PENULTIMATE APPROXIMATIONS FOR WEATHER AND CLIMATE EXTREMES. Rick Katz

Transcription

1 PENULTIMATE APPROXIMATIONS FOR WEATHER AND CLIMATE EXTREMES Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO USA Web site: Talk:

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3 R. A. Fisher,

4 L. H. C. Tippett,

5 Quotes Fisher and Tippett (1928): the case derived from the normal curve is peculiar for the extreme slowness with which the limiting form is approached even for samples of nearly a billion the penultimate form is still considerably different from the ultimate form

6 Outline (1) Motivation for Penultimate Approximations (2) Ultimate Extreme Value Theory (3) Penultimate Extreme Value Theory (4) Examples of Penultimate Approximations (5) Weather and Climate Extremes (6) Discussion

7 (1) Motivation for Penultimate Approximations Reasons for Lack of Attention -- Any benefit is automatic By always fitting generalized extreme value distribution (or generalized Pareto distribution) -- Improvement is not that great Reasons Deserve more Attention -- Improvement is large enough to matter -- Aid in physical interpretation

8 Maxima of Normally Distributed Random Variables -- Simulate independent pseudo random numbers from standard normal N(0, 1) distribution (Block size n = 100) -- Obtain maximum value (Repeat 40,000 times) -- Fit GEV distribution to sample of 40,000 maxima Estimated shape parameter: ξ (s.e ) Note: Ultimate approximation is ξ = 0

9 (2) Ultimate Extreme Value Theory Ultimate Extreme Value Theory X1, X2,..., Xn independent with cumulative distribution function (cdf) F Set Mn = max{x1, X2,..., Xn} -- Suppose that there exist constants μn and σn > 0 and such that Pr{(Mn μn) / σn x} G(x) as n Then G must a generalized extreme value (GEV) cdf; that is, G(x; μ, σ, ξ) = exp { [1 + ξ (x μ)/σ] 1/ξ } μ location, σ > 0 scale, ξ shape parameter

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11 Domain of Attraction -- Hazard rate (or failure rate ) hf (x) = F'(x) / [1 F(x)] Instantaneous rate of failure given survived until x Alternative expression: hf (x) = [log(1 F )]' (x)

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13 -- Von Mises sufficient condition F is in domain of attraction of Gumbel (i. e., ξ = 0) if (1/hF)' (x) 0 as x xf Here xf denotes upper endpoint for cdf F -- Expressions for normalizing constants (F in domain of attraction of Gumbel) μn = F 1 (1 1/n) Characteristic largest value σn = 1 / hf (μn)

14 (3) Penultimate Extreme Value Theory Penultimate Approximations -- Suppose cdf F in domain of attraction of Gumbel type (i. e., ξ = 0) -- Still preferable in nearly all cases to use GEV as approximate distribution for maxima (i. e., act as if ξ 0) -- Expression (as function of block size n) for shape parameter ξn Consider behavior of hazard rate for large block size n (Instead of ultimate limiting behavior)

15 -- Penultimate shape parameter (Block size n) ξn = (1/hF)' (μn) Here μn is characteristic largest value (centering constant) ξn can be viewed as derivative of σn (scaling constant) -- Shape parameter ξn 0 as block size n Rate of convergence/order of ultimate approximation: Typically depends on block size n in same way as ξn Rate of convergence/order of penultimate approximation: Relation more complex (sometimes square of ultimate rate)

16 (4) Examples of Penultimate Approximations Normal Distribution N(0, 1) -- Hazard rate hf (x) x, x large -- Characteristic largest value μn (2 log n) 1/2, n large -- Shape parameter ξn 1 / (2 log n) ξ (Recall obtained simulated value of ξ 0.094)

17 Stretched Exponential Distribution -- Traditional form of Weibull distribution (unit scale, shape parameter c) 1 F(x) = exp( x c ), x > 0, c > 0 -- Hazard rate hf (x) = c x c 1, x > 0 -- Characteristic largest value μn = (log n) 1/c -- Shape parameter ξn = (1 c) / (c log n) (i) c > 1 implies ξn 0 as n (i. e., Weibull type) (ii) c < 1 implies ξn 0 as n (i. e., Fréchet type)

