Assessment of uncertainty in computer experiments: from Universal Kriging to Bayesian Kriging. Céline Helbert, Delphine Dupuy and Laurent Carraro

Transcription

1 Assessment of uncertainty in computer experiments: from Universal Kriging to Bayesian Kriging., Delphine Dupuy and Laurent Carraro

2 Historical context First introduced in the field of geostatistics (Matheron, 1960). Recent use as a response surface for computer experiments (Sacks,1989 ; Santner,003). Prediction and uncertainty on prediction (Jones, 1998, Oakley, 004) Two different approaches among practitioners: Universal Kriging (UK) parameters are estimated (CV, ML) Bayesian Kriging (BK) parameters are random variables Goal : BK allows the interpretation of UK uncertainty as a prediction variance Application of BK : petroleum case study

3 Outline Universal Kriging limits Bayesian Kriging pro and con Case study

4 Universal Kriging

5 Assumptions : ( ) ( ) ( ) Probabilistic Context Y x = f x β + Z x where Z is a GP E ( Z ( x) ) ( ) ( ) = 0 Cov Z x Z x h R h (, + ) = σ ( θ ) T n values, Y = ( y1 y n ), are observed at points = ( ) Prediction and uncertainty : X x1 x n T

6 Assumptions : ( ) = ( ) β + ( ) Probabilistic Context Y x f x Z x where Z is a GP E ( Z ( x) ) ( ) ( ) = 0 Cov Z x Z x h R h (, + ) = σ ( θ ) T n values, Y = ( y1 y n ), are observed at points = ( ) Prediction and uncertainty : Case Simple Kriging : parameters are known SK T 1 ( ) = ( ) β + ( β ) Y x f x r R Y F 0 0 T ( 0 ) = σ ( 1 θ θ θ ) 1 σ SK x r R r θ θ X x1 x n T

7 Assumptions : ( ) = ( ) β + ( ) Probabilistic Context Y x f x Z x where Z is a GP E ( Z ( x) ) ( ) ( ) = 0 Cov Z x Z x h R h (, + ) = σ ( θ ) T n values, Y = ( y1 y n ), are observed at points = ( ) X x1 x n T Prediction and uncertainty : Case Simple Kriging : parameters are known SK T 1 ( ) = ( ) β + ( β ) Y x f x r R Y F 0 0 T ( 0 ) = σ ( 1 θ θ θ ) 1 σ SK x r R r θ θ Case Universal Kriging : parameters are estimated UK ( ) ( ) ˆ T 1 0 = 0 β + ˆ ˆ ( ˆ β θ θ ) Y x f x r R Y F 1 ( )( ) ( ( ) ) T 1 T 1 T 1 T 1 UK ( x 0 ) ˆ 1 r ˆ R ˆ r ˆ f ( x 0 ) r ˆ R ˆ F F R ˆ F f x 0 r ˆ R σ = σ + ˆ F θ θ θ θ θ θ θ θ T

8 Simple Kriging - example output SK data ( ( )) ( ) ( ) ( ) ( ) ( ) ( h ) E Y x = 0 Var Y x = 4 Corr Y x, Y x + h = exp 0.

9 Limits of Universal Kriging Difficulties due to estimation : flat Likelihood too few data to estimate covariance function and ranges experimental design sensibility Underestimation of uncertainty σ UK ( x 0 ) does not take into account uncertainties due the estimation of variance, σ, and range θ No probabilistic interpretation ( ) ( ) ˆ T 1 Y ( ˆ ) UK x0 f x0 β rˆ R ˆ Y Fβ θ θ ( ) = + E Y ( x0 ) Y T 0 = ˆ σ 1 ˆ ˆ ˆ +... θ θ θ Var ( Y ( x0 ) Y ) ( ) ( ) 1 σuk x r R r

10 Bayesian Kriging

11 Model of Bayesian Kriging Assumptions : β, θ, σ are Random Variables. Let π be the prior distribution. ( ) = ( ) + ( ) ( Z ( x) ) x D Y x β, θ, σ f x β Z x, with the same assumptions for T n values, = ( ), are observed at points = ( ) Y y1 y n X x x n 1 T Interpretation : A mixture of Gaussian Processes (Y(x) is not Gaussian) The weight of a given process in the mixture depends on its prior π.

