Uncertainty representation and propagation in chemical networks

Transcription

1 Uncertainty representation and propagation in chemical networks P. Pernot Laboratoire de Chimie Physique, Orsay

2 Plan 1 Introduction 2 Uncertainty propagation 3 Treatment of uncertainties for chemical rates 4 Conclusions

3 Uncertainties in a chemical model Structural uncertainties Model incompleteness Parametric uncertainties transport excitation processes reaction rates neutral-neutral ion-molecule

4 Uncertainties in a chemical model Structural uncertainties Model incompleteness Parametric uncertainties transport excitation processes reaction rates neutral-neutral ion-molecule ca. 680 Ion-molecule reactions in database

5 Uncertainties in a chemical model Structural uncertainties Model incompleteness Parametric uncertainties transport excitation processes reaction rates neutral-neutral ion-molecule ca. 680 Ion-molecule reactions in database

6 Parametric uncertainty sources

7 Uncertainty propagation as a tool Model predictions to be compared with observational data model predictions as virtual measurements : value + incert. Model assessment : significative discrepancies can be identified (model improvement) ; Sensitivity analysis : major prediction uncertainties can be analyzed to improve input parameters (new lab. experiments...).

8 Uncertainty propagation as a tool Model predictions to be compared with observational data model predictions as virtual measurements : value + incert. Model assessment : significative discrepancies can be identified (model improvement) ; Sensitivity analysis : major prediction uncertainties can be analyzed to improve input parameters (new lab. experiments...).

9 Uncertainty propagation as a tool Model predictions to be compared with observational data model predictions as virtual measurements : value + incert. Model assessment : significative discrepancies can be identified (model improvement) ; Sensitivity analysis : major prediction uncertainties can be analyzed to improve input parameters (new lab. experiments...).

10 Example : Cassini INMS Predicted ion mass spectrum with all uncertainty sources vs. Cassini s INMS (T5@1200km)

11 Example : Cassini INMS Predicted ion mass spectrum with all uncertainty sources vs. Cassini s INMS (T5@1200km)

12 Local Uncertainty Propagation 1 List all sources of uncertainty 2 Characterize uncertainty sources and estimate standard uncertainties 1 Type A : statistical analysis of a sample 2 Type B : all the rest u x from designed pdf 3 Combine standard uncertainties u 2 y = i ŷ = F (ˆx 1, ˆx 2,...) ( ) 2 y ux 2 x i + ( ) y i ˆx i x i i j ˆx i ( ) y x j ˆx j cov(x i, x j ) Guide to the expression of Uncertainty in Measurement (BIPM et al., 1995)

13 Local Uncertainty Propagation Guide to the expression of Uncertainty in Measurement (BIPM et al., 1995)

14 Global Uncertainty Propagation 1 List all sources of uncertainty 2 Represent uncertainty sources by pdf g {xi }({ξ i }) 3 Perform uncertainty propagation g y (η) = d {ξ i } δ(η f ({ξ i })) g {xi }({ξ i }) 4 Estimate uncertainties of model outputs from g y (η) Evaluation of measurement data - Supplement 1 to the GUM (BIPM et al., 2006)

15 Global Uncertainty Propagation Evaluation of measurement data - Supplement 1 to the GUM (BIPM et al., 2006)

16 Limitations of local and global UP Local approach, pb. with : asymmetrical pdfs (ex. lognormal) ; improved versions of standard formula nonlinear models (large uncertainties) ; nonlinear correlations (ex : prescribed sum). Most of the above apply to chemical networks of interest here...

17 Limitations of local and global UP Local approach, pb. with : asymmetrical pdfs (ex. lognormal) ; improved versions of standard formula nonlinear models (large uncertainties) ; nonlinear correlations (ex : prescribed sum). Most of the above apply to chemical networks of interest here... Global approach more complex (pdf design) grid-based methods (polynomial chaos, Galerkin...) curse of dimensionality pb. with positivity constraints sampling based methods pb. if model y = F (x) computer intensive

18 Sampling based uncertainty propagation Monte Carlo, Latin Hypercube Sampling (LHS)... g {x1,...,x n}(ξ 1, ξ 2,..., ξ n ) {}}{ F ξ (1) 1 ξ (1) 2... ξ n (1) η (1) ξ (2) 1 ξ (2) ξ n (2) η (2) ξ (m) 1 ξ (m) 2... ξ (m) n η (m) g y (η)

