Empirical Climate Prediction: Some Features of EOF ANALYSIS when used on its own or for Regression or CCA Slides taken from Willem Landman, Simon Mason, and Tony Barnston

1
Empirical Climate PredictionSome Features of
EOF ANALYSIS when used on its own orfor
Regression or CCASlides taken from Willem
Landman, Simon Mason,and Tony Barnston
2
A long-standing, simple method to do climate
prediction
Analogs Finding cases in the past that are
similar to the current climate state, and
predicting what happened, on average, in those
past cases. Variations in analog
forecasting --Number of analogs --Weighting
equal vs. by degree of similarity --using EOFs,
for climate state vector (Barnett
Preisendorfer 1978) --Inclusion of cases that are
opposite (mult. by 1 assumes linearity)
Advantage of analogs Nonlinearity is taken
into account Constructed analog Building an
analog for the present climate state from ALL
available cases, using a multiple regression
overfit (Van den Dool 1992). This makes analog
forecasting more like regression.
3
A related empirical method
to do climate prediction
Composites Based on a (Believed) Relevant
Dimension, Such as the ENSO state Advantage
Nonlinearity taken into account
4
Example GUAM ANNUAL RAIN
We can make a composite of these
NOTE POST-EL NINO YEARS IN RED
5
GUAM
6
Percent of Normal
GUAM
7
Average El Nino VS Mean Annual
GUAM
8
Probabilistic Composites Based on El
Nino Mason and Goddard (2001)
9


For Oct-Nov-Dec precipitation

http//iri.columbia.edu/climate/forecast//enso/ind
ex.html
10
Correlation between two variables Pearson
product-moment correlation
Correlation is a systematic relationship between
x and y When one goes up, the other tends to go
up also, or may tend to go down. Need
corresponding pairs of cases of x, y. Perfect
positive correlation is 1 Perfect negative
correlation is 1 No correlation (x and y
completely unrelated) is 0 Correlation can be
anywhere between 1 and 1. A relationship
between x and y may or may not be causal if
not, x and y may be under control of some third
variable. Correlation can be estimated visually
by looking at a scatterplot of dots on an x vs. y
graph.
11



o

o o
o
o o
o


o o Y
o o o
o
o

o o

o o

o

o
o


_______________________________________________

X correlation 0.8
12

o

o
o
o o
o o
o o
o
o o
o
o Y
o o o
o
o o o

o o o

o o

o

o o o
o

____________________________
___________________
X correlation 0.55
13
o o o o o
o o o oo o Y
o o o o o
o o o o o o o
o o o o o o
o o o o o o o
o o o o o o o
o o o o o o o o oo o
o _____________________________________
__
X correlation 0.0
14

Y

o
o
o o
o
o
_____________________________________
__
X correlation 1.0
15
o
o Y
o
o
o o
o
o
o
_____________________________________
__
X correlation 1.0
16
o
Y
o

o
o

o
_____________________________________
__ X correlation -1.0
17
o
o Y
o
o
o o
o
o
o
_____________________________________
__
X correlation undefined because SD of X
is zero (X doesnt change)
18

Y

o o o
o o o o o o o o o o


_______________________________
________
X correlation undefined because SD
of Y is zero (Y doesnt change)
19
o x
o x Y
o x
x o
x o o x
x
o x
o cor 0 x o
cor 1 ____________________________________
___
X correlation for all 18 points 0.707
correlation squared 0.5
When points having a perfect correlation are
mixed with an equal number of points having no
correlation, and the two sets have same mean and
variance for X and Y, correlation is 0.707.
Correlation squared (amount of variance
accounted for) is 0.5.
20
o
Y



o
o o o
oooooo o o
oooooo_o______________________
___________
X correlation 0.87 (due to one
outlier in upper right) If domination by one
case is not desired, can use the Spearman rank
correlation (correlation among ranks instead of
actual values).
21

Y

o

o


______________________________
___________
X correlation 1.0
(correlation between two points is always 1 or
-1, unless x is the same for both or y is
the same for both, in which case
correlation is undefined)
22