12 Equations of Bayesian Kriging ( SK SK ) ( ), β, θ, σ Ν ( ), σ ( ) Y x Y Y x x ( ( 0 ) ) ( 0 ) β, θ, σ ( ) ( ) E Y x Y E Y x Y, β, θ, σ π β, θ, σ Y dβ dθdσ = (,, Y ) π β θ σ = ( ; β, θ, σ ) π ( β, θ, σ ) L Y π ( Y )

13 Equations of Bayesian Kriging ( SK SK ) ( ), β, θ, σ Ν ( ), σ ( ) Y x Y Y x x ( ( 0 ) ) ( 0 ) β, θ, σ ( ) ( ) E Y x Y E Y x Y, β, θ, σ π β, θ, σ Y dβ dθdσ = Prediction : BK ( ) ( ) = ( ) Y x E Y x Y 0 0 (,, Y ) π β θ σ = ( ; β, θ, σ ) π ( β, θ, σ ) L Y π ( Y )

14 Equations of Bayesian Kriging ( SK SK ) ( ), β, θ, σ Ν ( ), σ ( ) Y x Y Y x x ( ( 0 ) ) ( 0 ) β, θ, σ ( ) ( ) E Y x Y E Y x Y, β, θ, σ π β, θ, σ Y dβ dθdσ = Prediction : BK ( ) ( ) = ( ) Y x E Y x Y Measure of uncertainty : 0 0 ( ) ( ) ( ) BK x 0 Var Y x 0 Y σ = Simulation of the distribution of Y(x 0 ) Y (,, Y ) π β θ σ = ( ; β, θ, σ ) π ( β, θ, σ ) L Y π ( Y )

15 Particular case of prior distribution Gaussian Case for β Prior distribution (θ and σ are constant) ( ) = Ν( µ, λσ) π β

16 Particular case of prior distribution Gaussian Case for β (θ and σ are constant) ( ) = Ν( µ, λσ) Prior distribution π β Posterior Gaussian distribution for β 1 T T ( β ) = µ + λσ ( λ Σ + σ θ ) ( µ ) 1 T T ( β ) λ λ ( λ σ θ ) E Y F F F R Y F Var Y = Σ ΣF FΣ F + R FΣ

17 Particular case of prior distribution Gaussian Case for β (θ and σ are constant) ( ) = Ν( µ, λσ) Prior distribution π β Posterior Gaussian distribution for β Posterior Gaussian distribution for Y(x 0 ) ( ( 0 ) ) ( 0 ) ( ( ) ) ( ) 1 ( )( µ ( λ σ ) ( µ )) T 1 T T T 1 E Y x Y = f x rθ Rθ F + ΣF FΣ F + Rθ Y F + rθ Rθ Y 1 T 1 T T T 1 1 ( ) ( ) ( ( ) ) T T θ θ λ λ λ σ θ θ θ σ ( θ θ θ ) Var Y x0 Y = f x0 r R F Σ ΣF FΣ F + R FΓ f x0 r R F + 1 r R r

18 Particular case of prior distribution Gaussian Case for β (θ and σ are constant) ( ) = Ν( µ, λσ) Prior distribution π β Posterior Gaussian distribution for β Posterior Gaussian distribution for Y(x 0 ) Particular case : λ + (non informative prior for β) 1 Posterior Gaussian distribution for β T 1 T 1 ( ) ( ) ˆ E β Y = F Rθ F F Rθ Y = β 1 T 1 Var β Y = σ F R F Var ˆ θ = β Posterior Gaussian distribution for Y(x 0 ) ( ) ( ) ( ) ( ) ( ML ) ( 0 ) = UK ( 0 ) ( 0 ) = σuk ( 0 ) E Y x Y Y x Var Y x Y x ML

19 Bayesian Kriging- difficulties The simulation of the posterior distribution of the parameters can be hard (Simulation by a Monte Carlo Markov chain method) The choice of the prior is difficult Case of a flat prior for β, θ and σ Roughly equivalent to maximize the likelihood function Advantages: the prediction variance takes all sources of uncertainty into account, the optimization problem disappears Case of an informative prior: which one? What impact? IDEA: to use a simplified simulation (faster) to derive prior information Example : petroleum field

