PRINCIPAL COMPONENT ANALYSIS(PCA) EOFs and Principle Components; Selection Rules

1. LECTURE 8 PRINCIPAL COMPONENT ANALYSIS(PCA) EOFs and Principle Components; Selection Rules Supplementary Readings: Wilks, chapters 9

2. WE’LL START OUT WITH AN EXAMPLE: 20th GLOBAL SURFACE TEMPERATURE RECORD

3. Surface Temperature Changes Climatic Research Unit (‘CRU’), University of East Anglia

4. EOF #1 EOFs for the five leading eigenvectors of the global temperature data from 1902-1980. The gridpoint areal weighting factor used in the PCA procedure has been removed from the EOFs so that relative temperature anomalies can be inferred from the patterns. 12% (88%) EOF #2 6% (3%) EOF #3 5% (1%) EOF #4 4% (1%) EOF #5 3% (0.5%)

5. FILTERING THROUGH PCA SURFACE TEMPERATURE RECORD FILTERED BY RETAINING PROJECTION ONTO WITH FIRST FIVE EIGENVECTORS

6. GLOBAL TEMPERATURE TREND PC #1 EOF #1

7. EL NINO/SOUTHERN OSCILLATION (ENSO) EOF #2 PC #2 Multivariate ENSO Index (“MEI”)

8. NORTH ATLANTIC OSCILLATION PC #3 EOF #3

9. NORTH ATLANTIC OSCILLATION PC #3 EOF #3

10. TROPICAL ATLANTIC “DIPOLE” PC #3 EOF #3

11. ATLANTIC MULTIDECADAL OSCILLATION PC #5 EOF #5

12. ATLANTIC MULTIDECADAL OSCILLATION PC #5 EOF #5

13. ATLANTIC MULTIDECADAL OSCILLATION PC #5 EOF #5

14. PCA as an SVD on the Data Matrix X

15. Recall from our earlier lecture the variance-covariance matrix A in the multivariate regression problem: The eigenvectors of A comprise an orthogonal predictor set (Principal Components Regression)

16. Let us return to the data matrix, (assume it has zero mean) Assume M>N (overdetermined; greater number of “equations” than “unknowns”) We can write Where U,V are unitary matrices (orthogonal matrices if X is real-valued), U is MxN, S is diagonal NxN, and V is NxN Singular Value Decomposition (SVD)

17. Typically, we are interested in the case N>M. A revisedoverdetermined problem can be obtained by redefining the problem: We can then write Where U, V are unitary matrices (orthogonal matrices if X is real-valued), U is NxM, S is diagonal MxM, and V is MxM Singular Value Decomposition (SVD)

19. V is a unitary matrix which diagonalizes XXT! Thus, S2 contains the eigenvalues of XXT There is a mathematical equivalence between taking the Singular Value Decomposition (SVD) of X, and finding the eigenvectors ofA=XXT

20. U contains as its columns the temporal patterns or Principal Components (“PC”s) corresponding to the M eigenvalues, which are the “right eigenvectors” of the SVD: V contains the as its columns the Spatial Pattern or Empirical Orthogonal Function (“EOF”) corrresponding to the M eigenvalues, which are the “left eigenvectors” of the SVD:

21. FILTERING WITH EIGENVECTORS We can filter the original data with a subset of M* eigenvectors:

22. Some Additional Considerations: • Standardization & Areal Weighting • Gappy Data • Frequency domain • “Rotation” • Selection Rules

23. SELECTION RULES How many eigenvectors do we consider significant? There is no uniquely defensible criterion... • Eigenvalue > 1/M • Break in slope in eigenvalue spectrum (“Scree” test) or log eigenvalue (“LEV”) spectrum • Eigenvalue lies outside expected distribution for M uncorrelated Gaussian time series of length N (Preisendorfer Rule N). This is an example of a Monte Carlo method • Rule N’ (take into account serial correlation)

24. SELECTION RULES Preisdendorfer Rule N

25. SELECTION RULES Asymptotic results of Preisendorfer Rule N for large sample size (N,M>100 or so) b=N/M

26. MATLAB EXAMPLE: NORTH ATLANTIC SEA LEVEL PRESSURE DATA 1899-1999