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Use of Latent Variables in the Parameter Estimation Process

Use of Latent Variables in the Parameter Estimation Process. Jonas Sjöblom Energy and Environment Chalmers University of Technology. NO X Reduction catalysis. ~mm. ~µm. ~nm. Introduction. Use of Latent Variables (LV). What is LV? How does it work?

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Use of Latent Variables in the Parameter Estimation Process

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  1. Use of Latent Variables in the Parameter Estimation Process Jonas Sjöblom Energy and Environment Chalmers University of Technology

  2. NOX Reduction catalysis ~mm ~µm ~nm Introduction

  3. Use of Latent Variables (LV) • What is LV? How does it work? • How can it be applied in the parameter estimation process? • 3 case studies • Why is it good? Outline

  4. x3=dY/d3 p1 x2=dY/d2 p2 x1=dY/d1 Latent Variable modelling • Reduces a data matrix (using projections) to new, few and independent components (Latent Variables). • Latent Variable (LV) Model: • P: loadings (linear combination of original variables) • T: scores (projections on the subspace defined by P) • # components: # linear independent directions • Different types of Latent Variable (LV) models: • Principal Components Analysis (PCA) • Partial Least Squares (PLS) What is LV modelling?

  5. Use of LV models Parameter Estimation Process Define model and model assumptions 3. Define ”experimental space” 2. Choice of experiments to perform Evaluate the Design by LV - model 1&2 Evaluate the Design (perform experiments) (experimental rank) 1. Fit parameters Satisfactory results? No! No! Yes! How can LV models be applied?

  6. NOX Storage and Reduction (NSR) Mechanism 62 parameters Poor experimental design Jacobian f/ used in gradient search ill-conditioned Local minima Application 1: LV models during the fitting process Objective: to improve parameter fitting by analysing parameter correlations and make parameters more orthogonal Ref: Sjoblom et al, Comput. Chem. Eng. 31 (2007) 307-317

  7. Parameter assessment • Jacobian f/ • Evaluated for ALL adjustable parameters (not only fitted ones) • Latent Variable (LV) method: • Partial Least Squares (PLS) using the Jacobian as "X" and f (Residual: simulated-observed gas phase concentrations) as “Y” • Outcomes: • Correlation structure ! • Number of independent directions (# parameters to fit) ! • Which parameters to choose ! (method 1) • Parameter fit in LV space (method 2) How can LV models be used? -appl.1

  8. X Y 0 , 3 0 k30_ Ea02 s-NO Ea29 k07_ k25_ k12_ 0 , 2 0 k22_ sCO2 Ea30 Ea34 k14_ k01_ 0 , 1 0 k10_ Ea20 k15_ Ea22 k20_ k27_ k16_ k26_ Ea19 k32_ k18_ Ea11 Ea24 Ea09 Eth3 k33_ k05_ Ea05 k03_ Ea04 Eth0 Ea03 Ea10 Ea25 Ea06 k35_ k17_ k28_ Ea28 Ea15 k24_ Ea14 0 , 0 0 Ea17 Ea31 Ea27 Ea16 Ea18 sNOx Ea33 k04_ w*c[2] Ea23 Ea26 k06_ - 0 , 1 0 k34_ Ea07 k02_ k09_ Ea35 k19_ k23_ Ea32 sNO2 k29_ k13_ - 0 , 2 0 k31_ k11_ k08_ - 0 , 3 0 k21_ - 0 , 2 0 - 0 , 1 0 0 , 0 0 0 , 1 0 0 , 2 0 0 , 3 0 w*c[1] LV example: "loading" plot How can LV models be used? -appl.1

  9. Results • Fitting results are comparable, but the fitting is more efficient (faster) due to fewer and more independent parameters, adopted for the data set at hand 90 90 500 Method I (9 selected parameters) Method II (fitting of 9 scores) Method “brute force” (all 62 parameters) How can LV models be used? -appl.1