19 Sampling based uncertainty propagation Monte Carlo, Latin Hypercube Sampling (LHS)... g {x1,...,x n}(ξ 1, ξ 2,..., ξ n ) {}}{ F ξ (1) 1 ξ (1) 2... ξ n (1) η (1) ξ (2) 1 ξ (2) ξ n (2) η (2) ξ (m) 1 ξ (m) 2... ξ (m) n η (m) g y (η)

20 Sampling based uncertainty propagation Monte Carlo, Latin Hypercube Sampling (LHS)... g {x1,...,x n}(ξ 1, ξ 2,..., ξ n ) {}}{ F ξ (1) 1 ξ (1) 2... ξ n (1) η (1) ξ (2) 1 ξ (2) 2... ξ n (2) η (2) ξ (m) 1 ξ (m) 2... ξ n (m) η (m) g y (η)

21 Sampling based uncertainty propagation Monte Carlo, Latin Hypercube Sampling (LHS)... g {x1,...,x n}(ξ 1, ξ 2,..., ξ n ) {}}{ F ξ (1) 1 ξ (1) 2... ξ n (1) η (1) ξ (2) 1 ξ (2) ξ n (2) η (2) ξ (m) 1 ξ (m) 2... ξ (m) n η (m) g y (η)

22 Sampling based uncertainty propagation Monte Carlo, Latin Hypercube Sampling (LHS)... g {x1,...,x n}(ξ 1, ξ 2,..., ξ n ) {}}{ F ξ (1) 1 ξ (1) 2... ξ n (1) η (1) ξ (2) 1 ξ (2) 2... ξ n (2) η (2) ξ (m) 1 ξ (m) 2... ξ n (m) η (m) g y (η)

23 Global UP : Titan photochemistry Bimodality of outputs PDF : out of reach of local UP Global UP provides extensive exploration of parameter space Hébrard. et al., JPPC (2006), PSS (2007)

24 PDF design type of PDF depends on the nature of the parameters discrete vs. continuous interval of definition ], + [, [0, + [, [a, b]... PDF of outputs depends on PFDs of inputs, tempered by Central Limit Theorems the joint PDF of all uncertain parameters can be factorized in groups of independent parameters g {xi } ({ξ i }) = ( ) g {xj } j k {ξ j } j k k

25 PDF design type of PDF depends on the nature of the parameters discrete vs. continuous interval of definition ], + [, [0, + [, [a, b]... PDF of outputs depends on PFDs of inputs, tempered by Central Limit Theorems the joint PDF of all uncertain parameters can be factorized in groups of independent parameters g {xi } ({ξ i }) = ( ) g {xj } j k {ξ j } j k k

26 PDF design type of PDF depends on the nature of the parameters discrete vs. continuous interval of definition ], + [, [0, + [, [a, b]... PDF of outputs depends on PFDs of inputs, tempered by Central Limit Theorems the joint PDF of all uncertain parameters can be factorized in groups of independent parameters g {xi } ({ξ i }) = ( ) g {xj } j k {ξ j } j k k

27 Parametric uncertainties of reaction rates Analysis of measured rate constants Example. Arrhenius ln k = ln A ϑ/t Data analysis provides {ln A, ϑ} Norm ` ln A, ϑ, Σ Σ : variance/covariance matrix sample of k from {ln A, ϑ} and stat. analysis or u 2 ln k = u 2 ln A + u 2 ϑ/t 2 2/T Cov (ln A, ϑ) Note : Cov (ln A, ϑ) generally large (>0.9), very important for UP!

