DETECTION AND ATTRIBUTION

Transcription

1 DETECTION AND ATTRIBUTION Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC , USA 41st Winter Conference in Statistics Statistics and Climate Storhonga, Sweden March

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9 Basic idea: How can we measure the agreement between an artificially generated climate signal from a climate model, and real data as measured by surface observation stations or satellites? Both observations and model data aggregated into both the observational and model-generated data are aggregated into grid cells, but several thousand of them. Example: Use 5 o latitude and longitude grid cells, implying 36 latitude classifications (from 90 o N to 90 o S), and similarly 72 longitude increments, giving a total of 2,592 grid cells. 9

10 Use l to denote the number of grid cells. One common kind of study is to generate m possible response patterns x k, k = 1,..., m using the climate model, each one an l-dimensional time series corresponding to the response to one particular kind of forcing factor. For example, x 1 may be the response due to the increase in carbon dioxide, x 2 the response due to changes in sulfate aerosols, x 3 the response due to change in solar forcing, x 4 the response due to volcanic eruptions. Assuming that the observed climate signal is a linear combination of the components due to different forcing factors, this suggests a model of the form y = Xβ + u, where y is l 1, X = ( x 1... x m ) is a l m matrix consisting of all the modeled responses to forcing factors, and β is the vector of coefficients which we are trying to estimate. 10

11 Here β 1 could be the main component of interest (the greenhouse gas effect), while the full family of regression coefficients β 1, β 2,... is needed to determine the influence of all the components. We say that the greenhouse gas signal is detected if we can calculate an estimate and standard error for β 1, that lead to rejection of the null hypothesis H 0 : β 1 = 0. Given that we have detected a signal, we then estimate all the coefficients β 1, β 2,... to attribute the observed climate change to the different components. 11

12 If the residual vector u has covariance matrix C N, then the optimal generalized least squares (GLS) estimator of β is and ˆβ = (X T CN 1 X) 1 X T CN 1 y, (1) Var{ˆβ} = (X T C 1 N X) 1. (2) The crucial difficulty is that C N is unknown and therefore we need some reliable method of estimating it. One way of deriving (1) and (2) from the standard ordinary least squares (OLS) regression equations is to define a matrix P so that P C N P T = I l, and then to write the regression equation in the form P y = P Xβ + P u where the covariance matrix of P u is I l. The OLS estimator for β is then ˆβ = (X T P T P X) 1 X T P T P y with covariance (X T P T P X) 1. But if P and C N are invertible we will have P T P = CN 1, which leads to (1) and (2). 12

13 Usual approach in climate studies: take model observations from runs of the GCM in stationary conditions without forcing factors, typically 1,000 2,000 years long. With observations Y N from N replications of unforced model runs centered to 0 mean, estimate Ĉ N = 1 N Y NY T N. (3) Difficulty with (3): typically l > N and so Ĉ N is a singular matrix. Even though this in itself is not a fatal objection, e.g. valid versions of (1) and (2) can be given involving generalized inverses, the real difficulty is that with so many degrees of freedom in the covariance function, the low-amplitude components of covariance are not reliably estimated, and therefore, any direct attempt to apply (2) by substituting Ĉ N for C N will give poor estimates for ˆβ and innaccurate estimates of uncertainty. 13

14 Solution: define a κ l transformation matrix P (κ) corresponding to the κ largest EOFs (or principal components) of Y N, so that P (κ) Ĉ N P (κ)t = I κ, and then define the OLS regression equation which leads to an estimator P (κ) y = P (κ) Xβ + P (κ) u, β = (X T P (κ)t P (κ) X) 1 X T P (κ)t P (κ) y (4) and estimated covariance (X T P (κ)t P (κ) X) 1. (5) 14

