Transcript Powerpoint

BioSS reading group
Adam Butler, 21 June 2006
Allen & Stott (2003) Estimating signal
amplitudes in optimal fingerprinting, part I:
theory. Climate dynamics, 21, 477-491.
1: Introduction
• Optimal fingerprinting: statistical methods
for climate change detection & attribution
• Attempt to assess the extent to which spatial
and temporal patterns in observed climate
data are related to corresponding patterns
within outputs generated by climate models
• Assume climate variability independent of
externally forced signals of climate change
“attribution of observed climate change to a given
combination of human activity and natural
influences… requires careful assessment of
multiple lines of evidence to demonstrate, within a
pre-specified margin of error, that the observed
changes are:
• unlikely to be due entirely to natural variability
• consistent with the estimated responses to the
given combination of anthropogenic and natural
forcing; and
• not consistent with alternative explanations of
recent climate change that exclude important
elements of the given combination of forcings.”
The current paper
• Optimal fingerprinting is just a particular
take on multiple regression
• The current paper attempts to deal with one
element of climate model uncertainty
• Does this by replacing Ordinary Least
Squares with Total Least Squares: a
standard approach to “errors-in-variables”
Model uncertainty
• A+S define sampling uncertainty to be “the variability in the model-simulated response
which would be observed if the ensemble of
simulations were repeated with an identical model
and forcing and different initial conditions…”
• They argue this limited definition is
difficult to generalise in practice...
Avoiding model uncertainty
• Restrict attention to mid c21 estimates signal-to-noise ratio by then so high that
inter-ensemble variation is unimportant
• Use a purely correlative approach
• Use a noise-free model such an energy
balance model to simulate response pattern
• Use a large number of ensemble runs
Problems
• Standard optimal fingerprinting uses OLS,
estimates can be severely biased towards
zero when errors in explanatory variables
• Bias particularly problematic when
estimating upper limits of uncertainty
intervals (Fig. 1)
2.1: Optimal fingerprinting
m
• Basic model:
y   x ii  0  X  
i 1
• “Pre-whitening”: find a matrix P such that
E(P P )  I
T
T
• Rank of P typically [much] smaller than
length of y
• Minimise
~
~
~
  y  X
T T ~
~
r ( )   P P where
2
• P is IID noise, so the solution is:
  X P PX  X P Py
~
T
T
-1
T
T
(ordinary least squares)
• Compute confidence intervals based on
standard asymptotic distributions…
2.2: Noise variance unknown
• Ignoring uncertainty in estimated noise
properties can lead to “artificial skill”
• Solution: base uncertainty analysis on sets
of noise realisations which are statistically
independent of those used to estimate P
• Obtain such realisations from segments of a
control run of a climate model
• Elements are not mutually independent…
3. Errors in variables
m
• Extended model: y   (x i  i )i  0
i 1
• Pre-whitening:
(Pυ υ P )  I
Z  PX , Py  , where 
T
(Pυ υ P )  I
T
1 i
T
0 0
• Seek to solve (Fig. 2)
Z
true
v  (Z   ) v  0
T
3.1: Total least squares:
estimation of 
• Seek to minimise:
~ T
~
~~
~
L( v )   tr{( Z  Z) (Z  Z)} / 2 for Zv  0
~ T
~ ~
T
~
~
s ( v )  v ( Z  Z) ( Z  Z) v
2
T T ~
2
T~
~
~
~
s ( v )  v Z Zv   (1  v v )
2
• Solution to the corresponding eigenequation
1 ( s 2 )
T ~
2~
 Z Zv   v
~
2 ( v)
s2
takes to be smallest eigenvalue of ZTZ
& takes ~
v as the corresponding eigenvector
• Use a singular value decomposition
• Can show that
2
2
smin ~  k  m
2
smin
“…in geometric terms minimising s2 is
equivalent to finding the mdimensional plane in an m/-dimensional
space which minimises the sum
squared perpendicular distance from
the plane to the k points defined by the
rows of Z…”
(Adcock, 1878)
3.2: Total least squares:
unknown noise variance
• If the same runs are used to derive P and to
construct CIs about estimates of  then
uncertainty will again be underestimated
• As in standard Optimal Fingerprinting, can
account for uncertainty in noise variance by
using a set of independent control runs…
3.3: Open-ended confidence
intervals
• The  quantify the ratio of the observed to
the model-simulated responses
• In TLS we estimate the angle of the slope
relating observations to model response
• Can obtain highly asymmetric confidence
intervals when transform back to  scale via
tan(slope)
- intervals can even contain infinity
4. Application to a chaotic system
• Non-linear system of Palmer & Lorenz,
which corresponds to low-order
deterministic chaos:
dX
dt
 X  Y  f 0 cos 
dY
dt
  XZ  rX  y  f 0 sin 
dZ
dt
 xy  bZ
Some properties of the Palmer model –
•
•
•
Radically different properties at differ
aggregation levels (Figs. 3 & 4)
Sign of response in X direction depends
on the amplitude of the forcing (Fig. 5)
Variability at fine resolution changes due
to forcing with a plausible amplitude, but
variability at coarser resolution does not…
A+S choose this system because:
• it is a plausible model of true climate “…Palmer (1999) observed that
climate change is a nonlinear system
which could also thouht of as a change in
the occupancy statistics of certain
preferred ‘weather regimes’ in response
to external forcing…”
• optimal fingerprinting may be expected
to have problems with the nonlinearity
•
Use the Palmer model to simulate 1) pseudo-observations y under a linear
increase in forcing from 0 to 5 units
2) spatio-temporal response patterns X
for a set of ensemble runs
3) The level of internal variability using
an unforced control run from the model
• Investigate performance of OLS and TLS,
for different numbers of ensembles and
different averaging periods (50Ld or 500Ld)
• Figure 6: look at the (true) hypothesis =1
• OLS consistently underestimates observed
response amplitude for small number of
ensembles
5. Discussion
• Promoted as an approach to attribution
problems when few ensembles are available
• Most relevant for low signal-to-noise ratio
• Linear: relies on assumption that forcing
does not change level of climate variability
• Good performance relative to OF with OLS
in simulations under deterministic chaos