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Outline

Title
Abstract
Figures
Introduction
The Shape of Experimental Data
The Three Basic Data Analytical Questions
Data Inhomogeneities
Results for Example 1 (Cuprum)
Better Approach-Principal Components Analysis
Summary of the Cuprum Application
Three-Way Data and Higher-Way Arrays
The Three Blocks of Spectral Data
Discussion of Carrageenan Example
Summary and Conclusions
References
All Topics
Chemistry
Analytical Chemistry

A chemometrics toolbox based on projections and latent variables

Profile image of Johan TryggJohan Trygg

2014, Journal of Chemometrics

https://doi.org/10.1002/CEM.2581
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15 pages

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Abstract

A personal view is given about the gradual development of projection methods-also called bilinear, latent variable, and more-and their use in chemometrics. We start with the principal components analysis (PCA) being the basis for more elaborate methods for more complex problems such as soft independent modeling of class analogy, partial least squares (PLS), hierarchical PCA and PLS, PLS-discriminant analysis, Orthogonal projection to latent structures (OPLS), OPLS-discriminant analysis and more. From its start around 1970, this development was strongly influenced by Bruce Kowalski and his group in Seattle, and his realization that the multidimensional data profiles emerging from spectrometers, chromatographs, and other electronic instruments, contained interesting information that was not recognized by the current one variable at a time approaches to chemical data analysis. This led to the adoption of what in statistics is called the data analytical approach, often called also the data driven approach, soft modeling, and more. This approach combined with PCA and later PLS, turned out to work very well in the analysis of chemical data. This because of the close correspondence between, on the one hand, the matrix decomposition at the heart of PCA and PLS and, on the other hand, the analogy concept on which so much of chemical theory and experimentation are based. This extends to numerical and conceptual stability and good approximation properties of these models. The development is informally summarized and described and illustrated by a few examples and anecdotes.

