EXTREME VALUE ANALYSIS FOR CLIMATE TIME SERIES. Rick Katz

Transcription

1 1 EXTREME VALUE ANALYSIS FOR CLIMATE TIME SERIES Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO USA Home page: Lecture: /staff/katz/docs/pdf/banffrwk.pdf

2 2 Quote Climate change undermines a basic assumption that historically has facilitated management of water supplies, demands, and risks. Stationarity is dead: Whither water management Milley et al. (Science, 2008)

3 3 Outline (1) Background (2) Statistics of Climate Extremes (3) Temporal Dependence / Scaling of Climate Extremes (4) Interpretation of Tail Behavior of Climate Extremes (5) Complex Extreme Climate Events (6) Economic Impact of Climate Extremes (7) Unified Approach (Extremes + Non-Extremes) (8) Risk Communication under Climate Change

4 4 (1) Background Use of Extremal Models in Climatology and Hydrology -- Stationarity Prevailing paradigm (historically until recently) -- Many early applications dealt with climate or hydrology Precipitation, max. temp., min. temp., pressure (Gumbel, Bull. Amer. Meteor. Soc., 1942)

5 5 -- Engineering design Estimating return level of floods: Design of dams Flood plain regulation ( 100-year flood )

6 6 Non-Stationarity -- Sources Cycles (annual, diurnal) Trends (global climate change) Physically-based covariates (e. g., El Niño phenomenon) -- Traditional approaches Ignore (e. g., via block maxima) Remove (e. g., from all data, not just extremes) Sub-divide (e. g., separate analysis by month)

7 7 (2) Statistics of Climate Extremes Tail Behavior -- Shape parameter ξ of GEV or GP distribution -- Bounded upper tail (i. e., ξ < 0) Daily minimum & maximum temperature Hourly or daily wind speed Pressure Pollutant concentration Sea level Wave height

8 8 -- Heavy upper tail (i. e., ξ > 0) Precipitation: Typical estimates for daily totals 0.1 < ξ < 0.15 (Numerical models of climate system: Output less heavy-tailed?) Streamflow: Estimates of ξ tend to be higher than for precipitation (Effect of integrating precipitation across water basin?)

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10 10 Cycles -- Diurnal & Annual Phase tends to be roughly consistent with that for overall distribution Amplitude / shape possibly differs: Issue if standardize entire data set to remove annual cycles in mean and standard deviation Physical appeal: Extremal model which reflects process through which extremes occur

11 11 Trends -- Example (Urban heat island) Summer minimum of daily temperature at Phoenix, AZ (i. e., block minima) min{x 1, X 2,..., X T } = max{ X 1, X 2,..., X T } Assume summer minimum temperature in year t has GEV distribution with location & scale parameters: μ(t) = μ 0 + μ 1 t, log σ(t) = σ 0 + σ 1 t, ξ(t) = ξ, t = 1, 2,... Likelihood ratio test for μ 1 = 0 (P-value < 10 5 ) Likelihood ratio test for σ 1 = 0 (P-value 0.366)

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13 13 Other Covariates (e. g., physically based) -- Phoenix example (continued) Magnitude of heat island effect ln(population) Z t denotes population in Phoenix in year t Given population Z t = z, assume conditional distribution of summer minimum temperature has GEV distribution with location parameter: μ(z) = μ 0 + μ 1 ln(z), σ(z) = σ, ξ(z) = ξ Likelihood ratio test for μ 1 = 0 (P-value < 10 5 )

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15 15 (3) Temporal Dependence / Scaling of Climate Extremes Temporal Dependence -- Theory Weak dependence has no effect on limit theorems Example: Stationary Gaussian process Autocorrelation function ρ n, with ρ n ln n 0 as n 0 Then limit theorems same as under independence (not even any adjustment to normalizing constants) Note: Includes long-memory processes (such as fractional ARIMA processes)

16 16 -- Asymptotic independence at high levels Lack of temporal clustering of extremes Pr{X t+1 > u X t > u} 0 as u For example, satisfied by Gaussian process -- Asymptotic dependence at high levels Need to adjust normalizing constants Temporal clustering of extremes ( extremal index θ, 0 < θ 1; 1/θ mean cluster size; θ = 1 under asymptotic independence) Some evidence for temperature (i. e., θ < 1) Not necessarily for precipitation (i. e., θ = 1?)

