1. EOT2 EOT1? EOT

2. Empirical Methods in Short-Term Climate Prediction
Huug van den Dool
Price: £49.95 (Hardback) ISBN-10: ISBN-13: Estimated publication date: November 2006
Oxford University Press
Foreword , Professor Edward Lorenz (Massachusetts Institute of Technology)
Preface
1. Introduction
2. Background on orthogonal functions and covariance
3. Empirical wave propagation
4. Teleconnections
5. Empirical orthogonal functions
6. Degrees of freedom
7. Analogues
8. Methods in short-term climate prediction
9. The practice of short-term climate prediction
10. Conclusion
References
Index

3. Chapter 8. Methods in Short-Term Climate Prediction 121 8
Chapter 8. Methods in Short-Term Climate Prediction Climatology Persistence Optimal climate normals Local regression Non-local regression and ENSO Composites Regression on the pattern level The time-lagged covariance matrix CCA, SVD and EOT LIM, POP and Markov Numerical methods Consolidation Other methods Methods not used 151 Appendix 1: Some practical space–time continuity requirements 152 Appendix 2: Consolidation by ridge regression 153

4. EOT-normal iteration Rotation EOF Laudable goal:
Q is diagonalized
Qa is not diagonalized
(1) is satisfied with αm orthogonal
Q tells about Teleconnections
Matrix Q with elements:
qij=∑ f(si,t)f(sj,t) t
iteration
Rotation
EOF
Both Q and Qa Diagonalized
(1) satisfied –
Both αm and em orthogonal
αm (em) is eigenvector of Qa(Q)
Laudable goal:
f(s,t)=∑ αm(t)em(s) (1) m
Discrete Data set f(s,t)
1 ≤ t ≤ nt ; 1 ≤ s ≤ ns
Arbitrary
state
Rotation
iteration
Matrix Qa with elements:
qija=∑f(s,ti)f(s,tj) s
EOT-alternative
Q is not diagonalized
Qa is diagonalized
(1) is satisfied with em orthogonal
Qa tells about Analogues
Fig.5.6: Summary of EOT/F procedures.

5. Iteration (as per power method)
EOT-normal
Q is diagonalized
Qa is not diagonalized
(1) is satisfied with αm orthogonal
Q tells about Teleconnections
Matrix Q with elements:
qij=∑ f(si,t)f(sj,t) t
iteration
Rotation
EOF
Both Q and Qa Diagonalized
(1) satisfied –
Both αm and em orthogonal
αm (em) is eigenvector of Qa(Q)
Laudable goal:
f(s,t)=∑ αm(t)em(s) (1) m
Discrete Data set f(s,t)
1 ≤ t ≤ nt ; 1 ≤ s ≤ ns
iteration
Arbitrary
state
Rotation
Iteration (as per power method)
Matrix Qa with elements:
qija=∑f(s,ti)f(s,tj) s
EOT-alternative
Q is not diagonalized
Qa is diagonalized
(1) is satisfied with em orthogonal
Qa tells about Analogues
Fig.5.6: Summary of EOT/F procedures.

6. Fig. 4.3 EV(i),the variance explained by single gridpoints in % of the total variance, using equation 4.3. In the upper left for raw data, in the upper right after removal of the first EOT mode, lower left after removal of the first two modes. Contours every 4%. The timeseries shown are the residual height anomaly at the gridpoint that explains the most of the remaining domain integrated variance.

7. Fig.4.4 Display of four leading EOT for seasonal (JFM) mean 500 mb height. Shown are the regression coefficient between the height at the basepoint and the height at all other gridpoints (maps) and the timeseries of residual 500mb height anomaly (geopotential meters) at the basepoints. In the upper left for raw data, in the upper right after removal of the first EOT mode, lower left after removal of the first two modes. Contours every 0.2, starting contours +/ Data source: NCEP Global Reanalysis. Period Domain 20N-90N

8. Fig.5.4 Display of four leading alternative EOT for seasonal (JFM) mean 500 mb height. Shown are the regression coefficient between the basepoint in time (1989 etc) and all other years (timeseries) and the maps of 500mb height anomaly (geopotential meters) observed in 1989, 1955 etc . In the upper left for raw data, in the upper right after removal of the first EOT mode, lower left after removal of the first two modes. A postprocessing is applied, see Appendix I, such that the physical units (gpm) are in the time series, and the maps have norm=1. Contours every 0.2, starting contours +/ Data source: NCEP Global Reanalysis. Period Domain 20N-90N

9. Fig 5.7. Explained Variance (EV) as a function of mode (m=1,25) for seasonal mean (JFM) Z500, 20N-90N, Shown are both EV(m) (scale on the left, triangles) and cumulative EV(m) (scale on the right, squares). Red lines are for EOF, and blue and green for EOT and alternative EOT respectively.

