PENULTIMATE APPROXIMATIONS FOR WEATHER AND CLIMATE EXTREMES Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO USA Email: rwk@ucar.edu Web site: www.isse.ucar.edu/staff/katz/ Talk: www.isse.ucar.edu/staff/katz/docs/pdf/penult16.pdf
R. A. Fisher, 1890 1962
L. H. C. Tippett, 1902 1985
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
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
(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
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 0.094 (s.e. 0.003) Note: Ultimate approximation is ξ = 0
(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
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)
-- 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)
(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)
-- 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)
(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) ξ100 0.109 (Recall obtained simulated value of ξ 0.094)
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)
-- GEV distribution fit to 40,000 simulated maxima
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: ξ100 0.550 Simulated estimate of ξ 0.609 (s.e. 0.005) (ii) n = 100, σ = 1: ξ100 0.221 Simulated estimate of ξ 0.248 (s.e. 0.004)
Cubed Gamma Distribution -- Random variable Y = X 3, where X has gamma dist. (shape parameter α)
(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
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)
-- Example calculation Set cv 1.5, b 4.25 So cl 1.5 / 4.25 0.35 Penultimate shape parameter for damage L: ξn = (1 cl) / (cl log n) For n = 50, ξ50 0.47 For n = 100, ξ100 0.40 Note: Example of apparent bounded upper tail for hurricane intensity, but apparent heavy upper tail for hurricane damage
(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: ξ100 0.109 Simulated estimate of ξ 0.097 (s.e. 0.004)
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
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 = ξ100 0.109 Simulate 40,000 block maxima with random block size (binomial dist.): Estimated shape parameter ξ 0.096 (s.e. 0.004) (as compared with ξ 0.097 for fixed block size n = 100) So effect is negligible
(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, ξ100 0.109 (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
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 0.095 (s.e. 0.004) (based on n = 200 or nθ = 100) Recall estimate of ξ 0.097 for n = 100 under temporal independence
(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)
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