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4th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA2003), April 2003, Nara, JapanAPPLICATION OF INDEPENDENT COMPONENT ANALYSIS TOCHEMICAL REACTIONSS.Triadaphillou, A. J. Morris and E. B. MartinCentre for Process Analytics and Control TechnologySchool of Chemical Engineering and Advanced MaterialsUniversity of Newcastle, Newcastle upon Tyne, NE1 7RU, UKSophia.Triadaphillou@ncl.ac.uk, julian.morris@ncl.ac.uk, e.b.martin@ncl.ac.ukdetailed understanding of the reaction mechanisms.Thus by-products may be produced that are unknownand information on their existence cannot bedetermined using existing spectral interpretationmethods. There is consequently a need for moreadvanced methods for the resolution of mixtures ofchemical reactions.The paper investigates a number of calibration freemethods for the resolution of mixtures. A simulateddata set from a first order reaction was generated fordifferent reaction rates. Techniques includedMultivariate Curve Resolution-Alternating LeastSquares (MCR-ALS) [2] and IndependentComponent Analysis (ICA) [3]. Specifically two ICAalgorithms were investigated, FastICA [4] and JointApproximate Diagonalization of Eigenmatrices(JADE) [5] with the results of JADE being used asan initial estimate in the MCR-ALS algorithm.Applications of ICA have previously been reportedin the areas of voice and sound separation,biomedical signal processing, financial time series,wireless communications and image featureextraction.ABSTRACTThe analysis of kinetic data monitored usingspectroscopic techniques and its resolution into itsunknown components is described. IndependentComponent Analysis (ICA) can be considered acalibration free technique with the outcome of theanalyses being the spectral profiles of the unknownspecies. This enables the realisation of qualitativeinformation concerning the identification of thenumber and type of components present within thereaction mixture over time. The ICA approaches ofFastICA and JADE and the calibration free techniqueof multivariate curve resolution-alternating leastsquares were applied to the mixture spectra of a firstorder synthetic reaction. For all approaches the signalwas successfully separated from the constituentcomponents.1.INTRODUCTIONReaction monitoring is a major challenge across theprocess industries. This form of monitoring typicallyinvolves the measurement and prediction of theconcentration of a number of components in achemical reaction and the determination of thenumber of components in the mixture. Furthermorethe qualitative information extracted from thespectral analysis is of importance in terms of processunderstanding.Traditionally calibration models are built to predictthe property of interest. However the success of theapproach depends on a number of factors.Calibration modelling is time consuming with thefinal model being sensitive to changes in processconditions. In addition it only provides quantitativeinformation about the property of interest with noinformation about side reactions and intermediates[1]. Moreover in industrial processes a number ofunknown components may be present in the mixturebecause of operational disturbances or a lack of2. DETERMINATION OF THE NUMBER OFCOMPONENTSThe first step is the estimation of the number ofcomponents in the mixture. This factor is related tothe amount of variance present as a result of sourcessuch as noise, background and baseline changes. Onemethod that has been applied for the identification ofthe number of latent variables [6] is PrincipalComponent Analysis (PCA). PCA has been widelyapplied for the decomposition of the covariancematrix. A data matrix representing I observations onJ variables can be decomposed as follows:D T VT879(1)

where D is the spectral response, (I J ) , V is theloadings matrix, ( J K ) and T is the scores matrix,( I K ). Each principal component is a linearcombination of the J variables and accounts for themain sources of variation. The current principalcomponent is mutually orthogonal to the set ofprincipal components previously calculated.Evolving Factor Analysis (EFA) [7] is an alternativeapproach to the estimation of the number ofcomponents in a mixture. It uses the idea of thesequential expanding window. A series of spectrafrom a reaction mixture are measured for differentwavelengths and these are arranged into a datamatrix. The order of the spectra provides additionalinformation on the behaviour of a chemical reaction.This is taken into account through the formation ofsub-matrices by adding rows to an initial top submatrix, top down, or by adding rows to an initialbottom sub-matrix, bottom up. The rank of thematrix increases every time a row is added and byanalysing these ranks as a function of the number ofadded rows, time widows are derived where aspecific number of principal components are present.This is the first step in EFA. The number of speciesinvolved is equal to the number of significanteigenvalues of the second moment matrix. Thus asnew absorbing species start to become significant,new factors/eigenvalues evolve. Hence as a newcomponent elutes in the overlapping peak, thepresence of an additional eigenvalue is required toexplain this variability. EFA thus takes advantage ofinformation, which is sometimes unused in the timedomain. In a reaction, the compound that appearsfirst in the spectra should also be the first todisappear.KR ( t , l) c i ( t )s i (l)(2)i 1where c i ( t ) is the concentration of component, i, attime, t, and s i (l) is the spectral response ofcomponent, i, at wavelength, l.An advantage of this method is that knowledge of theconcentration of all the compounds at the beginningof the kinetic process is not required. Tauler et al [9]developed a technique