15 Refinements (Allen and Tett 1999) 1. Estimate of covariance likely underestimate, alternative Var{ β} ˆ = (X T ĈN 1 X) 1 X T Ĉ 1 1 N C 1 N2 CN 1 X(X T Ĉ 1 1 N X) 1 1 Ĉ N1 estimate of C N from EOF construction, Ĉ N2 independent estimate from separate run of the climate model. 2. Correct for serial correlation 3. Effect of random variation in X? 4. How to choose κ? Proposed residual test r 2 = (y X β) T Ĉ 1 N (y X β) χ 2 κ m, Rejection of r 2 taken as indication that κ is too large. 15

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17 Human influence on climate has been detected in surface air temperature, sea level pressure, tropopause height, ocean heat content, etc...but not, so far, precipitation. Models predict... Anthropogenic forcing implies increase in global mean precipitation Increasing at high latitudes Decreasing at sub-tropical latitudes We compare increases in land precipitation with changes in 14 climate models. 17

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19 Trends over two periods Focus on 40 o S 70 o N Climate models: ANT (27 simulations from 8 models), anthropogenic signals only (GHGs+aerosols) ALL (50 simulations from 10 models), includes volcanic, solar NAT4 (15 simulations from 4 models), natural external forcing only Also look at ANT4, ALL4 (same as ANT, ALL but restricted to same 4 models as are in NAT4) 19

20 Methods Analyzed trends in annual zonal mean precipitation anomalies relative to Trends in observed and simulated precipitation compared using optimal fingerprint methodology. Linear trends from observations and averages of multi-model simulations exhibit important areas of consistency in spatial distribution of precipitation change. But Inconsistency on o N High level of uncertainty 20

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23 Estimated combined effects of anthropogenic and natural forcings by regressing observed trends on anthropogenic trends Response to ALL or ANT was detected in observed trends, but not response to NAT alone. To separate contribution from ANT and NAT, need twosignal attribution analysis Response to anthropogenic forcing separable from that due to natural forcings and internal variability The two responses can be reliably separated, though the Allen-Tett for internal residual consistency failed (underestimate of internal variability?) 23

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26 Detection of anthropogenic influence is robust... Little uncertainty about sign of trend Robust to different constructions of dataset (e.g. gridding, treatment of missing observations) Robust to use of signal patterns from subsets of models Consistent with our understanding of mechanisms but... Internal variability estimate is smaller than variability estimated from observations Discrepancy in magnitude of changes (generally, models underestimate observed trends) 26

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29 Methods (Allen and Stott 2003) y = m i=1 C N = E{uu T }. x i β i + u = Xβ + u, Prewhitening matrix P (κ N), E{P uu T P T } = I κ. Estimate β minimizes r 2 ( β) = (P X β P y) T (P X β P y) = ũ T P T P ũ, β = (X T P T P X) 1 X T P T P y = F y. Rows of F are distinguishing fingerprints. Also r 2 min χ2 κ m. 29

30 If we ignore uncertainty in estimate of noise variance, β N[β, V ( β)], V ( β) = (X T P T P X) 1. Equivalently, ( β β) T (X T P P T X)( β β) χ 2 m. Use to construct joint confidence intervals for β. Also compute (one-dimensional) confidence interval for any linear function c T β. 30

31 Accounting for uncertainty in estimated noise patterns Estimation of P based on a set of noise realizations Ŷ 1. Take a second independent sample Ŷ 2. v columns of Ŷ 2 not independent DF based on Ŷ 2 is v 2 < v. Typically use overlapping segments, but set v 2 to be 1.5 times the number of nonoverlapping segments r 2 min = κ ˆV ( β) = F T Ŷ 2 Ŷ T 2 F v 2, ( β β) T ˆV ( β) 1 ( β β) mf m,v2, i=1 (P ũũ T P T ) i,i (P Ŷ 2 Ŷ T 2 P T ) i,i /v 2 (κ m)f κ m,v2. Final distributional result approximate. 31