Figures (13)
Figure 1. Usually, data can be organized as a table, a matrix, here denoted X, with N rows=observations and K columns=variables. Provided the variation between the observations is moderate, X can be approximated by a bilinear model (latent variable model) with just a few components (A in number). The smaller the variation between the observations, the fewer components are needed to well approximate X. A residual matrix, E (not shown) comprises the difference between the data matrix, X, and the model, that is, E=X-T P’. The interpretation of the model is helped by the layers, tap, , appearing in the order of importance, with the first layer (a= 1) explaining more of the variation in X than the second layer (a=2), and so on. And the expansion is terminated when the components, layers, no longer explain significant parts of X as determined, for instance, by cross-validation (CV). The number
Figure 1. Usually, data can be organized as a table, a matrix, here denoted X, with N rows=observations and K columns=variables. Provided the variation between the observations is moderate, X can be approximated by a bilinear model (latent variable model) with just a few components (A in number). The smaller the variation between the observations, the fewer components are needed to well approximate X. A residual matrix, E (not shown) comprises the difference between the data matrix, X, and the model, that is, E=X-T P’. The interpretation of the model is helped by the layers, tap, , appearing in the order of importance, with the first layer (a= 1) explaining more of the variation in X than the second layer (a=2), and so on. And the expansion is terminated when the components, layers, no longer explain significant parts of X as determined, for instance, by cross-validation (CV). The number
Figure 3. A time series plot of the total analysis index (TAI). Nine sam- ples are marked for reference purposes (see text).
Figure 3. A time series plot of the total analysis index (TAI). Nine sam- ples are marked for reference purposes (see text).
Figure 6. Combined control chart with T2 and DModxX versus time order. Tolerance levels corresponding to 95% and 99% are given for T2, and the critical distance for DModX is plotted. Seven samples are marked for comparison with Figure 3. It is interesting to compare Figures 3 and 6, with the same seven samples (167, 195, 228, 338, 399, 577, and 611) marked. These are, according to the TAI scale, among the most impure copper samples (Figure 3). The combined control charts in Figure 6 also identifies these but differentiates them into two categories. Five samples are abnormal according to T? and two according to DModx. Thus, the former five samples (167, 195, 228, 399, and 577) conform to the dominating correlation struc- ture. This means that their impurity profiles have normal shape but are unacceptably large. The latter (338 and 611), however, break the normal correlation structure and hence indicates an abnormal type of impurity. This information is not available in overall aggregates, such as TAI. Further interpretation of the samples—in the model plane—from the center in the score plot (Figure 4). This can be used to construct elliptic tolerance regions for the scores, as displayed in Figure 4 and as a control chart in Figure 6.
Figure 6. Combined control chart with T2 and DModxX versus time order. Tolerance levels corresponding to 95% and 99% are given for T2, and the critical distance for DModX is plotted. Seven samples are marked for comparison with Figure 3. It is interesting to compare Figures 3 and 6, with the same seven samples (167, 195, 228, 338, 399, 577, and 611) marked. These are, according to the TAI scale, among the most impure copper samples (Figure 3). The combined control charts in Figure 6 also identifies these but differentiates them into two categories. Five samples are abnormal according to T? and two according to DModx. Thus, the former five samples (167, 195, 228, 399, and 577) conform to the dominating correlation struc- ture. This means that their impurity profiles have normal shape but are unacceptably large. The latter (338 and 611), however, break the normal correlation structure and hence indicates an abnormal type of impurity. This information is not available in overall aggregates, such as TAI. Further interpretation of the samples—in the model plane—from the center in the score plot (Figure 4). This can be used to construct elliptic tolerance regions for the scores, as displayed in Figure 4 and as a control chart in Figure 6.
Figure 5. Principal components analysis loading plot corresponding to Figure 4. Each single quality variable gives rise to one plotted point. The interpretation suggests that the first principal component reflects “average” impurity, because all loadings are positive. The second princi- pal component models deviations from this “average” impurity level.
Figure 5. Principal components analysis loading plot corresponding to Figure 4. Each single quality variable gives rise to one plotted point. The interpretation suggests that the first principal component reflects “average” impurity, because all loadings are positive. The second princi- pal component models deviations from this “average” impurity level.
Figure 4. Principal components analysis score plot with t2 versus t1 accounting for 67% of the total variation. Each copper sample is shown as a unique plot mark. Samples 111 and 302 are highlighted for reference purposes. The “acceptable” Hotelling’s T region is indicated by the ellipse (95% tolerance region).
Figure 4. Principal components analysis score plot with t2 versus t1 accounting for 67% of the total variation. Each copper sample is shown as a unique plot mark. Samples 111 and 302 are highlighted for reference purposes. The “acceptable” Hotelling’s T region is indicated by the ellipse (95% tolerance region).