Y


o o o ooo
ooo oo
oo oo oo
oo oo
______oo________________________oo____

X correlation 0 but there is a strong
nonlinear relationship
The Pearson correlation only detects linear
relationships
23

o Y
o
o
o o
o
o
o
____________________________________
___
X correlation 0.9 but there is an
exact nonlinear relationship such as y

24
How is linear correlation measured? First it is
necessary to have data for the two
variables whose correlation is to be computed.
For each case of an x value, there is an
associated y value. X Y Case 1 30 18 Case
2 28 22 Case 3 11 9 Case 4 3 7
.. . . .. . . Case n 50 27
25
Covariance and correlation between two variables
x itself x itself x vs. y
vs
The above formula defines the Pearson
product-moment correlation
26
Correlation between X and itself 1
So the expected (or mean) value of the square of
z is 1 That means that if x and y are equal,
their correlation is 1. If they are proportional,
their correlation also is 1. If they are the
negative of one another, correlation is 1. If
they are negatively proportional, correlation
also is 1. If the graph of y vs. x is any
straight line, their correlation Is either 1 or
1 depending on the sign of the slope.
This implies perfect predictability of y from x.
Exception If line is exactly horizontal
or exactly vertical, then either x or
y has SD0, and correlation is undefined.
27
Approximate Standard Error of a Zero Correlation
Coefficient (for example if X and Y are random
data)
Examples of and critical values
for 2-sided significance at 0.05 level for
various sample sizes n n
10 0.33 0.65 20 0.23 0.45
50 0.14 0.28
100 0.10 0.20 400
0.05 0.10
For small n, true values of are
slightly smaller.
Note For significance of a correlation, z-distrib
ution is used, rather than t-distribu- tion, for
any sample size.
x 1.96
28
Confidence intervals for a nonzero correlation
are smaller than those for zero correlation, and
are asymmetric such that the interval toward
lower absolute values is larger. (Uses the Fisher
R-to-Z transformation) Example for n100 and
sample correlation 0.35, 95 confidence
interval is 0.17 to 0.51. That is 0.35 minus
0.18, but 0.35 plus 0.16. (For zero
correlation, it is zero plus 0.20 and zero minus
0.20.)
29
A line in the x vs. y coordinate system has the
form y a bx a is y-intercept
b is slope Regression line is defined such
that the sum of squares of the errors (predicted
y vs. true y) is minimized. Such a line predicts
y from x such that For example, if
then y will be predicted to be half
as many SDs away from its mean as x. When
correlation between y and x is zero, the mean
of y will always be predicted, no matter what x
is. When we have no predictive info, the mean is
the best guess for minimizing the sum of squared
errors.
30
Simple regression prediction Now we incorporate
the actual units of x and y rather than the
standardized (z) version in SD units. This is the
raw numbers form of the same equation The
above equation dresses up the basic z
relationship by adjusting for (1) ratio of SD of
y to SD of x, and (2) the difference between the
mean of y and the mean of x.
is the slope (b) of
is the y-
the regression line
intercept
31
Standard error of estimate of regression
forecasts .is the standard deviation of the
error distribution, where the errors are St
Error of Estimate (of standardized y data, or
) St Error of Estimate (of actual y
data)
When cor 0, St Error of Estimate is same as the
SD of y. When cor 1, St Error of Estimate is 0
(all errors are zero).
32
Standard Error of Estimate vs. Correlation
Standard Error
of Estimate Correlation (as a
fraction of SD of the
predictand y ) 1.00
0.00 0.90 0.44
We need a very 0.80
0.60 high correlation to 0.70
0.71 get a low
standard 0.60 0.80
error of estimate 0.50
0.87 need cor 0.866
0.40 0.92 to
get SD of error 0.30 0.95
of half of SD or the 0.20
0.98 predicted
variable (y). 0.10 0.99
This has implications for 0.00
1.00 probability of middle
tercile!
33
Tercile probabilities for various correlation
skills and predictor signal strengths (in SDs).
Assumes Gaussian probability distri- bution.
Forecast (F) signal (Predictor Signal) x
(Correl Skill).
Correlation Skill Predictor Signal0.0 Predictor Signal 0.5 Predictor Signal 1.0 Predictor Signal 1.5 Predictor Signal 2.0
0.00 F signal 0.00 33 / 33 / 33 F signal 0.00 33 / 33 / 33 F signal 0.00 33 / 33 / 33 F signal 0.00 33 / 33 / 33 F signal 0.00 33 / 33 / 33
0.20 F signal 0.00 33 / 34 / 33 F signal 0.10 29 / 34/ 37 F signal 0.20 26 / 33 / 41 F signal 0.30 23 / 33 / 45 F signal 0.40 20 / 31 / 49
0.30 F signal 0.00 33 / 35 / 33 F signal 0.15 27 / 34 / 38 F signal 0.30 22 / 33 / 45 F signal 0.45 17 / 31 / 51 F signal 0.60 14 / 29 / 57
0.40 F signal 0.00 32 / 36 / 32 F signal 0.20 25 / 35 / 40 F signal 0.40 18 / 33 / 49 F signal 0.60 13 / 30 / 57 F signal 0.80 9 / 25 / 65
0.50 F signal 0.00 31 / 38 / 31 F signal 0.25 22 / 37 / 42 F signal 0.50 14 / 33 / 53 F signal 0.75 9 / 27 / 64 F signal 1.00 5 / 21 / 74
0.60 F signal 0.00 30 / 41 / 30 F signal 0.30 18 / 38 / 44 F signal 0.60 10 / 32 / 58 F signal 0.90 5 / 23 / 72 F signal 1.20 2 / 15 / 83
0.70 F signal 0.00 27 / 45 / 27 F signal 0.35 13 / 41 / 46 F signal 0.70 6 / 30 / 65 F signal 1.05 2 / 17 / 81 F signal 1.40 1 / 8 / 91
0.80 F signal 0.00 24 / 53 / 24 F signal 0.40 8 / 44 / 48 F signal 0.80 2 / 25 / 73 F signal 1.20 0 / 10 / 90 F signal 1.60 0 / 3 / 97
0.3 0.04
34
Spearman rank correlation
Rank correlation is simply the correlation
between the ranks of X vs. the ranks of Y,
treating ranks as numbers. Rank correlation
defuses outliers by not honoring original
intervals between the numbers corresponding to
adjacent ranks. Adjacent ranks only differ by
1. When there are outliers, or when the X and/or
Y data are very much non-normal, the Spearman
rank correlation should be computed in addition
to the standard correlation. Example of
conversion to ranks for X or for Y Original
numbers 2 9 189 3 21
7 Corresponding ranks 6 3 1 5 2 4
35
Multiple Linear Regression uses 2 or more
predictors
General form Let us take simplest multiple
regression case--two predictors Here, the bs
are not simply and ,
unless x1 and x2 have zero correlation with one
another. Any correl- ation between x1 and x2
makes determining the bs less simple. The bs
are related to the partial correlation, in which
the value of the other predictor(s) is held
constant. Holding other predictors constant
eliminates the part of the correlation due to the
other predictors and not just to the predictor at
hand. Notation partial correlation of y with
x1, with x2 held constant, is written
36
For 2 (or any n) predictors, there are 2 (or any
n) equations in 2 (or any n) unknowns to be
solved simultaneously. When n gt3 or so,
determinant operations are necessary. For case of
2 predictors, and using z values (variables
standardized by subtracting their mean and then
dividing by the standard deviation) for
simplicity, the solution can be done by hand.
The two equations to be solved simultaneously
are b1.2 b2.1(cor x1,x2)
cory,x1 b1.2(corx1,x2) b2.1
cory,x2 Goal is to find the two b
coefficients, b1.2 and b2.1
37
X1 Polar north Atlantic 500 millibar height X2
North tropical Pacific sea level pressure Y
Seasonal number of hurricanes in North Atlantic