32 Total Least Squares Early regression-based approaches to detection and attribution (e.g. Hegerl et al. (1996), Allen and Tett (1999)) relied on standard regression equations of the form Y = m j=1 β j X j + η (6) where Y is the observational record (e.g. a vector of trend in temperature means), X 1,..., X m are the signals from m climate models, and η is an error term. Instead of ordinary least squares, Allen and Stott (2003) proposed to fit (6) by total least squares, which allows for errors in the X j s as well as Y. Technique extended by Huntingford, Stott, Allen and Lambert (2006). I want to propose a (non-bayesian) alternative approach, stemming from Gleser (1981). 32

33 Gleser s formulation of errors in variables (EIV) x i = ( xi1 x i2 ) = ( ui1 u i2 ) + ( ei1 e i2 ), (7) x i1 and x i2 observations of dimensions p and r respectively, u i1 and u i2 are true unobserved signals, e i1 and e i2 noise with covariances σ 2 I p and σ 2 I r. Also MLE: choose B and u 11,..., u n1 to minimize Q = 1 2σ 2 (x i1 u i1 ) T (x i1 u i1 ) i u i2 = Bu i1. (8) i (x i2 Bu i1 ) T (x i2 Bu i1 ). (9) 33

34 Computing the MLE Choose B to minimize Q = 1 2σ 2 (x i2 Bx i1 ) T (I r + BB T ) 1 (x i2 Bx i1 ). (10) i (also called generalized least squares by Gleser) 34

35 Distribution Theory Assume estimator ˆB n based on n observations, U 1 the p n matrix whose columns are u 11,..., u n1. [Assumption A:] e i are i.i.d. random vectors with mean 0 and common covariance matrix σ 2 I p+r [Assumption C:] = lim n n 1 U 1 U1 T exists, positive definite. [Assumption E:] The cross-moments of the common distribution of the e i are identical, up to and including moments of order 4, to the corresponding moments of the multivariate normal distribution with the same mean and covariance matrix. Then the elements of the n 1/2 ( ˆB n B) have an asymptotic rpdimensional normal distribution with mean 0 and the covariance between the (i, j) and (i, j ) elements is given by σ 2 [σ 2 1 (I p + B T B) ] jj [I r + BB T ] ii. 35

36 Application to Climate I Identify x i2 with Y (single observation, dimension r) Identify x i1 with (X 1,..., X m ) T (dimension rm). Also B = ( β 1 I r β 2 I r... β m I r ). (11) GLSE chooses β 1,..., β m to minimize S = (Y j β j X j ) T (Y j β j X j ) 1 + βj 2. (12) Equivalent to Allen and Stott (2003). In principle, we could use Gleser s theory to approximate the asymptotic (co-)variances of the estimators. However, here n =

37 Application to Climate II Apply Gleser s formulation to individual years. Assume an EOF rotation applied so that x i2 is r-dimensional observation for year i, x i1 is rm-dimensional vector of model runs for year i. Assume: x i2 = u i2 + e i2 (r 1), x i1 = u i1 + e i1 (p 1), e i2 independent with mean 0 and covariance matrix σ 2 I r, e i1 independent with mean 0 and covariance matrix σ 2 I p, u i2 = Bu i1 where B = ( ) β 1 I r β 2 I r... β m I r. GLSE chooses β 1,..., β p to minimize Q = 1 2σ 2 i (Y i j β j X ij ) T (Y i j β j X ij ) 1 + βj 2. (13) or let B be completely arbitrary??? (Needs rm < n, so we really would have to use a low-order EOF decomposition for this.) 37

38 Comments / Caveats Gleser s theorem not directly applicable for various reasons, but could adapt. Covariance matrix C now needed for individual years, not linear trends. Simpler than earlier discussion. Could also examine whether forced model runs are consistent with estimated covariances. As presently formulated, doesn t allow for (temporal) autocorrelation. 38

39 Detection and Attribution Analysis for Precipitation We re-analyze the data from the paper of Zhang et al., Nature 2007, discussed earlier. 39

