STATISTICAL MODELING OF HOT SPELLS AND HEAT WAVES

Transcription

1 STATISTICAL MODELING OF HOT SPELLS AND HEAT WAVES Rick Katz Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO USA Joint work with Eva Furrer Home page: Talk: /staff/katz/docs/pdf/heatuvic.pdf Paper: Furrer et al., 2010: Climate Research, V. 43, pp

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3 Outline (1) Background / Motivation (2) Statistical Model for Simple Extreme Temperature Events (3) Statistical Model for Hot Spells under Stationarity (4) Statistical Model for Hot Spells with Trends (5) Heat Wave Simulator (6) Extensions

4 (1) Background / Motivation Heat waves -- Extreme event -- Yet extreme value theory rarely applied Definition -- Ambiguous -- Focus on hot spells instead (for now) Goal -- Devise simple model (only use univariate extreme value theory) -- Simple enough to incorporate trends

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8 (2) Statistical Model for Simple Extreme Temperature Events Point process approach -- Combines two processes (i) Occurrence of extreme event Poisson process for exceedance of high threshold (ii) Severity of extreme event Generalized Pareto (GP) distribution for excess over threshold -- Select high threshold u -- Decluster (i. e., identify clusters & model cluster maxima)

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11 Condition for no clustering at high levels Pr{X t + k > u X t > u} 0 as u, k = 1, 2,... Clustering at high levels (Extremal index θ, 0 θ 1) (i) θ = 1 No clustering at high levels (independent or Gaussian process) (ii) 0 θ < 1 Clustering at high levels Interpretations: (i) Mean cluster length 1 / θ (ii) Effective sample size (proportion)

12 Daily maximum temperature -- Strong evidence of clustering at high levels (i. e., extremal index θ < 1) Ferro-Segers (2003) intervals estimator of extremal index: Does not require identification of clusters Interexceedance times (Coefficient of variation converges to simple function of θ) -- Runs declustering (Parameter r = 1, 2,...) r = 1 corresponds to defining clusters as run of consecutive exceedances of threshold

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14 (3) Statistical Model for Hot Spells under Stationarity Model clusters (instead of declustering) -- Rate of occurrence of clusters Modeled as Poisson process (rate parameter λ) -- Intensity of cluster Cluster maxima modeled as generalized Pareto (GP) distribution with shape parameter ξ and scale parameter σ -- Distribution of cluster length? -- Model of dependence among excesses within cluster?

15 Statistical modeling of clusters -- Cluster length Geometric distribution for cluster length with mean 1/θ (Need to truncate?) Alternative: Discrete analogue of GP (Zipf distribution) -- Dependence of excesses Conditional GP distribution for temporal dependence of excesses within cluster Advantage: Requires only univariate extreme value theory (not multivariate)

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18 -- Let Y 1, Y 2,..., Y k denote excesses over threshold 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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20 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

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24 (4) Statistical Model for Hot Spells with Trends Incorporation of non-stationarity / climate change -- Cluster rate Trend in mean of Poisson rate parameter λ(t), year t -- Cluster length Trend in mean of geometric distribution 1/θ(t), year t -- Cluster maxima (or first excess) Trend in scale parameter of GP distribution σ(t), year t

25 Procedure for fitting trends -- Cluster rate and cluster maxima (or first excess) Obtain from fitting trend in point process intensity (indirectly if use GEV parameterization) -- Cluster length Use generalized linear model (glm) (negative binomial family)

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29 (5) Heat Wave Simulator Stochastic simulation of hot spells (i) Simulate number of clusters from Poisson distribution (ii) Conditional on number of clusters, generate dates of start of clusters from uniform distribution (iii) Generate cluster lengths from geometric distribution (assigned in manner to minimize need to truncate) (iv) Generate excess for first day of cluster from GP distribution (v) Generate excesses for subsequent days from conditional GP model

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31 Hot spells to heat waves -- Merge clusters -- Minimum cluster length -- Higher threshold (example on next slide) -- Cluster functionals Measures of heat wave intensity (e. g., mean or total excess) -- Trends in hot spell characteristics Convert to corresponding trends in heat wave characteristics

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34 (6) Extensions More realistic treatment of hot spells/heat waves -- Seasonality (at least allow for trend in length of heat wave season) -- Multiple thresholds -- Apparent temperature (instead of maximum temperature) -- Minimum temperature (in addition to maximum temperature)

35 Mortality -- Covariates Hot spell model could motivate more realistic covariates (Instead of using lagged temperature variables) Blocking patterns -- Use blocking index as covariate (with/without trend variable)

36 Resources Statistics of Weather and Climate Extremes -- Extremes Toolkit cran.r-project.org/web/packages/extremes/