0.20 (x1,y)
0.40 (x2,y)
0.30
(x1,x2) ??
Example Prediction
one pre- dictor vs the other
Simultaneous equations to be solved
b1.2 (0.30)b2.1
0.20 (0.30)b1.2 b2.1
0.40 Solution Multiply 1st equation by 3.333,
then subtract second equation from first
equation. This gives (3.033)b1.2 0
0.267 So b1.2 0.088 and use
this to find that b2.1 0.374 Regression
equation is Zy (0.088)zx1 (0.374)zx2
38
Multiple correlation coefficient R
correlation between predicted y and actual y
using multiple regression. The b coefficients are
for standardized (z) X1, X2, and Y.
In example above,
0.408 Note this
is only very slightly better than using the
second predictor alone in simple regression. This
is not surprising, since the first predictors
total correlation with y is only 0.2, and it is
correlated 0.3 with the second predictor, so that
the second predictor already accounts for some of
what the first predictor has to offer. A decision
would probably be made concerning whether it is
worth the effort to include the first predictor
for such a small gain. Note the
multiple correlation can never decrease when more
predictors are added.
39
Multiple R is usually inflated somewhat compared
with the true relationship, since additional
predictors fit the accidental variations found in
the test sample. Adjustment (decrease) of R for
the existence of multiple predictors gives a less
biased estimate of R Adjusted R
n sample size

