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Grey modeling approaches to investigate chemical processes

Grey modeling approaches to investigate chemical processes. Romà Tauler 1 and Anna de Juan 2 IIQAB-CSIC 1 , UB 2 Spain E-mail: rtaqam@iiqab.csic.es. Grey modeling approaches to investigate chemical processes.

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Grey modeling approaches to investigate chemical processes

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  1. Grey modeling approaches to investigate chemical processes Romà Tauler1 and Anna de Juan2 IIQAB-CSIC1, UB2 Spain E-mail: rtaqam@iiqab.csic.es

  2. Grey modeling approaches to investigate chemical processes • Introduction to chemical modeling: white (hard), black (soft) and grey modeling in chemistry • Multivariate Curve Resolution as a grey modeling method • Grey modeling applications using MCR-ALS

  3. Modeling approaches Hard ModelingWhite Modeling Models based on Physical/Chemical Laws Soft ModelingBlack Modeling Empirical Models with no knowledge/assumptions about the Physical/chemical laws of the system (usually non-linear) Models with no assumptions about the physical/chemical model but with assumptions about the measurement model (usually multivariate and linear) Soft+Hard Modeling ? Grey Modeling? Mixed Models partially using information about physical/ /chemical laws

  4. Chemical model(variation of compound contribution) s 1 PROCESS Known Too complex Unknown MIXTURE Non-existent = s n S c c 1 n D C s s s 1 2 n = + + ... + c c c 1 2 n D = + + ... + D D1 D2 Dn Chemical multicomponent systems. Structure Measurement model (variation of the instrumental signal) Simple additive linear model (Factor Analysis tools)

  5. Hard (White) Modeling • Data modeling and data fitting in chemical sciences has been traditionally done by hard modeling techniques. • They are based on physical/chemical models which are already known (or assumed, proposed,...) • The parameters of these model are not known and they are estimated by least squares curve fitting • This approach may be also called white modeling and it is valid for well known phenomena and laboratory data, where the variables of the model are under control during the experiments and only the phenomena under study affect the data.

  6. Hard (White) Modeling Find the optimal parameters of the Model , 

  7. Hard (White) Modeling Case 1 Kinetic Systems: Yij = Aij measured absorbances of sample/solution i wavelength j Measurement model assumptions: Chemical Model assumptions: Defining the residuals: Finding the best model and its parameters

  8. Hard (White) Modeling Case 2 Solution Equilibria: Yij = Aij measured absorbances of sample/solution i wavelength j. Measurement model assumptions: Chemical Model assumptions: Defining the residuals: Finding the best model and its parameters

  9. guess parameters, k0 calculate residuals, r(k0) and the sum of squares, ssq yes <  end; display results mp=0 no > mp / 3 mp=0 mp5 calculate Jacobian J calculate shift vector k, and k0 = k0 + k ssqold <> ssq Hard (White) Modeling The Newton-Gauss-Levenberg/Marquardt (NGL/M) algorithm

  10. Soft (Black) Modeling • In soft (black) modeling no physical model is assumed. • In some cases a linear measurement model is assumed (factor analysis methods) • In other cases dependencies among variables and sources of variation are considered to be non linear (neural networks, genetic algorithms, …) • The goal of these methods is the explanation of data variance using the minimal or softer assumptions about data

  11. J J J VT N D E U  I + I I N << I or J Example of Soft (Black) Modeling Factor Analysis/Principal Component Analysis Bilinear Model D = U VT + E Unique solutionsbut without physical meaning Constraints: U orthogonal, VT orthonormal VTin the direction of maximum variance N

  12. Hard (white)- vs. Soft (black)-modelling Pros HM • Well defined behaviour model (useful chemical information). • Unique solutions. • Reduced number of parameters to be optimised (e.g., K, k,..) • Pros SM • No explicit model is required. • Information on the process or signal may be used (constraints). • May help to set or to validate a physicochemical model.

  13. Hard (white)- vs. Soft (black)-modelling Cons HM • The underlying model should be correct and completely known. • No variations other than those related to the model should be present in the data set. Cons SM • Ambiguous solutions. • Does not provide directly physicochemical (kinetic or thermodynamic,...) information.

  14. Hard (white)- vs. Soft (black)-modelling Use HM • The variation of the system is completely described by a reliable physicochemical model. • Clean reaction systems (kinetic or thermodynamic processes) Use SM • The model describing the variation of the data is too complex, unknown or non-existent. • Images. • Chromatographic data. • Macromolecular processes.

  15. Grey (hard+soft) modeling • Mixed systems with hard-modelable and soft-modelable parts are proposed • Hard-model: kinetic process, equilibrium reaction..... • Soft-model: interferent, background, drift, unknown.... • Introducing a hard-model part decreases the ambiguity related to pure soft-modeling methods and gives additional information (parameters). • Introducing a soft-model part, may help to clarify the nature of the physicochemical model and give more reliable results.

