On the maximum likelihood training of gradient-enhanced spatial Gaussian processes

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

Resumé

Spatial Gaussian processes, alias spatial linear models or Kriging estimators, are a
powerful and well-established tool for the design and analysis of computer experiments in a multitude
of engineering applications. A key challenge in constructing spatial Gaussian processes is the training
of the predictor by numerically optimizing its associated maximum likelihood function depending on
so-called hyper-parameters. This is well understood for standard Kriging predictors, i.e., without
considering derivative information. For gradient-enhanced Kriging predictors it is an open question
of whether to incorporate the cross-correlations between the function values and their partial deriva-
tives in the maximum likelihood estimation. In this paper it is proved that in consistency with the
model assumptions, both the autocorrelations and the aforementioned cross-correlations must be
considered when optimizing the gradient-enhanced predictor’s likelihood function. The proof works
by computational rather than probabilistic arguments and exposes as a secondary effect the connec-
tion between the direct and the indirect approach to gradient-enhanced Kriging, both of which are
widely used in applications. The theoretical findings are illustrated on an academic example as well
as on an aerodynamic engineering application.
OriginalsprogEngelsk
TidsskriftS I A M Journal on Scientific Computing
Vol/bind35
Udgave nummer6
Sider (fra-til)A2554–A2574
Antal sider21
ISSN1064-8275
StatusUdgivet - 2013
Udgivet eksterntJa

Fingeraftryk

Spatial Process
Kriging
Gaussian Process
Maximum likelihood
Maximum Likelihood
Predictors
Gradient
Cross-correlation
Likelihood Function
Engineering Application
Maximum likelihood estimation
Autocorrelation
Hyperparameters
Computer Experiments
Aerodynamics
Spatial Model
Maximum Likelihood Estimation
Value Function
Derivatives
Linear Model

Citer dette

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title = "On the maximum likelihood training of gradient-enhanced spatial Gaussian processes",
abstract = "Spatial Gaussian processes, alias spatial linear models or Kriging estimators, are apowerful and well-established tool for the design and analysis of computer experiments in a multitudeof engineering applications. A key challenge in constructing spatial Gaussian processes is the trainingof the predictor by numerically optimizing its associated maximum likelihood function depending onso-called hyper-parameters. This is well understood for standard Kriging predictors, i.e., withoutconsidering derivative information. For gradient-enhanced Kriging predictors it is an open questionof whether to incorporate the cross-correlations between the function values and their partial deriva-tives in the maximum likelihood estimation. In this paper it is proved that in consistency with themodel assumptions, both the autocorrelations and the aforementioned cross-correlations must beconsidered when optimizing the gradient-enhanced predictor’s likelihood function. The proof worksby computational rather than probabilistic arguments and exposes as a secondary effect the connec-tion between the direct and the indirect approach to gradient-enhanced Kriging, both of which arewidely used in applications. The theoretical findings are illustrated on an academic example as wellas on an aerodynamic engineering application.",
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On the maximum likelihood training of gradient-enhanced spatial Gaussian processes. / Zimmermann, Ralf.

I: S I A M Journal on Scientific Computing, Bind 35, Nr. 6, 2013, s. A2554–A2574.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - On the maximum likelihood training of gradient-enhanced spatial Gaussian processes

AU - Zimmermann, Ralf

PY - 2013

Y1 - 2013

N2 - Spatial Gaussian processes, alias spatial linear models or Kriging estimators, are apowerful and well-established tool for the design and analysis of computer experiments in a multitudeof engineering applications. A key challenge in constructing spatial Gaussian processes is the trainingof the predictor by numerically optimizing its associated maximum likelihood function depending onso-called hyper-parameters. This is well understood for standard Kriging predictors, i.e., withoutconsidering derivative information. For gradient-enhanced Kriging predictors it is an open questionof whether to incorporate the cross-correlations between the function values and their partial deriva-tives in the maximum likelihood estimation. In this paper it is proved that in consistency with themodel assumptions, both the autocorrelations and the aforementioned cross-correlations must beconsidered when optimizing the gradient-enhanced predictor’s likelihood function. The proof worksby computational rather than probabilistic arguments and exposes as a secondary effect the connec-tion between the direct and the indirect approach to gradient-enhanced Kriging, both of which arewidely used in applications. The theoretical findings are illustrated on an academic example as wellas on an aerodynamic engineering application.

AB - Spatial Gaussian processes, alias spatial linear models or Kriging estimators, are apowerful and well-established tool for the design and analysis of computer experiments in a multitudeof engineering applications. A key challenge in constructing spatial Gaussian processes is the trainingof the predictor by numerically optimizing its associated maximum likelihood function depending onso-called hyper-parameters. This is well understood for standard Kriging predictors, i.e., withoutconsidering derivative information. For gradient-enhanced Kriging predictors it is an open questionof whether to incorporate the cross-correlations between the function values and their partial deriva-tives in the maximum likelihood estimation. In this paper it is proved that in consistency with themodel assumptions, both the autocorrelations and the aforementioned cross-correlations must beconsidered when optimizing the gradient-enhanced predictor’s likelihood function. The proof worksby computational rather than probabilistic arguments and exposes as a secondary effect the connec-tion between the direct and the indirect approach to gradient-enhanced Kriging, both of which arewidely used in applications. The theoretical findings are illustrated on an academic example as wellas on an aerodynamic engineering application.

KW - design and analysis of computer experiments

KW - gradient-enhanced Kriging

KW - Gaussian process

KW - maximum likelihood

M3 - Journal article

VL - 35

SP - A2554–A2574

JO - S I A M Journal on Scientific Computing

JF - S I A M Journal on Scientific Computing

SN - 1064-8275

IS - 6

ER -