How To Solve A Save Problem With Time T

Transcription

1 Bayesian Functional Data Analysis for Computer Model Validation Fei Liu Institute of Statistics and Decision Sciences Duke University Transition Workshop on Development, Assessment and Utilization of Complex Computer Models SAMSI, May 14nd, 2007 Joint work with Bayarri, Berger, Paulo, Sacks,...

2 Outline Computer model with functional output The methodology Basis representation Bayesian Analysis The case study example The road load data The analysis Results with Gaussian bias Results with Cauchy bias Dynamic Linear models Summary and On-going Work

4 Computer model validation Question: Does the computer model adequately represent the reality?

5 A solution SAVE (Bayarri et al., 2005) treats time t as another input. - Prior: ( ) y M (, ) GP Ψ (, )θ L 1, λ M cm ((, ), (, )). - Fast surrogate: ( y M (z, t) Data N m(z, t), V ) (z, t).

6 Limitations Treating t as another input, - is only applicable to low dimensional problem. - need smoothness assumption. Need new methodology to deal with: - Complex computer models. - High dimensional irregular functional output. - Uncertain (field) inputs.

8 Notation y F r (v i, δ ij ; t) = y M (v i, δ ij, u ; t) + y B u (v i, δ ij ; t) + e ijr (t)

9 Notation yr F (v i, δ ij ; t) = y M (v i, δ ij, u ; t) + yu B (v i, δ ij ; t) + e ijr (t) Inputs to the code. v i : configuration. δ ij : unknown inputs. u : calibration parameters. Simulator. Bias function. Reality y R. Field runs. Measurement errors.

15 Basis expansion Represent the functions as: y M (z j ; t) = W k=1 w M k (z j)ψ k (t), j = 1,..., n M ; y B u (v i, δ ij ; t) = W k=1 w B k (v i, δ ij )ψ k(t), i = (1,..., m), j = (1,..., n f i ) ; y F r (v i, δ ij ; t) = W k=1 w F kr (v i, δ ij )ψ k(t), r = 1,..., r ij.

16 Thresholding Reduce computational expense, y M (z j ; t) W 0 k=1 w M k (z j)ψ k (t), j = 1,..., n M ; y B u (v i, δ ij ; t) W 0 k=1 w B k (v i, δ ij )ψ k(t), i = (1,..., m), j = (1,..., n f i ) ; y F r (v i, δ ij ; t) W 0 k=1 w F kr (v i, δ ij )ψ k(t), r = 1,..., r ij.

17 Emulators For each k = 1,..., W 0, ) wk (µ M ( ) GP M k, σ 2M k Corr k (, ). Corr k (, ): power exponential family, Corr k (z, z ) = exp ( p d=1 ) β (M k ) d z d z k ) d α(m d. Modular approach: ( wk M (z) Data, ˆµM k, ˆσ 2M k, ˆα M k, ˆβ M ) ( k N m k M (z), V ) k M (z).

18 Bayesian analysis For each k = 1,..., W 0, wkr F (v i, δ ij ) = w k R(v i, δ ij ) + ɛ ijkr, ɛ ijkr N ( 0, σk 2F ), w R k (v i, δ ij ) = w M k (v i, δ ij, u ) + w B k (v i, δ ij ). w B k (v i, δ ij ) = b k(v i ) + ɛ b ijk

19 Bayesian analysis For each k = 1,..., W 0, wkr F (v i, δ ij ) = w k R(v i, δ ij ) + ɛ ijkr, ɛ ijkr N ( 0, σk 2F ), w R k (v i, δ ij ) = w M k (v i, δ ij, u ) + w B k (v i, δ ij ). w B k (v i, δ ij ) = b k(v i ) + ɛ b ijk Data: - w ijk F = 1 rij r ij r=1 w kr F (v i, δ ij ). - S 2 ijk = r ij r=1 (w F kr (v i, δ ij ) w F ijk )2. - m k ( ), V k ( ).

20 Bayesian analysis For each k = 1,..., W 0, wkr F (v i, δ ij ) = w k R(v i, δ ij ) + ɛ ijkr, ɛ ijkr N ( 0, σk 2F ), w R k (v i, δ ij ) = w M k (v i, δ ij, u ) + w B k (v i, δ ij ). w B k (v i, δ ij ) = b k(v i ) + ɛ b ijk The bias, ( ) b k (v i ) GP µ B k, σ2b k CorrB k (, ), ɛ b ijk N(0, σ2bɛ k ), with correlation, ( Corr B k (v, v p v ) = exp d=1 β (B k ) d v d v d α(b k ) d ).