  10. Application 2: Model-based DoE for precise parameter estimation • "Simple" but realistic system: • NO-oxidation on Pt • Model from Olsson et.al. (1999) • Using simulated data (noise added) as experiments • Objective: • How to find the experiments that enable precise estimation of the kinetic parameters Ref: Sjoblom et al, Comput . Chem. Eng 32 (2008) 3121-3129

  11. Define model and model assumptions Define experimental space D - optimal Choice of experiments to perform Choice of experiments to perform using X or T from LV - model Evaluate the Design by LV - model Evaluate the Design by LV- model (experimental rank) , fit analyze Satisfactory results? No! No! Yes! Experiment assessment • Jacobian f/ • Evaluated for ALL "possible" experiments (3 iterations) • Latent Variable (LV) method: • Principal Component Analysis (PCA) of J (unfolded 3 way matrix) • D-optimal design to select experiments • Outcomes: • Correlation structure ! • Number of independent directions (# parameters to fit) ! • Which experiments to choose ! How can LV models be applied? -appl.2

  12. Results • Overcomes dimensional reduction of the Fischer information matrix: by use of PCA (LV model of unfolded 3-way matrix) • Almost perfect fit was obtained but parameter values were different (J not full rank) • Using X (as is) or an LV approximation of X performs equally well • but becomes more efficient since it requires less experiments • The LV model gives additional information of the dimensionality of selected experiments before they are performed. How can LV models be applied? -appl.2

  13. Application 3: Extended Sensitivity Analysis for targeted Model Improvements CO2, DH NO, H2 • H2 assisted HC-SCR over Ag-Al2O3 • Detailed model (23 reactions, heat balance) • Acceptable fit, but still significant Lack-of-Fit • Objectives: • Verify (falsify) model assumptions • Get indications on how to improve model fit N2 C8H18 O O NO3 O NO2 NO2 CH2 Thesis available at: http://publications.lib.chalmers.se/records/fulltext/92706.pdf Refs: Creaser et al. Appl.Catal.B 90 (2009) 18-28, Sjöblom PhD Thesis (2009) Chalmers

  14. Experimental • Sensitivity analysis of 62 model parameters (not only fitted ones, not only kinetic parameters) • 46 kinetic parameters • 10 mass and heat transport parameters • 6 other parameters • Scaled local sensitivities • Unfold 3-way matrix to size n x pk, where n=26025 time points, p=62 parameters and k=5 responses • Univariate analysis as well as LV modelling How can LV models be applied? -appl.3

  15. LV-model and univariate measures • PCA model • Scores plot • Loadings plot • 25 components • Univariate table data • Confidence intervals • Sensitivity average, std, max • Correlations How can LV models be applied? -appl.3

  16. Sensitivity Analysis results (examples) • Mass transfer model needs attention • Include diffusivities in fitting? • Include internal mass transport? • Targeted transients? • Heat transfer model needs attention • Improve/extend temperature measurements? • Consider additional sensors (HC, H2)? • Modify heat transfer model? • Targeted experiments? (For more details, see poster) How can LV models be applied? -appl.3

  17. Factors for successful parameter estimation Model assumptions • Ability to master different parts of the process • The model (assumptions) • The available data (experiments) • The parameter values (which to fit) • Ability to “change focus” in the process as the fit develops ”experimental space” Choice of experiments Evaluate the Design Fit parameters New PhD project: “Improved methods for parameter estimation” Advertisement out now! Application dead line 20th sept 2009 http://www.chalmers.se/chem/EN/news/vacancies/positions/phd-student-position-in8778 Happy? No! No! Yes! Why are LV models good?

  18. LV Components:Few, New & linearly Independent • Few: Improved efficiency • Linear: Non-linear systems, LV models provide more robust linearisations • Independent: Orthogonal sensitivities fulfils statistical requirements Why are LV models good?

  19. Conclusions • The LV concept is a viable way in the Parameter estimation process • Widely applicable • during fitting, DoE, evaluation • Proven more efficient (due to fewer dimensions) • Superior? Yet to be “proven”...

  20. End Acknowledgements The Swedish Research council for financial support The Competence Centre for Catalysis (KCK) for good collaboration Derek Creaser & Bengt Andersson for fruitful supervision Thank you for your attention!

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