20 Application Simulator : flow simulator - 3DSL 3 inputs on [-1,1] lmultkz (permeability) krwmax (relative permeability), lbhp (low bottom hole pressure) Output: Field oil production total after 7000 days Problem : uncertainty analysis Method : metamodel and its uncertainty (Bayesian Kriging)

21 3DSL /degraded simulations Idea : using a faster simulator to get prior information degraded simulations (NODESMAX, DTMAX, DVPMAX etc)

22 3DSL /degraded simulations Correlation Coefficient ALL lmultkz krwmax lbhp 3DSL /Degraded DSL /Degraded Note : calculations carried out on a grid of 1331=11 3 points

23 3DSL /degraded simulations 3.4 x 107 LMULTKZ 3.5 x 107 KRWMAX 3.45 x 107 LBHP FOPT DSL Degraded 1 Degraded

24 4 different strategies runs UK no info BK BK info1 BK info time UK no info BK BK info1 BK info 3DSL DSL Deg. 1 X X 4 X Deg. 1 X X 4 X Deg. X X X 4 Deg. X X X 3 Total Info 1 = No info BK on the 4 runs of Degraded 1 provides information on trend, variance and correlation

25 4 different strategies UK no info BK BK info1 BK info RMSE* Average standard deviation proportion of outside points* 40% 19% 8% 4% Note : calculations carried out on a grid of 1331=11 3 points Accuracy of prediction: N N i = 1 ( ( ) ( )) i i 1 RMSE = Y x Yˆ x Accuracy of uncertainty: ASD 1 N σ Yˆ i N i= 1 = N i= 1 ( x ) 1 PR = 1 N σ ( ) ˆ( ) > ( ) Y x Y x x i i Yˆ i

26 Conclusion Advantages of Bayesian kriging In the case of a non informative prior for β: ( ( 0 ) ) = UK ( 0 ) ( ) = σ E Y x Y Y x ( 0 ) UK ( 0 ) Var Y x Y x Good estimation of the prediction variance which takes into account all sources of uncertainty: on β, σ and θ. Weakness of Bayesian kriging MCMC simulations Choice of the prior

27 Universal Kriging - example θ = 0. (known) ˆ σ = 0.86 (σ = 4) ˆ β = 1.09 (β = 0) output SK UK data

28 Experimental design sensibility likelihood optimization : f(θ) = -log(l(θ)) x Variability on θ θ Identification pb

29 Prior and posterior distributions 0,35 0,19 0,78 0,68 0,70 0,68 0,4 0,61 Std(. Y) 0,97 0,48,5 1,69-0,58-0,63 0,00-0,56 E(. Y) Deg. 1,30 0,31 1,93 0,88 0,53 0,88 0,57 0,78 Std(. Y),71 0,76 3,57,1-0,64-0,5-0,3-0,40 E(. Y) Deg. 1 teta3 teta teta1 sigma beta3 beta beta1 beta0 1,73 0,8,00 0,80 0,4 1,05 0,55 1,06 Std(. Y) 4,36 1,08 3,86,64-0,35 0,34 0,54-0,57 E(. Y) no info 0,8 0,13 0,64 0,1 0,40 0,44 0,4 0,39 Std(. Y) 1,45 0,69,89 0,97-0,36-0,63 0,16-0,40 E(. Y) info 0,88 0,19 1,16 0,40 0,3 0,64 0,30 0,55 Std(. Y) 3,3 0,87 3,74 1,88-0,47-0,41 0,19-0,41 E(. Y) info 1 teta3 teta teta1 sigma beta3 beta beta1 beta0 17 points Prior

30 Range variation sensibility 4 beta = 1.1 sigma = 0.94 teta = beta = 1.1 sigma = 0.98 teta = beta = 4.4 sigma = 66 teta =

31 Sensibility to range variation 4 beta = 1.1 sigma = 0.98 teta = beta = 4.4 sigma = 66 teta = beta = 1.1 sigma = 0.94 teta = Vraisemblance en fonction des paramètres d'échelle θ θ 1