18 -- GEV distribution fit to 40,000 simulated maxima

19 Lognormal Distribution -- Random variable Y has lognormal distribution (shape parameter σ) Y = exp(σx), σ > 0, where X N(0, 1) ξn = [σ (2 log n) 1/2 1] / (2 log n) ξn 0 as n -- Examples (Simulation of 40,000 maxima) (i) n = 100, σ = 2: ξ Simulated estimate of ξ (s.e ) (ii) n = 100, σ = 1: ξ Simulated estimate of ξ (s.e )

20 Cubed Gamma Distribution -- Random variable Y = X 3, where X has gamma dist. (shape parameter α)

21 (5) Weather and Climate Extremes (i) Transformations -- Use advocated to improve accuracy of extreme value approximations (Wadsworth et al. 2010) Straightforward to express effect of transformation on hazard rate -- Commonly used in weather and climate applications for other purposes

22 Example: Economic Damage from Hurricanes -- Hurricane intensity measure by wind speed V V assumed to have stretched exponential dist. (with shape par. cv > 1) -- Damage function Express damage L as function of V Power transformation typically assumed L V b, b > 0 (e. g., b = 3 argued on physical basis, in practice b > 3) So L has stretched exponential dist. (with shape par. cl = cv / b)

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24 -- Example calculation Set cv 1.5, b 4.25 So cl 1.5 / Penultimate shape parameter for damage L: ξn = (1 cl) / (cl log n) For n = 50, ξ For n = 100, ξ Note: Example of apparent bounded upper tail for hurricane intensity, but apparent heavy upper tail for hurricane damage

25 (ii) Extreme High Precipitation -- Physical/statistical argument (Wilson & Toumi 2005) Extreme high precipitation should have stretched exponential distribution with shape parameter c = 2/3 -- Simulation of 40,000 maxima with block size n = 100 from stretched exponential distribution with c = 2/3: ξ Simulated estimate of ξ (s.e )

26 Issue of Random Number of Wet Days -- Ultimate extreme value theory Maximum of random number of random variables N(n) Assume N(n) / n π (in dist.), 0 < π < 1 Note: N(n) could be generated by Markov chain Limiting distribution of MN(n) same as that of Mnπ (i. e., GEV distribution with same shape parameter, but adjusted location and scale parameters) -- Penultimate approach Replace n with nπ in expression for penultimate shape parameter ξn

27 Simulation Study for Random Block Size -- Stretched exponential distribution (c = 2/3) with random block size N E(N) = 100 (i.e., n = 365, π 0.274) -- Check penultimate approximation of ξnπ = ξ Simulate 40,000 block maxima with random block size (binomial dist.): Estimated shape parameter ξ (s.e ) (as compared with ξ for fixed block size n = 100) So effect is negligible

28 (iii) Temporal Dependence Gaussian Process (i. e., lack of clustering at high levels ) -- No effect on rate of convergence of penultimate approx. (Cohen 1982) -- Simulation study (40,000 block maxima, block size n = 100) First-order autoregressive process [AR(1)] Gaussian process with autocorrelation coefficient Φ Recall for Φ = 0, ξ (Simulated value of ξ 0.094) Result: Clear dependence on Φ is evident (Penultimate approximation combined with pre-asymptotic effect) Dependence on Φ slowly diminishes for large block size n

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30 Running Maxima (i. e., clustering at high levels) -- Effective adjustment to block size through extremal index θ θ 1 / (mean cluster length) -- Running maxima of order two (extremal index θ = 1/2) Choose cdf F 1/2 so that F stretched exponential distribution (shape parameter c = 2/3) Ultimate extreme value theory: suggests using ξn = (1 c) / [c log (nθ)] Simulation study (40,000 block maxima) indicates works fine: e. g., estimate of ξ (s.e ) (based on n = 200 or nθ = 100) Recall estimate of ξ for n = 100 under temporal independence

31 (6) Discussion Block Maxima Approach -- Focus (for simplicity) Peaks over Threshold Approach -- Similar results for fitting generalized Pareto distribution to excesses over high threshold -- Issue of choice of threshold Set u = F 1 (1 1/n)

32 Relevance of Penultimate Approximations -- Important to keep in mind that still only an approximation -- Preferable to view extremes from lens of penultimate theory, rather than only ultimate extreme value theory Interpretation of Extreme Value Analyses -- Circumstances are more complex than ultimate extreme value theory would suggest -- Makes any interpretation based solely on ultimate theory potentially suspect

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