Inferring and predicting global temperature trends

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1 Craig Ansley, NZ & Piet de Jong, Macquarie University, Sydney June 29, 2012

2 Temperature records GISS, CRU and UAH C year = 3859 monthly observations Many other time series like this available of differing lengths, frequency, relatedness (tree thickness), missing data properties, validity (urban heat islands), comprehensiveness etc. (Selection bias)

3 Temperature trends Monthly with northern/southern hemisphere seasonality issues removed Different starts for the 3 series all end at 2010 All standardized on same (arbitrary) level even though they are centred differently (see GLS later) Noisy but each appears to suggest (global) warming Decreasing volatility? Evidence of this sort is basis for $trillions proposed taxes/expenditure on climate change mitigation strategies What really is the trend/slope out into the future and associated confidence intervals?

4 Overview Time series analysis done on such climatic time series is typically inadequate an open to much criticism Many, many people laying in on such analyses similar to stock market chartism See for example the various websites: climateaudit.org, realclimate.org etc etc for the bitterness of the controversies. Most statistical analyses (even by the climate scientists ) ignore the time series literature in particular as well as the broader statistical literature (significance, selection biases, preprocessing issues often ignored/swept away). We aim to partly critique existing approaches and provide a more proper approach. Proper approach is based on the ideas and methods initiated, motivated, inspired and developed by Andrew Harvey. Area is controversial, politicised, at times bitter. Warrants attention from proper time series analysts.

5 Mannian smoothing after Mann (2004) This may be typical what is goes in climatic time series analysis. To smooth a climate series: Use a Butterworth filter to smooth the time series Where the filter runs into either end of the series just mirror the time series horizontally about t =0ort = n. To preserve trend/slope, mirror the mirrored series vertically about y = y n. ( Double mirroring ) Result is a smoothing procedure, unhampered by end effects, that properly resolves the trend Why Butterworth? What, are the statistical properties of such a procedure? Voodoo smoothing? ỹ n = y n always! How do these smoothing procedures relate to proper time series procedures?

6 Example series: Northern Hemisphere Temperature Index Mann(2008) Annual: 1850 Current C correlation year lag 1 1 correlation correlation lag lag

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9 Harvey & Trimbur (2003)

10 Connection between Butterworth and Components Model summary Components Model m order of differencing q = σ 2 ζ /σ2 Butterworth m Slope of the gain function Determine window half width: 40 years? λ 0 equal to reciprocal of window half width Determine q = {2sin(λ 0 /2)} 2m

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15 Note: End effects handled properly above but improperly by Mann etc

16 Busetti Harvey (2007) Model Suppose stochastic trend (m = 2) + AR(1)+WN y t = µ t + a t + λ t, µ t+1 = µ t + s t, t (0,σ 2 ) s t+1 = s t + η t, a t+1 = ρ a t + θ ν t, η t,ν t (0,σ 2 ) Note: λ and θ are noise to signal parameters. σ 2 is scale parameter: not of interest Imagine model holds for 3 series with µ t, a t common and t specific with λ equal to θ times e r1, e a + b t or e r 3 µ t to be inferred/predicted: inference/prediction is pooled Each series shifted with starting conditions to level each series on the common µ t.how?

17 Forecasts + Error Bounds based on KF/SF slope level 16 C C/month year year Pooled estimator: > 3 series unlikely to narrow bounds No distinction between inferring and prediction End effects handled properly through use of KF/SF MIT Professor Richard Lindzen: no statistically significant warming since 1995 MIT President Susan Hockfield: climate change accelerating.

18 Technicalities: starts and shifts KF/SF dealt with all estimation/prediction issues including the problem of end effects (vs Mannian smoothing) KF/SF dealt with different starting points through use of different starting conditions: Estimates of shifts and initial conditions Shift CRU GISS UAH Jan level Jan slope estimate std. dev Estimation in terms of ln(θ) andln(λ i )=ln(θ)+r i, i =1, 3 and r 2 = a + bt allowing for changing volatility.

19 Model parameter estimates parameter vector ψ AR GISS CRU UAH AR coeff. ln(θ) r 1 a b r 3 ρ estimate std dev correlation matrix Estimates and cov matrix derived from KF/SF based likelihood evaluation Parameter estimate uncertainty used to simulate future scenarios

20 Predicted 2060 global temperature temperature c : change in temp exceedence probability Figure: Forecast February 2060 temperature. The left panel plots the forecast versus the estimate of the 2010 slope. Different points correspond to different choices of ψ. The right panel indicates exceedence probabilities with the flatter curve assuming ψ = ˆψ and the steeper curve factoring in the uncertainty regarding ψ.

21 Is the model adequate? Stress tests level slope p-value p-value year year Figure: Left panel display p values of estimated shocks introduced to the level while the right hand side is p values for the slope. Largest p values for shocks to s t all occur in the period indicating the model is under most stress from rapidly increasing temperatures during this period. Little evidence that shocks are warranted to the slope post 1977.

22 Conclusions real time series analysts should step up to the plate Professor Andrew Harvey has been instrumental in inspiring, developing, applying, exploring, justifying a very useful and powerful approach to time series analysis rooted in the statistical paradigm This approach is largely if not completely ignored by the climate science community The approach is completely applicable and useful for the extensive set of climate time series More should be done to muscle in on this territory.