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18 18 Scaling / Aggregation -- Apparent conflict with extreme value theory Precipitation extremes: Shape parameter tends to decrease with aggregation over time (e. g., hourly vs. daily total amounts) Example: Regional analysis of precipitation extremes in Texas Extreme value theory: Shape parameter should be invariant with respect to aggregation

19 Potential explanation: Penultimate extreme value theory 19

20 20 Role of log-log linearity in extreme value theory? δ = h / h' (change in time scale), μ' = μ + [σ' (1 δ ξ )] / ξ, σ' = σ δ ξ

21 21 (4) Interpretation of Tail Behavior of Climate Extremes Ultimate Extreme Value Theory -- GEV distribution as limiting distribution of maxima X 1, X 2,..., X T with common distribution function F M T = max{ X 1, X 2,..., X T } Penultimate Extreme Value Theory -- Suppose 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)

22 22 -- Expression for shape parameter ξ T Hazard rate (or failure rate ): H(x) = F'(x) / [1 F(x)] Then one choice of shape parameter is: ξ T = (1/H)' (x) x=u(t) where the characteristic largest value u(t) = F 1 (1 1/T) Here ξ T 0 as block size T

23 23 Stretched Exponential Distribution -- Traditional form of Weibull distribution (Bounded below) 1 F(x) = exp( x c ), x > 0, c > 0 where c is shape parameter (unit scale parameter) Note: Weibull extremal type is reflected version -- Shape parameter for penultimate approximation is: ξ T (1 c) / (c log T ) (i) Superexponential (c > 1) ξ T 0 as T (ii) Subexponential (c < 1) ξ T 0 as T

24 24 Apparent Heavy Tail -- Precipitation (i) Penultimate approximation Fréchet type of GEV can be obtained with F stretched exponential distribution (Shape parameter c < 1) (ii) Physical argument Wilson & Toumi (2005) gave heuristic argument for universal shape parameter of c = 2/3 for stretched exponential distribution for extreme high precipitation

25 25 -- Simulation experiment Generate observations with stretched exponential distribution (with shape parameter c = 2/3) Use block size of T = 100 to simulate maxima M 100 (Corresponds to daily precipitation occurrence rate about 27%, ignoring variation in number of wet days) Penultimate approximation: Should produce GEV shape parameter of ξ Fitted GEV distribution (40,000 replications): Obtained estimate of ξ

26 26 -- Aggregation Issue Apparent decrease in shape parameter of GEV or GP distribution Stretched exponential should be capable of resolving (at least qualitatively) Simulation experiment: Sum of two independent stretched exponentials (each with c = 2/3) Use block size of T = 100 to simulate maxima M 100 Fitted GEV dist. (40,000 replications): Estimate ξ But note that precipitation really a random sum

27 27 (5) Complex Extreme Climate Events Climate/Hydrologic Extremes -- Many events have complex structure (e. g., spells) Definition of Heat wave / Hot spell -- Recall runs de-clustering algorithm -- More complex definition (e. g., multiple thresholds)? Lack of Use of Extremal Models -- e. g., paper by Meehl & Tebaldi (Science, 2004) Example (European heat wave, 2003)

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29 29 Model clusters (instead of declustering) -- Let Y 1, Y 2,..., Y k denote excesses within given cluster / spell -- Model conditional distribution of Y 2 given Y 1 as GP distribution with scale parameter σ depending on Y 1 : e. g., σ(y) = σ 0 + σ 1 y, given Y 1 = y > 0 Hold shape parameter constant ξ(y) = y Similar model for conditional distribution of Y 3 given Y 2 (etc.) -- Drawbacks Need to identify link function σ(y) Unconditional distribution of Y 2 no longer exactly GP

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31 31 Conditional distribution of Y 2 given Y 1 = y -- Conditional mean [increases with σ(y)] E(Y 2 Y 1 = y) = σ(y) / (1 ξ), ξ < 1 -- Conditional variance (increases with mean) Var(Y 2 Y 1 = y) = [E(Y 2 Y 1 = y)] 2 / (1 2 ξ), ξ < 1/2 -- Conditional quantile function F 1 [p; ξ, σ(y)] = [σ(y) / ξ ] [(1 p) ξ 1], 0 < p < 1 Increases more rapidly with σ(y) for higher p

