotation or a diffusion from the center of the disk. We introduce so-called polar GPs defined on the space of polar coordinates. Their kernels are obtained as a combination of a kernel for the radius and a kernel for the angle, based on either chordal or geodesic distances on the circle. Their efficiency is illustrated on two industrial applications. We further consider the problem of designing experiments on the disk. Two new Latin hypercube designs are obtained, by defining a valid maximin criterion for polar coordinates. Finally, an extension of the whole methodology to higher dimensions is investigated

Additive Covariance Kernels for High-Dimensional Gaussian Process Modeling

http://afst.cedram.org/afst-bin/fitem?id=AFST_2012_6_21_3_481_0National audienceGaussian process models -also called Kriging models- are often used as mathematical approximations of expensive experiments. However, the number of observation required for building an emulator becomes unrealistic when using classical covariance kernels when the dimension of input increases. In oder to get round the curse of dimensionality, a popular approach is to consider simplified models such as additive models. The ambition of the present work is to give an insight into covariance kernels that are well suited for building additive Kriging models and to describe some properties of the resulting models