EOT2 EOT1 EOT

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EOT2EOT1?EOT
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Empirical Methods in Short-Term Climate
Prediction Huug van den Dool Price 49.95
(Hardback)ISBN-10 0-19-920278-8ISBN-13
978-0-19-920278-2Estimated 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
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Chapter 8. Methods in Short-Term Climate
Prediction 1218.1 Climatology 1228.2
Persistence 1248.3 Optimal climate normals
1268.4 Local regression 1298.5 Non-local
regression and ENSO 1358.6 Composites 1388.7
Regression on the pattern level 1398.7.1 The
time-lagged covariance matrix 1398.7.2 CCA, SVD
and EOT2 1418.7.3 LIM, POP and Markov 1458.8
Numerical methods 1468.9 Consolidation 1478.10
Other methods 1498.11 Methods not used
151Appendix 1 Some practical spacetime
continuity requirements 152Appendix 2
Consolidation by ridge regression 153
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EOT-normal Q is diagonalized Qa is not
diagonalized (1) is satisfied with am 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 am and em orthogonal am (em) is eigenvector
of Qa(Q)
Laudable goal f(s,t)? am(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.
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EOT-normal Q is diagonalized Qa is not
diagonalized (1) is satisfied with am 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 am and em orthogonal am (em) is eigenvector
of Qa(Q)
Laudable goal f(s,t)? am(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.
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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.
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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 /-
0.1. Data source NCEP Global Reanalysis. Period
1948-2005. Domain 20N-90N
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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 norm1. Contours every
0.2, starting contours /- 0.1. Data source NCEP
Global Reanalysis. Period 1948-2005. Domain
20N-90N
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Fig 5.7. Explained Variance (EV) as a function of
mode (m1,25) for seasonal mean (JFM) Z500,
20N-90N, 1948-2005. 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.
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EOT2 (1) and (2) satisfied am and ßm orthogonal
(homo-and-heterogeneous Cff (t0), Cgg(t0) 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 t)/nt t
Laudable goals f(s,t) ? am(t)em(s) (1) m
g(s,t t) ? ßm(t t)dm(s)
(2) m constrained by a connection between a and
ß and/or e and d.
Discrete Data set f(s,t) 1 t nt 1 s
ns Discrete Data set g(s,t t) 1 t nt 1
s ns
Alt Cross Cov Matrix Cafgwith elements caij
? f(s,ti)g(s,tj t)/ns s
SVD Somewhat like EOT2a, but two maps, and
(heterogeneously) orthogonal time series.
EOT2-alternative (1) and (2) satisfied em and dm
orthogonal am and ßm heterogeneously orthogonal
Caff (t0), Cagg(t0) and Cafg diagonalized Two
time series, one map.
Fig.x.y Summary of EOT2 procedures.
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EOT2 (1) and (2) satisfied am and ßm
orthogonal Cff (t0), Cgg(t0) 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 t)/nt t
Laudable goals f(s,t) ? am(t)em(s) (1) m
g(s,t t) ? ßm(t t)dm(s)
(2) m Constrained by a connection between a and
ß and/or e and d.
Discrete Data set f(s,t) 1 t nt 1 s
ns Discrete Data set g(s,t t) 1 t nt 1
s ns
Alt Cross Cov Matrix Cafgwith elements caij
? f(s,ti)g(s,tj t)/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 (t0), Cagg(t0) and Cafg
diagonalized Two time series, one map.
Fig.x.y Summary of EOT2 procedures.
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• CCA
• Make a square M Qf-1 Cfg Qg-1 CfgT
• E-1 M E diag ( ?1 , ?2, ?3, ?M)
• ? cor(m)sqrt (?m)

SVD 1) UT Cfg V diag (s1 , s2, ,
sm) Explained Squared Covariance s2m

• Assorted issues
• Prefiltering f and g , before calculating Cfg
• Alternative approach complicated when domains for
  f and g dont match
• Iteration and rotation CCA ?gt EOT2-normal SVD
  ?gt EOT2-alternative ???

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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)

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En dan nu iets geheel anders..
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                  T-obs T-CA   diff
1991-2006   .52C  .10C   .42C     100.00  (all
gridpoints north of 20N and anomalies relative
to 61-90.) 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.
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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 1991-2006 
1.65C   .82C   .83C      62.53   (at points
where T-obs anomaly is positive) 1991-2006
-1.35C -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
0ltdlt1) 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 delta0.37
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I am now building a Table as follows Month   
Damping    Delta 1         0.64         0.37
2         0.67         0.27 3        
0.67         0.33 4         0.58         0.15
5         0.57         0.19 6        
0.56         0.31 7         0.48         0.33
8         0.50         0.22 9 (sep) 0.51       
0.47
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