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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 ii 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 ) T ~ 2~ Z Zv v ~ 2 ( v) 2 s2 takes to be smallest eigenvalue of ZTZ & takes ~ v as the corresponding eigenvector • Use a singular value decomposition • Can show that 2 smin s 2 min ~ 2 k m “…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