Functional-input metamodeling: an application to coastal ... · functional input points within the...
Transcript of Functional-input metamodeling: an application to coastal ... · functional input points within the...
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Functional-input metamodeling: an applicationto coastal flood early warning
Jose Betancourt
Advisor:Co-advisor:
Prof. Thierry KleinProf. Francois Bachoc
PhD thesis defense - Applied MathematicsJune 08, 2020
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Outline
Problem description
Chapter 2. First metamodeling experiences
Chapter 3. ACO based structural optimization
Chapter 4. Consolidation of funGp R package
Chapter 5. Structural optimization in RISCOPE
Chapter 6. Asymptotics of trans-Gaussian Processes
Perspectives and contributions
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Problem description
RISCOPE project framework: Coastal flooding early warning
Objective: provide fast-running statistical models to replace theslow-running computer code → Metamodeling
hours to days few seconds
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Research problem
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Research problem
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Research problem
Questions
Functional inputs
what functional inputs are worth keeping active?
what dimension reduction method is ideal?
what is a suitable projection dimension?
Gaussian process metamodel
what is a suitable kernel?
what is a convenient distance to measure similarities betweenfunctional input points within the kernel function?
→ We call these, structural parameters of the metamodel
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Chapter 2. First metamodeling experiences
First functional-input metamodeling attempts
Proof of concept for goal-oriented structural optimization
Development of first R routines
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Ch.2 - Simplified numerical code
Inputs
37 time instants
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Ch.2 - Metamodeling setting
We split each functional input into a scalar and a functionalrepresentation, x and f , respectively.
We observe a deterministic code of the form
(x,f) 7→ fcode (x,f)
• x the vector of ds scalar inputs(x(i) ∈ R,∀i = 1, . . . , ds
)• f the vector of df functional inputs
(f (i) : Ti ⊂ R→ R, ∀i = 1, . . . , df
)
We see fcode as a realization of a Gaussian process Y
Y ∼ GP(m, k),
• m the mean function • k the covariance function.
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Ch.2 - Ingredients of our Gp metamodel
1. Dimension reduction: project f (i) : Ti ⊂ R→ R onto a space oflower dimension pi .
f (i)(t) ≈ Π(f (i))
(t) =
p∑r=1
αrBr(t)
which projection method?
B-splines (Muehlenstaedt, Fruth, & Roustant, 2017)
PCA (Lataniotis, Marelli, & Sudret, 2018)
Wavelets (Cohen, Daubechies, & Feauveau, 1992)
PLS (Papaioannou, Ehre, & Straub, 2019)
which dimension?
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Ch.2 - Ingredients of our Gp metamodel
2. Covariance function
k(Y (x,f), Y (x, f)) = σ2 R(‖x− x‖L2,θs
)R(‖f − f‖L2,θf
)Distance: measure the similarity between points
For scalars: ‖x− x‖L2,θs =
√√√√√ ds∑i=1
∥∥x(i) − x(i)∥∥2(θ(i)s
)2
For functions:∥∥Π(f)−Π
(f)∥∥S,θf
:=
√√√√√√ df∑i=1
pi∑r=1
(α(i)r − α(i)
r
)2(θ(i)f,r
)2
∥∥Π(f)−Π(f)∥∥D,θf
:=
√√√√√√√√√ df∑i=1
∫Ti
(pi∑r=1
(α(i)r − α(i)
r
)B(i)r (t)
)2
dt
(θ(i)f
)2
classical
Nanty et al. (2016)
Muehlenstaedt et al. (2017)
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Ch.2 - Ingredients of our Gp metamodel
Features to define:
State of functional inputs• active • inactive
Projection dimension• {1, . . . , 37}
Projection method• B-splines • PCA
Distance for functions• ‖·‖S,θf • ‖·‖D,θf
Features fixed:
State of scalar inputs: active
Kernel function: Matern 5/2
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Ch.2 - First exploration approach
Based on Response Surface Methodology (Montgomery, 1997; Pulido et al., 2012)
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Ch.2 - Results for RISCOPE
Best structural configuration found
Best sample Worst sample
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Ch.2 - Results for analytic case from Muehlenstaedt et al. (2017)
Best structural configuration found
Best sample Worst sample
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Ch.2 - Fitting in both applications
Fit of sorted output
RISCOPE Analytic
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Chapter 2. Results and conclusions
Results
Outstanding metamodels in all the experiments
Ideal projection method and dimension varied with the application
‖·‖D,θf consistently outperformed ‖·‖S,θf
Conclusions
Effective tool for structural optimization
The optimization must be performed each time
Intensive exploration of more features and levels might lead to policies
A more efficient exploration approach would be required
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Chapter 3. ACO based structural optimization
More efficient exploration of the solution space
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Ch.3 - Biological inspiration
The double bridge experiment (Goss et al., 1989)
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Ch.3 - Decision network for structural optimization
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Ch.3 - Decision network for structural optimization
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Ch.3 - Decision network for structural optimization
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Ch.3 - Decision network for structural optimization
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Ch.3 - Decision network for structural optimization
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Ch.3 - Decision network for structural optimization
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Ch.3 - Decision network for structural optimization
local pheromone update→
← global pheromone update
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Ch.3 - Decision network for structural optimization
case of inactive functional input
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Ch.3 - Evolution of the heuristic
(a) Black-box 1. (b) Black-box 2. (c) Black-box 3.