28 Arrhenius UP example : N( 2 D) +C 2 H 4 ln A E a/r (K) Correl Bayesian ± ±

29 Arrhenius UP example : N( 2 D) +C 2 H 4 ln A E a/r (K) Correl Sato et al. (1999) ± ± 50 n/a

30 Arrhenius UP example : N( 2 D) +C 2 H 4 ln A E a/r (K) Correl Bayesian ± ±

31 Arrhenius UP example : N( 2 D) +C 2 H 4 Uncertainty factors

32 Choice of pdf for evaluated rates Distributions implementing positivity constraint Preferred value k 0 i, estimated uncertainty factor F 0 i k i Lognormal(k 0 i, F 0 i ) P(k 0 i /(2F 0 i ) < k i < k 0 i 2F 0 i ) 0.95 k 0 i is median of pdf, not mean (Stewart and Thompson, 1994) modelers truncate to 2 Fi 0 or 3 Fi 0 to avoid exotic rates Preferred interval k i Loguniform(ki min, ki max ) no preferred value within an interval (dispersed experimental data) k min i = k 0 i /(2F 0 i ), k max i = k 0 i (2F 0 i )

33 Choice of pdf for evaluated rates Distributions implementing positivity constraint Preferred value k 0 i, estimated uncertainty factor F 0 i k i Lognormal(k 0 i, F 0 i ) P(k 0 i /(2F 0 i ) < k i < k 0 i 2F 0 i ) 0.95 k 0 i is median of pdf, not mean (Stewart and Thompson, 1994) modelers truncate to 2 Fi 0 or 3 Fi 0 to avoid exotic rates Preferred interval k i Loguniform(ki min, ki max ) no preferred value within an interval (dispersed experimental data) k min i = k 0 i /(2F 0 i ), k max i = k 0 i (2F 0 i )

34 Lognormal vs. Loguniform Output pdf depends on the place of the species in the network Might have an impact on sensitivity analysis Carrasco. et al., PSS (2007)

35 Parametric uncertainties of branching ratios Multi-pathway reactions k i,j = k i b i,j ; b i,j = 1 Reaction rates and branching ratios are mostly measured by different experiments/techniques larger uncertainties for branching ratios (more difficult to measure than rates). In such cases, it is better to keep an explicit separation of uncertainty sources more pertinent sensitivity analysis (key parameters) ; easier to manage the sum rule wrt. uncertainties ; T-dependence of k i different from b i,j safer update of databases when new branching ratios or rate available. j

36 Parametric uncertainties of branching ratios Multi-pathway reactions k i,j = k i b i,j ; b i,j = 1 Reaction rates and branching ratios are mostly measured by different experiments/techniques larger uncertainties for branching ratios (more difficult to measure than rates). In such cases, it is better to keep an explicit separation of uncertainty sources more pertinent sensitivity analysis (key parameters) ; easier to manage the sum rule wrt. uncertainties ; T-dependence of k i different from b i,j safer update of databases when new branching ratios or rate available. j

37 PDFs for branching ratios Distributions implementing the sum rule Preferred values and precision {b i,j } Diri ({α i,j }) j b α i,j 1 i,j No preference {b i,j } Diri (1, 1,..., 1) ex : low-t extrapolation for ion-molecule reactions Preferred intervals {b i,j } Diun ({ b min i,j, bi,j max }) Carrasco et al., PSS (2007)

38 Example : N CH 4 CH + 3 CH + 2 N 2 H % 100% 100% Standard elicitation 1 10 % < 30 % < 100 % 0 Carrasco et al., PSS (2007)

39 Example : N CH 4 CH + 3 CH + 2 N 2 H % 10% 10% Improved elicitation according to Nicolas (PhD Thesis, 2002) Carrasco et al., PSS (2007)

40 Example : N CH 4 CH + 3 CH + 2 N 2 H + 1/3 1/3 1/3 100% 100% 100% Full uncertainty Carrasco et al., PSS (2007)

41 Branching ratios and the sum rule I 1 + M 1 P 1 ; k 1, b 11 I 1 + M 1 P 2 ; k 1, b 12 I 1 + M 2 P 3 ; k 2 I 2 + M 2 P 3 ; k 3 [M i ] [I i ] F k F b

42 Branching ratios and the sum rule Uncorrelated partial rates : b 11 = 0.33 ± 0.12, b 12 = 0.67 ± 0.12

43 Branching ratios and the sum rule Correlated partial rates : {b 11, b 12 } Diri (15, 30)

44 Branching ratios and the sum rule Uncorrelated partial rates : b 11 b 12 Unif (0, 1)

45 Branching ratios and the sum rule Correlated partial rates : {b 11, b 12 } Diri (1, 1)

46 Shopping for rate uncertainty What have existing databases to offer to MCUP-aware modelers? udfa 06 k i (T ) = α i (T /300) β i exp ( γ i /T ) The accuracy is described by a letter - A, B, C, D, E - where the errors are < 25%, < 50%, within a factor of 2, within an order of magnitude, and highly uncertain, respectively. No T-dependence of uncertainty No pdf proposed

