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## Douglas N. Rutledge

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**Outer Product Analysis (OPA)studying the relations among**sets of variables measured on the same individuals Douglas N. Rutledge**Some publications on Outer Product Analysis**Infrared spectroscopy and outer product analysis for quantification of fat, nitrogen, and moisture of cocoa powder A. Vesela, A. S. Barros, A. Synytsya, I. Delgadillo, J. Copıkova, M. A. Coimbra Analytica Chimica Acta 601 (2007) 77–86 Multi-way analysis of outer product arrays using PARAFAC D. N. Rutledge, D. Jouan-Rimbaud Bouveresse Chemometrics and Intelligent Laboratory Systems 85 (2007) 170–178 Image processing of outer-product matrices – a new way to classify samples. Examples using visible/NIR/MIR spectral data B. Jaillais, V. Morrin, G. Downey Chemometrics and Intelligent Laboratory Systems xx (2006) xxx–xxx**Some publications on Outer Product Analysis**Variability of cork from Portugese Quercus suber studied by solid state 13C-NMR and FTIR spectroscopies M.H. Lopes, A.S. Barros, C. Pascoal Neto, D. Rutledge, I. Delgadillo, A. M. Gil Biopolymers (Biospectroscopy) 62 (5) (2001) 268–277 Outer Product Analysis of electronic nose and visible spectra: application to the measurement of peach fruit characteristics C. di Natale, M. Zude-Sasse, A. Macagnano, R. Paolesse, B. Herold, A. D'Amico Analytica Chimica Acta 459 (2002) 107–117 Determination of the degree of methylesterification of pectic polysaccharides by FT-IR using an outer product PLS1 regression A.S. Barros, I. Mafra, D. Ferreira, S. Cardoso, A. Reis, J.A. Lopes de Silva, I. Delgadillo, D.N. Rutledge, M.A. Coimbra Carbohydrate Polymers 50 (2002) 85–94**Some publications on Outer Product Analysis**Enhanced multivariate analysis by correlation scaling and fusion of LC/MS and 1H NMR data J. Forshed, R. Stolt, H. Idborg, S. P. Jacobsson Chemometrics and Intelligent Laboratory Systems 85 (2007) 179–185 Outer-product analysis (OPA) using PCA to study the influence of temperature on NIR spectra of water B. Jaillais, R. Pinto, A.S. Barros, D.N. Rutledge Vibrational Spectroscopy 39 (2005) 50–58 Outer-product analysis (OPA) using PLS regression to study the retrogradation of starch B. Jaillais, M.A. Ottenhof, I.A. Farhat, D.N. Rutledge Vibrational Spectroscopy 40 (2006) 10–19**Principal Components Analysis (PCA)**Calculate the covariance matrix, C, of the original data matrix, X Covariance Matrix : C p p Cij = cov(i,j)**xi**1 p p 1 Ci = xiT . xi xiT p p Calculate the matrix of individual covariances between variables of one data set, X • Mutual weighting of each signal by the other: • if intensities simultaneously high in the two domains, the product is higher; • if intensities simultaneously low in the two domains, the product is lower; • if one intensity high and the other low, the product tends to an intermediate value**Calculate all the individual covariance matricesof a single**matrix, X For n samples, one gets n Outer Product matrices Group them together one under the other in the form of a cube of individual matrices of covariances among variables 1 p p p 1 1 1 . . n 1 . n p 1 A cube of symmetrical matrices**Decomposition of the « Mean » OP matrix by SVD≡**Principal Components Analysis Calculate the mean of the individual covariance matrices to have a :matrix of mean covariances p 1 p p p 1 1 1 1 . . n p 1 . n 1 p 1 Decomposition of the column-mean matrix by SVD Principal Components Analysis**SVD applied to the initial data matrix, X**X(n,p) = U(n,r) S(r,r) VT(r,p) S : diagonal matrix of singular values V : loadings matrixU*S : scores matrix**SVD applied the covariance matrix, XTX**or column-means of the Outer Product cube XTX(p,p) = V(p,r) S2(r,r) VT(r,p) S2 : diagonal matrix of eigenvalues V : loadings matrix X*V : scores matrix**Application of « Mean » Outer Product Analysis to real**data (1) Lignin-starch mixtures by TD-NMR D.N. Rutledge, Food Control, (2001) 12(7), 437-445**Decomposition of the matrix by SVD**(Principal Components Analysis) V Loadings on PC2, PC3 & PC4 X*V Scores on PC2, PC3 & PC4**Application of « Mean » Outer Product Analysis to real**data (2) Retrogradation of starch by X-ray diffraction Diffraction Rayons X B. Jaillais, M.A. Ottenhof, I.A. Farhat, D.N. Rutledge, Vib. Spec. (2006), 40, 10–19.