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Extreme Value TheoryFitting Models
Investigation of an Automated Approach to ThresholdSelection for Generalized Pareto
Kate R. Saunders
Supervisors: Peter Taylor & David Karoly
University of Melbourne
April 8, 2015
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Extreme Value TheoryFitting Models
Outline
1 Extreme Value Theory
2 Fitting Models
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Extreme Value TheoryFitting Models
Problem
What are the climate processes that drive extreme rainfall?(El Nino Southern Oscillation, Interdecadal Pacific Oscillation)
How do these drivers differ at different timescales; sub-daily, daily,consecutive day totals?
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Extreme Value TheoryFitting Models
Data
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Extreme Value TheoryFitting Models
Extreme Value Theory
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Extreme Value TheoryFitting Models
Block Maxima
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Extreme Value TheoryFitting Models
Block Maxima
Let X1, X2, ... , Xn be a sequence of i.i.d. random variables withdistribution function F . Define Mn = max{X1, X2, . . . , Xn}.
(Xi might be daily rainfall observations and M365 the annual maximumrainfall.)
Pr(Mn ≤ x) = Pr(X1 ≤ x , . . . ,Xn ≤ x)
= Pr(X1 ≤ x)× · · · × Pr(Xn ≤ x)
= F (x)n.
As n → ∞, the distribution of the Mn converges to a generalisedextreme value distribution.
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Extreme Value TheoryFitting Models
Generalized Extreme Value Theorem (Fisher-Tippett-Gnendenko)
If there exists sequences of constants {an > 0} and {bn} such that
Pr
(Mn − bn
an≤ z
)→ G (z) as n→∞
for a non-degenerate distribution function G , then G is a member of theGeneralized Extreme Value family
G (z) = exp
−[
1 + ξ
(z − µσ
)]−1
ξ
defined on {z : 1 + ξ(z − µ)/σ > 0}, where ∞ < µ <∞, σ > 0 and−∞ < ξ <∞.
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Extreme Value TheoryFitting Models
Leveraging more data
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Extreme Value TheoryFitting Models
Generalized Pareto Distribution
Let X1, X2, ... , Xn be a sequence of iid random variables with marginaldistribution function F .
Pr{X > u + y |X > u} =1− F (u + y)
1− F (u)y > 0.
If F satisfies Generalized Extreme Value Theorem then for a large enoughthreshold u, the distribution function of (X − u) conditional on X > u isthe GPD.
Generalized Pareto Distribution - Picklands (1975)
H(y) = 1−(
1 +ξy
σ
)−1/ξdefined on {y : y > 0} and (1 + ξy/σ > 0) where, σ = σ + ξ(u − µ).
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Extreme Value TheoryFitting Models
Dependence
Rainfall observations are dependentHeavy rainfall yesterday effects the probability of heavy rain todayHeavy rainfall a year ago doesn’t
Extreme Value Theory extends to stationary series with weak longrange dependence
However, for processes with short range dependence extremes occurin clusters
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Extreme Value TheoryFitting Models
Clusters
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Extreme Value TheoryFitting Models
Dependent Series
Let {Xi}i≥1 be a stationary series and {X ∗i }i≥1 be an independent seriesof variables with the same marginal distribution.
Define Mn = max{X1, . . . , Xn} and M∗n = max{X ∗1 , . . . , X ∗n }. Undersuitable regularity conditions,
Pr
{(M∗n − bn)
an≤ z
}→ G (z),
as n→∞ for normalizing sequences {an > 0} and {bn}, where G is anon-degenerate distribution functions, if and only if
Pr
{(Mn − bn)
an≤ z
}→ G θ(z),
for a constant θ such that 0 < θ ≤ 1.
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Extreme Value TheoryFitting Models
Extremal Index
θ = {Limiting mean cluster size}−1 ∈ (0, 1]
θ = 0.5⇒ 2 observations per cluster on average.
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Extreme Value TheoryFitting Models
Fitting Models
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Extreme Value TheoryFitting Models
Fitting Models
Select a threshold
Decluster the data for independent observations
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Extreme Value TheoryFitting Models
Declustering
BlocksPartition the observation sequence into blocks of length, bAssume extreme observations within the same block belong to thesame same cluster.
RunsSpecify a run length, KAssume extreme observations with an inter-exceedance time of lessthan K belong to the same cluster.
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Extreme Value TheoryFitting Models
Intervals
The limiting process of exceedance times is compound Poisson forstationary series (Hsing et al. 1988).
Ferro and Segers (2003) showed the limiting distribution ofinter-exceedance times is a mixture distribution with weight θ,
Tθ(t) = (1− θ)ε0 + θ · θ exp(−θt),
where ε0 is a degenerate distribution, Tθ is the distribution of arrivaltimes of exceedances at threshold u.
By equating moments a non-parametric estimator can be found for θ.
The largest θ(N − 1) inter-exceedance times can be interpreted as betweencluster arrivals.
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Extreme Value TheoryFitting Models
Fitting Models
→ Select a threshold
Decluster the data for independent observations
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Extreme Value TheoryFitting Models
Mean Residual Life Plots
For sufficiently high thresholds, as the threshold increases the expectedexceedance above the threshold should grow linearly.
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Extreme Value TheoryFitting Models
Parameter Stability Plots
Parameter estimates of (modified) scale and shape parameters should beconstant for the range of valid thresholds.
