Maximum Covariance Analysis
Canonical Correlation Analysis
FIG. 6.
The
CCA mode 1 for (a)
SLP
and (b)
SST
.
The pattern
in (a) is scaled
by
[max(u
)

min
(u)]/2, and (b)
by
[max
(
v
)

min(
v
)
]/2
.
Contour interval
is 0.5
mb
in (a) and
0.5C in
(b).
Hsieh, W. (2001) Nonlinear
Canonical
Correlation Analysis of the Tropical Pacific
Climate
Variability Using
a Neural Network
Approach, J. Climate, 14, 2528

2539
Hsieh, W. (2001) Nonlinear
Canonical
Correlation Analysis of the Tropical Pacific
Climate
Variability Using
a Neural Network
Approach, J. Climate, 14, 2528

2539
FIG.
12. The
CCA mode
2
for (a)
SLP
and (b)
SST
.
Contour interval
is
0.2
mb
in (a) and
0.2C
in
(b).
Statistical downscaling
Von
Storch
, H., and F. W.
Zwiers
(2002) Statistical analysis in
climate research, Cambridge University Press.
14.3.3 North Atlantic SLP and
Iberian Rainfall
: Analysis and
Historic
Reconstruction
In
this example, winter (DJF) mean
precipitation from a
number of rain gauges on the Iberian Peninsula is related to
the air

pressure field over the North Atlantic. CCA was used
to obtain a pair of canonical correlation pattern estimates
(Figure 14.4), and corresponding time series of canonical
variate
estimates. These strongly correlated modes of
variation (the estimated canonical correlation is 0.75)
represent about 65% and 40% of the total variability of
seasonal mean SLP and Iberian Peninsula precipitation
respectively.
The two patterns represent a simple physical mechanism:
when SLP mode 1 has a strong positive coefficient,
enhanced cyclonic circulation
advects
more maritime air
onto the Iberian Peninsula so that precipitation in the
mountainous northwest region (
precip
mode 1) is increased.
Since the canonical correlation is large,
the results
of the
CCA can be used to forecast
or specify
winter mean
precipitation on the
Iberian peninsula
from North Atlantic
SLP.
14.3.3 North Atlantic SLP and
Iberian Rainfall
: Analysis and
Historic
Reconstruction
The analysis described above was
performed with
the 1950

80 segment of a data set
that extends
back to 1901. Since
the 1901

49
segment is
independent of that used to 'train'
the
model,
it can be used to validate the model
Figure
14.5 shows both the specified and
observed winter
mean rainfall averaged over all
Iberian stations
for this
period. The overall upward
trend and
the low

frequency
variations in
observed precipitation
are well reproduced by
the
indirect method
indicating the usefulness of the
technique as
well as the reality of both the trend
and the
variations in the Iberian winter precipitation
.
14.3.4
North Atlantic SLP and
Iberian Rainfall
:
Downscaling of GCM
output
This regression approach has an interesting
application in climate change
studies. GCMs
are
widely used to assess the
impact that
increasing
concentrations of greenhouse
gases might
have on
the climate system. But, because
of their
resolution,
GCMs do not represent the
details of
regional climate
change well. The
minimum scale
that a GCM is able to
resolve is the
distance between
two
neighboring
grid
points whereas
the skillful
scale is generally accepted
to be four
or more
grid lengths. The minimum scale in
most climate
models in the mid 1990s is of the
order
of
250

500 km so that the
skillful
scale is at
least
1000

2000 km
.
Thus the scales at which GCMs
produce useful
information does not match the scale
at which
many
users, such as hydrologists,
require information
.
Statistical downscaling
is a possible
solution to this
dilemma. The
idea is
to build a statistical model from
historical observations
that relates large

scale
information that
can be well simulated by GCMs to
the desired
regional scale information that can not be
simulated. These models are then applied to
the
large

scale model output.
The following steps must be
taken.
1. Identify
a regional climate variable R of
Interest
2. Find
a climate variable
L
that:
•
controls
R in the sense that there is
a
statistical
relationship between
R
and
L
of
the form
R = G(
L,
a
) + e
in which
G(
L,
a
)
represents a
substantial fraction
of the total variance of R.
Vector
a
contains parameters
that can be used
to
adjust the fit
•
is
reliably simulated in a climate model.
3. Use
historical realizations
of
(
R, L
)
to estimate
a
.
4. Validate
the fitted model on
independent
historical data
5. Apply
the validated model to GCM
simulated
realizations
of
L
.
This
are
exactly
the steps taken in the Atlantic SLP
and Iberian precipitation analysis.
A statistical model was constructed
that related
Iberian rainfall
R
to North
Atlantic SLP
L
through a
simple linear
functional.
The
adjustable parameters a
consisted of
the
canonical correlation
patterns.
These parameters
were estimated from 1950 to
1980 data. Observations
before 1950
were used
to validate
the model
.
The downscaling
model
was
applied to
the output
of a
2xC0
2
experiment
performed
with a
GCM.
Figure 14.6 compares the '
downscaled’ response
to doubled C0
2
with the model's
grid point
response. The latter suggests that there
will be
a
marked decrease in precipitation over
most of
the
Peninsula whereas the downscaled
response is
weakly positive. The downscaled
response is
physically more reasonable than the
direct
response
of the model.
In relatively
high

dimensional x and y spaces, among
the many
dimensions and using correlations
calculated with
relatively small
samples, CCA can often
find
directions of high correlation
but with
little variance, thereby extracting
a spurious
leading
CCA mode
, as illustrated.
Figure: With the ellipses denoting the data clouds
in the two
input spaces
, the dotted lines illustrate
directions with little variance but
by chance
with
high correlation (as illustrated by the perfect
order in
which the
data points 1, 2, 3 and 4 are
arranged in the x and y spaces).
Since CCA finds
the
correlation
of the data points
along the dotted lines to
be higher
than that
along the dashed lines (where the data points a,
b,
c and
d in the x

space are ordered as b, a, d and
c in the y

space),
the dotted
lines are chosen as
the
first
CCA mode.
Maximum covariance analysis (MCA),
looks
for
modes
of maximum
covariance instead of
maximum correlation,
and would select the
dashed lines over the dotted lines since the
length of the lines
do count
in the covariance
but not in the
correlation.
It can be shown that the MCA problem can be
derived from CCA by pre

filtering the data using
Principal Components (EOFs) of the data.
But a more straightforward derivation is obtained
using a different normalization before using the
method of Lagrange multipliers.
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