k number of predictors
40
Sampling variability of a simple (x, y)
correlation coefficient around zero when
population correlation is zero is approximately
In multiple regression the same approximate
relationship holds except that n must be further
decreased by the number of predictors additional
to the first one. If the number of predictors
(xs) is denoted by k, then the sampling
variability of R around zero, when there is no
true relationship with any of the predictors, is
given by It is easier to get a given multiple
correlation by chance as the number of predictors
increases.
41
Partial Correlation is correlation between y and
x1, where a variable x2 is not allowed to vary.
Example in an elemen- tary school, reading
ability (y) is highly correlated with the
childs weight (x1). But both y and x1 are really
caused by something else the childs age (call
x2). What would the correlation be between weight
and reading ability if the age were held
constant? (Would it drop down to zero?)
A similar set of equations exists for the second
predictor.
42
Suppose the three correlations are reading
vs. weight reading vs. age weight vs.
age The two partial correlations come out to
be Finally, the two regression weights (for
zs) turn out to be
Weight is seen to be a minor factor compared with
age.
43
The means and the standard deviations of three
data sets (y, x1, x2) are y Jul-Aug-Sep Sahel
rainfall (mm) mean 230 mm, SD 88 mm x1 Tropical
Atlantic/Indian ocean SST mean 28.3 degr C, SD
1.7 C x2 Deforestation (percent of initial)
mean 34, SD 22 Suppose that Cor(x1,y) -0.52
Cor(x2,y) -0.37 Cor(x1,x2)0.50 If
regression equation in SD units is Zy
-0.447-zx1 -0.147zx2 After
simplification, final form will be y
coeff x1 coeff x2 constant (here, both coeff
lt0) b1 b2
44
We now compute the multiple correlation R, and
the standard error of estimate for the multiple
regression. Using the two individual correlations
and the b terms Cor(x1,y) -0.52 Cor(x2,y)
-0.37 Cor(x1,x2)0.50 Regression equation is
Zy -0.447 zx1 -0.147 zx2
0.535
The deforestation factor helps the prediction
accuracy only slightly. If there were less
correlation between the two predictors, then the
second predictor would be more valuable.
Standard Error of Estimate
0.845 In
physical units it is (0.845)(88 mm) 74.3 mm
45
Let us evaluate the significance of the multiple
correlation of 0.535. How likely could it have
arisen by chance alone? First we find the
standard error of samples of 50 drawn from a
population having no correlations at all, using 2
predictors
For n50 and k2 we get
0.145 For a 2-sided z test at the 0.05 level,
we need 1.96(0.145) 0.28 This is easily
exceeded, suggesting that the combination of
the two predictors (SST and deforestation) do
have an impact on Sahel summer rainfall. (Using
SST alone in simple regression, with cor0.52,
would have given nearly the same level of
significance.)
46
Example problem using this regression
equation Suppose that a climate change model
predicts that in year 2050, the SST in the
tropical Atlantic and Indian oceans will be 2.4
standard deviations above the means given for the
50-year period of the preceding problem. (It is
now about 1.6 standard deviations above that
mean.) Assume that land use practices (percentage
deforestation) will be the same as they are now,
which is 1.3 standard deviations above the mean.
Under this scenario, using the multiple regression
relationship above, how many standard
deviations away from the mean will Jul-Aug-Sep
Sahel rainfall be, and what seasonal total
rainfall does that correspond to?
47
The problem can be solved either in physical
units or in standard deviation units, and then
the answer can be expressed in either (or both)
kinds of units afterward. If solved in physical
units, the values of the two predictions in
SD units (2.4 and 1.3) can be converted to raw
units using the means and standard deviations of
the variables provided previously, and the raw
units form of the regression equation would be
used. If solved in SD units, the simpler
equation can be used Zy -0.447zx1
-0.147zx2 The zs of the two predictors,
according to the scenario given, will be 2.4 and
1.3, respectively. Then Zy -0.447(2.4)
0.147(1.3) -1.264. This is how many SDs away
from the mean the rainfall would be. Since the
rainfall mean and SD are 230 and 88 mm,
respectively, the actual amount predicted is
230 1.264(88) 230 111.2 118.8 mm.
48
A problem in multiple regression
Colinearity When the predictors are highly
correlated with one another in multiple
regression, a condition of colinearity exists.
When this happens, the coefficients of two highly
correlated predictors may have opposing signs,
even when each of them has the same sign of
simple correlation with the predictand. (Such
opposing signed coefficients minimizes squared
errors.) Issues and problems with this are (1) it
is counterintuitive, and (2) the coefficients are
very unstable, such that if one more sample is
added to the data, they may change
drastically. When colinearity exists, the
multiple regression formula will often still
provide useful and accurate predictions.
To eliminate colinearity, predictors that are
highly correlated can be combined into a single
predictor.
49
EOF Analysis,and its use in Regressionor in
CCA
50
Empirical Orthogonal Functions (EOFs) (closely
related to Principal Components, and Factor
Analysis) Identifying preferred patterns within
many variables
51
Suppose we have a long time record of data for a
field variable, such as temperatures at many
locations. Examples in climate science average
temperature data for a 40-year period
across across much of globe, at grid points 5
degrees latitude and 5 degrees longitude apart
(over 2,000 grid points) Sea surface temperature
data for a 40-year period over much of the
globes oceans, on a 4 degree grid
(again, roughly 2,000 grid points). Time
1 grid1 grid2..grid2000 Time 2 grid1
grid2..grid2000 .. ..
.. .. .. .. .. ..
.. Time 40 grid1 grid2..grid2000
52
Climate sciences often deal with data that have
high dimensionality such as collections of
spatially distributed time series like the
temperature observations or SST observations.
Because such observations are not entirely random
and are often related to eacg other, the
information contained in such datasets can often
be compressed down to a few spatial patterns that
cluster stations/ grid points that are strongly
related. EOF is an exploratory analysis
technique designed to perform such a compression
in an objective way, without any prior knowledge
of the relationships linking the observations or
underlying physical processes. It expresses the
data in a smaller set of new variables defined
through a linear combination of the original
ones. The desired result is a limited collection
of patterns, called EOF modes, that are
sufficient to reconstruct a good approximation
of the original data and also easy to visualize
and recognize. Although such modes sometimes
repre- sent known physically phenomena, they are
not designed to isolate only physical
mechanisms. EOF should be always thought of
only as an efficient statistical compression
tool.
53

