bijgevoegde stelling Temporal dependencies for natural events - - PowerPoint PPT Presentation

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

Arthur Charpentier

bijgevoegde stelling Temporal dependencies for natural events

Promoters: Jan Beirlant (KUL) & Michel Denuit (UCL)

Katholieke Universiteit Leuven, June 2006.

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Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

Modeling daily windspeed and temperature

Consider some stationary time series (Xt)t∈Z, and define γX(h) = cov(Xt, Xt−h) and its Fourier transform fX(ω) = 1 2π

• h∈Z

γX(h) cos(ωh). Define autocorrelations as ρX(h) = γX(h)/γX(0). Long rang dependence:

h∈Z |ρX(h)| = ∞.

Haslett & Rasftery (1989), ARFIMA processes for windspeed (1 − L)dΦ(L)Xt = Θ(L)εt, where d ∈ (0, 1). GARMA processes, Hosking (1981, 1984), (1 − 2uL + L2)−λΦ(L)Xt = Θ(L)εt, where λ ∈ (0, 1/2), |u| < 1. ω = cos−1(u) is related to the seasonality of the series (e.g. ω = 2π/365). For daily windpspeed ˆ λ = 0.126 and ˆ σ2 = 4.472. 2

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF 500 1000 1500 2000 2500 3000 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF

Figure 1: Autocorrelations, daily windspeed in Ireland. 3

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

10 20 30

Daily windspeed in Ireland

date x$V1 1965 1970 1975 200 400 600 800 1000 0.0 0.4 0.8 Lag ACF

Autocorrelation of daily time series

0.0 0.1 0.2 0.3 0.4 0.5 5 20 200 frequency spectrum

Series: x Smoothed Periodogram

bandwidth = 0.000434

Figure 2: Daily windspeed in Ireland. 4

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

15 knots 20 knots 25 knots 2 days GARMA 47.7% 8.5% 0.3% seasonal ARMA 36.9% 3.6% 0.0% ratio 0.775 0.421 0.117 3 days GARMA 46.3% 8.0% 0.3% seasonal ARMA 27.7% 1.4% 0.0% ratio 0.597 0.176 0.010 4 days GARMA 45.3% 7.6% 0.3% seasonal ARMA 20.6% 0.5% 0.0% ratio 0.454 0.062 0.001 Table 1: Probabilities to have strong wind during consecutive days. 5

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

10 20 30 40 50 60 100 120 140 160 180 200 220

Minimal daily temperature in Paris (1997−2003)

July and August Temperature in ’0.1 °C 10 20 30 40 50 60 100 150 200 250 300

Maximal daily temperature in Paris (1997−2003)

July and August Temperature in ’0.1 °C

Figure 3: Daily temperature in Paris, years 1997 to 2003. 6

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

−10 −5 5 10 15 20

Daily Minimum Temperatures in Paris

date Temperature (°C) 1900 1920 1940 1960 1980 2000

Daily minima in Paris − detrended (in °C)

Time x 10000 20000 30000 −20 10 200 400 600 800 1000 −0.5 0.5 Lag ACF

Autocorrelation of daily time series

0.0 0.1 0.2 0.3 0.4 0.5 1 100 frequency spectrum

Series: x Smoothed Periodogram

bandwidth = 7.63e−05

Figure 4: Trend of the series, and analysis of the series of residuals. 7

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

Daily minima in Paris − detrended (in °C)

Time x.des 10000 20000 30000 −15 10 20 40 60 80 100 0.0 0.4 0.8 Lag ACF

Autocorrelation of residuals

0.0 0.1 0.2 0.3 0.4 0.5 0.5 20.0 frequency spectrum

Series: x Smoothed Periodogram

bandwidth = 7.63e−05

Figure 5: Residuals (εt)t∈Z. 8

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

−4 −2 2 4 −15 −10 −5 5 10

QQ plot of residuals (Gaussian)

Theoretical Quantiles Sample Quantiles −4 −2 2 4 −15 −10 −5 5 10

QQ plot of residuals (Student)

Theoritical Quantiles Sample quantiles

Figure 6: Residuals (εt)t∈Z, versus Gaussian and t-distribution. 9

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0

Distribution function of the period of return

Years before next heat wave 4 consecutive days exceeding 24 degrees GARMA + Gaussian noise ARMA + t noise ARMA + Gaussian noise 50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0

Distribution function of the period of return

Years before next heat wave 11 consecutive days exceeding 19 degrees GARMA + Gaussian noise ARMA + t noise ARMA + Gaussian noise 50 100 150 200 0.000 0.005 0.010 0.015 0.020 0.025 0.030

Density of the period of return

Years before next heat wave 4 consecutive days exceeding 24 degrees GARMA + Gaussian noise ARMA + t noise ARMA + Gaussian noise 50 100 150 200 0.000 0.005 0.010 0.015 0.020 0.025 0.030