Figure 7. Relationship between measured and estimated lambda proportions for the 102 samples of the training set (left). Partial least squares (PLS) predictions for the 26 samples of the test set (right). The similarity of root mean square error of estimation, root mean square error of cross-validation, and root mean square error of prediction indicates that the PLS model has sound explanatory and predictive powers of the lambda content in the sam- ples. Similar results are obtainable using PLS to predict the remaining four carrageenan constituents (no plots shown). Partial least squares trees (PLS-Tress) are constructed similarly to regression trees [44] with the difference that one searches the score space of X instead of individual X-variables. This allows tree-type models to be applied to complex data matrices with many and collinear, even incomplete, variables [45]. Other To minimize the effect of the baseline variation, the spectral data were preprocessed using standard normal variate transform filtering [41]. Thereafter, the spectral data were Pareto scaled, and the Y-block (comprising the proportions of the five carra- geenan species) was UV-scaled.
Figure 7. Relationship between measured and estimated lambda proportions for the 102 samples of the training set (left). Partial least squares (PLS) predictions for the 26 samples of the test set (right). The similarity of root mean square error of estimation, root mean square error of cross-validation, and root mean square error of prediction indicates that the PLS model has sound explanatory and predictive powers of the lambda content in the sam- ples. Similar results are obtainable using PLS to predict the remaining four carrageenan constituents (no plots shown). Partial least squares trees (PLS-Tress) are constructed similarly to regression trees [44] with the difference that one searches the score space of X instead of individual X-variables. This allows tree-type models to be applied to complex data matrices with many and collinear, even incomplete, variables [45]. Other To minimize the effect of the baseline variation, the spectral data were preprocessed using standard normal variate transform filtering [41]. Thereafter, the spectral data were Pareto scaled, and the Y-block (comprising the proportions of the five carra- geenan species) was UV-scaled.
Figure 8. Hierarchical model structure used when separating Y-predictive and Y-orthogonal spectral variation. J means a predictive component expressing joint information. Ux means an orthogonal-in-X component expressing structure that is unique to X. Uy means an orthogonal-in-Y component expressing structure that is unique to Y. Figure 8 shows that the first base level OPLS model between the NIR block and the Y-block contains four predictive compo- nents expressing the joint information. Additionally, the model contains 2 orthogonal components accounting for systematic structure in the NIR data that is unrelated to the Y-block (the five concentration variables). The corresponding results for the
Figure 8. Hierarchical model structure used when separating Y-predictive and Y-orthogonal spectral variation. J means a predictive component expressing joint information. Ux means an orthogonal-in-X component expressing structure that is unique to X. Uy means an orthogonal-in-Y component expressing structure that is unique to Y. Figure 8 shows that the first base level OPLS model between the NIR block and the Y-block contains four predictive compo- nents expressing the joint information. Additionally, the model contains 2 orthogonal components accounting for systematic structure in the NIR data that is unrelated to the Y-block (the five concentration variables). The corresponding results for the
Figure 9. Scores (left) and loadings (right) of the first two components of the top level OPLS. The design structure of the underlying design is easy to see. In the loading plot, $M15 refers to the near infrared base level model, $M16 to the infrared model, and $M17 to the Raman model.
Figure 9. Scores (left) and loadings (right) of the first two components of the top level OPLS. The design structure of the underlying design is easy to see. In the loading plot, $M15 refers to the near infrared base level model, $M16 to the infrared model, and $M17 to the Raman model.
Figure 10. Score plot of the two orthogonal scores of the lower level near infrared model (upper left). Score plot of the two first orthogonal scores of the lower level infrared model (upper right). Score plot of the two first orthogonal scores of the lower level Raman model (lower left). Score plot of the two scores of the top level PCA model (lower right). The same observations (samples) are marked throughout all four plots.
Figure 10. Score plot of the two orthogonal scores of the lower level near infrared model (upper left). Score plot of the two first orthogonal scores of the lower level infrared model (upper right). Score plot of the two first orthogonal scores of the lower level Raman model (lower left). Score plot of the two scores of the top level PCA model (lower right). The same observations (samples) are marked throughout all four plots.
Figure 11. Score line plot of t1 of the hierarchical principal components analysis model. Systematic shifts between the different sampling days are obvious.
Figure 11. Score line plot of t1 of the hierarchical principal components analysis model. Systematic shifts between the different sampling days are obvious.
Figure 12. Group contribution plot between the day 3 and day 4 samples. This plot confirms that the “jump” seen in t1 arises because of anomalies in the near infrared and infrared spectra.
Figure 12. Group contribution plot between the day 3 and day 4 samples. This plot confirms that the “jump” seen in t1 arises because of anomalies in the near infrared and infrared spectra.
Figure 13. Line plot of po1—the loading of the first orthogonal-in-X component—of the lower level near infrared model. This plot suggests the region around 1970 nm to contain a large fraction of the noncorrelating variation.
Figure 13. Line plot of po1—the loading of the first orthogonal-in-X component—of the lower level near infrared model. This plot suggests the region around 1970 nm to contain a large fraction of the noncorrelating variation.

