Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic - PowerPoint PPT Presentation

Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data ETH Zurich, March 23 rd 2009 Julien Billeter julien.billeter@chem.ethz.ch ETH Zürich, Institute for Chemical and Bioengineering, Safety and Environmental Technology Group, Zürich, Switzerland. Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Content Part 1 Chemometrics and kinetic hard-modelling Part 2 Uncertainties and error propagation Part 3 Rank deficiency and spectral validation Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Part 1 Chemometrics and kinetic hard-modelling Part 2 Uncertainties and error propagation Part 3 Rank deficiency and spectral validation Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

What is Chemometrics? � Chemometrics is the chemical discipline that uses mathematical and statistical methods to: � (a) design or select optimal measurement procedures and experiments, and � (b) provide maximum chemical information by analyzing chemical data Matthias Otto (2007) From: Chemometrics – Statistics and Computer Application in Analytical Chemistry Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Beer’s law nw nw nw ns nt nt ns nt x + A R Y C = ( nt x nw ) ( ns x nw ) ( nt x ns ) ( nt x nw ) x = + + ⎯⎯ → k A B C 1 + ⎯⎯ → k A C D 2 Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Direct fitting = Y C A Beer’s law Modelled part = = C function ( ) k A function ( ) θ modelled modelled Linear counter part + + = = � Implicit calibration A C Y [1] C YA [2] modelled modelled + + = = A ( C Y ) � Explicit calibration C ( YA ) calibration calibration 2 2 − − Least squares fitting min Y CA min Y C A modelled modelled θ k 2 − + = + = min C C ( ) − − ( ) 1 1 T T T T [1] C C C C modelled [2] A A AA k Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Indirect (inverse) fitting = = C Y B A B Y Inverse model C A Modelled part = = C function ( ) k A function ( ) θ modelled modelled Linear counter part + = + = � Implicit calibration B Y C B A Y C modelled A modelled + + = = B ( Y C ) � Explicit calibration B ( AY ) C calibration A calibration 2 2 − − Least squares fitting min A B Y min C YB modelled A modelled C θ k 2 − min C C modelled k Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Modelling concentrations (C) or spectra (A) ? Modelling C Modelling A Based on a first-principles model Based on the modelling of (the kinetic rate law) A using Gaussian functions Depends on a limited number Depends on a large number of of kinetic parameters, parameters θ , which are e.g. rate constants k (1 x nr ) difficult to determine Requires a subsequent modelling of C to determine kinetic parameters, e.g. rate constants k (1 x nr ) Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Fitting method used in Parts 2 and 3 � Direct fitting by modelling concentrations ( C ) 2 ( ) − min Y C k A modelled k � using implicit and explicit calibration of the pure component spectra ( A ) ( ) + = � Implicit calibration: A C k Y modelled (i.e. calibration-free) ( ) + = � Explicit calibration: A C Y calibration (also called Strategy 2) Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Modelling concentration profiles using the rate law A B C D A B C D A B C D ⎧ ⎪ + ⎯⎯ → k A B C -1 -1 1 0 1 1 0 0 1 0 0 1 0 ⎪ - [ ] = ⎨ = k k k 1 0 1 0 -1 0 -1 1 0 0 0 1 ⎪ + 1 2 ⎯⎯ → k A C D ⎪ 2 ⎩ P ( nr x ns ) E ( nr x ns ) N ( nr x ns ) nr = 2 kinetic rate laws ns = 4 concentration profiles (System of ODE) ⎧ ⎪ ⎧ d x ⎪ d c d x d x d x d x ⎪ ⎪ = e e e e = t,1 1 1 = + =− ⋅ − ⋅ ⎪ k c c c c k c c t, A t,1 t,2 t,1 t,2 1,1 1,2 1,3 1,4 n n 1 1 ⎪ ⎪ 1 t, A t, B t, C t, D 1 t, A t, B ⎪ ⎪ 1,1 2,1 d t d t d t d t d t d t ⎪ ⎨ ⎪ d c d x d x d x d x ⎪ ⎪ d x ⎪ = + =− ⋅ + ⋅ t, B t,1 t,2 t,1 t,2 ⎪ n n 1 0 = e e e e = ⎪ t,2 1 1 k c c c c k c c ⎪ 2,1 2,2 2,3 2,4 ⎪ 1,2 2,2 ⎪ d t d t d t d t d t 2 t, A t, B t, C t, D 2 t, A t, C ⎨ ⎪ ⎩ d t ⎪ d c d x d x d x d x ⎪ = + = ⋅ − ⋅ t, C t,1 t,2 t,1 t,2 n n 1 1 ⎪ ⎪ 1,3 2,3 Numerical integration d t d t d t d t d t ⎪ ⎪ d c d x d x d x d x ⎡ ⎤ = ⎣ ⎪ C c c c c = + = ⋅ + ⋅ t, D t,1 t,2 t,1 t,2 ⎦ ⎪ n n 0 1 t, A t, B t, C t, D ⎪ 1,4 2,4 ⎪ ⎩ d t d t d t d t d t The System of ODE is integrated with initial ⎡ ⎤ = ⎣ concentrations c c c c c ⎦ 0 0, A 0, B 0, C 0, D Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Modelling concentration profiles using the rate law A B C D A B C D A B C D ⎧ ⎪ + ⎯⎯ → k A B C -1 -1 1 0 1 1 0 0 1 0 0 1 0 ⎪ - [ ] = ⎨ = k k k 1 0 1 0 -1 0 -1 1 0 0 0 1 ⎪ + 1 2 ⎯⎯ → k A C D ⎪ 2 ⎩ P ( nr x ns ) E ( nr x ns ) N ( nr x ns ) nr = 2 kinetic rate laws ns = 4 concentration profiles (System of ODE) ⎧ ⎪ ⎧ d x ⎪ d c d x d x d x d x ⎪ ⎪ = e e e e = t,1 1 1 = + =− ⋅ − ⋅ ⎪ k c c c c k c c t, A t,1 t,2 t,1 t,2 1,1 1,2 1,3 1,4 n n 1 1 ⎪ ⎪ 1 t, A t, B t, C t, D 1 t, A t, B ⎪ ⎪ 1,1 2,1 d t d t d t d t d t d t ⎪ ⎨ ⎪ d c d x d x d x d x ⎪ ⎪ d x ⎪ = + =− ⋅ + ⋅ t, B t,1 t,2 t,1 t,2 ⎪ n n 1 0 = e e e e = ⎪ t,2 1 1 k c c c c k c c ⎪ 2,1 2,2 2,3 2,4 ⎪ 1,2 2,2 ⎪ d t d t d t d t d t 2 t, A t, B t, C t, D 2 t, A t, C ⎨ ⎪ ⎩ d t ⎪ d c d x d x d x d x ⎪ = + = ⋅ − ⋅ t, C t,1 t,2 t,1 t,2 n n 1 1 ⎪ ⎪ 1,3 2,3 Numerical integration d t d t d t d t d t ⎪ ⎪ d c d x d x d x d x ⎡ ⎤ = ⎣ ⎪ C c c c c = + = ⋅ + ⋅ t, D t,1 t,2 t,1 t,2 ⎦ ⎪ n n 0 1 t, A t, B t, C t, D ⎪ 1,4 2,4 ⎪ ⎩ d t d t d t d t d t The System of ODE is integrated with initial ⎡ ⎤ = ⎣ concentrations c c c c c ⎦ 0 0, A 0, B 0, C 0, D Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data