32 32 (6) Economic Impact of Climate Extremes Damage Functions -- Implications for tail behavior -- Nordhaus (2010) uses damage-intensity function for hurricanes Damage (Intensity) β where β 9 (Theory suggests β = 3) -- Effect of power transformation on tail (in penultimate sense) Shape parameter of stretched exponential: c > 1 but c* = c / β < 1 Bounded tail for wind speed (c > 1), but apparent heavy tail for damage (c* < 1)

33 33 Insurance / Reinsurance Applications -- Total damage (e. g., annual) Random sum representation (Embrechts et al.: Bread and butter of insurance mathematics ) (i) Variation in number of extreme events Poisson distribution natural model (ii) Variation in damage from individual events Lognormal distribution commonly used (But should allow for GP tail)

34 Damages adjusted for inflation & changes in societal vulnerability 34

35 35 (7) Unified Approach (Extremes + Non-Extremes) Hybrid Approach to Modeling Entire Distribution -- Example of daily precipitation intensity Given gamma distribution (fit to all data) Replace with GP distribution above high threshold How to tie together two pdfs at threshold u? Adjust scale parameter σ of GP distribution σ = 1 / H(u) where H is hazard rate for gamma distribution

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37 37 Unified Treatment of Covariates -- Alternatives to focusing solely on extremes Trends: Estimate on basis of all data (extreme & non-extreme)? Seasonality: Remove seasonal cycle from all data before modeling extremes? Geophysical covariates: Alternative modeling procedure such as quantile regression?

38 38 (8) Risk Communication Under Climate Change Return Period / Return Level (e. g, 100-year flood ) -- Return level x(p) with return period T (under stationarity) Pr{X > x(p)} = p, where p = 1/T (i) Length of time T for which expected number of events = 1 1 = Expected no. events = T p, so T = 1/p (ii) Expected waiting time (assume temporal independence) Waiting time W has geometric distribution: Pr{W = k } = (1 p) k 1 p, k = 1, 2,..., E(W) = 1/p = T

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40 40 Period of rapid urbanization (starting about 1970) Effects of land-use changes on runoff within watershed

41 41 One Approach (under non-stationarity) -- Effective return level (i. e., conditional quantiles) Permit return level to vary from one time period to next (Hold probability of occurrence constant) -- Like moving flood plain from one year to next Impractical for many long-term planning purposes

42 42 Alternative Approach (under non-stationarity) -- Retain one of two interpretations under stationarity Notation: Let p t (u) = Pr{X t > u}, year t = 1, 2,... (i) Expected number of events -- Let N T (u) denote number of events during t = 1, 2,..., T E[N T (u)] = p 1 (u) + p 2 (u) + + p T (u) Given specified return period T: Set E[N T (u)] = 1 & solve for return level u Then u satisfies p 1 (u) + p 2 (u) + + p T (u) = 1 In other words, average of T probabilities is 1/T

43 Frequency-Based Return Level (u 871 cfs for T = 50 yr) 43

44 44 (ii) Expected waiting time Assume temporal independence -- Let W(u) denote first time t that X t > u Pr{W(u) = k } = { t=1,k 1 [1 p t (u)] } p k (u), k = 1, 2,... Given specified return period T: Set E[W(u)] = T Solve for return level u Not possible to obtain analytical expression for E[W(u)] (or for threshold u )

45 Waiting Time-Based Return Level (u 871 cfs for T = 50 yr) 45

46 46 Note: Two definitions would generally produce different return levels Abandon Concept? -- Provide information (e. g., in form of table) on probability of one or more events (for given threshold) as function of length of time Decision-theoretic justification -- Examples on internet (under stationarity) Educate consumers about need to purchase flood insurance (tables of probability of 100-year flood within next, e. g., 30 years)

47 47 Resources Statistics of Weather and Climate Extremes -- Application of statistics of extremes to weather & climate Extremes Toolkit (extremes) -- Open source software in R with GUIs Developed by Eric Gilleland S functions from Stuart Coles