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References (55)

  1. Kowalski BR, Bender CF. Pattern recognition. 1. A powerful approach to interpreting chemical data. J. Am. Chem. Soc. 1972; 94: 5632-5638.
  2. Kowalski BR, Bender CF. Pattern recognition. 11. Linear and nonlinear methods for displaying chemical data. J. Am. Chem. Soc. 1973; 95: 686-692.
  3. Jöreskog KG. Herman Wold, ed.s, systems under indirect observation, parts I and II. Cartigny proceedings, North-Holland, Amsterdam, 1982.
  4. Mathematics and statistics in chemistry, proceedings NATO Adv. Study inst. On Chemometrics, Cosenza, Italy, Sept. 1983. (BR Kowalski, Ed.), D.Reidel, Dordrecht, The Netherlands, 1984.
  5. Geladi P, Kowalski B. Partial least-squares regression-a tutorial. Anal. Chim. Acta 1986; 185: 1-17.
  6. Gerlach RW, Kowalski BR, Wold, H. Partial Least Squares modeling with latent variables. Anal. Chim. Acta 1979; 112: 417-21.
  7. Hofstadter D, Sander E. The forgotten fuel of our minds. New Sci. 2013; 218: 30-33.
  8. Wold S, Martens H, Wold H. The multivariate calibration problem in chemistry solved by the PLS method, matrix pencils. Lect. Notes Math. (Springer Verlag) 1983; 973: 286-293.
  9. Spiegelman CH, McShane MJ, Goetz MJ, Motamedi M, Yue QL, Coté GL. Theoretical justification of wavelength selection in PLS calibration: development of a new algorithm. Anal. Chem. 1998; 70(1):35-44.
  10. Horst P. Factor Analysis of Data Matrices. Holt, Rinehart, and Winston, Inc., New York, 1965.
  11. Wold H. Estimation of principal components and related models by iterative least squares. In Multivariate Analysis, Krishnaiah PR (eds.). Academic Press: New York, 1966; 391-420.
  12. Andersson M. A comparison of nine PLS1 algorithms. J. Chemometr. 2009; 23: 518-529.
  13. Wold S. Cross validatory estimation of the number of components in factor and principal components models. Technometrics 1978; 20: 397-405.
  14. Wiklund S, Nilsson D, Eriksson L, Sjöström M, Wold S, Faber K. A randomization test for PLS component selection, Svante Wold special issue. J. Chemometr. 2007; 21: 427-439.
  15. Martens H, Naes T. Multivariate calibration H. Martens, T. Naes, Multivariate Calibration. Wiley: New York, 1989. ISBN 0 471 90979 3
  16. Kvalheim OM. Latent-structure decompositions (projections) of multivariate data. Chemom. Intell. Lab. Syst. 1987; 2: 283-290.
  17. Kvalheim OM, Karstang TV. Interpretation of latent-variable regres- sion models. Chemom. Intell. Lab. Syst. 1989; 7: 39-51.
  18. Palm JH. The computation of stereographic projections for non- cubic crystals. Z. Kristallogr. 1966; 123: 388-390.
  19. Wold S. Chemica Scripta: A theoretical foundation of extrather- modynamic relationships (linear free energy relationships). Chemica Scripta 1974; 5: 97-106.
  20. Figure 13. Line plot of po1-the loading of the first orthogonal-in-X component-of the lower level near infrared model. This plot suggests the region around 1970 nm to contain a large fraction of the noncorrelating variation.
  21. Hammett LP. The effect of structure upon the reactions of organic com- pounds, benzene derivatives. J. Am. Chem. Soc. 1937; 59: 96-103.
  22. Tauler R, Smilde A, Kowalski BR. Selectivity, local rank, three-way data analysis and ambiguity in multivariate curve resolution. J. Chemometr. 1995; 9: 31-58.
  23. Tauler R, Kowalski B. Multivariate curve resolution applied to spectral data from multiple runs of an industrial process. Anal. Chem. 1993; 65: 2040-2047.
  24. Trygg J. Prediction and spectral profile estimation in multivariate calibration. J. Chemometr. 2004; 18: 166-172.
  25. Wakeling IN, Macfie HJF. A robust PLS procedure. J. Chemometr. 1992; 6: 189-198.
  26. Wikström C, Albano C, Eriksson L, Friden H, Johansson E, Nordahl Å, Rännar S, Sandberg M, Kettaneh-Wold N, Wold S. Multivariate process and quality monitoring applied to an electrolysis process. Part I: Process supervision with multivariate control charts. Chemom. Intell. Lab. Syst. 1998; 42: 221-231.
  27. Miller P, Swanson RE, Heckler C. Contribution plots: a missing link in multivariate quality control. App. Math. Comp. Sci. 1998; 8: 775-792. ISSN 0867-857X
  28. Wold S. Pattern recognition by means of disjoint principal compo- nent models. Pattern recognition 1976; 8: 127-139.