Modelling concentration profiles using the rate law A B C D A B C D A B C D ⎧ ⎪ + ⎯⎯ → k A B C -1 -1 1 0 1 1 0 0 1 0 0 1 0 ⎪ - [ ] = ⎨ = k k k 1 0 1 0 -1 0 -1 1 0 0 0 1 ⎪ + 1 2 ⎯⎯ → k A C D ⎪ 2 ⎩ P ( nr x ns ) E ( nr x ns ) N ( nr x ns ) nr = 2 kinetic rate laws ns = 4 concentration profiles (System of ODE) ⎧ ⎪ ⎧ d x ⎪ d c d x d x d x d x ⎪ ⎪ = e e e e = t,1 1 1 = + =− ⋅ − ⋅ ⎪ k c c c c k c c t, A t,1 t,2 t,1 t,2 1,1 1,2 1,3 1,4 n n 1 1 ⎪ ⎪ 1 t, A t, B t, C t, D 1 t, A t, B ⎪ ⎪ 1,1 2,1 d t d t d t d t d t d t ⎪ ⎨ ⎪ d c d x d x d x d x ⎪ ⎪ d x ⎪ = + =− ⋅ + ⋅ t, B t,1 t,2 t,1 t,2 ⎪ n n 1 0 = e e e e = ⎪ t,2 1 1 k c c c c k c c ⎪ 2,1 2,2 2,3 2,4 ⎪ 1,2 2,2 ⎪ d t d t d t d t d t 2 t, A t, B t, C t, D 2 t, A t, C ⎨ ⎪ ⎩ d t ⎪ d c d x d x d x d x ⎪ = + = ⋅ − ⋅ t, C t,1 t,2 t,1 t,2 n n 1 1 ⎪ ⎪ 1,3 2,3 Numerical integration d t d t d t d t d t ⎪ ⎪ d c d x d x d x d x ⎡ ⎤ = ⎣ ⎪ C c c c c = + = ⋅ + ⋅ t, D t,1 t,2 t,1 t,2 ⎦ ⎪ n n 0 1 t, A t, B t, C t, D ⎪ 1,4 2,4 ⎪ ⎩ d t d t d t d t d t The System of ODE is integrated with initial ⎡ ⎤ = ⎣ concentrations c c c c c ⎦ 0 0, A 0, B 0, C 0, D Chemometric Methods for the Kinetic Hard-modelling of Spectroscopic Data