1 / 34

PCA for analysis of complex multivariate data

PCA for analysis of complex multivariate data. 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

Download Presentation

PCA for analysis of complex multivariate data

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related