Martin Roth: A spatial peaks-over-threshold model in a nonstationary climate

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Martin Roth: A spatial peaks-over-threshold model in a nonstationary climate

  1. 1. A spatial peaks-over-threshold model in a nonstationary climate Martin Roth (KNMI / EURANDOM)joint work with A. Buishand (KNMI), G. Jongbloed (TU Delft), A. Klein Tank (KNMI), and H. v. Zanten (TU Eindhoven) 16 February 2011
  2. 2. Goals estimate site specific quantiles / return levels assess the temporal trends in these quantiles reduce the estimation uncertainty by spatial pooling Inspired by the work of M. Hanel, A. Buishand and C. Ferro (2009) for the block maxima data.
  3. 3. Data gridded daily precipitation data of the winter (DJF) season over the Netherlands (E-OBS v. 5.0) for 1950 – 2010 some disadvantages but convenient, and problems might be smaller for the Netherlands, because of the high station density (a) mean winter maxima (b) event on December 3, 1960
  4. 4. Components of the Model GP distribution non-stationary Index flood IF POT model non-stat. threshold
  5. 5. GEV/GP distribution Generalized Extreme Value (GEV) distribution P (M ≤ x ) = Hξ ∗ ,σ∗ ,µ∗ (x ) −1/ξ ∗  exp − 1 + ξ ∗ x −∗ µ∗ , ξ ∗ = 0,  σ = µ∗ exp − exp(− x −∗ ) , ξ ∗ = 0,  σ Generalized Pareto (GP) distribution P (Y ≤ y |Y ≥ 0) = Gξ, σ (y )  −1/ξ 1 − 1 + ξy , ξ = 0,  = σ 1 − exp − y , ξ = 0, σ
  6. 6. GEV/GP Relation Assume GP distribution for the excesses and that the exceedance times follow a Poisson process, then the following relationship between the parameters of the GP and the GEV distribution can be shown: u − σ (1 − λ ξ ), ξ = 0, µ∗ = ξ u + σ ln(λ), ξ = 0, (1) σ∗ = σλξ , ξ ∗ = ξ, where u is the threshold in the POT model, and λ is the mean number of exceedances in one winter.
  7. 7. Index flood for POT I The index flood method assumes that all site specific distributions are identical apart from a site specific scaling factor, the index flood1 , i.e. for exceedances we get Xs P ≤ x |Xs ≥ us = ψ (x ) ∀s ∈ S , (2) ηs where Xs is a random variable representing the site-specific daily precipitation, us is the site specific threshold, ηs is the index flood and ψ does not depend on site s. 1 Hosking and Wallis (1997)
  8. 8. Index flood for POT II Index flood equals threshold We have ψ(us /ηs ) = P (Xs ≤ us |Xs ≥ us ) = 0 ∀s ∈ S and because ψ has a continuous distribution it follows that us /ηs has to be the lower endpoint of the support of ψ for every s ∈ S . This can be only true, if the index flood is a multiple of the threshold. Without loss of generality we can set ηs = us . Index flood also for the excesses Ys P ≤ y |Ys ≥ 0 = ψ (y ) ∀s ∈ S , (3) ηs where ψ(y ) := ψ(y + 1) is independent of site s.
  9. 9. Index flood for POT III Site specific threshold The τ-th quantile (τ >> 0.9) of the daily precipitation amounts is a natural choice for a site specific threshold. ⇒ λs will be approximately constant over the region. Restriction on the GP parameters The distribution of the scaled excesses has the following form: Ys P ≤ y |Ys ≥ 0 = Gξ s , us (y ) ≡ ψ(y ). σ (4) ηs s Therefore we have: σs ≡ γ, ξs ≡ ξ ∀s ∈ S . (5) us We refer to γ as the dispersion coefficient.
  10. 10. Index flood for BM Assuming constant λ, γ and ξ gives for the GEV parameters: ∗ ξs ≡ ξ λξ ∗ σ∗ γ −1 − 1 (1 − λ ξ ) , ξ =0 γs := s = ∗ 1 ξ ≡ γ∗ µs γ−1 −log(λ) , ξ = 0. Therefore the transformed parameters fullfil the IF assumption for BM data2 . This does not apply for the IF model for POT data proposed by Madsen and Rosbjerg (1997). 2 Hanel, Buishand and Ferro (2009)
  11. 11. Nonstationary Threshold The threshold is determined as the 0.96 linear regression quantile3 : (c) Mean of the threshold for the 1950–2010 pe- (d) Trend in the threshold for the 1950–2010 riod in mm period in % 3 Koenker (2005), Kysel´, Picek and Beranov´ (2010) y a
  12. 12. Nonstationary Version of the IF Model IF restrictions on the GP parameters σs (t ) ξ s (t ) ≡ ξ (t ), ≡ γ (t ). us (t ) Quantile estimates −1 α qα (s, t ) = us (t ) + Gξ (t ),σs (t ) 1 − λ γ (t ) us ( t ) · 1 − ξ (t ) [ 1 − ( λ ) − ξ (t ) ] α , ξ (t ) = 0, = us (t ) · 1 + γ(t ) ln(λ/α) , ξ (t ) = 0. Note the factorization into a time and site dependent index flood and a quantile function, which depends on time only.
  13. 13. Composite Likelihood simplified (not true) likelihoods4 (e.g. independence likelihood5 or pairwise likelihood) allows to assess for spatial dependence specify a certain structure for the parameters, e.g. ¯ γ(t ) = γ1 + γ2 · (t − t ), ξ (t ) = ξ 1 . maximize: S T I (θ ) = ∑ ∑ log fγ(t )us (t ),ξ (t ) (ys (t )) , s =1 t =1 ys (t )≥0 where fσ,ξ (y ) is the density of the GP distribution. 4 Varin, Reid and Firth (2011) 5 Chandler and Bate (2007)
