Extreme Value Theory
Fitting Models
Investigation of an Automated Approach to Threshold
Selection for Generalized Pareto
Kate R. Saunders
Supervisors: Peter Taylor & David Karoly
University of Melbourne
April 8, 2015
Extreme Value Theory
Fitting Models
Outline
1
Extreme Value Theory
2
Fitting Models
Extreme Value Theory
Fitting Models
Problem
What are the climate processes that drive extreme rainfall?
(El Ni˜
no Southern Oscillation, Interdecadal Pacific Oscillation)
How do these drivers differ at different timescales; sub-daily, daily,
consecutive day totals?
Extreme Value Theory
Fitting Models
Data
Extreme Value Theory
Fitting Models
Extreme Value Theory
Extreme Value Theory
Fitting Models
Block Maxima
Extreme Value Theory
Fitting Models
Block Maxima
Let X1 , X2 , ... , Xn be a sequence of i.i.d. random variables with
distribution function F . Define Mn = max{X1 , X2 , . . . , Xn }.
(Xi might be daily rainfall observations and M365 the annual maximum
rainfall.)
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 generalised
extreme value distribution.
Extreme Value Theory
Fitting Models
Generalized Extreme Value Theorem (Fisher-Tippett-Gnendenko)
If there exists sequences of constants {an > 0} and {bn } such that
Mn − bn
≤ z → G (z) as n → ∞
Pr
an
for a non-degenerate distribution function G , then G is a member of the
Generalized Extreme Value family



