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

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

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

EOFs for the five leading eigenvectors of the global temperature data from 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%) 6% (3%) 5% (1%) 4% (1%) 3% (0.5%) EOF #1 EOF #2 EOF #3 EOF #4 EOF #5

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

GLOBAL TEMPERATURE TREND EOF #1 PC #1

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

NORTH ATLANTIC OSCILLATION EOF #3 PC #3

EOF #3 PC #3 NORTH ATLANTIC OSCILLATION

TROPICAL ATLANTIC “DIPOLE” EOF #3 PC #3

ATLANTIC MULTIDECADAL OSCILLATION EOF #5 PC #5

EOF #5 PC #5 ATLANTIC MULTIDECADAL OSCILLATION

EOF #5 PC #5 ATLANTIC MULTIDECADAL OSCILLATION

PCA as an SVD on the Data Matrix X

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)

Let us return to the data matrix, (assume it has zero mean) 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) Assume M>N (overdetermined; greater number of “equations” than “unknowns”)

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) Typically, we are interested in the case N>M. A revised overdetermined problem can be obtained by redefining the problem:

V is a unitary matrix which diagonalizes XX T ! There is a mathematical equivalence between taking the Singular Value Decomposition (SVD) of X, and finding the eigenvectors of A=XX T Thus, S 2 contains the eigenvalues of XX T

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:

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

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

How many eigenvectors do we consider significant? 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) There is no uniquely defensible criterion... SELECTION RULES

Preisdendorfer Rule N SELECTION RULES

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

MATLAB EXAMPLE: NORTH ATLANTIC SEA LEVEL PRESSURE DATA