• How EOF modes are defined from a dataset.
• First, a complete intercorrelation matrix is
  computed
• 1 2 .. 2000
• -------------------------------------------------
  ----
• 1 1.00 0.81 .. -0.13
• 2 0.81 1.00 .. 0.07
• 2000 -0.13 0.07.... 1.00

54
Then, using the cross-correlation matrix, a
procedure is used to identify which grid points
best form a coherent clusterpoints that vary
similarly or oppositely from one another. This
information leads to the formation of a linear
combination of all the grid points. In this
combination, each gridded value will be assigned
a weight (positive or negative), something like
the weights assigned to the predictors in
multi- ple regression. The pattern of these
weights often shows up, visually, as a coherent
(non-random) pattern in the spatial domain. Such
a pattern of weights is an EOF loading
pattern (technically, it is called an
eigenvector). By multiplying the values at the
grid points for one particular time by their
loading weights, and adding them all up, we
get the amplitude (or temporal score) for that
time. Times whose original data assume that
pattern have high ( or -) scores.
55
EOF analysis is performed by inputting the
correlation matrix to a procedure called
eigenvalue/eigenvector analysis. It involves
solving a large set of linear equations. Grid
points having high correlations with the most
other grid points ( or -) participate most
strongly. Each EOF pattern that emerges explains
a certain percent- age of the total variance of
all the grid points over time. This percentage of
variance explained is maximized. The first EOF
mode gathers the most variance, and then the
second EOF mode works on what remains after
all the variability associated with the first
mode is removed. Often, after 2 to 6 modes have
been defined, the coherent portion of the total
variability is exhausted, and further modes just
work on the remaining incoherent noise. When
this happens, the loading patterns start
looking random and physically meaningless, and
the amounts of additional variance explained
become small.
56