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Ch.3 - Relative quality of selected models
(a) Black-box 1. (b) Black-box 2. (c) Black-box 3.
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Ch.3 - Cross validation (LOO)
(a) Black-box 1. (b) Black-box 2. (c) Black-box 3.
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Chapter 3. Remarks
What our algorithm does: smart selection of configurations to testWhat is doesn’t do: faster evaluation of one configuration
Conclusions
Better use the time based stopping condition
The new algorithm allows for a wider exploration of features and levels
Desirable to make this development publicly accessible (R package?)
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Chapter 4. Consolidation of funGp R package
Gaussian Process Regression for Scalar andFunctional Inputs with funGp
The in-depth tour
Authors: Jose Betancourt, Francois Bachoc, Thierry Klein.Contributors: Deborah Idier, Jeremy Rhomer.
Official manual for funGp 0.1.0 (2020), downloadable from CRAN and GitHub.
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Ch.4 - funGp R package
• Scalar-, Functional-, and Hybrid-input Gaussian Process Regression
Construction
Prediction
Simulation
Plotting
Structural optimization
LOOCV or Hold-out V
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Ch.4 - funGp R package
• Scalar-, Functional-, and Hybrid-input Gaussian Process Regression
Construction
Prediction
Simulation
Plotting
Structural optimization
LOOCV or Hold-out V
Efficient exploration of metamodel configurations
Selection of models with superior predictability
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Ch.4 - funGp R package
fgpm(sIn, fIn, sOut)
fgpm(sIn, fIn, sOut, basis)
fgpm(sIn, fIn, sOut, kernel)
fgpm(sIn, fIn, sOut, basis, dimen, dist, kernel, n.strt, cl)
fgpm factory(sIn, fIn, sOut)
fgpm factory(sIn, fIn, sOut, constr, tlim, n.iter, ACOSetup, cl)
Metamodeling
Structural optimization
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Ch.4 - funGp R package
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Chapter 4. Synthesis
funGp in brief
Regression with scalar and functional inputs based on a non-parametric model
Built-in dimension reduction
Boosted performance through parallelization
Modular-oriented implementation for easy extension
Comprehensive manual available as pre-print in HAL
Open source, available from CRAN and GitHub
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Chapter 5. Structural optimization in RISCOPE - an update
+
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Ch.5 - Full numerical code
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Ch.5 - Full numerical code
Q2loocv for each of the 13 output variables
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Ch.5 - Full numerical code
LOO cross-validation for Ysurf max
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Ch.5 - Full numerical code
5-fold cross-validation for Ysurf max
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Ch.5 - Full numerical code
5-fold cross-validation for Ysurf max
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Chapter 5. Conclusions so far
The ACO algorithm allowed to achieve superior results the 13 metamodels→ robustness of the method to variations in the input-output relationship
Promising results for most of the outputs→ expected to improve when adding Ysurf max as input for the other metas
Results achieved with funGp in 10 minutes→ expected to improve when allowed to run for more time
The solutions found by ACO were non-trivial
fgpm call delivered by ACO:
fgpm(sIn[,c(1,2,3,4,6,8)], fIn[c(4,5)], Ysurf max,
basis = c("B-splines", "PCA"), dimen = c(3,3),
dist = c("byindex", "byindex"), kernel = "gauss")
→ potential loss of predictability when we make selections by hand→ better to optimize
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Chapter 6. Asymptotics of trans-Gaussian processes
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Ch.6 - Motivation
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Ch.6 - Objective of the study
Two asymptotic frameworks for covariance parameter estimation
fixed-domain: observation points dense in a bounded domainincreasing-domain: number of observation points is proportional todomain volume → unbounded observation domain
Two estimators for covariance parameters estimation
Maximum likelihood (ML)Cross validation (CV)
Under increasing domain asymptotics
Assumption that observations come from Gaussian field:ML and CV consistent and asymptotically normalWithout the Gaussianity assumption: no results so far
Objective: to extend the asymptotic results to observationsfrom a non-Gaussian random field
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Ch.6 - Framework
Transformed Gaussian process
Unobserved Gaussian process Z on Rd, with zero mean and covariancefunction kZ
Fixed (and unknown) non-linear transformation function T
Observed transformed Gaussian process Y defined by Y (s) = T (Z(s)) forany s ∈ Rd. Assumed zero mean. Covariance function kY
Sequence of observation locations (si)i∈N, si ∈ Rd, i ∈ N
Non-Gaussian observation vector y = (Y (s1), ..., Y (sn))> with covariancematrix R = (kY (si − sj))i,j=1,...,n, for n ∈ N
Covariance parameters
Parametric set of stationary covariance functions {kY,θ; θ ∈ Θ} on Rd,with Θ a compact set of Rp
θ = (σ2, ψ), with σ2 and ψ the variance and correlation parameters, resp.
kY = kY,θ0 , with θ0 ∈ Θ
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Ch.6 - Covariance parameter estimators
The maximum likelihood estimator of θ0 is
θML ∈ argminθ∈Θ
Lθ,
with Lθ =1
n
(log(det(Rθ)) + y>R−1
θ y)
The cross validation estimator of ψ0 is
ψCV ∈ argminψ∈S
CVψ,
with CVψ =1
n
n∑i=1
(yi − yi,ψ)2
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Ch.6 - Base theorems
Bounds on the elements of inverse covariance matrices (Theorem 1)
Let (si)i∈N be the sequence of observation locations, with si ∈ Rd
Let R be the covariance matrix of the non-Gaussian observations
Then we have ∣∣∣(R−1)i,j∣∣∣ ≤ Csup
1 + |si − sj |d+τ,
with Csup <∞, d the input dimension and 0 < τ <∞
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Ch.6 - Base theorems
CLT for quadratic forms of trans-Gps (Theorem 2)
Let y be the n-element vector of non-Gaussian observations.
Let (An)n∈N be a sequence of matrices (each n× n). Assume
|(An)i,j | ≤Csup
1 + |si − sj |d+Cinf, ∀n∈N, ∀i,j=1,...,n
Let Vn = 1ny>Any. Let Ln be the distribution of
√n(Vn − E[Vn])
As n→∞, dw(Ln,N [0, nVar(Vn)]
)→ 0
In addition, (nVar(Vn))n∈N is bounded
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Ch.6 - Main results
Maximum likelihood (Theorems 3 and 4, resp.)
Consistency: θMLp→ θ0 as n→∞.
Asymptotic normality:
Let Lθ0,n be the distribution of√n(θML − θ0). Using Theorem 2,
as n→∞, dw
(Lθ0,n,N [0,M−1θ0 Σθ0M
−1θ0
])→ 0,
with (Mθ0)i,j = (1/n) tr(R−1θ0
(∂Rθ0/∂θi)R−1θ0
(∂Rθ0/∂θj))
(Σθ0)i,j = Cov(√n ∂Lθ0/∂θi,
√n ∂Lθ0/∂θj), i, j = 1, . . . , p.
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Ch.6 - Main results
Maximum likelihood (Proofs of Theorems 3 and 4, resp.)
Steps for consistency:1. Show that Var(Lθ)→ 0 as n→∞2. Show that supθ∈θ
∣∣∂Lθ
∂θ
∣∣ = Op(1) by finding a suitable bound for it.
