GEOS 597e: Spatiotemporal Data Analysis Workshop

Homework 6: EOF rotation
Last updated 10/16/06. To be completed prior to class session Weds., Oct. 18th.

Introduction: This week we'll analyze our EOF  calculations from Homework 4 and 5:  of the EOF structures that pass our Rule N test for significance with respect to white noise processes, how does rotation of the patterns affect our interpretation of the results?
  1. Simple example.  Consider the matrix
F = [ -1 2 -1; -2 0 2; 3 0 -3]

whose rows are variables (e.g. gridpoints) and whose columns are observations.
    1. Code and find the varimax rotation of the EOFs of F following the algorithm suggested by Jennrich (2001):
      1. Use matlab to find the covariance matrix R of F, the eigenvector matrix E of R, the eigenvalue matrix L corresponding to E, and the principal components C.  Based on your prework reading, how do you expect the rotated EOFs to differ from the unrotated EOFs?
      2. Let ns and m be the number of rows and columns in E.  Define an initial transition matrix T as the identity matrix with dimension m, and an initial rotation B=ET.  Define an initial maximization criterion D and set it to zero.
      3. Within a loop of 300 iterations,
        1. Calculate G = ET*(ns*B.3 - B*(diag(diag(BTB)))).  This is the matrix form of the derivative of Kaiser's (1958) varimax criterion given by Neudecker (1981): K=trace(ns*trace((B.2)T(B.2))-trace(BTB))).   Note that B.x is shorthand for taking each element of B to the xth power (this is called the Hadamard product).
        1. Find the singular value decomposition G=USVT.
        1. Calculate the transition matrix T = UVT .
        1. Calculate the new rotation B=ET.
        1. Calculate the new value of the maximization criterion as the sum of the diagonal elements of S (you can use trace for this).
        1. Calculate the absolute value of the fractional change in D relative to your prior estimate of D.  This is a tolerance for accepting convergence of the algorithm on a solution B.  If this fractional change is less than 0.01, break the loop.  If not, continue looping.   How many iterations did it take? 
      4. Once you have converged to a stable rotation, compare the rotated EOF matrix B with the unrotated EOF matrix E.  How is B altered from E?  Now set the tolerance to  eps.   How many loops did it take?  Did the structure of B continue to take the form you expected, based on just a few iterations?  Check your rotation matrix B and transition matrix T against these results. 
      5. Calculate the rotated principal components Cr = BTF.  Is B orthogonal?  Is T orthogonal?  Is (TTT)-1 = I?  Is Cr orthogonal?
    1. Plot amplitudes vs. station for your unrotated and rotated EOFS on a single plot.  (Hint: use a different color for each EOF/rEOF pair, and use dashed and solid lines connecting different symbols, to differentiate EOF from rEOF).  In addition to axis labels and title, use a legend to label your plot.  How are the rEOFs different from the EOFs?
  1. SST EOF rotation.  Now use the varimax code you've just developed to rotate the subset of EOFs which you found significant at the 95% level by Rule N in HW5.
    1. Plot the first four unrotated and rotated EOFs (use the same plotting routines from HW4 and HW5; you should have four pairs of pairs, perhaps on two pages of 4 as in HW4/5) so you can compare the rotated and unrotated EOF patterns.  How are they similar?  How are they different?
    2. Plot the first four PCs corresponding to the unrotated and rotated EOFs (using four panels, each with the two time series to be compared).  How are they similar?  How are they different?
  1. Food for thought.  Now read through Dommenget and Latif (2002).
    1. What is the analytical observation illustrated in the first three figures, that motivated the authors to write this paper?
    2. What is the strategy used to resolve the discrepancies between results of the different flavors of pattern analysis?  Why is it useful?  Why is it overly simplistic?
    3. How is it possible to obtain a dipole with anticorrelated nodes from EOF analysis of a domain having a basin-wide, same-signed pattern?
    4. In what situation (e.g. general structure of the physical modes) would it make most sense to use
    5. What is the author's recommendation when using spatiotemporal analyses to develop physical interpretations for patterns?  Would you add any other suggestions to this recommendation?
  1. Products.  Please be sure you've handed in a copy of your answers to Prework 6.  Please write "Prework 6" and your name on it.  Turn in a printed copy of your hw6 script.  Turn in your written answers to questions 1, 2b, and 3a-e, and your printed plots for questions 1b and 2b.
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