− 1 
 
z −µ
ξ
G (z) = exp − 1 + ξ


σ


defined on {z : 1 + ξ(z − µ)/σ > 0}, where ∞ < µ < ∞, σ > 0 and
−∞ < ξ < ∞.
Extreme Value Theory
Fitting Models
Leveraging more data
Extreme Value Theory
Fitting Models
Generalized Pareto Distribution
Let X1 , X2 , ... , Xn be a sequence of iid random variables with marginal
distribution 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 enough
threshold u, the distribution function of (X − u) conditional on X > u is
the GPD.
Generalized Pareto Distribution - Picklands (1975)
ξy −1/ξ
H(y ) = 1 − 1 +
σ
˜
defined on {y : y > 0} and (1 + ξy /˜
σ > 0) where, σ
˜ = σ + ξ(u − µ).
Extreme Value Theory
Fitting Models
Dependence
Rainfall observations are dependent
Heavy rainfall yesterday effects the probability of heavy rain today
Heavy rainfall a year ago doesn’t
Extreme Value Theory extends to stationary series with weak long
range dependence
However, for processes with short range dependence extremes occur
in clusters
Extreme Value Theory
Fitting Models
Clusters
Extreme Value Theory
Fitting Models
Dependent Series
Let {Xi }i≥1 be a stationary series and {Xi∗ }i≥1 be an independent series
of variables with the same marginal distribution.
Define Mn = max{X1 , . . . , Xn } and Mn∗ = max{X1∗ , . . . , Xn∗ }. Under
suitable regularity conditions,
∗
(Mn − bn )
Pr
≤ z → G (z),
an
as n → ∞ for normalizing sequences {an > 0} and {bn }, where G is a
non-degenerate distribution functions, if and only if
(Mn − bn )
Pr
≤ z → G θ (z),
an
for a constant θ such that 0 < θ ≤ 1.
Extreme Value Theory
Fitting Models
Extremal Index
θ = {Limiting mean cluster size}−1 ∈ (0, 1]
θ = 0.5 ⇒ 2 observations per cluster on average.
Extreme Value Theory
Fitting Models
Fitting Models
Extreme Value Theory
Fitting Models
Fitting Models
Select a threshold
Decluster the data for independent observations
Extreme Value Theory
Fitting Models
Declustering
Blocks
Partition the observation sequence into blocks of length, b
Assume extreme observations within the same block belong to the
same same cluster.
Runs
Specify a run length, K
Assume extreme observations with an inter-exceedance time of less
than K belong to the same cluster.
Extreme Value Theory
Fitting Models
Intervals
The limiting process of exceedance times is compound Poisson for
stationary series (Hsing et al. 1988).
Ferro and Segers (2003) showed the limiting distribution of
inter-exceedance times is a mixture distribution with weight θ,
Tθ (t) = (1 − θ)0 + θ · θ exp(−θt),
where 0 is a degenerate distribution, Tθ is the distribution of arrival
times 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 between
cluster arrivals.
Extreme Value Theory
Fitting Models
Fitting Models
→ Select a threshold
Decluster the data for independent observations
Extreme Value Theory
Fitting Models
Mean Residual Life Plots
For sufficiently high thresholds, as the threshold increases the expected
exceedance above the threshold should grow linearly.
Extreme Value Theory
Fitting Models
Parameter Stability Plots
Parameter estimates of (modified) scale and shape parameters should be
constant for the range of valid thresholds.
Extreme Value Theory
Fitting Models
Alternative
Set the threshold according to a high quantile of non-zero observations
Eg. 90th percentile.
Is this an appropriate threshold?
Is our model is misspecified?
Suggested approach by S¨
uveges and Davison et al. (2010) is to test the
threshold, u, and run parameter, K pair for model misspecification.
Extreme Value Theory
Fitting Models
Log-Likelihood
Limiting distribution of inter-exceedance times:
Tθ (t) = (1 − θ)0 + θ2 exp(−θt),
Log-Likelihood (strictly positive inter-exceedance times):
N−1
X
i=1
I(ti =0)
log (1 − θ)
2
I(ti >0)
(θ exp(θti )
=
N−1
X
2I(ti > 0) log(θ) − θti ,
i=1
i
where ti = NT
n , n is the total number of observations and N is the
number of exceedances.
ˆ tends to 1 suggesting
However as n gets large our estimate, θ,
independence.
Extreme Value Theory
Fitting 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
X
I(ci = 0) log(1 − θ) + 2I(ci > 0) log(θ) − θci
i=1
Approach used in Fukutome et al. (2014) and S¨
uveges and Davison
(2010).
Test combinations of threshold, u, and run parameter, K , for
misspecification of the likelihood function. Select the (u, K ) pair that
maximizes the number of independent clusters.
Extreme Value Theory
Fitting 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 {`00 (θ; cj } is equal to the
variance of the score vector, J(θ) = Var {`0 (θ; cj )}.
Test the hypothesis:
D(θ) = J(θ) − I (θ),
where H0 : D(θ) = 0 and H1 : D(θ) 6= 0.
Extreme Value Theory
Fitting Models
Empirically:
ˆ =
IN−1 (θ)
N−1
−1 X 00 ˆ
` (θ; cj )
(N − 1)
j=1
ˆ =
JN−1 (θ)
N−1
X
1
ˆ cj )2
`0 (θ;
(N − 1)
j=1
ˆ = JN−1 (θ)
ˆ − IN−1 (θ)
ˆ
DN−1 (θ)
ˆ =
VN−1 (θ)
N−1
X
2
1
0
−1 0 ˆ
ˆ
ˆ
ˆ
dj (θ; cj ) − DN−1 (θ)IN−1 (θ) ` (θ; cj )
(N − 1)
j=1
ˆ is the sample variance of DN−1 (θ).
ˆ
where VN−1 (θ)
Extreme Value Theory
Fitting Models
Model Misspecification
Theorem: (Information Matrix Test - Whyte 1982) If the assumed
model `(θ; ci ) contains the true model for some θ = θ0 , then as n → ∞,
p
w
ˆ −
(i)
(N − 1)DN−1 (θ)
→ N(0, V (θ0 )),
a.s.
ˆ ) −−→ V (θ0 ), and VN−1 (θ)
ˆ is non-singular for sufficiently
(ii) VN−1 (θN−1
large N,
(iii) Then the Information Matrix Test statistic,
ˆ 0 VN−1 (θ)
ˆ −1 DN−1 (θ)
ˆ is asymptotically χ2 distributed.
(N − 1)DN−1 (θ)
1
Extreme Value Theory
Fitting Models
Example: AR(2)
Yi = 0.95Yi−1 − 0.89Yi−2 + Zi where Zi ∼ GP(1, 1/2) and n = 8000.
100 simulations
Extreme Value Theory
Fitting Models
Adjusting inter-exceedance times
Common to assume stationarity by enforcing seasonal blocking.
Collapse inter-exceedance times across seasonal blocks using the
memoryless property of the exponential for fitting.
Extreme Value Theory
Fitting Models
Results: Gatton, South East Queensland
Extreme Value Theory
Fitting Models
Results: Oenpelli, Northern Territory
Extreme Value Theory
Fitting Models
Summary
Shown how to check if the threshold and run parameter selected
violate the assumptions of the model
Given confidence to threshold selection in the absence of a hard and
fast rule and in the presence of subjectivity
Extreme Value Theory
Fitting Models
References
Ferro, C. and Segers, J. (2003). Inference for clusters of extreme values. Journal
of the Royal Statistical Society: Series B (Statistical Methodology), 65(2),
pp.545-556.
Fukutome, S., Liniger, M. and Sveges, M. (2014). Automatic threshold and run
parameter selection: a climatology for extreme hourly precipitation in
Switzerland. Theoretical and Applied Climatology.
Hsing, T., H¨
usler, J. and Leadbetter, M. (1988). On the exceedance point
process for a stationary sequence. Probability Theory and Related Fields, 78(1),
pp.97-112.
S¨
uveges, M. and Davison, A. (2010). Model misspecification in peaks over
threshold 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.
Extreme Value Theory
Fitting Models
ANZAPW 2015: Barossa Valley, South Australia
This work has been supported by the ARC through the Laureate
Fellowship FL130100039.
Questions?
Extreme Value Theory
Fitting Models
Results: Kalamia, Far North Queensland
Extreme Value Theory
Fitting Models
Results: Yamba, New South Wales