3. Proceed as in the proof of Proposition 3.1 in Bachoc (2014).
Steps for asymptotic normality:1. Expand the first derivative of Lθ as a Taylor series around θ0.
2. Evaluate at θML, equal to zero, solve for (θML − θ0) and multiply by√n.
3. Apply Slutsky lemma.
3.1 Check convergence in probability of√n[−∂2Lθ0/∂θ
2]−1
V .
3.2 Check convergence in distribution of√n[−∂2Lθ0/∂θ
2]−1
∂Lθ0/∂θ.
4. Show that the elements of the asymptotic covariance matrix are bounded.
4.1 Show that elements of M−1θ0
and Σθ0 are bounded.
4.2 Conclude that lim supn→∞ λ1(M−1θ0
Σθ0M−1θ0
) < +∞.
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Ch.6 - Main results
Cross Validation (Theorems 5 and 6, resp.)
Consistency: ψCVp→ ψ0 as n→∞.
Asymptotic normality:
Let Qψ0,n be the distribution of√n(ψCV − ψ0). Using Theorem 2,
as n→∞, dw
(Qψ0,n,N [0, N−1ψ0
Γψ0N−1ψ0
])→ 0,
with (Nθ0)i,j = a long to write expression (provided in the manuscript)
(Γψ0)i,j = Cov(√n ∂CVψ0/∂ψi,
√n ∂CVψ0/∂ψj), i, j = 1, . . . , p.
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Ch.6 - Main results
Joint estimator (Theorem 7)
Consistency: follows from Theorems 3 and 5.
Asymptotic normality:
Let Qθ0,n be the distribution of√n
(θML − θ0ψCV − ψ0
). Using Theorem 2,
as n→∞, dw
(Qθ0,n,N [0, D−1θ0 Ψθ0D
−1θ0
])→ 0,
with Dθ0 = a long to write expression depending on Mθ0 and Nψ0 (provided
in the manuscript)
Σθ0 = Cov(√n ∂Lθ0/∂θ, ∂CVψ0/∂ψ).
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Chapter 6. Conclusions and extension
Conclusions
Covariance parameters of transformed Gaussian processescan be properly estimated by ML and CV
Both consistent and asymptotically normal
Same rate of convergence√n for ML and CV
Same rate of convergence√n as for Gaussian processes
Explicit expressions for the asymptotic covariance matrix
Foreseen extension
Same questions but estimating the transform parameters
potential improvement in predictionspossibility to account for boundary constraints
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Perspectives and contributions
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Ongoing work and perspectives
RISCOPE till 2021! (Jose till August) - short term perspectives
Metamodels for the Decision Support System→ ACO allowed to run for a longer period→ Ysurf max as input for the other metamodels
Analysis of patterns in the record of explored metamodels
More levels of structural parameters→ other projection methods
Beyond RISCOPE - long term perspectives
ACO extensions
More levels of structural parameters→ other distances, kernelsOther structural parameters→ type of mean functionFunctional inputs in larger dimensions→ fields or imagesFunctional outputs→ time as input, ensamble of metas, cokriging
Strongly skewed data→ corrective DOE, transformed Gp
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Scientific production
Published papers
Gaussian process metamodeling of functional-input code for coastal flood hazardassessment, Reliability Engineering & System Safety, 2020.Betancourt, Bachoc, Klein, Idier, Pedreros & Rohmer (2020)
Asymptotic properties of the maximum likelihood and cross validation estimators fortransformed Gaussian processes, Electronic Journal of Statistics.Bachoc, Betancourt, Furrer, & Klein (2020)
Conference paper
Toward a user-based, robust and fast running method for coastal flooding forecast,early warning, and risk prevention, Journal of coastal research, proceedings from theInternational Coastal Symposium (ICS) 2020.Idier, Aurouet, Bachoc, Baills, Betancourt, Durand, Mouche, Rohmer, Gamboa, Klein,Lambert, Le Cozannet, Le Roy, Louisor, Pedreros & Veron (2020)
Software
R package funGp, Available on CRAN and GitHub since 2020.Authors: Betancourt, Bachoc, Klein Contributors: Idier, Rohmer
funGp user manual: Gaussian process regression for scalar and functional inputs withfunGp - The in-depth tour. Available as pre-print from HAL (2020).
