arXiv subjects

François Bachoc

Publications and source records attributed to François Bachoc.

69 records · Page 4Linked to original sources

arXiv2018

A supermartingale approach to Gaussian process based sequential design of experiments

Gaussian process (GP) models have become a well-established frameworkfor the adaptive design of costly experiments, and notably of computerexperiments. GP-based sequential designs have been found practicallyefficient for various objectives, such as global optimization(estimating the global maximum or maximizer(s) of a function),reliability analysis (estimating a probability of failure) or theestimation of level sets and excursion sets. In this paper, we studythe consistency of an important class of sequential designs, known asstepwise uncertainty reduction (SUR) strategies. Our approach relieson the key observation that the sequence of residual uncertaintymeasures, in SUR strategies, is generally a supermartingale withrespect to the filtration generated by the observations. Thisobservation enables us to establish generic consistency results for abroad class of SUR strategies. The consistency of several popularsequential design strategies is then obtained by means of this generalresult. Notably, we establish the consistency of two SUR strategiesproposed by Bect, Ginsbourger, Li, Picheny and Vazquez (Stat. Comp.,2012)---to the best of our knowledge, these are the first proofs ofconsistency for GP-based sequential design algorithms dedicated to theestimation of excursion sets and their measure. We also establish anew, more general proof of consistency for the expected improvementalgorithm for global optimization which, unlike previous results inthe literature, applies to any GP with continuous sample paths.

stat.ML↗

arXiv2018

A Gaussian Process Regression Model for Distribution Inputs

Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding stochastic processes. We prove that the Gaussian processes indexed by distributions corresponding to these kernels can be efficiently forecast, opening new perspectives in Gaussian process modeling.

stat.ML↗

arXiv2017

Uniformly valid confidence intervals post-model-selection

We suggest general methods to construct asymptotically uniformly valid confidence intervals post-model-selection. The constructions are based on principles recently proposed by Berk et al. (2013). In particular the candidate models used can be misspecified, the target of inference is model-specific, and coverage is guaranteed for any data-driven model selection procedure. After developing a general theory we apply our methods to practically important situations where the candidate set of models, from which a working model is selected, consists of fixed design homoskedastic or heteroskedastic linear models, or of binary regression models with general link functions. In an extensive simulation study, we find that the proposed confidence intervals perform remarkably well, even when compared to existing methods that are tailored only for specific model selection procedures.

math.ST↗

arXiv2017

Finite-dimensional Gaussian approximation with linear inequality constraints

Introducing inequality constraints in Gaussian process (GP) models can lead to more realistic uncertainties in learning a great variety of real-world problems. We consider the finite-dimensional Gaussian approach from Maatouk and Bay (2017) which can satisfy inequality conditions everywhere (either boundedness, monotonicity or convexity). Our contributions are threefold. First, we extend their approach in order to deal with general sets of linear inequalities. Second, we explore several Markov Chain Monte Carlo (MCMC) techniques to approximate the posterior distribution. Third, we investigate theoretical and numerical properties of the constrained likelihood for covariance parameter estimation. According to experiments on both artificial and real data, our full framework together with a Hamiltonian Monte Carlo-based sampler provides efficient results on both data fitting and uncertainty quantification.

stat.ML↗

arXiv2017

Nested Kriging predictions for datasets with large number of observations

This work falls within the context of predicting the value of a real function at some input locations given a limited number of observations of this function. The Kriging interpolation technique (or Gaussian process regression) is often considered to tackle such a problem but the method suffers from its computational burden when the number of observation points is large. We introduce in this article nested Kriging predictors which are constructed by aggregating sub-models based on subsets of observation points. This approach is proven to have better theoretical properties than other aggregation methods that can be found in the literature. Contrarily to some other methods it can be shown that the proposed aggregation method is consistent. Finally, the practical interest of the proposed method is illustrated on simulated datasets and on an industrial test case with $10^4$ observations in a 6-dimensional space.

stat.ML↗

arXiv2017

Valid confidence intervals for post-model-selection predictors

We consider inference post-model-selection in linear regression. In this setting, Berk et al.(2013) recently introduced a class of confidence sets, the so-called PoSI intervals, that cover a certain non-standard quantity of interest with a user-specified minimal coverage probability, irrespective of the model selection procedure that is being used. In this paper, we generalize the PoSI intervals to post-model-selection predictors.

math.ST↗

arXiv2016

Maximum likelihood estimation for a bivariate Gaussian process under fixed domain asymptotics

We consider maximum likelihood estimation with data from a bivariate Gaussian process with a separable exponential covariance model under fixed domain asymptotic. We first characterize the equivalence of Gaussian measures under this model. Then consistency and asymptotic distribution for the microergodic parameters are established. A simulation study is presented in order to compare the finite sample behavior of the maximum likelihood estimator with the given asymptotic distribution.

math.ST↗

arXiv2016

On the smallest eigenvalues of covariance matrices of multivariate spatial processes

There has been a growing interest in providing models for multivariate spatial processes. A majority of these models specify a parametric matrix covariance function. Based on observations, the parameters are estimated by maximum likelihood or variants thereof. While the asymptotic properties of maximum likelihood estimators for univariate spatial processes have been analyzed in detail, maximum likelihood estimators for multivariate spatial processes have not received their deserved attention yet. In this article we consider the classical increasing-domain asymptotic setting restricting the minimum distance between the locations. Then, one of the main components to be studied from a theoretical point of view is the asymptotic positive definiteness of the underlying covariance matrix. Based on very weak assumptions on the matrix covariance function we show that the smallest eigenvalue of the covariance matrix is asymptotically bounded away from zero. Several practical implications are discussed as well.