**Decomposition of the matrix by SVD**Principal Components Analysis V Loadings on PC2 X*V Scores on PC2**p x p**1 n n-PLS, n-PCA (ANOVA) … « Unfold » Outer Product Analysis Analyse the unfolded individual covariance matrices Unfold the cube to form a matrix p 1 p p 1 1 1 . . n 1 . n p 1**X3**X2 3 x PCA p X1 q n n x q n x p p x q p p n Different data unfolding schemes X**Application of unfolded OP to real data (1)**Lignin-starch mixtures by TD-NMR unfolded OP matrix (X1) p x p n**Decompose the unfolded OP matrix (X1) by SVD**(Unfold-PCA) p p V Refolded Loadings of X1on PC2, PC3 & PC4 U*S Scores of X1 on PC2, PC3 & PC4**Decompose the unfolded OP matrices (X1 & X2) by SVD**(Unfold-PCA) U*S Scores of X2 on PC2, PC3 & PC4 U*S Scores of X1 on PC2, PC3 & PC4**Decomposition of the matrix by SVD**(Column-mean PCA) V Loadings on PC2, PC3 & PC4 X*V Scores on PC2, PC3 & PC4**Decompose the unfolded OP matrices (X1 & X2) by SVD**Unfold-PCA U*S Scores of X2 on PC4 U*S Scores of X1 on PC4**Decompose the unfolded OP matrix (X2) by SVD**Unfold-PCA p n V Refolded Loadings of X2(X3) on PC4 U*S Scores of X2 (X3) on PC4**Application of unfolded OP to real data (2)**Starch retrogradation by XRD unfolded OP matrix (X1) p x p n**Decompose the unfolded OP matrix (X1) by SVD**Unfold-PCA p p V Refolded Loadings of X1 on PC2 U*S Scores of X1 on PC2**Decompose the unfolded OP matrices (X1 & X2) by SVD**Unfold-PCA U*S Scores of X2 on PC2 U*S Scores of X1 on PC2**Decompose the unfolded OP matrix (X2) by SVD**Unfold-PCA p n V Refolded Loadings of X2 on PC2 U*S Scores of X2 on PC2**« Multi-way » Outer Product Analysis**Group them together, one under the other, in the form of a cube of individual matrices of covariances among variables p 1 p p 1 1 1 . . n 1 . n p 1 Decomposition of the cube PARAFAC**PARAFAC – Parallel Factor Analysis**3-way data X (n,q,p) : F is the number of Factors used in the PARAFAC model. This model minimises the sum of squared residuals. p p q q 1 = + + … k n 1 F n n 1 1 q R. Bro, Chemometrics and Intelligent Laboratory Systems, (1997), 38, 149-171**PARAFAC applied to OP cube**Starch retrogradation by XRD Time Loadings on the 1° mode (samples)**Starch Data : PARAFAC Model**Loadings on the 2° mode (XRD) Loadings on the 3° mode (XRD)**Comparaison PARAFAC / SVD**OP-PARAFAC SVD on XTX= PCA**yi**1 q 1,1 1, q 1 Ci = xiT . yi xi p p, q Calculate the matrix of individual covariancesbetween variables of 2 different matrices X & Y**n**n Signal 2 Signal 1 n q p p q Visualisation of the Outer Product cube =**Calculate the matrix of covariances of 2 matrices X & Y**For n samples, one gets n Outer Product matrices Group them together in the form of a “cube” Calculate the column-mean of the individual covariance matrices to give the matrix of covariances between the 2 groups of variables Apply SVD q 1 p p q 1 1 1 1 . . n p 1 . n 1 q 1 n OP (p, q) matrices 1 “cube” (n, p, q) 1 mean matrix (p, q)**Decompose the « Mean » OP matrix by SVD≡ Tucker**Analysis Analyse the links between 2 tables of data, X & Y Singular Value Decomposition of the matrix of covariances between the 2 groups of variables (1/n)XTY That decomposition of the matrix (1/n)XTY corresponds to looking for successive pairs of variables (th = Xah , uh = Ybh ) where : - covariance between thet uh maximal, - axes ah orthogonal - axes bh orthogonal L. Tucker, Psychometrika, (1958), 23, 111-136**SVD applied to the covariance matrix, XTY**or column-means of the Outer Product cube XTY = VXS VYT S: diagonal matrix of singular values VX et VY: X & Y loadings matrices X*VX : scores of X Y*VYT: scores of Y**Application of « Mean » Outer Product Analysis to real**data (3) Complexation between TPP & Cu TD-NMR Vis D.N. Rutledge, A.S. Barros, F. Gaudard, Mag. Res. in Chemistry, 35 (1997), 13–21**SVD on matrix of column-means of Outer Product (Tucker**Analysis) [uRMN, sRMN_Vis, vVis] = svd (meanRMN_Vis,'econ'); uRMN vVis**SVD on matrix of column-means of Outer Product (Tucker**Analysis) sRMN = RMN xuRMN / (uRMN' xuRMN); sVis = Visx vVis / (vVis' x vVis); sRMN sVis**Application of unfolded OP to real data (3)**Complexation between TPP & Cu unfolded OP matrix (X1) p x q n**Decompose the unfolded OP matrix (X1) by SVD**Unfold-PCA q p V Loadings of X1 on PC1 & PC2 U*S Scores of X1 on PC1 & PC2**Decompose the unfolded OP matrices (X1 & X2) by SVD**Unfold-PCA U*S Scores of X2 on PC1 & PC2 U*S Scores of X1 on PC1 & PC2**Decompose the unfolded OP matrix (X2) by SVD**Unfold-PCA p n V Loadings of X2 on PC1 & PC2 U*S Scores of X2 on PC1 & PC2**Decompose the unfolded OP matrices (X1 & X3) by SVD**Unfold-PCA U*S Scores of X3 on PC1 & PC2 U*S Scores of X1 on PC1 & PC2**Decompose the unfolded OP matrix (X3) by SVD**Unfold-PCA q n V Loadings of X3 on PC1 & PC2 U*S Scores of X3 on PC1 & PC2**« Multi-way » Outer Product Analysis**For n samples, one gets n Outer Product matrices Group them together, one under the other, in the form of a cube of individual matrices of covariances among variables q 1 p p 1 1 1 . . n 1 . n q 1 Decomposition of the cube PARAFAC