40 Datasets Latitude-band averages 75 years 11 ten-degree latitude bands (from 40 o S to 70 o N) One observational dataset Several forced model runs under different forcings (ANT, ALL, NAT) Control runs typically 6 control runs of 1000 years broken up into 75-year segments 40

41 Traditional D&A Analysis 1. For each of 11 latitude bands, linear trend is calculated by OLS, including control runs 2. Covariance matrix from control run used to define the weights for EOF decomposition 3. Truncate the EOFs, e.g. by Allen-Tett test (anticipate 5 EOFs retained 4. Same formation of linear trends and EOFs applied to forced model runs to develop a set of comparison vectors. 5. Optimal weights for D&A defined by regressing observational data EOFs on forcing model EOFs. Calculate standard errors, tests of significance, etc. 41

42 Estimation of the Covariance Matrix The data we have available consists of 17 runs of 75 years data from control runs of HADCM3. The data are 11-dimensional because of the 11 latitude bands. The covariance matrix C is defined to be the covariance matrix of estimators of linear trends over the 75 years. Possible methods of estimating C include: 42

43 Method 1 Calculate linear trends for each of the 11 latitude bands and each of the 17 runs; then estimate C (say, as Ĉ 1 ) as the sample covariance matrix Ĉ 1 = (B B) T (B B) (14) 16 where B is the matrix of linear trend estimates and B is the mean of B over all rows. 43

44 Method 2 Suppose Y ijk is observation in year i from latitude band j and control model run k. Suppose that, for each i and k, {Y ijk, j = 1,..., 11} is an independent vector from N[0, V ], where V is the covariance matrix associated with a single year of control data. Linear trend estimator for region j and model run k given by B jk = 75 i=1 for weights w i. Then for any two regions j, j, Cov(B jk, B j k ) = 75 w i Y ijk (15) i=1 If ˆV is sample covariance matrix, estimate wi 2 V jj. (16) Ĉ 2 = ( w 2 i )ˆV. (17) 44

45 Method 3 Correct for autocorrelation in method 2. Assume h-order autocovariances negligible for h > H some given H. Then we can modify (16) to Cov(B jk, B j k ) = H h= H 75 i=1 w i w i+h Cov(Y ijk, Y (i+h)j k ) (18) where w i+h is defined to be 0 if i + h < 1 or i + h > 75. If ˆv hjj denotes sample covariance of Y ijk and Y (i+h)j k (computed from all control years of data) then this implies a covariance estimator Ĉ 3 with entries Ĉ 3 (j, j ) = H h= H 75 i=1 w i w i+hˆv hjj. (19) 45

46 Looking for autocorrelations in the control runs Sample autocorrelations and partial autocorrelations were calculated from the control data. They were considered statistically significant if outside the range ± 2 n These results suggest that significant autocorrelations do not persist beyond lag 2. However, as a test, the estimator Ĉ 3 was computed for each of H = 1, 2, 3, 4, 5. 46

47 Testing one estimated covariance matrix against another Advantage of Ĉ 1 makes no assumption at all about the autocorrelation structure, but it s based on a small sample (17). Ĉ 2 or Ĉ 3 far larger sample size, but maybe incorrect assumptions. Suggests a hypothesis test that of either H 0 : C = Ĉ 2 or H 0 : C = Ĉ 3, treating Ĉ 1 as a sample covariance matrix. Korin (1968) proposed likelihood ratio test statistic { ( ) } Σ0 L = N log p + tr(sσ 1 S 0 ) (20) where S is a p p sample covariance matrix with N degrees of freedom, and Σ 0 is the null hypothesis value of the true covariance matrix. Sampling distributions by Bartlett correction to the χ 2 test, an alternative approximation proposed by Korin, or simulation. 47

48 Application Ĉ 1 against Ĉ 2 (as null hypothesis): reject under Bartlett s approximation, accept under Korin s or simulation (p-values.047,.077,.11) but maybe not a clear-cut result. However for Ĉ 1 against Ĉ 3 (as null hypothesis), with any of H = 1, 2, 3, 4, 5, accept null hypothesis. As a specific example, for H = 2, Bartlett s, Korin s and the simulation test yield p- values 0.33, 0.42 and Conclusion: matter. Ĉ 3 seems fine, and the actual value of H doesn t 48