  16. Grey modeling approaches to investigate chemical processes • Introduction to chemical modeling: white (hard), black (soft) and grey modeling in chemistry • Multivariate Curve Resolution as a grey modeling method • Grey modeling applications using MCR-ALS

  17. PROCESS The composition changes in a continuous evolutionary manner. E.g. chemical reactions, processes, HPLC-DAD. The composition changes with a non-random pattern variation. E.g. environmental data, spectroscopic images. MIXTURE The composition changes with a random pattern variation. E.g. Series of independent samples. Multivariate Curve Resolution (MCR) A tool to analyse (resolve) changes in composition and response in multicomponent systems. Goal Knowing the identity and contribution of each pure compound (entity) in the process or in the mixture.

  18. Mixed information  tR D Wavelengths Multivariate Curve Resolution Pure component information s1  sn ST c c 1 n C Retention times Pure signals Compound identity Pure concentration profiles Chemical model Process evolution Compound contribution

  19. Multivariate Curve Resolution methods D = CST + E • Investigation of chemical reactions (kinetics, equilibria, …) using multivarite measurements (spectrometric,...) • Industrial processes (blending, syntheses,…). • Macromolecular processes. • Biochemical processes (protein folding). • Spectroscopic images. • Mixture Analysis (in general) • Hyphenated separation techniques (HPLC-DAD, GC-MS, CE-DAD,...). • Environmental data (model of pollution sources) • ……………..

  20. N J J J ST K D E C  I + I I N << I or J Multivariate Curve Resolution Bilinear Model: Factor Analysis Model D = C ST + E Non-unique solutionsbut with physical meaning (rotational/ intensity ambiguities are present) Constraints: C and STnon-negative C or STscaled (normalization, closure) Other constraints (unimodality, local rank, selectivity, previous knowledge... )

  21. Multivariate Curve resolution Alternating Least Squares MCR-ALS Extension to multiple data matrices NC D1 C1 ST NR1 column-, spectra profiles D2 C2 NR2 D3 C3 NR3 row-, concentration profiles NM = 3 D C column-wise augmented data matrix quantitative information Z

  22. Advantages of matrix augmetation(multiway data) • Resolution local rank conditions are achieved in many situations for well designed experiments (unique solutions!) • Rank deficiency problems can be more easily solved • Unique decompositions are easily achieved for trilinear data (trilinear constraints) • Constraints (local rank/selectivity and natural constraints) can be applied independently to each component and to each individual data matrix. J,of Chemometrics 1995, 9, 31-58 J.of Chemometrics and Intell. Lab. Systems, 1995, 30, 133

  23. Multivariate Curve Resolution – Alternating Least Squares (MCR-ALS) Data exploration • Determination of the number of components (i.e. by SVD) • Building of initial estimates (C or ST) Input of external information as CONSTRAINTS • Iterative optimisation of C and/or ST by Alternating Least Squares (ALS) subject to constraints. • Check for satisfactory CST data reproduction. Fit and validation The aim is the optimal description of the experimental data using chemically meaningful pure profiles.

  24. An algorithm for Bilinear Multivariate Curve Resolution Models : Alternating Least Squares (MCR-ALS) C and STare obtainedby solving iteratively the two LS equations: • Optional constraints (local rank, non-negativity, unimodality,closure,…) are applied at each iteration • Initial estimates of C or S are obtained from EFA or from pure variable detection methods.

  25. Constraints Definition Any chemical or mathematical feature obeyed by the profiles of the pure compounds in our data set. • C and ST can be constrained differently. • The profiles within C and S can be constrained differently. Constraints transform resolution algorithms into problem-oriented data analysis tools

  26. C* Cc 0.3 0.35 0.25 0.3 0.2 0.25 0.15 0.2 0.1 0.15 0.05 0.1 0 0.05 -0.05 -0.1 0 0 10 20 30 40 50 0 10 20 30 40 50 Retention times Retention times Concentration profiles spectra Soft constraints Non-negativity

  27. C* Cc 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 Reaction profiles Chromatographic peaks Voltammograms 0 0 5 10 15 20 25 30 35 40 45 50 0 Retention times 0 5 10 15 20 25 30 35 40 45 50 Retention times Soft constraints Unimodality

  28. Soft constraints Concentration selectivity/local rank constraint Selectivity/local rank Cc < threshold values C* 0.35 0.35 0.3 0.3 0.25 0.25 We know that this region is not rank 3, but rank 2! 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 Retention times Retention times

  29. Concentration correlation constraint (multivariate calibration) ST ALS C D Updated Select b, b0 Local model cALS b, b0

  30. Trilinearity Constraint (flexible to every species) Extension of MCR-ALS to multilinear systems D1 D2 D3 Substitution of species profile D Selection of species profile Trilinearity Constraint Unique Solutions! C Folding species profile Unfolding species profile loadings 1st score Loadings give the relative amounts! PCA, SVD 1st score gives the common shape = ST C R.Tauler, I.Marqués and E.Casassas. Journal of Chemometrics, 1998; 12, 55-75

  31.  = ctotal C* Mass balance Cc 0.35 0.35 0.3 0.3 0.25 ctotal 0.25 ctotal 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 pH pH Closed reaction systems Hard modeling constraints Hard modeling: Mass balance or Closure constraint

  32. Physicochemical model 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 pH Hard modeling constraints Hard modeling: Mass action law and rate laws C Ccons Kinetic processes Equilibrium processes pH

  33. Grey modeling using MCR-ALS soft + hard modeling constraints • The hard model is introduced as a new and essential constraint in the soft-modelling resolution process. • It is applied in a flexible manner, as the soft-modelling constraints. • To some or to all process profiles. • To some or to all matrices in a three-way data set. • Different hard models can be applied to different matrices in a three-way data set.

