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MODEGAT 2009-09-14
Chalmers University of Technology
Use of Latent Variables in the Parameter Estimation Process
Jonas Sjöblom
Energy and Environment
Chalmers University of Technology
MODEGAT 2009-09-14
Chalmers University of Technology
NOX Reduction catalysis
BART
Ea
BA Aekr
Introduction
~mm ~µm ~nm
MODEGAT 2009-09-14
Chalmers University of Technology
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
MODEGAT 2009-09-14
Chalmers University of Technology
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)
'ˆ TPXX
x1=dY/d1
x2=dY/d2
x3=dY/d3
p1
p2
What is LV modelling?
MODEGAT 2009-09-14
Chalmers University of Technology
Parameter Estimation Process
How can LV models be applied?
Define model and model assumptions
Define ”experimental space”
Fit parameters
Satisfactory results?Yes!
Evaluate the Design by LV-model (experimental rank)
No!No!
Evaluate the Design (perform experiments)
Choice of experiments to perform
Use of LV modelsUse of LV models
1.
2.
3.
1&2
MODEGAT 2009-09-14
Chalmers University of Technology
Application 1: LV models during the fitting process
• NOX Storage and Reduction (NSR) Mechanism– 62 parameters
• Poor experimental design
• Jacobian f/ used in gradient search– ill-conditioned– Local minima
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
MODEGAT 2009-09-14
Chalmers University of Technology
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:1. Correlation structure !2. Number of independent directions (# parameters to fit) !3. Which parameters to choose ! (method 1)4. Parameter fit in LV space (method 2)
How can LV models be used? -appl.1
MODEGAT 2009-09-14
Chalmers University of Technology
LV example: "loading" plotXY
k02_
k03_k04_
k05_
k06_
k07_
k08_
k09_
k10_
k11_
k12_
k13_
k14_
k15_k16_
k17_k18_
k19_
k20_
k21_
k22_
k23_
k24_
k25_
k26_k27_
k28_
k29_
k30_
k31_
k32_k33_
k34_
k35_
Ea02
Ea03 Ea04Ea05Ea06
Ea07
Ea09Ea10
Ea11
Ea14Ea15Ea16Ea17Ea18
Ea19Ea20 Ea22
Ea23
Ea24Ea25
Ea26Ea27Ea28
Ea29
Ea30
Ea31
Ea32
Ea33
Ea34
Ea35
Eth0Eth3
s-NO
sNO2
sNOx
sCO2
w*c
[2]
-0,30
-0,20
-0,10
0,00
0,10
0,20
0,30
-0,20 -0,10 0,00 0,10 0,20 0,30
w*c[1]
k01_
How can LV models be used? -appl.1
MODEGAT 2009-09-14
Chalmers University of Technology
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
Method I (9 selected parameters)
Method II(fitting of 9 scores)
Method “brute force”(all 62 parameters)
90 90 500
How can LV models be used? -appl.1
MODEGAT 2009-09-14
Chalmers University of Technology
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
*
2*2
*
)(2)(2
)(2)()(
)()(2
)()(
7
8
5
6
3
4
1
2
g
r
rads
ads
r
radsg
ads
r
rg
ads
r
rg
NONO
NOONO
OO
NONO
Ref: Sjoblom et al, Comput . Chem. Eng 32 (2008) 3121-3129
MODEGAT 2009-09-14
Chalmers University of Technology
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
Define model and model assumptions
Define experimental space
D-optimalChoice of experiments to performusing X or T from LV-model
fit, analyze
Satisfactory results?Yes!
Evaluate the Design by LV-model (experimental rank)
No!No!
Evaluate the Design by LV- model
Choice of experiments to perform
MODEGAT 2009-09-14
Chalmers University of Technology
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.
sr
m
r
m
srs JJM '
1 1
Define model and model assumptions
Define “experimental space”
D-optimal Choice of experiments to performusing X or T from LV-model
fit, analyze
Satisfactory results?Yes!
Evaluate the Design by LV-model (experimental rank)
"novel" use of LV-models
No!No!
