Robust Extremes in Chaotic Deterministic SystemsRenato Vitolo, Mark P. Holland, Christopher A.T. Ferro

School of Engineering, Computing and Mathematics, University of Exeter (www.secam.ex.ac.uk/xcs/).

The concept

A chaotic deterministic system fρ has robust extremesunder observable φ when the associated extreme valuestatistics depend smoothly on control parameter ρ.

Results

Robustness of extremes:

1. depends on system fρ and on observable φ;

2. allows improved estimates by pooling data and

3. improved prediction of (non-stationary) return levels.

Phenomenology

Robustness of extremes depends on the system fρ and onthe observable φ.

Illustration for Lorenz63 model with σ = 10, β = 8

3:

x = σ(y − x), y = x(ρ − z) − y, z = xy − βz. (1)

For ρ = 28: (1) has robust strange attractor [1]. Let

φ1(x, y, z) = x, φ2(x, y, z) = 1 − |x − 5|0.25. (2)

Generate time series of length 10n units (recorded every0.5) and extract maxima over blocks of 1000 time units.Fit generalised extreme value (GEV) distribution:

G(x; µ, σ, ξ) = exp

[

−

(

1 + ξx − µ

σ

)−1/ξ

+

]

. (3)

30 40 50 60

−0.

25−

0.21

−0.

17 φ1 = x

φ2 = 1 − |x − 5|0.25

ξ

ρ

FIG. 1: Maximum likelihood estimates of ξ for ob-servables φ1 and φ2 in (2), for ρj = 27 + j, j = 0, 1, . . .

φ1: for small ρ (≈ 28), ξ varies smoothly with ρ. Non-linear scaling of attractor shape of ξ(ρ). Discontinuousfor large ρ due to hyperbolicity loss (folds in return map).

φ2: ξ(ρ) = 0.25 is constant even under hyperbolicity loss.

Rigorous proof available for 1D Lorenz maps.

−3 −2 −1 0 1 2 3 4

−3

−2

−1

01

23

4

−3 −2 −1 0 1 2 3 4

ρ=

58

ρ = 59 ρ = 60

FIG. 2: Discontinuity in upper tail at ρ = 59 for φ1

(quantile-quantile plot of 104 maxima from Fig 1, foreach ρ = 58, 59, 60).

Pooling dataRobustness of extremes enhanced precision of GEV

estimators.17

.417

.617

.8

n=5n=8

0.4

0.5

0.6

27.0 27.4 27.8

−0.

4−

0.2

0

n=5 pooledn=8

27.0 27.4 27.8

µσ

ξ

ρρ

FIG. 3: Left: GEV parameter estimates for (1) withn = 5 (dashed, 95% conf. int. in gray) and n = 8(solid). Right: “pooled” estimates with n = 5(dashed, 95% conf. int. in gray).

Large n: GEV parameter estimates → smooth functionsof ρ (solid lines in Fig 3, n = 8).Small n: wild oscillations around “true values” (dashedlines, left column in Fig 3, n = 5).

Enhanced precision: “pooling” short series (n = 5)Given robust extremes, information can be pooled fromnearby ρ reduction in uncertainty due to parameter estimation,cfr. grey bands in Fig 3. Assume functional forms

µ(ρ) = µ0 + µ1ρ, σ(ρ) = σ0 + σ1ρ, ξ(ρ) = ξ0.

ξ constant in ρ: approximation, only valid locally.Estimate (µ0, µ1, σ0, σ1, ξ0) by maximum likelihood.

Prediction & non-stationarity

Robustness of extremes interpreting and predictingnon-stationary extremes.

Robust extreme windspeeds are found in a simple two-layer quasi-geostrophic model [2]:smooth dependence of windspeed return levels wrt baro-clinic forcing parameter TE in stationary case.

10 15 20 25 30

2040

6080

100

120

100−yr return levels

win

dspe

ed (

m/s

)

TE

FIG. 4: 100-year windspeed return levels at centre oflower layer, for different values of TE (non-pooledGEV fits, stationary case).

Introduce linear time trend in QG model:

TE(t) = (T 0

E − 1) + t ∆TE, t ∈ [0, t0], ∆TE = 2/300yrs.

Ansatzen: adiabatic + slow trend.

1. trend speed ∆TE is sufficiently small wrt samplingtime for upper tail of windspeed distribution;

2. non-stationary extremes remain close (locally in time)to those of stationary system for “frozen” TE(t).

=⇒ robustness of extremes wrt control parameter trans-lates to smooth change of extremes wrt time.

We adopt the Generalized Additive Models for Location,Scale and Shape (GAMLSS) [3]:

1. response distribution is GEVwith constant ξ and cubicsmoothing spline for (µ, σ) with identity link;

2. split sequence of yearly maxima into training and testset (years 1-2250 and 2251-3000);

3. fit non-stationary GEV-GAMLSS to training set;

4. compute time-dependent quantiles and compare totraining and test set.

0 500 1000 1500 2000 2500 3000

2040

6080

100

120

Time (years)

Win

dspe

ed (

m/s

)

FIG. 5: Points: observed yearly windspeed max-ima during training (black) and test (blue) periods.Curves: time-dependent estimated quantiles fromGEV-GAMLSS.

quantiles 0.4 2 10 25 50 75 90 98 99.6training 0.5 2.3 9.5 25.2 50.3 76.1 90.0 98.0 99.6test 0.7 3.6 14.3 30.5 51.7 74.7 91.3 97.7 99.3

Fraction of points below the estimated quantile curves inFig. 5 (corresp. to top row) during training (green, centrerow) and test (red, bottom row) periods.

Illustrates potential for predicting return levels in a non-stationary system exhibiting robust extremes.

References

[1] Morales, C. A., Pacifico, M. J., and Pujals, E. R. Proc.Amer. Math. Soc. 127(11), 3393–3401 (1999).

[2] Felici, M., Lucarini, V., Speranza, A., and Vitolo, R. J.Atmos. Sci. 64(7), 2137–2158 Jul (2007).

[3] Rigby, R. A. and Stasinopoulos, D. M. J. Roy. Statist.Soc. Ser. C 54(3), 507–554 (2005).