Generalized Additive Modelling for Sample Extremes: An Environmental Example

Transcription

1 Generalized Additive Modelling for Sample Extremes: An Environmental Example V. Chavez-Demoulin Department of Mathematics Swiss Federal Institute of Technology Tokyo, March 2007

2 Changes in extremes? Likely to be slow in environmental applications May be difficult to detect because of noise Aim to combine the point process approach to exceedances with smoothing methods to give a flexible exploratory approach to modelling changes in extremes

3 Stations in Swiss Alps

4 Winter temperatures at 21 Swiss stations height height height height height height height height height 5 10 Excesses height height height height height height height height height height height height Year

5 Swiss winter temperatures by year Excesses Day

6 Summary Climate change and extremes? Need flexible models Mix threshold approach to extremal modelling, semiparametric smoothing, and bootstrap Brief description of the threshold method Implementation of spline smoothers Application to the Swiss Alps data Discussion

7 Traditional Method The mathematical foundation of EVT is the class of extreme value limit laws X 1, X 2,..., are independent random variables with common distribution function F and M n = max {X 1,..., X n } for suitable normalising constants a n > 0 and b n, we{ seek a limit } law satisfying P Mn b n a n x = F n (a n x + b n ) G(x)

8 There are only 3 fundamental types of extreme value limit laws that can be combined into a simple GEV distribution H(x) = exp { ( 1 + κ x µ ψ ) 1/κ + } The parameters < µ <, ψ > 0 and < κ < are resp. the location, scale and shape parameters

9 r-largest Extremes M n,..., M n r : the r-largest observations among X 1,..., X n to get more information about the extremes than the max alone The asymptotic joint distribution of M1 n,..., Mr n at m { n 1,..., mn r ( is ) } 1/κ exp 1 + κ mr n µ ψ ( ) 1/κ 1 r j=1 1 ψ 1 + κ mi n µ ψ + which forms a likelihood for the parameters

10 In case m years of data are available, the likelihood is constructed from the r-largest values in each year, considering data for different years as independent, an overall likelihood is simply the product of such terms, for all years Choice of r; bias if r is too large

11 Threshold method Treat occurrences of events over (or under) threshold u as Poisson process Number of exceedances N over u follows homogeneous Poisson process, rate λ Exceedance sizes W j = Y j u are random sample from GPD 1 (1 + κw/σ) 1/κ + if κ 0 G(w) = 1 exp( w/σ) if κ = 0 where σ and κ are scale and shape parameters

12 Use orthogonal parametrization κ, ν = σ(1 + κ) below Log likelihood for data splits into two parts l(λ, κ, σ) = l N (λ) + l W (κ, ν)

13 Semiparametric model Generalize previous approach Take λ to be time-varying, where λ = exp { x T α + f(t) } Take exceedances to be GPD with κ = x T β + g(t), ν = exp { x T η + s(t) }

14 f, g and s are smooth functions of time t, and parameters can also depend on ordinary covariates Penalize roughness of f, g and s through second derivatives Other link functions possible

15 Penalized log likelihoods For rate of exceedances λ, maximize l N (λ) 1 2 γ λ f (t) 2 dt, equivalent to fitting standard generalized additive model For sizes of exceedances, maximize l W {κ(β, g), ν(η, s)} 1 2 γ κ g (t) 2 dt 1 2 γ ν s (t) 2 dt

16 If g, s are cubic splines, equivalent to maximizing l W {κ(β, g), ν(η, s)} 1 2 γ κg T Kg 1 2 γ νs T Ks over β, η, g, s and leads to generalized ridge regression Parameters γ λ, γ κ and γ ν control smoothness of f, g and s

17 Methodology Choose forms for λ, κ and ν and fit Choose smoothing parameters γ λ etc using AIC Use likelihood ratio statistics/aic for model comparisons

18 When model correct, residuals R j = ˆκ 1 j log {1 ˆκ j W j (1 ˆκ j )ˆν j } are approximately independent unit exponential variables

19 Bootstrap uncertainty assessment Need model-robust assessment of uncertainty Clustering across stations must be taken into account Use bootstrap, either resampling the R j computed from undersmoothed curves added to oversmoothed curves or resample seasons within blocks Either yields percentile confidence intervals/pointwise bands

20 Alpine winter temperatures Fitted intensity log ˆλ = ˆα 0 + ˆf(d, 4) + ˆq(t, 2) at Vattis for (left) and for January 1 from (right) lambda lambda Day Year

21 Fitted model and 20-year return level log ˆλ = ˆα 0 + ˆf(d, 4) + ˆq(t, 2), ˆκ = ˆβ (h 1000) ˆβ 1, log ˆν = ˆη 0 + ˆη 2 t+ŝ(d, 4) Rheinfelden Vattis Arosa Temperatures & 20-year return level Year Temperatures & 20-year return level Year Temperatures & 20-year return level Year

22 Discussion Inhomogeneous Poisson process λ depends on time but not location Shape parameter κ varies with altitude exceedances at higher stations have shorter tails Scale parameter ν depends on time but not on altitude Increase since 1985 is consistent with the supposed effect of climate change but also with short-term fluctuations (decrease from !)

23 Conclusion Exceedances over/under thresholds widely-used approach with natural interpretation exceedance times modelled using existing code (GAM) Smoothing extremes by penalized log likelihood convenient and rapid exploration technique highlights features of underlying distribution

24 References Chavez-Demoulin, V. and Davison, A. C. (2005) Applied Statistics Chavez-Demoulin, V., Embrechts, P. (2004) Smooth Extremal Models in Finance and Insurance. Journal of Risk and Insurance. Davison, A. C. and Smith, R. L. (1990) JRSS,B Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance Springer Green, P. J., Silverman, B. W. (1994)