  29. Hinz DC. Process analytical technologies in the pharmaceutical industry: the FDA's PAT initiative. Anal. Bioanal. Chem. 2006; 384: 1036-1042.
  30. Kourti T, MacGregor JF. Multivariate SPC methods for process and product monitoring. J. Qual. Tech. 1996; 28: 409-428.
  31. Wise BM, Veltkamp DJ, Davis B, Ricker NL, Kowalski BR. "Principal components analysis for monitoring the west valley liquid fed ce- ramic melter," Waste Management '88 Proceedings, pps. 811-818, Tucson AZ 1988.
  32. Hassani S, Martens H. El Mostafa Qannari, Achim Kohler, degrees of freedom estimation in principal component analysis and consensus principal component analysis. Chemom. Intell. Lab. Syst. 2012; 118: 246-259.
  33. Kolda TG, Orthogonal tensor decompositions. SIAM J. Matrix Anal. Appl. 2001; 23(1):243-255.
  34. Geladi P, Wold S, Esbensen K. Image-analysis and chemical infor- mation in images. Anal. Chim. Acta 1986; 191: 473-480.
  35. Hoerl AE, Kennard RW. Ridge regression: biased estimation for nonorthogonal problems. Technometrics 1970; 12: 55-67.
  36. Wold S, Ruhe A, Wold H, Dunn WJ. The collinearity problem in linear- regression, the partial least squares (PLS) approach to generalized inverses. SIAM J. Sci. Stat. Comp. 1984; 5: 735-743.
  37. Wold S, Albano C, Dunn III WJ, Esbensen K, Hellberg S, Johansson E, Sjöström M. Pattern recognition: finding and using regularities in multi-variate data (H.Martens and H Russwurm Jr, Ed.s). Proc. IUFOST Conf. Food Research and Data Analysis, 1983, pp. 147-188.
  38. Sjöström M, Wold S, Lindberg W, Persson JA, Martens H. A multivar- iate calibration problem in analytical chemistry solved by partial least squares models in latent variables. Anal. Chim. Acta. 1983; 150: 61-70.
  39. Trygg J, Wold S. Orthogonal projections to latent structures (O-PLS). J. Chemometr. 2002; 16: 119-128.
  40. de Jong S. SIMPLS: an alternative approach to partial least squares regression. Chemom. Intell. Lab. Syst. 1993; 18: 251-263.
  41. Dyrby M, Petersen RV, Larsen J, Rudolf B, Norgaard L, Engelsen SB, Towards on-line monitoring of the composition of commercial carrageenan powders. Carbohydr. Polym. 2004; 57: 337-348.
  42. Barnes RJ, Dhanoa MS, Lister SJ, Standard normal variate transforma- tion and de-trending of near-infrared diffuse reflectance spectra, Appl. Spectrosc., 1989, 43: 772-777.
  43. Sjöström M, Wold S, Söderström B. PLS Discriminant Plots in Pattern Recognition in Practice II. Elsevier Science Publ. B. V: Holland, 1986; 461-470.
  44. Bylesjö B, Rantalainen M, Cloarec O, Nicholson JK, Holmes E, Trygg J. OPLS discriminant analysis, combining the strengths of PLS-DA and SIMCA classification. J. Chemometr. 2007; 20: 341-351.
  45. Eriksson L, Trygg J, Wold S. PLS-Trees, a top-down clustering approach. J. Chemometr. 2009; 23: 569-580.
  46. Wold S, Sjöström M, Eriksson L. PLS in Chemistry, The Encyclopedia of Computational Chemistry. Wiley: Chichester, 1999; 2006-2020.
  47. Macgregor JF. Online statistical process control. Chem. Eng. Prog. 1988; 84: 21-31.
  48. Eriksson L, Toft M, Johansson E, Wold S, Trygg J. Separating Y- predictive and Y-orthogonal variation in multi-block spectral data. J. Chemom. 2006; 20(8-10):352-361.
  49. Wold S, Kettaneh N, Tjessem K. Journal of Chemometrics, Hierarchi- cal multiblock PLS and PC models for easier model interpretation and as an alternative to variable selection. 1996, 10, 463-482.
  50. Fisher RA. The Design of Experiments. Oliver and Boyd: Edinburgh, 1935. ISBN 0-02-844690-9
  51. Box GEP, Hunter WG, Hunter JS. Statistics for Experimenters: An Intro- duction to Design, Data Analysis, and Model Building. Wiley, 1978. ISBN 978-0471093152
  52. Wold S, Sjöström M, Carlson R, Lundstedt T, Hellberg S, Skagerberg B, Wikström C, and Öhman J. Multivariate Design. Analytica. Chemica. Acta. 1986; 191: 17-32.
  53. Carlson R. Design and Optimization in Organic Synthesis. Elsevier, 1991. ISBN: 978-0-444-89201-0
  54. Katkevica D, Trapencieris P, Boman A, Kalvins I, Lundstedt T. The Nenitzescu reaction: an initial screening of experimental conditions for improvement of the yield of a model reaction. J. Chemometr. 2004; 18: 183-187.
  55. Tukey JW. Exploratory Data Analysis. Addison-Wesley, 1977. ISBN 0-201-07616-0