• What EOF analysis provides
• A set of EOF loading patterns (eigenvectors)
• A set of corresponding amplitudes (temporal
  scores)
• A set of corresponding variances accounted for
• (from the eigenvalues)
• Often, the EOF analysis allows a set of hundreds
• or thousands of variables (like grid points) to
  be
• compacted into just 3 to 6 EOF variables (modes)
• that account for two-thirds or more of the
  original
• variance. These modes capture coherent
  variations.
• More than one field can be input to EOF analysis.

57
EOF mode 1 Global 500 mb height, JFM 1950-2004
loading pattern
temporal scores (amplitude) for mode 1
58
EOF mode 2 Global 500 mb height, JFM 1950-2004
loading pattern
temporal scores (amplitude) for mode 2
59
EOF mode 3 Global 500 mb height, JFM 1950-2004
loading pattern
temporal scores (amplitude) for mode 3
60
EOF mode 1 Global 500 mb height, JFM 1950-2004
loading pattern
temporal scores (amplitude) for mode 1
61
Using correlation matrix
Mean SSTs for JFM, 1950-2002 Since all grid
point data are standardized, only coherent
relationships matter, and differing grid point
variances do not affect the pattern.
62
Using covariance matrix
Mean SSTs for JFM, 1950-2002 Differing variances
count, and grid points in extra- tropics (having
high variance) receive more weight.
63
EOF analysis of JFM SST history using correlation
matrix Amount of variance explained by EOF
modes 1 to 12
EOF Mode number
64
Principal component regression EOF
amplitude time series, instead of raw (original)
variables, can be used as predictors for a
multiple regression. The EOF time series
represent scores with respect to the EOF loading
patterns, which contain large portions of the
variance of the original variables. The EOF
time series therefore may be an efficient way to
include the information of many raw predictors at
once. This depends, however, on whether or not
the pattern is relevant to what is being
predicted. (Sometimes, grid points that matter
may not be part of the main EOF Patterns, such as
SST points right along a coastline.)
65
Interpreting EOFs EOFs are sometimes difficult
to interpret physically. The weights are defined
to maximize the variance, which may not
necessarily maximize the physical
interpretability. With spatial data (including
climate data) the interpretation becomes even
more difficult because there are geometric
controls on the correlations between the data
points.
66
Buell patterns Imagine a rectangular domain in
which all the points are strongly correlated with
their neighbors.
67
Buell patterns The points in the middle of the
domain will have the strongest average
correlations with all other points, simply
because their average distance to all other grids
is the smallest.
68
Buell patterns The points in the corners of the
domain will have the weakest average correlations
with all other points, simply because their
average distance to all other grids is the
greatest.
Mode 2 will repre-sent points with weak
correlations between distant grids, because their
variance has not yet been ex-plained. A dipole
pattern appears. The axis of the dipole is
deter-mined by the domain shape it is along the
long dimension.
69
Buell patterns Are these real, or are they
related to the domain shape? (They may be both
together.)
First EOF of Indian Ocean SST during Oct-Nov-Dec,
for several decades
70

• Buell patterns
• Domain shape dependency can create these
  influences
• the first EOF frequently indicates positive
  loadings with strongest values in the center of
  the domain
• the second PC frequently indicates negative
  loadings on one side and positive loadings on the
  other side, with axis along the longest dimension
  of the domain.
• Similar kinds of problems can occur when using
• gridded data with converging longitudes, or
  simply with longitude spacing different from
  latitude spacing
• station data (middle stations vs. edge
  stations).

