EVA Tutorial #2 PEAKS OVER THRESHOLD APPROACH. Rick Katz

Transcription

1 1 EVA Tutorial #2 PEAKS OVER THRESHOLD APPROACH Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO USA Home page: Lecture:

2 2 Outline (1) Generalized Pareto Distribution (2) Return Levels (3) Choice of Threshold (4) Declustering (5) Peaks over Threshold/Point Process Representation (6) Point Process Approach under Stationarity (7) Point Process Approach under Nonstationarity

3 3 (1) Generalized Pareto Distribution Analogue to max stability -- X random variable Y = X u excess over high threshold u, conditional on X > u Then Y has an approximate generalized Pareto (GP) distribution for large u with cdf: H(y; σ*, ξ) = 1 [1 + ξ (y/σ*)] 1/ξ, y > 0, 1 + ξ (y/σ*) > 0 σ* > 0 scale parameter [alternative notation σ*(u)] ξ shape parameter (same interpretation as for GEV dist.)

4 4 (i) ξ = 0 (exponential type) Light tail

5 5 (ii) ξ > 0 (Pareto type) Heavy tail

6 6 (iii) ξ < 0 (beta type) Bounded tail [ y < σ* / ( ξ) or x < u + σ* / ( ξ) ]

7 7 Connection between GP & GEV -- Maximum M n u if none of X t 's exceeds u, t = 1, 2,..., n Scaling -- Memoryless property of exponential dist. (scale parameter σ*) Pr{Y > y + y Y > y } = Pr{Y > y} = exp[ (y/σ*)], y, y > 0 -- Stability of GP distribution Lose memoryless property (need to rescale) Suppose excess Y over threshold u has exact GP[ξ, σ*(u)] dist. Excess over a higher threshold u > u has GP[ξ, σ*(u )] dist.: σ*(u ) = σ*(u) + ξ(u u), u > u

8 8 Hurricane damage (1995 US$) -- Adjusted data (Remove trends in societal vulnerability),

9 9 Fit GP distribution to damage from individual storms -- Use threshold of u = 6 billion $ -- Parameter estimates and standard errors Parameter Estimate (Std. Error) Scale σ * bill. $ (1.817 bill. $) Shape ξ bill. $ (0.341 bill. $)

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12 12 (2) Return Levels GP distribution -- Quantile of GP distribution (Invert cdf) x(p) = H 1 (1 p; σ*, ξ) = (σ* / ξ) (p ξ 1), 0 < p < 1 -- Complication Need to take into account exceedance rate of threshold (to provide return level corresponding to interpretable return period)

13 13 Hurricane example -- Lack of structure in data file Reasonable to input average number of damaging hurricanes per year (in place of no. of obs. per year ) 144 / hurricanes per year Estimated 20-yr return level: 17.6 billion $ 95% confidence interval for 20-year return level (based on profile likelihood): 12.2 billion $ < x(0.05) < 35.6 billion $

14 14 (3) Choice of Threshold Invariance of GP above threshold -- Same shape parameter ξ -- Modify scale parameter as threshold u varies σ*(modified) = σ*(u) ξ u -- Check for stability in parameter estimates as vary threshold Trade-off -- Better GP approximation for higher threshold -- More reliable estimation for lower threshold -- Lack of automatic procedure

15 Hurricane damage example 15

16 16 (4) Declustering Declustering procedures -- Runs declustering Clusters separated by at least r consecutive observations below threshold (r = 1, 2,...) Model cluster maxima (instead of individual cluster members) Concept of extremal index θ, 0 < θ 1 1/θ mean cluster size θ = 1 corresponds to lack of clustering at high levels Degree of clustering at high levels increases as θ decreases

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18 18 Phoenix minimum temperature -- Lower tail vs. upper tail Model lower tail as upper tail after negation -- Fit GP distribution to declustered data Threshold u = 73 F & use runs declustering (r = 1) De-clustering Parameter Estimate (Std. Error) None Scale σ* F (0.303 F) r = F (0.501 F) None Shape ξ (0.049) r = (0.079) Mean cluster size: 262 / days (estimate of θ 0.44)

