PCA for analysis of complex multivariate data

1. PCA for analysis of complex multivariate data

2. Interpretation of large data tables by PCA • In industry, research and finance the amount of data is often very large • Little information is available a priori • There is a need for methods based on few assumptions and which can give a simple and easily understandable overview • Overall broad interpretation • Ideas for further analyses • Generating hypotheses • PCA is such a method!!!!

3. PCA used for • Interpretation • Pre-processing for regression • Classification • SPC • Noise reduction • Pre-processing for other statistical analyses

4. Examples of use in industry • Process monitoring • Sensory analysis (tasting etc.) • Product development and quality control • Rheological measurements • Process prediction • Spectroscopy (NIR and other)

5. Examples of use outside industry • Psychology • Food science • Information retrieval systems • Consumer studies, marketing

6. PCA • Compresses the information • Finds the directions with most variability • Projects the information down on these dimensions • Presents the information in simple plots • Scores plot • Projection of data onto subspace • Loadings plot • Plot of relation between original variables and subspace dimensions

7. Data structure for PCA, data matrix Rows are objects, ”samples” Columns are variables

8. Scatter plots, vectors • Vector x=( x1,x2,…xK) • Can be plotted. If several vectors are plotted it is called a scatter plot

9. X=(x1,x2,x3) x3 x2 x1

10. Principal component analysis Variables Data Matrix X Scores plot PCA Loadings plot Other results Objects

11. X3 PC 1 PC 2 X2 X1 X

12. PCA model Model X=TPT + E The matrix X is modelled as components (systematic effects) plus residuals, E (noise)

13. The main plots • Scores plot • For interpreting relations among samples • Loadings plot • For interpreting relations among variables • Explained variance plot

14. PC2 25% t2 t1 PC1 70% Scores plot/projection (T)

15. pc2 x2 pc1 x1 x3 Loadings plot

16. Loadings plots • Usually 2-dimensional • For spectroscopy and other continuous measurements, 1-dimensional plots are used.

17. Guidelines for how to interpret the plots • Variables which are close have high correlation • Samples which are close are similar • Variables on opposite side of origin have negative correlation • Objects on the right are dominated by variables to the right and so on….

18. Variance pr. component • Sum of the variances of the original x-variables is equal to the sum of the variances of the scores. • We cantalk about variance pr. component and explained variance (in %) pr. component • Can be presented in a cumulative way (or not)

19. Explained variance 100% 50% 1 3 2 No. of components Cumulative plot (in % or absolute units)

20. Explained variance Bar plots can also be used 1.0 0.5 1 2 3 Number of components Non-cumulative plot (in % or absolute units)

21. Sensory analysis of sausages Goals of the analysis • Investigate the possibility of using dairy ingredients in sausages • Type and concentration • Focus on sensory properties • Investigate the interaction of diary ingredients with other ingredients and process parameters • Characterise the differences among the dairy ingredients used in sausages

22. Sensory analysis of sausages • Factorial design in 4 variables • 5 dairy ingredients • Na caseinate • Na caseinate (high viscosity) • Skim milk • Whey protein • Demineralised whey powder • 3 concentration levels • 1%, 3% and 5% • 2 starch levels • 2% and 4% • 2 cooking temperatures • 76 and 82 degrees C. Published: Baardseth et al, J. Food Science.

23. Variables/attributes used • Graininess • Stickiness • Firmness • Juiciness • Fatness • Elasticity • Colour hue • Colour intensity • Whiteness • Meat taste • Off-taste • Rancidity • Smokiness

24. 70%

25. Loadings and scores Scores split up according to ingredient on next slide

26. Na caseinate Below average • Na caseinate (high viscosity) • Skim milk Can also be done using colours Above average • Whey protein • Demineralised whey powder

27. We have got information about • Which samples that are similar • Which variables that are similar or very different • Which samples that are characterised by which variables • Which design variables that are most important for variation • Differences among the ingredients

28. Pre-processing • If variables are in very different units, it may be advantageous to standardise the variables prior to PCA • Xnew=Xold/std(X) for each variable • Be aware of noise!! Can be tested by ANOVA or replicates.

29. Variables of different types Difficult to compare Standard deviations pH Temp Viscosity Water content

30. Pre-processing • In spectroscopy usually not done • Very important if measurements from different instruments are used together

31. Outlier detection • Outliers may always be present • Influence the solution • New information? • Important to detect them

32. Tools for outlier detection • Residuals = • Plot residuals pr. object • Compute sum of squared residuals pr. object • Leverage, distance to mean within space (Mahalanobis distance)

33. x3 PCA plane e Leverage point ”normal samples” x1 x2

34. Validation • Plots, how natural is the solution: Relate to knowledge and design. • Steep increase of explained variance • Can also use cross-validation • Leave out one sample and test on the rest. Repeat for all samples. Compute explained prediction variance.