71
EOF input can be correlation matrix or covariance
matrix. Covariance matrix should NOT be
used... (1) When variances differ greatly. For
example Eq. Indian Ocean SST variance ltlt Eq.
Pacific Ocean SST variance. Indian Ocean will
have only very small influence on results. (2)
When units are different. For example SSTs
combined with 200 hPa geopotential heights (SSTs
would have very low weight, almost no influence).
72
How many modes should be used when doing EOF
analysis? Ways to determine the answer

• Proportion of variance
• SCREE test (shape of eigenvalue curve)
• Average eigenvalue (Guttman Kaiser)
• Sensitivity tests (used in statistical modeling)
• Monte Carlo tests determines when results
  become only random
• Visual inspection of mode patterns

73
The SCREE test
//
74
Monte Carlo Exercises Find where the red
and blue curves intersect.
75
Extended EOFs
Can capture time evolution. Example AMJJASOND
JFM tropical SSTs are combined as predictor
fields for EOFs. They are temporally stacked
SST data. Resulting modes will show evolutionary
or steady-state features of SSTs
76
Rotation The weights are redefined for an
alternative criteria. Varimax rotation maximizes
the variance of the loading weights across the
domain. The way this occurs, is there are some
very high weights, and a large number of weights
close to zero. Consequently, the patterns are
more localized. The variances of the first few
principal components are reduced after rotation,
and the curve of variance explained is flatter
than the curve for the original (unrotated) modes.
77
Rotation of EOFs Finding different directions of
axes

• Rotation can be helpful when Buell patterns exist
  (Richman 1986)
• Rotation provides a more simple, localized
  structure AO pattern become NAO pattern
• Same amount of variation explained after
  rotation, for a given truncation choice
• Two types of rotation
• Orthogonal varimax
• Oblique

78
Prediction of any element Y individually using
multiple regression
Predictors can
be elements of X
OR
Predictors
can be EOFs of X
(principal component regression)
x1 x2 x3 x4 x5 x6 x7 x8 x9 x1.

x2
x3
x4
x5
x6
x7
x8

x9

Intercorrorrelation matrix for X elements
? EOFs of X
79
Introduction to CCA CCA is like EOF
analysis, except that there are TWO data sets (X
and Y, or predictor and predictand), and the
input matrix (correlation or covariance)
contains cross-dataset coefficients. Only (an X
element with a Y element no X-to-X or Y-to-Y).
Both X and Y can be time-extended or contain
multiple fields.
80
Analyses of correlation matrix of X and Y fields
9 elements of X and Y
x1 x2 x3 x4 x5 x6 x7 x8 x9 y1
y2 y3 y4 y5 y6 y7 y8 y9 x1.
x2
x3

x4
x5
x6
x7
x8
x9

- - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - -
- - y1
y2
y3
y4

y5
y6
y7
y8
y9

CCA of X vs Y
EOFs of X
Joint EOFs of X and Y
CCA of X vs Y
EOFs of Y
81
CCA mode 1
X
Y
X tropical Pacific SST, SON season Y Indian
Ocean SST, DJF season
82
CCA mode 2
X
Y
X tropical Pacific SST, SON season Y Indian
Ocean SST, DJF season
83
Another CCA example, using tropical Pacific SST
for two separated months July (X) and December
(Y)
X
July and December 1950 1999 sea-surface
temperatures
Y
84
Buell Patterns affect CCA too, not just EOFs
85
Two possible predictor designs in CCA
1. Observational predictor design
X is observed earlier predictors, such as
the field of governing SST
Y is rainfall pattern prediction for a
region of interest
2. Model MOS design
X is dynamical model prediction of rainfall
pattern around a region of interest
1. is a purely statistical forecast system 2. Is
a dynamical forecast corrected by a statistical
adjustment
86
Some simple matrix algebra
Rows are samples (time) Columns are variable
If
and a b c 0, and d e f 0, then
3 is number of cases for unbiased estimate, 2
would be used.
87
In general, matrix multiplication gives
So, if a b c 0, and d e f 0, then
88
If X contains data expressed as anomalies, then
If X contains data expressed as standardized
anomalies, then