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Extreme Value TheoryFitting Models
Alternative
Set the threshold according to a high quantile of non-zero observationsEg. 90th percentile.
Is this an appropriate threshold?Is our model is misspecified?
Suggested approach by Suveges and Davison et al. (2010) is to test thethreshold, u, and run parameter, K pair for model misspecification.
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Extreme Value TheoryFitting Models
Log-Likelihood
Limiting distribution of inter-exceedance times:
Tθ(t) = (1− θ)ε0 + θ2 exp(−θt),
Log-Likelihood (strictly positive inter-exceedance times):
N−1∑i=1
log((1− θ)I(ti=0)(θ2 exp(θti )
I(ti>0))
=N−1∑i=1
[2I(ti > 0) log(θ)− θti
],
where ti = NTin , n is the total number of observations and N is the
number of exceedances.
However as n gets large our estimate, θ, tends to 1 suggestingindependence.
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Extreme Value TheoryFitting Models
Log-Likelihood
Adjustment of the inter-exceedance times using the run parameter K :
ci = max{ti − K , 0}
Log-likelihood:
`(θ; ci ) =N−1∑i=1
[I(ci = 0) log(1− θ) + 2I(ci > 0) log(θ)− θci
]Approach used in Fukutome et al. (2014) and Suveges and Davison
(2010).
Test combinations of threshold, u, and run parameter, K , formisspecification of the likelihood function. Select the (u,K ) pair thatmaximizes the number of independent clusters.
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Extreme Value TheoryFitting Models
Model Misspecification
If a parametric model is misspecified then there is no θ such that g = f (θ),where g is the true model and f is the misspecified parametric model.
For a well specified model,the Fisher’s information matrix, I (θ) = E{`′′(θ; cj} is equal to thevariance of the score vector, J(θ) = Var{`′(θ; cj)}.
Test the hypothesis:D(θ) = J(θ)− I (θ),
where H0 : D(θ) = 0 and H1 : D(θ) 6= 0.
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Extreme Value TheoryFitting Models
Empirically:
IN−1(θ) =−1
(N − 1)
N−1∑j=1
`′′(θ; cj)
JN−1(θ) =1
(N − 1)
N−1∑j=1
`′(θ; cj)2
DN−1(θ) = JN−1(θ)− IN−1(θ)
VN−1(θ) =1
(N − 1)
N−1∑j=1
[(dj(θ; cj)− D
′N−1(θ)IN−1(θ)−1`
′(θ; cj)
)2]
where VN−1(θ) is the sample variance of DN−1(θ).
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Extreme Value TheoryFitting Models
Model Misspecification
Theorem: (Information Matrix Test - Whyte 1982) If the assumedmodel `(θ; ci ) contains the true model for some θ = θ0, then as n→∞,
(i)√
(N − 1)DN−1(θ)w−→ N(0,V (θ0)),
(ii) VN−1( ˆθN−1)a.s.−−→ V (θ0), and VN−1(θ) is non-singular for sufficiently
large N,
(iii) Then the Information Matrix Test statistic,(N − 1)DN−1(θ)′VN−1(θ)−1DN−1(θ) is asymptotically χ2
1 distributed.
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Extreme Value TheoryFitting Models
Example: AR(2)
Yi = 0.95Yi−1 − 0.89Yi−2 + Zi where Zi ∼ GP(1, 1/2) and n = 8000.100 simulations
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Extreme Value TheoryFitting Models
Adjusting inter-exceedance times
Common to assume stationarity by enforcing seasonal blocking.
Collapse inter-exceedance times across seasonal blocks using thememoryless property of the exponential for fitting.
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Extreme Value TheoryFitting Models
Results: Gatton, South East Queensland
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Extreme Value TheoryFitting Models
Results: Oenpelli, Northern Territory
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Extreme Value TheoryFitting Models
Summary
Shown how to check if the threshold and run parameter selectedviolate the assumptions of the model
Given confidence to threshold selection in the absence of a hard andfast rule and in the presence of subjectivity
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Extreme Value TheoryFitting Models
References
Ferro, C. and Segers, J. (2003). Inference for clusters of extreme values. Journalof the Royal Statistical Society: Series B (Statistical Methodology), 65(2),pp.545-556.
Fukutome, S., Liniger, M. and Sveges, M. (2014). Automatic threshold and runparameter selection: a climatology for extreme hourly precipitation inSwitzerland. Theoretical and Applied Climatology.
Hsing, T., Husler, J. and Leadbetter, M. (1988). On the exceedance pointprocess for a stationary sequence. Probability Theory and Related Fields, 78(1),pp.97-112.
Suveges, M. and Davison, A. (2010). Model misspecification in peaks overthreshold analysis. The Annals of Applied Statistics, 4(1), pp.203-221.
White, H. (1982). Maximum Likelihood Estimation of Misspecified Models.
Econometrica, 50(1), p.1.
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Extreme Value TheoryFitting Models
ANZAPW 2015: Barossa Valley, South Australia
This work has been supported by the ARC through the LaureateFellowship FL130100039.
Questions?
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Extreme Value TheoryFitting Models
Results: Kalamia, Far North Queensland
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Extreme Value TheoryFitting Models
Results: Yamba, New South Wales