  34. C 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 pH pH Grey modeling using MCR-ALS CSM CHM soft model (non-negativity) physicochemical model (mass action law, rate law) Ccons HM SM Kinetic processes Equilibrium processes

  35. Grey modeling using MCR-ALS • Select the soft-modelled profiles to be constrained (CSM). • Non-linear fit of the selected profiles according to the hard model selected. • Update the soft-modelled profiles CSM.by the fitted CHM. min(ssq(CSM-CHM)) ssq=f(CSM, model, parameters)

  36. Grey modeling approaches to investigate chemical processes • Introduction to chemical modeling: white (hard), black (soft) and grey modeling in chemistry • Multivariate Curve Resolution as a grey modeling method • Grey modeling applications using MCR-ALS

  37. Grey modeling approaches to investigate chemical processes Examples: • Getting kinetic and analytical information from mixed systems (drift and interferents) • Using a physicochemical model to decrease resolution ambiguity and getting analytical information • pH induced transitions in hemoglobin

  38. Kinetic process + drift Model 1 1 C A k1 = k2 = 1 0.5 0.5 Concentration D D Concentration Time a d Kinetic process + interferent B drift 0 0 0 5 10 0 5 10 1 4 Time x 10 4 D 3 0.5 Concentration i i 2 Absorbance interf. B A C 1 0 0 5 10 d 0 Time 0 50 100 Wavelengths Grey modeling applications using MCR-ALS Example 1 Getting kinetic information from mixed systems (drift and interferents) consecutive irreversible Anna de Juan, Marcel Maeder, Manuel MartÍnez, Romà Tauler Analytica Chimica Acta 442 (2001) 337–350;

  39. Kinetic model Kinetic process + drift/interferent A, B, C HM Drift, inter SM CHM = f(k1, k2)

  40. Kinetic process + drift Kinetic process + interferent 4 x 10 a) 1 4 a) x 10 3 0.8 1 3 0.8 0.6 2 Concentration Absorbance 2 0.6 0.4 Concentration (a.u.) Absorptivities (a.u.) 1 0.4 0.2 1 0.2 0 0 0 2 4 6 8 10 0 20 40 60 80 100 Time Wavelength channel 0 0 b) 0 2 4 6 8 10 0 20 40 60 80 100 4 x 10 Time Wavelength channel 1 b) 4 x 10 3 1 4 0.8 0.8 0.6 3 2 Concentration Absorbance 0.6 0.4 2 Concentration (a.u.) 1 0.4 0.2 1 0.2 0 0 0 2 4 6 8 10 0 20 40 60 80 100 0 0 Time Wavelength channel 0 2 4 6 8 10 0 20 40 60 80 100 Time Grey modeling applications using MCR-ALS Example 1 Getting kinetic information from mixed systems (drift and interferents) HM HSM

  41. Anna de Juan, Marcel Maeder, Manuel MartÍnez, Romà Tauler Chemometrics and Intelligent Laboratory Systems 54 2000 123–141

  42. Grey modeling applications using MCR-ALS Example 2. Using a physicochemical model to decrease resolution ambiguity. Getting analytical information. • Chemical problem: multiequilibria systems • Quantitation of an analyte (H2A) in the presence of an interferent (H2B). • Measurements FT-IR monitored pH titrations • H2A (malic acid) • H2B (tartaric acid) Highly overlapped concentration profiles

  43. Data set  Standard H2A pH Sample H2A/H2B Too ambiguous SM solutions Quantitation fails pH Grey modeling applications using MCR-ALS Example 2. Using a physicochemical model to decrease resolution ambiguity. Getting analytical information. Too correlated concentration profiles Too overlapped spectra

  44. 2.5 0.9 0.8 2 Fresh solution 0.7 0.6 1.5 0.5 0.4 1 0.3 0.2 0.5 0.1 0 0 350 400 450 500 550 600 650 700 2 3 4 5 6 7 8 9 10 pH Wavelengths (nm) 1 0.8 After 24 hours 0.6 0.4 0.2 0 350 400 450 500 550 600 650 700 Wavelengths (nm) Grey modeling applications using MCR-ALS Example 3 Time effect on pH induced transitions in hemoglobin Time effect on pH transitions (UV) SM 1,2 Heme group unbound 3 Native 4 Heme bound (change in coordination) 1 SM 0.8 0.6 0.4 0.2 0 3 4 5 6 7 8 9 10 pH

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