math.ST↗

arXiv2015

Asymptotic analysis of covariance parameter estimation for Gaussian processes in the misspecified case

In parametric estimation of covariance function of Gaussian processes, it is often the case that the true covariance function does not belong to the parametric set used for estimation. This situation is called the misspecified case. In this case, it has been shown that, for irregular spatial sampling of observation points, Cross Validation can yield smaller prediction errors than Maximum Likelihood. Motivated by this observation, we provide a general asymptotic analysis of the misspecified case, for independent and uniformly distributed observation points. We prove that the Maximum Likelihood estimator asymptotically minimizes a Kullback-Leibler divergence, within the misspecified parametric set, while Cross Validation asymptotically minimizes the integrated square prediction error. In a Monte Carlo simulation, we show that the covariance parameters estimated by Maximum Likelihood and Cross Validation, and the corresponding Kullback-Leibler divergences and integrated square prediction errors, can be strongly contrasting. On a more technical level, we provide new increasing-domain asymptotic results for independent and uniformly distributed observation points.

math.ST↗

arXiv2015

Improvement of code behaviour in a design of experiments by metamodeling

It is now common practice in nuclear engineering to base extensive studies on numerical computer models. These studies require to run computer codes in potentially thousands of numerical configurations and without expert individual controls on the computational and physical aspects of each simulations.In this paper, we compare different statistical metamodeling techniques and show how metamodels can help to improve the global behaviour of codes in these extensive studies. We consider the metamodeling of the Germinal thermalmechanical code by Kriging, kernel regression and neural networks. Kriging provides the most accurate predictions while neural networks yield the fastest metamodel functions. All three metamodels can conveniently detect strong computation failures. It is however significantly more challenging to detect code instabilities, that is groups of computations that are all valid, but numerically inconsistent with one another. For code instability detection, we find that Kriging provides the most useful tools.

stat.CO↗

arXiv2015

Optimal configurations of lines and a statistical application

Motivated by the construction of confidence intervals in statistics, we study optimal configurations of $2^d-1$ lines in real projective space $RP^{d-1}$. For small $d$, we determine line sets that numerically minimize a wide variety of potential functions among all configurations of $2^d-1$ lines through the origin. Numerical experiments verify that our findings enable to assess efficiently the tightness of a bound arising from the statistical literature.

math.NA↗

arXiv2014

Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes

Covariance parameter estimation of Gaussian processes is analyzed in an asymptotic framework. The spatial sampling is a randomly perturbed regular grid and its deviation from the perfect regular grid is controlled by a single scalar regularity parameter. Consistency and asymptotic normality are proved for the Maximum Likelihood and Cross Validation estimators of the covariance parameters. The asymptotic covariance matrices of the covariance parameter estimators are deterministic functions of the regularity parameter. By means of an exhaustive study of the asymptotic covariance matrices, it is shown that the estimation is improved when the regular grid is strongly perturbed. Hence, an asymptotic confirmation is given to the commonly admitted fact that using groups of observation points with small spacing is beneficial to covariance function estimation. Finally, the prediction error, using a consistent estimator of the covariance parameters, is analyzed in details.

math.ST↗

arXiv2014

Hastings-Metropolis algorithm on Markov chains for small-probability estimation

Shielding studies in neutron transport, with Monte Carlo codes, yield challenging problems of small-probability estimation. The particularity of these studies is that the small probability to estimate is formulated in terms of the distribution of a Markov chain, instead of that of a random vector in more classical cases. Thus, it is not straightforward to adapt classical statistical methods, for estimating small probabilities involving random vectors, to these neutron-transport problems. A recent interacting-particle method for small-probability estimation, relying on the Hastings-Metropolis algorithm, is presented. It is shown how to adapt the Hastings-Metropolis algorithm when dealing with Markov chains. A convergence result is also shown. Then, the practical implementation of the resulting method for small-probability estimation is treated in details, for a Monte Carlo shielding study. Finally, it is shown, for this study, that the proposed interacting-particle method considerably outperforms a simple-Monte Carlo method, when the probability to estimate is small.

math.ST↗

arXiv2013

Cross Validation and Maximum Likelihood estimations of hyper-parameters of Gaussian processes with model misspecification

The Maximum Likelihood (ML) and Cross Validation (CV) methods for estimating covariance hyper-parameters are compared, in the context of Kriging with a misspecified covariance structure. A two-step approach is used. First, the case of the estimation of a single variance hyper-parameter is addressed, for which the fixed correlation function is misspecified. A predictive variance based quality criterion is introduced and a closed-form expression of this criterion is derived. It is shown that when the correlation function is misspecified, the CV does better compared to ML, while ML is optimal when the model is well-specified. In the second step, the results of the first step are extended to the case when the hyper-parameters of the correlation function are also estimated from data.

math.ST↗

arXiv2013

Calibration and improved prediction of computer models by universal Kriging

This paper addresses the use of experimental data for calibrating a computer model and improving its predictions of the underlying physical system. A global statistical approach is proposed in which the bias between the computer model and the physical system is modeled as a realization of a Gaussian process. The application of classical statistical inference to this statistical model yields a rigorous method for calibrating the computer model and for adding to its predictions a statistical correction based on experimental data. This statistical correction can substantially improve the calibrated computer model for predicting the physical system on new experimental conditions. Furthermore, a quantification of the uncertainty of this prediction is provided. Physical expertise on the calibration parameters can also be taken into account in a Bayesian framework. Finally, the method is applied to the thermal-hydraulic code FLICA 4, in a single phase friction model framework. It allows to improve the predictions of the thermal-hydraulic code FLICA 4 significantly.

stat.AP↗