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BioSS reading groupAdam 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
• Basic model:
• “Pre-whitening”: find a matrix P such that
• Rank of P typically [much] smaller than length of y
Xxm
iiiy
10
IPPE )( TT
• Minimise
• P is IID noise, so the solution is:
(ordinary least squares)
• Compute confidence intervals based on standard asymptotic distributions…
~~ where~~)~
(2 XyPP TTr
PyPXPXPX TTTT ~ -1
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
• Extended model:• Pre-whitening:
• Seek to solve (Fig. 2)
m
iiiiy
10)( x
IPP
IPPPyPXZ
)(
)( where, ,
00
1TT
TTi
υυ
υυ
0)(true vZvZ
3.1: Total least squares: estimation of
• Seek to minimise:
)~~1(~~)~(
~)~
()~
(~)~(
~~for 2/)}
~()
~{(tr)~(
22
2
vvvZZvv
vZZZZvv
0vZZZZZv
TTT
TT
T
s
s
L
• Solution to the corresponding eigenequation
takes s2 to be smallest eigenvalue of ZTZ & takes as the corresponding eigenvector
• Use a singular value decomposition
• Can show that
vvZZv
~~)~(
)(
2
1 22
Ts
v~2mins
22min ~ mks
“…in geometric terms minimising s2 is equivalent to finding the m-dimensional 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:
bZxy
fyrXXZ
fYX
dtdZ
dtdY
dtdX
sin
cos
0
0
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