19 (5) Peaks over Threshold/Point Process Representation 19

20 20 (i) Poisson-GP Model Peaks over Threshold or Partial Duration Series Poisson process for exceedance of high threshold -- Event X t > u Rate parameter λ > 0 Number of events in [0, T] has Poisson distribution with parameter λt GP distribution for excess over threshold -- Excess Y t = X t u, given X t > u Parameters ξ & σ*

21 21 (ii) Point process representation -- Subsumes Poisson-GP model -- GEV parameterization (connection to GEV) Can relate parameters of GEV(μ, σ, ξ) to parameters of point process (λ, σ*, ξ): (i) Shape parameter ξ identical (ii) ln λ = (1/ξ) ln[1 + ξ(u μ)/σ] (iii) σ* = σ + ξ(u μ) Note: Need time scale parameter h to take into account block size for GEV (e. g., h 1/ for daily data & annual maxima)

22 22 (6) Point Process Approach under Stationarity Stationarity -- Poisson-GP and point process approaches equivalent Can obtain same parameter estimates (indirectly through use of relationships between two parameterizations) -- Point process More convenient to quantify total uncertainty Easier to interpret (eliminate dependence of parameters on threshold)

23 23 Fort Collins daily precipitation Estimation Method Parameter Estimate (i) Poisson-GP Rate λ 10.6 per yr. (u = in) Scale σ* in Shape ξ (ii) Point process Location μ (u = in, Scale σ in h = 1/365.25) Shape ξ Verify that two sets of parameter estimates are consistent ln λ = (1/ξ) ln[1 + ξ(u μ)/σ] [vs. ln(10.6) 2.361] σ* = σ + ξ(u μ) (vs )

24 24 (7) Point Process Approach under Nonstationarity Poisson-GP model -- Introduce nonstationarity separately into two components Occurrence: Poisson process with rate parameter λ(t) Excess: GP distribution with parameters σ*(t) & ξ(t) Point process approach -- Introduce nonstationarity in GEV parameters μ(t), σ(t), ξ(t) Threshold u(t) can be time varying as well

25 25 Fort Collins precipitation example -- Threshold u = in (could be time varying as well) Length of year T days -- Poisson-GP model (i) Annual cycle in Poisson rate parameter ln λ(t) = λ 0 + λ 1 sin(2πt / T) + λ 2 cos(2πt / T) Parameter Estimate (Std. Error) Rate: λ λ (0.045) λ (0.049) LRT for λ 1 = λ 2 = 0 (P-value 0)

26 26 (ii) Annual cycle in scale parameter of GP distribution ln σ*(t) = σ 0 * + σ 1 * sin(2πt / T) + σ 2 * cos(2πt / T) Parameter Estimate (Std. Error) LRT Scale: σ 0 * σ 1 * (0.048) σ 1 * = σ 2 * = 0 σ 2 * (0.069) (P-value < 10 5 ) Shape ξ Q-Q plot: Transform non-stationary GP to exponential dist. ε t = [1/ξ(t)] ln{1 + ξ(t) [Y t / σ*(t)]}

27 27 -- Point process approach (u = in, h = 1/365.25) Annual cycles in location & scale parameters of GEV distribution: μ(t) = μ 0 + μ 1 sin(2πt / T) + μ 2 cos(2πt / T) ln σ(t) = σ 0 + σ 1 sin(2πt / T) + σ 2 cos(2πt / T) Parameter Estimate (Std. Error) LRT Location: μ in μ in (0.031 in) μ 1 = μ 2 = 0 μ in (0.043 in) (P-value 0) Scale: σ σ (0.028) σ 1 = σ 2 = 0 σ (0.034) (P-value 0) Shape ξ 0.182

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