Stochastic Models for Weather and Hydroclimatic
Extremes
Workshop on metrics and methodologies of estimation of extreme climate events
27-29 September 2010,UNESCO headquarters, Paris, France
Anna K. Panorska
University of Nevada Reno, USA
with
Tomasz J. Kozubowski and Fares Qeadan
“From the tropics to the arctic, climate and weather
have powerful direct and indirect impacts on human life.
Extremes of heat and cold can cause potentially fatal
illnesses, .... Other weather extremes, such as heavy
rains, floods, and hurricanes, also have severe impacts
on health. Approximately 600 000 deaths occurred
worldwide as a result of weather-related natural
disasters in the 1990s.”
The WHO, 12 November, 2008
1
Outline
•
MOTIVATION: Extreme hydroclimatic and
weather events, e.g. drought, flood, or heat
waves.
•
•
Examples: Paris heat wave, August 2003.
Great Drought of the 1930s in the USA, THE
“DUST BOWL”.
•
STOCHASTIC MODELS: First joint model for
duration, magnitude and maximum of events
•
EXAMPLE: Back to the “Dust Bowl” and Paris
heat wave
•
Summary
2
From observations to events
precipitation
Original precipitation data in
standard deviation units plotted as
difference from a threshold.
Precipitation “events” or
“episodes”.
2
2
1.5
1.5
1
1
0.5
0.5
0
-0.5
1897
1905
1913
1921
-1
1929
1937
1945
0
1897
-0.5
1905
1913
1921
1929
1937
1945
-1
-1.5
-1.5
-2
year
Threshold -2
Episodes: wet and dry
An episode is a period with the process staying consecutively above/below threshold:
e.g. drought, flood, heat wave, etc.
Threshold for “dry” or “wet” depends on the definition of the episode (e.g. drought).
Data: Dendroclimatic (western juniper) reconstruction of precipitation from 300 BC to AD 2001
in the Walker River watershed (California/Nevada). California Climate Division 3.
3
Climatological/weather Events: Visualizing Intuition
Positive events
Start with a process:
X1 , X 2 ,
, Xk ,
X1 , X 2 ,
DURATION = the
number of years/time
periods in one event :
random = N.
, Xk ,
Negative events
threshold
Duration=N
MAGNITUDE = area of the
shaded region = random
sum of the series values
for one event.
N
Magnitude=X= X i
i 1
MAXIMUM = max observation
during an event = random
maximum of the series values
for one event.
Max/Peak=Y= max X i
i 1 N
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Kozubowski and Panorska (2005), (2008), Biondi, Kozubowski and Panorska (2005), Biondi et al. (2008).
RANDOM VECTOR
N
( N ,  X i , max X i )  ( N , X , Y )
i 1
1i  N
GOAL: Construct a mathematically natural, that is reflecting the
construction of (N, X, Y), model(s) for the JOINT distribution of
(duration, magnitude X and maximum Y) of events.
Notable properties of the random vector (N, X, Y):
• All components are related/dependent;
•The sum and maximum are of random number of random
observations;
•The joint behavior of X and Y is not trivial.
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CONSTRUCTION OF THE MODEL FOR
(N, X, Y) = (duration, magnitude, maximum)
Hierarchical approach:
1.
Specify distribution of N
2.
Given N=n, find conditional distr.
f ( x, y | n)
of
n
(X, Y|N=n) =
( X i , max X i )
i 1
i 1, , n
3. Get the joint distribution of (N, X, Y) as
f (n, x, y)  f ( x, y | n) f N (n)
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History of the joint distribution of magnitude and
maximum
Only asymptotics for large n were done for exponential observations.
 n