21 Bayesian analysis For each k = 1,..., W 0, wkr F (v i, δ ij ) = w k R(v i, δ ij ) + ɛ ijkr, ɛ ijkr N ( 0, σk 2F ), w R k (v i, δ ij ) = w M k (v i, δ ij, u ) + w B k (v i, δ ij ). w B k (v i, δ ij ) = b k(v i ) + ɛ b ijk Unknowns: Inputs: u and δ ij. Bias: {bk ( )}, wk B(v i, δ ij ). Model: wk M(v i, δ ij, u ). Reality: {wk R(v i, δ ij )} Hyper-parameters: σ 2B k, σ2bɛ k, σk 2F,µB k.

28 The road load data 1 Time history data at T = time points. Inputs include: - One configuration v = x nom. - Seven characteristics x = (x 1, x 2,..., x 7 ), x = x nom + δ. - Two calibration parameters u = (u 1, u 2 ).

29 The road load data 2 64 design points in 9-d space for computer model runs. {y M (z, t), z D M, t 1,..., T }, z = (x, u). 7 field replicates. {y F r (x, t), r (1,..., 7)}, x = x nom + δ.

30 0 5 y 10 The road load data d 50 60

31 Wavelet representation Represent the time history data by wavelets: y M (z j ; t) = y F r (x ; t) = W wk M (z j)ψ k (t), j = 1,..., 64; k=1 W wkr F (x )ψ k (t), r = 1,..., 7. k=1 Apply hard thresholding to reduce the computational expense. keep W 0 = 289 nonzero coefficients.

32 Emulators For each wavelet coefficient, wk (µ M ( ) GP k, 1 λ M k Corr k (, ) ). Corr k (, ): power exponential family, Corr k (z, z ) = exp ( p d=1 β (k) d z d z α(k) k d Response Surface: ) ( (w k M (z) Data, ˆµ k, ˆλ M k, ˆα k, ˆβ k N m k (z), V ) k (z). ).

33 Bayesian analysis For each wavelet coefficient, w R i (x ) = w M i (x, u ) + b i (x ), i = 1,..., W, w F ir (x ) = w R i (x ) + ɛ ir, r = 1,..., f. The bias, ( ) b i (x ) N 0, τj 2 or j: wavelet resolution level. ( ) b i (x ) C 0, τj 2. The error, ( ) ɛ ir N 0, σi 2.

34 Prior distributions The Input/Uncertainty Map, Parameter Type Variability Uncertainty Damping 1 Calibration Uncertain 15% Damping 2 Calibration Uncertain 15% Stiffness 1 Manufacturing Uncertain 10% Stiffness 2 Manufacturing Uncertain 10% Front-rebound 1 Manufacturing Uncertain 7% Front-rebound 2 Manufacturing Uncertain 8% Sprung-Mass Manufacturing Uncertain 5% Unsprung-Mass Manufacturing Uncertain 12% Body-Pitch-Inertia Manufacturing Uncertain 8% Calibration: uniform over specified range. Manufacturing: truncated normals over the ranges. π(σi 2 ) 1/σi 2, π(τj 2 {σi 2 }) 1. τj σ2 j

35 Bias function (under Gaussian bias) b h (t) = W bi h ψ i (t), h = 1,..., N i=1 Load Time

36 Reality (under Gaussian bias) y Rh (u h, x h, t) = W ( i=1 w Mh i (u h, x h ) + b h i ) ψ i (t), h = 1,..., N Load Time

37 Bias function (under Cauchy bias) Load Bias function MCMC (90% Tolerance bounds) Time

38 Reality (under Cauchy bias) Load Time

40 Dynamic Linear models

41 Model the computer model run at z by multivariate TVAR (West and Harrison, 1997), y M (z, t) = p φ t,j y M (z, t j) + ɛ M t (z), } {{ } j=1 GP(0, σ 2 t Corr(, )). Interpolator with forecasting capability. Computation is done by Forward filtering backward sampling algorithm. Predict at untried input z = 0.5.

42 Result

44 Summary We developed a Bayesian functional data analysis approach to computer model validation. The approach automatically take into account model discrepancy and model calibration. Unknown (field) inputs can be incorporated. Bias functions are modeled using hierarchical structures when needed.

45 On-going work: Time-dependent parameters Time-dependent parameters (Reichert, 2006) yt F = yt M (z F + φ z t ) +φ F t + ɛ F t, φ z t Stochastic process, } {{ } y M t (z F + φ z t ) y M t (z F ) + y M t (z F )φ z t. Build emulators for y M t (z F ) and y M t (z F ).

46 On-going work : Spatio-Temporal Outputs Figure: Regional Air Quality forecast (RAQAST) Over the U.S. (Provided by Serge et. al)

47 The model, y F (s, δ ; t) = y M (s, δ, u ; t) + b(s, δ ; t) + ɛ F t. - s: location (configuration). - u: calibration parameters. - δ: meteorology. - t: time. Spatial interpolation, y M (s, δ, u; t) = k K a k (s, δ, u)ψ k (t). Forecasting, b(s, δ; t) = p φ t,j b(s, δ; t j) + ɛ b t (s, δ). j=1

48 Thank you!