49 On the other hand, where we see real differences among the estimators is in computing their inverses. As an example, the diagonal entries of Ĉ1 1 are: The corresponding values for Ĉ 1 3 with H = 0 5 are: H=0: H=1: H=2: H=3: H=4: H=5: Evidently, there is much more stability in any of the versions of Ĉ3 1 (raising the question that maybe no dimension reduction is actually needed??) 49

50 Results Applied to D&A 50

51 Analysis using full data matrix (11 PCs) Analyses using Ĉ 1, Ĉ 2, Ĉ 3 (H = 2) Models ˆβ SE t statistic P-value ALL ANT NAT Models ˆβ SE t statistic P-value ALL ANT NAT Models ˆβ SE t statistic P-value ALL ANT NAT

52 Analysis using 5 PCs Analyses using Ĉ 1, Ĉ 2, Ĉ 3 (H = 2) Models ˆβ SE t statistic P-value ALL ANT NAT Models ˆβ SE t statistic P-value ALL ANT NAT Models ˆβ SE t statistic P-value ALL ANT NAT

53 Conclusions (preliminary) Using the full data matrix, there is a lot of discrepancy between the results based on Ĉ 1 and either of Ĉ 2 or Ĉ 3, but the results comparing Ĉ 2 and Ĉ 3 are quite close Using 5 PCs, there is much better agreement between the three covariance models, but results are quite different from those using all 11 PCs Based on all 11 PCs with either Ĉ 2 or Ĉ 3, we may claim to have attribution for the all forcings model, but not for either ANT or NAT. 53

54 More on Detection and Attribution Bayesian approaches: Levine and Berliner (J. Clim 12, , 1999) Levine, Berliner and Shea (J. Clim 13, , 2000) Berliner and Kim (J. Clim 21, , 2008) Detection and attribution applied to multi-model ensembles: Lopez, Tebaldi, New, Stainforth, Allen and Kettleborough (J. Clim 19, , 2006) 54

55 References Allen MR and Stott PA (2003), Estimating signal amplitudes in optimal fingerprinting. Part I: Theory. Clim. Dyn. 21, Allen MR and Tett SFB (1999), Checking for model consistency in optimal fingerprinting. Climate Dynamics 15, Berliner LM, Levine RA and Shea DJ (2000), Bayesian Climate Change Assessment. Journal of Climate 13, Berliner LM and Kim Y (2008), Bayesian design and analysis for superensemble-based climate forecasting. Journal of Climate 21, Gleser LJ (1981), Estimation of a multivariate Errors in Variables regression model: Large sample results. Annals of Statistics 9, Hasselmann K (1997), Multi-pattern fingerprint method for detection and attribution of climate change. Climate Dynamics 13,

56 Hegerl GC and North GR (1997), Comparison of statistical optimal approaches to detecting anthropogenic climate change. Journal of Climate 10, Hegerl G, von Storch H, Hasselmann K, Santer BD, Cubasch U and Jones PD (1996), Detecting greenhouse-gas-induced climate change with an optimal fingerprint method. Journal of Climate 9, Levine RA and Berliner LM (1999), Statistical Principles for Climate Change Studies. Journal of Climate 12, Lopez A, Tebaldi C, New M, Stainforth D, Allen M and Kettleborough J (2006). Two approaches to quantifying uncertainty in global temperature changes. Journal of Climate 19, Santer BD, Taylor KE, Wigley TML, Johns TC, Jones PD, Karoly DJ, Mitchell JFB, Oort AH, Penner JE, Ramaswamy V, Schwarzkopf MD, Stouffer RJ and Tett S. (1996), A search for human influences on the thermal structure of the atmosphere. Nature 382,