to reconstruct theconcentration profiles in reactions. For this approach,the compound windows were found by connectingthe line of the compound that first appears with theline of the last compound that appears. Both linesare combined in a single figure from which theconcentration windows can be reconstructed. Theseprofiles of the eigenvalues can be considered as afirst rough estimate of the concentration profiles.Based on the Beer Lambert law, the aim of MCRALS is the optimal decomposition of the data matrixD into the product of two smaller matrices, C , thatrelates to the concentrations and S T that denotes thespectral profiles:D CS T E(3)E is the error-related matrix. The decomposition ofdata matrix D is performed by iterative optimization.The error in the raw data set is minimised using thefollowing equations under suitable constraints for Cand S T :min D CS T(4)min D CS T(5)ST3. MULTIVARIATE CURVE RESOLUTIONALTERNATE LEAST SQUARES (MCR-ALS)CMCR-ALS is an iterative resolution methoddeveloped by Tauler. It has been applied to mixturedynamic processes that have been monitored throughspectroscopic techniques and also to other chemicaldata whose instrumental responses obey BeerLambert law. Beer’s law states that the spectralresponse of the components is independent of timeand concentration [8]. In the case of reactionmonitoring, the spectroscopic response, R, is afunction of two variables, the time, t, and the spectralwavelength, l.As a result, a mixture of Kcomponents gives a response:A description of the MCR-ALS algorithm can befound in the work of De Juan et al, [10], [11]4. INDEPENDENT COMPONENT ANALYSISAn alternative calibration free resolution method thatis considered is Independent Component Analysis(ICA). ICA can be used to identify the spectralprofile of each species in a mixture, i.e. to identifythe unknown components. ICA is a method designedto offer a solution to the Blind Source Separationproblem, i.e. separate the source signals from the880

mixture observations. ICA can be considered as anextension to PCA in that while PCA finds principalcomponents that are uncorrelated and that are linearcombinations of the observed variables, ICA isdesigned to extract components that are independentand that constitute the observed variables.Basically an ICA model is a “statistical latentvariable model” in the sense that it describes how theobserved data are generated by a process of mixingthe recorded signals, s i . The signals s i arestatistically mutually independent by definition andare called independent components. The basicproblem is:d i a i1s1 a i 2s 2 a in s n i 1, , nelements of ŝ , i.e. they have non-Gaussiandistributions. Non-gaussianity can be measured byeither kurtosis or negentropy.Two ICA methodologies were evaluated, FastICAand Joint Approximate Diagonalization ofEigenmatrices (JADE). JADE is a cumulative-basedbatch algorithm for source separation. Specifically itis a method that uses higher-order cumulative tensorsthat are a generalisation of the covariance matrix. Forthis family of methods, the fourth-order cumulativetensor is used to make the fourth-order cumulantszero or as small as possible. JADE computes theeigenvalue decomposition of a symmetric matrix.5. EXPERIMENTALGENERATING THE SYNTHETIC DATA SET(6)where d i are the observed random variables that aremodeled as a linear combination of n randomvariables, s i , and a ij , i, j 1, , n are realA first-order synthetic data set was generated byconsidering a starting material A to which a reagentwas added. As a result A is converted to product Bwith a specific rate constant, k1, and B is convertedcoefficients that are assumed to be unknown. It isalso assumed that each mixture d i and eachindependent component s i are random variables andnot true time signals or time series. Equation 6 can berewritten in vector matrix notation:d AsK1(7)k1 0.8k1 0.8k1 0.8where d is a column random vector whose elementsare d i , s is a column random vector whoseelements are si and A is a matrix with elements a ij .k2 0.8k2 0.08k2 0.008Experiment 1Experiment IIExperiment IIIThe data was obtained from the following kineticequations:The statistical estimation problem concentrates ontwo aspects, first under what conditions can themodel be estimated and secondly what can beestimated. In practice both the mixing coefficients,a ij , and the independent components, s i , could be[A]i [A]0 exp( k1t i )[B]iestimated using the observed variables d i . Forsimplicity it is assumed that d is a pre-whitenedvector, i.e. all its components are uncorrelated andtheir variances are equal to unity. An alternative wayto express the ICA model is:sˆ WdK2to product C with reaction rate, k2, i.e. A B C .Several experiments were performed for differentrate constants. For the data set generated from thefirst order synthetic reaction, the reaction rates tookthe values: [A]0 k 1k 2 k1(exp( k 1 t i ) exp( k 2 t i ))[C]i [A]0 [A ]i [B]i(9)(10)(11)where [A ]0 is the initial concentration of A, and[A]i , [B]iand [C]i are the concentrations of A, Band C respectively at time point i.Fig. 2 shows the concentration profiles of the threecomponents for the reaction for experiments I, II andIII respectively. The spectral profiles for thecomponents are the same for all experiments sincethe same data was used in all three experiments andare presented in Fig. 1 (lower right hand-side).Based on the data decomposition, it is expected thatthe pure spectra of the components should be the(8)where ŝ is the estimate of s, d i is the observedrandom variables and W is a separating matrix whichhas to be estimated. W can be defined as the weightmatrix of a two-layered feed-forward network whereŝ is the output and d is the input. The network isconstrained to have statistically independent881