10. Laudable goals: α and β and/or e and d.
EOT2
(1) and (2) satisfied
αm and βm orthogonal (homo-and-heterogeneous
Cff (τ=0), Cgg(τ=0) and Cfg diagonalized
One time series, two maps.
CCA
Very close to EOT2, but
two, maximally correlated, time series.
Cross Cov Matrix Cfgwith elements:
cij= ∑ f(si,t)g(sj,t+ τ)/nt t
Laudable goals:
f(s,t) = ∑ αm(t)em(s) (1) m
g(s,t +τ) = ∑ βm(t +τ)dm(s) (2)
constrained by a connection between
α and β and/or e and d.
Discrete Data set f(s,t)
1 ≤ t ≤ nt ; 1 ≤ s ≤ ns
Discrete Data set g(s,t +τ)
1 ≤ t ≤ nt ; 1 ≤ s ≤ n’s
Alt Cross Cov Matrix Cafgwith elements:
caij= ∑ f(s,ti)g(s,tj+ τ)/ns s
SVD
Somewhat like EOT2a, but
two maps, and
(heterogeneously) orthogonal time series.
EOT2-alternative
(1) and (2) satisfied
em and dm orthogonal
αm and βm heterogeneously orthogonal
Caff (τ=0), Cagg(τ=0) and Cafg diagonalized
Two time series, one map.
Fig.x.y: Summary of EOT2 procedures.

11. Laudable goals: α and β and/or e and d.
EOT2
(1) and (2) satisfied
αm and βm orthogonal
Cff (τ=0), Cgg(τ=0) and Cfg diagonalized
One time series, two maps.
CCA
Very close to EOT2, but
two, maximally correlated, time series.
Cross Cov Matrix Cfgwith elements:
cij= ∑ f(si,t)g(sj,t+ τ)/nt t
Laudable goals:
f(s,t) = ∑ αm(t)em(s) (1) m
g(s,t +τ) = ∑ βm(t +τ)dm(s) (2)
Constrained by a connection between
α and β and/or e and d.
Discrete Data set f(s,t)
1 ≤ t ≤ nt ; 1 ≤ s ≤ ns
Discrete Data set g(s,t +τ)
1 ≤ t ≤ nt ; 1 ≤ s ≤ n’s
Alt Cross Cov Matrix Cafgwith elements:
caij= ∑ f(s,ti)g(s,tj+ τ)/ns s
SVD
Somewhat like EOT2a, but
two maps, and
(heterogeneously) orthogonal time series.
EOT2-alternative
(1) and (2) satisfied
em and dm orthogonal
Caff (τ=0), Cagg(τ=0) and Cafg diagonalized
Two time series, one map.
Fig.x.y: Summary of EOT2 procedures.

12. Make a square M = Qf-1 Cfg Qg-1 CfgT E-1 M E =diag ( λ1 , λ2, λ3,… λM)
CCA:
Make a square M = Qf-1 Cfg Qg-1 CfgT
E-1 M E =diag ( λ1 , λ2, λ3,… λM)
 cor(m)=sqrt (λm)
SVD:
1) UT Cfg V =diag (σ1 , σ2, … , σm)
Explained Squared Covariance = σ2m
Assorted issues:
Prefiltering f and g , before calculating Cfg
Alternative approach complicated when domains for f and g don’t match
Iteration and rotation: CCA > EOT2-normal; SVD > EOT2-alternative ???

13. Keep in mind EV (EOF/EOT) and EOT2 Squared covariance (SC) in SVD
SVD singular vectors of C
CCA eigenvectors of M
LIM complex eigenvectors of L (close to C)
MRK no modes are calculated (of L)

14. En dan nu iets geheel anders…..

20.                   T-obs T-CA   diff    .52C  .10C   .42C     %  (all gridpoints north of 20N and anomalies relative to ) We are warmer by 0.52C, and the circulation explains only 0.10.  So has T850 risen 0.42 ±error due to factor X ??? The plain difference CA minus Obs has a problem, namely that CA (like a regression) 'damps'. (The degree of damping is related to the success of the specification as measured by a correlation). Obs minus CA thus tends to have the sign of the observed anomaly (which is warm these days).  A reasonable explanatory equation reads :    T_obs   =  T_CA *  1/d   + delta, where delta is the true temperature change we are seeking and the 1/d factor undoes the damping of the specification method.

21. Two unknowns (d(amping) and delta), and I make two equations by doing an averaging separately for gridpoints where T850 was observed above (62.5% of the cases in Jan) and below the mean (37.5%).  (This is a shortcut for a regression thru thousands of points (years times gridpoints))/ For instance, in January we find:                  T-obs T-CA  diff   1.65C   .82C   .83C      62.53%   (at points where T-obs anomaly is positive) C -1.09C  -.26C      37.47%        (at points where T-obs anomaly is negative) Notice that cold anomalies, although covering less area, are also damped. I assume that damping (d; 0<d<1) is the same for cold and warm anomalies. Substituting the T-obs and T-CA twice we get two equations and find (for January): damp= 0.64 and delta=0.37

22. I am now building a Table as follows: Month Damping Delta 1 0. 64 0
I am now building a Table as follows: Month    Damping    Delta 1         0.64                  0.67                  0.67                  0.58                  0.57                  0.56                  0.48                  0.50         (sep) 0.51        0.47