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Summary of contributions
Metamodels for Decision Support System in coastal flood early warningMost of them promising
Novel approach for optimization of structural parametersPertinency made evident for all the 13 output variables
R package integrating the metamodeling techniques used in RISCOPEExtends the state of the art in various ways
First asymptotic results for transformed Gaussian processesProperties of ML and CV in a common yet unexplored setting
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Functional-input metamodeling: an applicationto coastal flood early warning
Jose Betancourt
Advisor:Co-advisor:
Prof. Thierry KleinProf. Francois Bachoc
PhD thesis defense - Applied MathematicsJune 08, 2020
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Bibliography I
Bachoc, F. (2014). Asymptotic analysis of the role of spatial sampling for covariance parameterestimation of Gaussian processes. Journal of Multivariate Analysis, 125, 1-35.
Bachoc, F., Betancourt, J., Furrer, R., Klein, T., et al. (2020). Asymptotic properties of themaximum likelihood and cross validation estimators for transformed gaussian processes.Electronic Journal of Statistics, 14(1), 1962–2008.
Betancourt, J., Bachoc, F., & Klein, T. (2020a). funGp: Gaussian process models for scalar andfunctional inputs. https://github.com/djbetancourt-gh/funGp. GitHub.
Betancourt, J., Bachoc, F., & Klein, T. (2020b, April). fungp: Gaussian process models forscalar and functional inputs. https://CRAN.R-project.org/package=funGp. (Rpackage version 0.1.0)
Betancourt, J., Bachoc, F., & Klein, T. (2020c, April). Gaussian process regression for scalarand functional inputs with fungp - the in-depth tour. RISCOPE project.https://hal.archives-ouvertes.fr/hal-02536624. (hal-02536624)
Betancourt, J., Bachoc, F., Klein, T., & Gamboa. (2020, April). Technical report: Ant colonybased model selection for functional-input gaussian process regression. Ref. D3.b(WP3.2), RISCOPE project. https://hal.archives-ouvertes.fr/hal-02532713.(hal-02532713)
Betancourt, J., Bachoc, F., Klein, T., Idier, D., Pedreros, R., & Rohmer, J. (2020). Gaussianprocess metamodeling of functional-input code for coastal flood hazard assessment.Reliability Engineering & System Safety, 198, 106870.
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Bibliography II
Cohen, A., Daubechies, I., & Feauveau, J.-C. (1992). Biorthogonal bases of compactlysupported wavelets. Communications on pure and applied mathematics, 45(5), 485–560.
Goss, S., Aron, S., Deneubourg, J.-L., & Pasteels, J. M. (1989). Self-organized shortcuts in theargentine ant. Naturwissenschaften, 76(12), 579–581.
Idier, D., Aurouet, A., Bachoc, F., Baills, A., Betancourt, J., Durand, J., . . . Vron, A. (2020).Toward a user-based, robust and fast running method for coastal flooding forecast, earlywarning, and risk prevention. Journal of coastal research, 95, 11–15.
Lataniotis, C., Marelli, S., & Sudret, B. (2018). Extending classical surrogate modelling toultrahigh dimensional problems through supervised dimensionality reduction: adata-driven approach. arXiv preprint arXiv:1812.06309.
Montgomery, D. C. (1997). Response surface methods and other approaches to processoptimization. Design and analysis of experiments.
Muehlenstaedt, T., Fruth, J., & Roustant, O. (2017). Computer experiments with functionalinputs and scalar outputs by a norm-based approach. Statistics and Computing, 27(4),1083–1097.
Nanty, S., Helbert, C., Marrel, A., Perot, N., & Prieur, C. (2016). Sampling, metamodeling,and sensitivity analysis of numerical simulators with functional stochastic inputs.SIAM/ASA Journal on Uncertainty Quantification, 4(1), 636–659.
Papaioannou, I., Ehre, M., & Straub, D. (2019). Pls-based adaptation for efficient pcerepresentation in high dimensions. Journal of Computational Physics, 387, 186–204.
Pulido, H. G., De la Vara Salazar, R., Gonzalez, P. G., Martınez, C. T., & Perez, M. d. C. T.(2012). Analisis y diseno de experimentos. New York: McGraw-Hill.
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