  14. 14. Asymptotic Normality ˆ θI is asymptotically normal with mean θ and covariance matrix G −1 (θ ) Godambe (sandwich) information G ( θ ) = H ( θ )J −1 ( θ )H ( θ ) H (θ ) is the expected negative Hessian of I ( θ, Y ) Fisher information or sensitivity matrix J (θ ) is the covariance matrix of the score θ I ( θ, Y ) referred to as variability matrix In the independent case we have J (θ ) = H (θ ) ⇒ G (θ ) = H (θ )
  15. 15. Composite Information Criteria Composite likelihood adaptations of the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) AIC ˆ = −2 I (θI , Y ) + 2 dim(θ ), BIC ˆ = −2 I (θI , Y ) + log(n) dim(θ ), where dim(θ ) is an effective number of parameters, given by dim(θ ) = tr H (θ )G (θ )−1 , which is the true number of parameters in the case of independence.
  16. 16. Composite Likelihood Ratio (CLR) Test Cl adaptation of the likelihood ratio test W =2 I ˆ θM1 ; y − I ˆ θM0 ; y . The asymptotic distribution of W is given by a linear combination of independent χ2 variables, and can be determined using the Godambe information. Bootstrap transform to standard exponentials using the full model M1 sample monthly blocks of the whole region transform the sampled data back using the nested model M0
  17. 17. Composite Likelihood Ratio (CLR) Test Cl adaptation of the likelihood ratio test W =2 I ˆ θM1 ; y − I ˆ θM0 ; y . The asymptotic distribution of W is given by a linear combination of independent χ2 variables, and can be determined using the Godambe information. Bootstrap transform to standard exponentials using the full model M1 sample monthly blocks of the whole region transform the sampled data back using the nested model M0
  18. 18. Application I – Models and Information Criteria Table: IF models used Model dispersion γ shape ξ no trend γ1 ξ1 trend in dispersion ¯ γ1 + γ2 ∗ (t − t ) ξ1 trend in shape γ1 ¯ ξ 1 + ξ 2 ∗ (t − t ) Table: Information criteria for the IF models Model AIC BIC no trend 78387.28 78715.59 trend in dispersion 78435.60 78880.41 trend in shape 78333.28 78748.95
  19. 19. Application I – Models and Information Criteria Table: IF models used Model dispersion γ shape ξ no trend γ1 ξ1 trend in dispersion ¯ γ1 + γ2 ∗ (t − t ) ξ1 trend in shape γ1 ¯ ξ 1 + ξ 2 ∗ (t − t ) Table: Information criteria for the IF models Model AIC BIC no trend 78387.28 78715.59 trend in dispersion 78435.60 78880.41 trend in shape 78333.28 78748.95
  20. 20. Application II – Shape parameter Figure: shape parameter for different models (dotted – constant, dashed – linear trend, solid red – 20 year window estimates)
  21. 21. Application III – Significance Tests Trend in the GP parameters Table: p-values of the CLR-test against the IF model without trend Model asymptotic bootstrap trend in dispersion 82.9% 81.3% trend in shape 26.7% 12.2% Index flood assumption We compare the model without trend in the parameters with a model with site specific dispersion coefficient and common shape parameter with the bootstrap. We obtain a p-value of 0.103, i.e. the IF assumption must not be rejected.
  22. 22. Application IV – Uncertainty Figure: Estimated return levels and 95% pointwise confidence bands of the excesses distribution at the gridbox around De Bilt in 1980 (black – at site, red – IF)
  23. 23. Conclusions positive trends in the threshold are observed this leads to an increase in the scale parameter no change in the dispersion coefficient negative trend in the shape parameter not significant the uncertainty in the excesses distribution is cut in half
  24. 24. Further Research apply the method to climate model data validity of the bootstrap might be questionable use of pairwise likelihood could further decrease the uncertainty assess the uncertainty in the threshold and excess distribution together
  25. 25. Literature R.E. Chandler and S. Bate (2007), Inference for clustered data using the independence loglikelihood. Biometrika, 94(1):167–183. M. Hanel, T. A. Buishand, and C. A. T. Ferro (2009), A nonstationary index flood model for precipitation extremes in transient regional climate model simulations. J. Geophys. Res., 114(D15):D15107. J. R. M. Hosking and J. R. Wallis (1997), Regional frequency analysis. Cambridge University Press. Koenker, R. (2005) Quantile Regression. Cambridge University Press Kysel´, J., J. Picek, and R. Beranov´ (2010) y a Estimating extremes in climate change simulations using the peaks-over-threshold with a nonstationary threshold. Global and Planetary Change 72, 55-68 H. Madsen and D. Rosbjerg (1997), The partial duration series method in regional index-flood modeling. Water Resour. Res., 33(4):737–746. C. Varin, N. Reid, and D. Firth (2011), An overview of composite likelihood methods. Statistica Sinica, 21:5–42.

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