Empirical Orthogonal Functions

1
Empirical Orthogonal Functions
Andy Jacobson and Brad Holcombe July 2006
2
Variance and Covariance

• Notes
• E() is expectation (mean)
• M is the number of obs
• N is the number of stations (locations with
  data)
• All vectors are arranged in columns unless
  transposed.
• If computing var/cov by hand, you must remove
  the mean of the data at each station.
• Degrees of freedom M are reduced by one because
  of the computation of the mean.
• D is the data matrix
• The spatial dimensions of your input data must
  be unwrapped 2-D grids must be laid out as a
  1-D row in D
• C is the covariance matrix.

3
Eigenvalue Decomposition

• Two covariate time series, x and y.
• Generated from two uncorrelated random number
  sequences by multiplying by a specified
  covariance matrix.

4
Eigenvalue Decomposition

• Notes
• C is the covariance matrix
• E is the matrix of eigenvectors
• ? is the diagonal matrix of eigenvalues

5
Eigenvalue Decomposition

• Notes
• C is the covariance matrix
• E is the matrix of eigenvectors
• ? is the diagonal matrix of eigenvalues

6
Eigenvalue Decomposition

• Notes
• C is the covariance matrix
• E is the matrix of eigenvectors
• ? is the diagonal matrix of eigenvalues

7
Eigenvalue Decomposition

• Notes
• C is the covariance matrix
• E is the matrix of eigenvectors
• ? is the diagonal matrix of eigenvalues

8
Eigenvalue Decomposition
9
Eigenvalue Decomposition

• Notes
• C is the covariance matrix
• E is the matrix of eigenvectors
• ? is the diagonal matrix of eigenvalues

10
Eigenvalue Decomposition

• Notes
• C is the covariance matrix
• E is the matrix of eigenvectors
• ? is the diagonal matrix of eigenvalues

11
Eigenvalue Decomposition

• Notes
• C is the covariance matrix
• E is the matrix of eigenvectors
• ? is the diagonal matrix of eigenvalues

12
EOFs

• Notes
• EOF terminology is not well defined. Conflicting
  definitions are common in the literature.
• The projection of each eigenvector onto the data
  gives a time series of scores.
• A is the matrix of score time series and has
  dimensions M x N
• Principal components often refers to the
  scores.
• PCA, however, is sometimes taken to mean an
  eigen decomposition of the correlation matrix.
• The observations from any given time can be
  recovered with a weighted sum of the eigenvector
  scores from that time (row)

13
EOF Examples

1. Retrieving the magnitude and time series of two
   static patterns
2. Effects of noise, missing data, and few data
3. Interpretation
4. Propagating features
5. NAO example
6. Exercise ENSO