Density of the period of return

Years before next heat wave 11 consecutive days exceeding 19 degrees GARMA + Gaussian noise ARMA + t noise ARMA + Gaussian noise

Figure 7: Survival distributions and densities of time before the next heat wave event (optimistic scenario). 10

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0

Distribution function of the period of return

Years before next heat wave 4 consecutive days exceeding 24 degrees GARMA + Gaussian noise ARMA + t noise ARMA + Gaussian noise 50 100 150 200 0.0 0.2 0.4 0.6 0.8 1.0

Distribution function of the period of return

Years before next heat wave 11 consecutive days exceeding 19 degrees GARMA + Gaussian noise ARMA + t noise ARMA + Gaussian noise 50 100 150 200 0.000 0.005 0.010 0.015 0.020 0.025 0.030

Density of the period of return

Years before next heat wave 4 consecutive days exceeding 24 degrees GARMA + Gaussian noise ARMA + t noise ARMA + Gaussian noise 50 100 150 200 0.000 0.005 0.010 0.015 0.020 0.025 0.030

Density of the period of return

Years before next heat wave 11 consecutive days exceeding 19 degrees GARMA + Gaussian noise ARMA + t noise ARMA + Gaussian noise

Figure 8: Survival distributions and densities of time before the next heat wave event (pessimistic scenario). 11

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

• heat wave, type (A): 11 consecutive days with temperature exceeding

19◦ C, short memory short memory long memory short tail noise heavy tail noise short tail noise

• ptimistic

88 years 69 years 53 years pessimistic 79 years 54 years 37 years Table 2: Periods of return (expected value, in years) before the next heat wave similar with August 2003 (type (A)). 12

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

• heat wave, type (B): 3 consecutive days with temperature exceeding

24◦ C, short memory short memory long memory short tail noise heavy tail noise short tail noise

• ptimistic

115 years 59 years 76 years pessimistic 102 years 51 years 64 years Table 3: Periods of return (expected value, in years) before the next heat wave similar with August 2003 (type (B)). 13

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

Modeling flood events

Two classical approaches, based on annualized maxima,

• Hurst (1951), long memory Gaussian process
• Gumbel (1958), i.i.d. observations Gumbel distributed

Based on 100 years of observations, not enough to assess if observations are, or not, independent: impact on return periods. Another idea, Todorovic & Zelenhasic (1970) and Todorovic & Rousselle: consider flood event as a marked Poisson process. Financial ACD (Autoregressive Conditional Duration) approach of Engle & Russell (1998). 14

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 1.0 lag autocorrelation

600 observations

5 10 15 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 lag autocorrelation

87 observations

Figure 9: Autocorrelation function of the Nile annual maxima series. 15

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

Let Xi denote observed durations (Xi = Ti − Ti−1), and define Hi = σ(X1, ...., Xi−1). Assume that        Xi = Ψi · εi, where (εi) i.i.d. (Exponential or Weibull) E(Xi|Hi) = Ψi Ψi = ω + p

k=1 αkXi−k + q k=1 βkΨi−k,

i.e. Xi = ω +

max{p,q}

• k=1

(αk + βk) −

q

• k=1

βkηi−k + ηi, where ηi = Xi − Ψi = Xi − E(Xi|Hi−1). Two duration model in Engle & Lunde (2003): time between two flood events, and duration of the flood. Marked process, with peak of the flood and volume. Distribution of residuals (εi): exponential in finance, mixed Weibull in hydrology. 16

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

Figure 10: Two durations model 17

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

1000 2000 3000 4000 5000 6000 7000 2000 4000 6000 8000 10000 Theoretical volume Empirical observed volume

Theoretical vs. observed volume

Figure 11: Relationship between peak, volume and duration of a flood. 18

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 0.4 0.5 0.6

density(x = ttt)

N = 243 Bandwidth = 0.3112 Density

Figure 12: Kernel density estimation of residuals 19

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 simulation donnees

Figure 13: Empirical distribution function of the data vs. Weibull mixture. 20

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

10 20 30 40 50 0.00 0.02 0.04 0.06 0.08 0.10

Density of the period of return

Years before next decennial flood

Figure 14: Flood duration distribution. 21

Arthur CHARPENTIER - PhD Thesis Defense - Bijgevoegde stelling.

Structure of the bijgevoegde stelling

The bijgevoegde stelling is based on several papers,

Bouëtte, J.C., Chassagneux, J.F., Sibaï, D., Terron, R. & Charpentier, A. (2006). Windspeed in Ireland: long memory or seasonal effect ? Stochastic Environmental Research and Risk Assessment. 20, 141 - 151. Charpentier, A. (2006). 2003 heat wave and its return period. submitted to Weather. Charpentier, A. and Sibaï, D. (2006). Dynamic flood modelling: Combining Hurst and Gumbel’s approach. submitted to Journal of Hydrology.

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