1. PATTERN RECOGNITION : PRINCIPAL COMPONENTS ANALYSIS Richard Brereton r.g.brereton@bris.ac.uk

2. NEED FOR PATTERN RECOGNITION Exploratory data analysis e.g. PCA Unsupervised pattern recognition e.g. Cluster analysis Supervised pattern recognition e.g. Classification

3. Case study Coupled chromatography in HPLC : profile

4. Time : rows Wavelength : columns MULTIVARIATE DATA

5. DATA MATRICES The rows do not need to correspond to elution times in chromatography they can be any type of sample Blood sample Wood Chromatograms Samples from a reaction mixture Chromatographic columns

6. The loadings do not need to correspond to spectral wavelengths they can be any type of sample NMR peak heights Atomic spectroscopy measurements of elements Chromatographic intensities Concentrations of compounds in a mixture Results of chromatographic tests

7. Return to example of chromatography. Rows : elution times Columns : wavelengths

8. Chemical factors : X = C.S + E

9. It would be nice to look at the chemical factors underlying the chromatogram. We can use mathematical methods to do this.

10. ABSTRACT FACTORS : PRINCIPAL COMPONENTS

11. X = T. P + E = C. S + E T are called scores: these correspond to elution profile P are called loadings : these correspond to spectra Ideally the “size” of T and P equals the number of compounds in the mixture. This “size” equals the number of principal components, e.g. 1, 2, 3 etc. Each PC has an associated scores vector (column of T), and loadings vector (column of P).

12. Scores T Data X I J I J A Loadings P A PCA

13. Hence if the original data matrix is dimensions 30  28 (or I  J) (= 30 elution times and 28 wavelengths - or 30 blood samples and 28 compound concentrations - or 30 chromatographic columns and 28 tests) and if the number of PCs is denoted by A, then the dimensions of T will be 30  A, and the dimensions of P will be A  28.

15. A major reason for performing PCA is data simplification. Often datasets are very complex, it is possible to make many measurements, but only a few underlying factors. “See the wood from the trees”. Will look at this in more detail later.

16. SCORES AND LOADINGS HAVE SPECIAL MATHEMATICAL PROPERTIES Scores and loadings are orthogonal. What does this mean? Loadings are normalised. What does this mean?

17. PCA is an abstract concept. Theory. Non-mathematical Spectrum recorded at different concentrations and several wavelengths; wavelength 6 versus 9 : six spectra.

19. Each spectrum becomes ONE POINT IN 2 DIMENSIONAL SPACE (2D = 2 wavelengths) Spectra Fall on a straight line which is the FIRST PRINCIPAL COMPONENT The line has a DIRECTION often called the LOADINGS corresponding to the SPECTRAL CHARACTERISTICS Each spectrum has a DISTANCE along the line often called the SCORES corresponding to CONCENTRATION

20. EXTENSIONS TO THE IDEA 1.Measurement error 2.Several wavelengths 3.Several compounds

21. Best fit straight line - statistics Two PCs - the second relates to the error around the straight Measurement error

22. Several wavelengths Now no longer a point in 2 dimensional space. Typical spectrum. Several thousand wavelengths The number of dimensions equals the number of wavelengths. The spectra still fall (roughly) on a straight line. A point in 1000 dimensional space.

23. Several compounds Two compounds, two wavelengths. A B

24. RANK AND EIGENVALUE How many PCs describe a dataset? Often unknown How many compounds in a series of mixtures? How many sources of pollution? How many compounds in a reaction mixture? Sometimes just statistical concept. Sometimes mixture of physical and chemical factors, e.g. a reaction mixture : compounds, temperature etc.

25. EVERY PRINCIPAL COMPONENT HAS A CORRESPONDING EIGENVALUE The eigenvalue equals the sum of squares of the scores vector for each PC. The more important the PC the bigger the eigenvalue. The sum of squares of the eigenvalues of a matrix should never exceed that of the original matrix. The sum of squares of all significant PCs should approximate to that of the original matrix.

26. RESIDUAL SUM OF SQUARES : decreases as the number of eigenvalues increases. Log eigenvalue versus component number. Cut off? 0 1 2 3 4 5 1234567

27. SEVERAL OTHER APPROACHES FOR THE DETERMINATION OF NUMBER OF EIGENVALUES.

28. SUMMARY SO FAR PCA Principal components – how many? Scores Loadings Eigenvalues

29. GRAPHIC DISPLAY OF PCS SCORES PLOT PC2 VERSUS PC1

30. SCORES AGAINST TIME PC1 AND PC2 VERSUS TIME

31. LOADINGS PLOT PC2 VERSUS PC1

32. FOR REFERENCE : pure spectra

33. LOADINGS AGAINST WAVELENGTH PC1 AND PC2 VERSUS WAVELENGTH

34. BIPLOTS : SUPERIMPOSING SCORES AND LOADINGS PLOTS

35. MANY OTHER PLOTS Not only PC2 versus 1, also PC3 versus 1, PC3 versus 2 etc. 3D PC plots, 3 axes, rotation etc. Loadings and scores sometimes presented as bar graphs, not always a sequential meaning. Plots of eigenvalues against component number

36. DATA SCALING AND PREPROCESSING Influences appearance of plots Column centring – common in traditional statistics Standardisation of columns – subtract mean and divide by standard deviation. If data of different types or absolute scales this is an essential technique Row scaling – to constant total

37. ANOTHER EXAMPLE Grouping of elements from fundamental properties using PCA.

38. Step 1 : standardise the data. Why? On different scales.

39. PERFORM PCA : Choose the first two PCs Scores plot

40. Loadings plot

41. SUMMARY Many types of plot from PCA. Interpretation of the plots. Preprocessing important.