Evaluate the Design by LV-model
Choice of experiments to perform
How can LV models be applied? -appl.2
MODEGAT 2009-09-14
Chalmers University of Technology
Application 3: Extended Sensitivity Analysis for targeted Model Improvements
• 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
Refs: Creaser et al. Appl.Catal.B 90 (2009) 18-28, Sjöblom PhD Thesis (2009) Chalmers
Thesis available at: http://publications.lib.chalmers.se/records/fulltext/92706.pdf
NO2NO3
CH2
NO2O O
O
C8H18
CO2, H
N2
NO, H2
MODEGAT 2009-09-14
Chalmers University of Technology
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
),,()(*3
),,( tconf
t locallevellevel
adj βysy
β
β
y
y
ββyS
MODEGAT 2009-09-14
Chalmers University of Technology
LV-model and univariate measures
How can LV models be applied? -appl.3
• PCA model– Scores plot
– Loadings plot
– 25 components
• Univariate table data– Confidence intervals
– Sensitivity average, std, max
– Correlations -0,10
-0,05
-0,00
0,05
0,10
-0,10 -0,08 -0,06 -0,04 -0,02 0,00 0,02 0,04 0,06 0,08 0,10
p[2]
p[1]
GasSensitivity-thesis3.M1 (PCA-X), all paramsp[Comp. 1]/p[Comp. 2]Colored according to Var ID (Primary)
R2X[1] = 0,281539 R2X[2] = 0,209535
CO2CO_NO2NO_T__
SIMCA-P 11.5 - 2009-09-08 09:40:37-0,10
-0,05
-0,00
0,05
0,10
-0,10 -0,08 -0,06 -0,04 -0,02 0,00 0,02 0,04 0,06 0,08 0,10
p[2]
p[1]
GasSensitivity-thesis3.M1 (PCA-X), all paramsp[Comp. 1]/p[Comp. 2]Colored according to Var ID (Primary)
R2X[1] = 0,281539 R2X[2] = 0,209535
CO2CO_NO2NO_T__
SIMCA-P 11.5 - 2009-09-08 09:40:37
-20
-10
0
10
20
-50 -40 -30 -20 -10 0 10 20
t[2]
t[1]
GasSensitivity-thesis3.M1 (PCA-X), all paramst[Comp. 1]/t[Comp. 2]Colored according to Obs ID (Primary)
R2X[1] = 0,281539 R2X[2] = 0,209535 Ellipse: Hotelling T2 (0,95)
exp1exp2exp3exp4exp5
SIMCA-P 11.5 - 2009-09-08 09:53:58-20
-10
0
10
20
-50 -40 -30 -20 -10 0 10 20t[2
]
t[1]
GasSensitivity-thesis3.M1 (PCA-X), all paramst[Comp. 1]/t[Comp. 2]Colored according to Obs ID (Primary)
R2X[1] = 0,281539 R2X[2] = 0,209535 Ellipse: Hotelling T2 (0,95)
exp1exp2exp3exp4exp5
SIMCA-P 11.5 - 2009-09-08 09:53:58
MODEGAT 2009-09-14
Chalmers University of Technology
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
MODEGAT 2009-09-14
Chalmers University of Technology
• 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
Why are LV models good?
Factors for successful parameter estimation
Model assumptions
”experimental space”
Fit parameters
Happy?Yes!No!No!
Evaluate the Design
Choice of experiments
New PhD project: “Improved methods for parameter estimation”Advertisement out now! Application dead line 20th sept 2009http://www.chalmers.se/chem/EN/news/vacancies/positions/phd-student-position-in8778
MODEGAT 2009-09-14
Chalmers University of Technology
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?
MODEGAT 2009-09-14
Chalmers University of Technology
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”...
MODEGAT 2009-09-14
Chalmers University of Technology
End
AcknowledgementsThe Swedish Research council for financial
supportThe Competence Centre for Catalysis (KCK)
for good collaborationDerek Creaser & Bengt Andersson for fruitful
supervision
Thank you for your attention!