47 Shopping for rate uncertainty What have existing databases to offer to MCUP-aware modelers? osu k i (T ) = α i (T /300) β i exp ( γ i /T ) F i = 1.25,1.5, 2.0 or 10.0 No T-dependence of uncertainty No pdf proposed

48 Shopping for rate uncertainty What have existing databases to offer to MCUP-aware modelers? Anicich (ion-molecule, JPL 2003) Global rate k i ± f i (f i in percent) Branching ratios {b ij } j=1,n No uncertainty on branching ratios No T-dependence on properties and uncertainties No pdf proposed

49 Shopping for rate uncertainty What have existing databases to offer to MCUP-aware modelers? IUPAC - NASA/JPL k i (T ) = k 0 i exp ( E i /T ) F i (T ) = F 0 i exp `g i 1 T 1 T 0 ; T 0 = 298 K F i is an expanded uncertainty, CI 95% The assignment of the uncertainties is a subjective assessment of the evaluators. They are not determined by a rigorous statistical analysis of the database. No pdf proposed

50 Shopping for rate uncertainty What have existing databases to offer to MCUP-aware modelers? Hébrard et al. (JPPC, 2006 ; PSS, 2007) k i (T ) = α i (T /300) β i exp ( γ i /T ) F i (T ) = F i (300 K) exp `g i 1 1 T 300 F i is a standard uncertainty, CI 67% Both uncertainty factors, F i (300 K) and g i, do not necessarily result from a rigorous statistical analysis of the available data. No pdf proposed, but log k i = log k i (T ) + ɛ log F i (T ); ɛ Norm(0, 1), in Hébrard et al. (PSS, 2007)

51 Conclusions Global UP is necessary for chemical networks Monte Carlo UP is now recognized as a standard tool in metrology Utility of MCUP depends on adapted PDFs, with a correct description of inputs uncertainty amplitude (not too small, not too large...) distribution shape (to a minor degree, but we need more experiences in SA to conclude) a structure reflecting experimental uncertainty sources necessary step to exp.-oriented sensitivity analysis Existing databases have not be designed with MCUP in mind they can be updated and improved along these proposed lines...

52 Conclusions Global UP is necessary for chemical networks Monte Carlo UP is now recognized as a standard tool in metrology Utility of MCUP depends on adapted PDFs, with a correct description of inputs uncertainty amplitude (not too small, not too large...) distribution shape (to a minor degree, but we need more experiences in SA to conclude) a structure reflecting experimental uncertainty sources necessary step to exp.-oriented sensitivity analysis Existing databases have not be designed with MCUP in mind they can be updated and improved along these proposed lines...

53 Conclusions Global UP is necessary for chemical networks Monte Carlo UP is now recognized as a standard tool in metrology Utility of MCUP depends on adapted PDFs, with a correct description of inputs uncertainty amplitude (not too small, not too large...) distribution shape (to a minor degree, but we need more experiences in SA to conclude) a structure reflecting experimental uncertainty sources necessary step to exp.-oriented sensitivity analysis Existing databases have not be designed with MCUP in mind they can be updated and improved along these proposed lines...

54 Conclusions Global UP is necessary for chemical networks Monte Carlo UP is now recognized as a standard tool in metrology Utility of MCUP depends on adapted PDFs, with a correct description of inputs uncertainty amplitude (not too small, not too large...) distribution shape (to a minor degree, but we need more experiences in SA to conclude) a structure reflecting experimental uncertainty sources necessary step to exp.-oriented sensitivity analysis Existing databases have not be designed with MCUP in mind they can be updated and improved along these proposed lines...

55 Remerciements N. Carrasco (SA, Verrières) S. Plessis & Ch. Alcaraz (LCP, Orsay) M. Dobrijevic (LAB, Bordeaux) E. Hébrard (LISA, Crétail) M. Banaszkiewicz (SRC, Warsaw) R. Thissen & O. Dutuit (LPG, Grenoble) V. Vuitton & R. Yelle (LPL, Tucson) CNRS CNES EuroPlaNet Programme National de Planétologie