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Novel manipulations of the well-established multivariate calibration models namely; partial least square regression (PLSR) and support vector regression (SVR) are introduced in the presented comparative study. Two preprocessing methods comprising first derivatization and orthogonal projection to latent structures (OPLS) are implemented prior to modeling with PLSR and SVR. Quantitative determination of pyridostigmine bromide (PR) in existence of its two associated substances; impurity a (IMP A) and impurity b (IMP B); was utilized as a case study for achieving comparison. A series consisting of 16 mixtures with numerous percentages of the studied compounds was applied for implementation of a 3 factor 4 level experimental design. Additionally, a series consisting of 9 mixtures was employed in an independent test to verify the predictive power of the suggested models. Significant improvement of predictive abilities of the two studied chemometric models was attained via implementation of OPLS processing method. The root mean square error of prediction RMSEP for the test set mixtures was employed as a key comparison tool. About PLSR model, RMSEP was found 0.5283 without preprocessing method, 1.1750 when first derivative data was used and 0.2890 when OPLS preprocessing method was applied. With regard to SVR model, RMSEP was found 0.2173 without preprocessing method, 0.3516 when first derivative data was used and 0.1819 when OPLS preprocessing method was applied.

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Chemometrics: theory and application
Wagner Freitas

1977

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Exploration of multivariate chemical data by projection pursuit
Philip Hopke

Chemometrics and Intelligent Laboratory Systems, 1992

Glover, D.M. and Hopke, P.K., 1992. Exploration of multivariate chemical data by projection pursuit.

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PCA: The Basic Building Block of Chemometrics
Christophe Cordella

Analytical Chemistry, 2012

Famous for his contributions to the foundation of modern statistics (2 test, linear regression, principal components analysis), Karl Pearson (1857-1936), English mathematician, taught mathematics at London University from 1884 to 1933 and was the editor of The Annals of Eugenics (1925 to 1936). As a good disciple of Francis Galton (1822-1911), Pearson continues the statistical work of his mentor which leads him to lay the foundation for calculating correlations (on the basis of principal component analysis) and will mark the beginning of a new science of biometrics. 2 Harold Hotelling: American economist and statistician (1895-1973). Professor of Economics at Columbia University, he is responsible for significant contributions in statistics in the first half of the 20th century, such as the calculation of variation, the production functions based on profit maximization, the using the t distribution for the validation of assumptions that lead to the calculation of the confidence interval.

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Principal Component Analysis - Multidisciplinary Applications
Erica Nascimento
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