  X i , max X i 
1i  n
 i 1

Appropriate normalization
 X
, Y 
Normal
Extreme value distribution
independent
Levy 1953; D.Z. Arov and A. A. Bobrov 1960; Chow and Teugels 1978; Woodroofe and Zinn
7
1981; Maller and Resnick 1983; Peter J. Haas 1992.
KEY NEW RESULT: PDF of the Bivariate Distribution of the
Random Sum and Max for any n
n
Let Xi, i=1, …, n, n ≥ 2, be iid exp(β). The joint pdf of (X, Y)=
is given explicitly by
( X i , max X i )
i 1
i 1, , n
f ( x, y)   ne x H ( x, y, n), where
n(n  1)
k
n2
s 1
(
x

sy
)
(

1
)
for n  2, ( x, y )  S k

 ( s  1)!(n  s )!
 s 1
H ( x, y , n )  
1
for n  1, ( x, y )  S 0

0
otherwise.


and S0={(x,y): x=y>0} while
S k  {( x, y )  R 2 : 0 
1
1
x  y  x}, k  1,2, , n  1.
k 1
k
This new distribution is called BGGE(β, n) for bivariate distribution with
gamma and generalized exponential marginals.
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Qeadan, Kozubowski, Panorska (2010), Communication in Statistics: Theory and Methods, to appear.
Densities
n=2
n=5
n=25
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Covariance Structure
(X, Y)~ BGGE(β, n). EX=n/ β, EY=(ψ(n+1)+γ)/ β, corr(X, Y) is
n 
 (n  1)  
n
 n '(n  1)
6
2
,
where ψ and ψ’ are the digamma and
polygamma functions, respectively,
and γ is the Euler constant.
NOTE: ρn→0, but very slowly.
Thus, approximation of (X, Y) with independent X and Y is not good even for
large n. E.g. ρ500 =0.24
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New Stochastic Model: Joint Distribution of the
Duration, Magnitude and Maximum of an Episode
N-random N-Geo(p), Xi’s iid exp(β) ind. of N.
 N

Yi , N 
 X, Y, N     Xi , max
1i  N
 i 1

TETLG( ,p)
Trivariate distribution with exponential, truncated logistic and geometric
marginals1.
The pdf of (X, Y, N) is
f ( x, y, n)   ne x p(1  p)n1 H ( x, y, n),
where function H is given in the pdf of the (X,Y|N=n)
1
Kozubowski, Panorska, Qeadan (2010), in final review.
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Why bother with the stochastic models?
• Can be used to estimate probabilities or figure out properties
of events not yet observed.
• Incorporate the dependence between duration, magnitude and
maximum of the events.
• Represent the process in a mathematically natural way.
• Can be extended to include the spatial dimension.
• Can use them as framework for precise formulation of
definitions, probabilities, etc., for natural events.
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EXAMPLE 1: THE “DUST BOWL”
The great drought of the 1930s in the US, the 'Dust Bowl' impacted millions
of people in many states. It had terrifying effects on the economy and
the natural environment.
Magnitude=X=7.76
Maximum=Y= 1.04
Maximum=Y= 1.04
• Probability of a drought longer or larger than the 'Dust Bowl' is 0.08;
• Probability of a drought longer and larger than 'Dust Bowl' is 0.06.
• Conditional probability of a drought with at least 'Dust Bowl'’s magnitude
given that duration is 11 years is 0.46
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EXAMPLE 2: Paris heat wave of the 2003
In August 2003, France experienced an extreme heat wave, that resulted in an
estimated 14,802 deaths*.
Definition of a hot event: consecutive observations above the 33oC.
•Probability of a hot event
with the same duration of 11
days is 0.000075;
Max=64
Magnitude=479
N=11 days
Zero level=330C
•Conditional probability of a
heat wave with at least that
magnitude given that
duration is 11 days is about
5.5e-4.
•Probability of a heat wave
longer than 6 days and
larger than 100 (98th
percentile of magnitudes) is
0.005.
Note: Maximum Y = 64deg C*10 (really 39.4oC)
*Dhainaut et al. 2004., Data from http://eca.knmi.nl/, station ID 104969
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SUMMARY
Given a process X1, X2, X3, …. fluctuating around a threshold.
Positive events
Suppose Xi’s are iid exp(β),
and the process
fluctuates according to a
geometric model.
2
1.5
precip
1
threshold
0.5
0
-0.5 1920
1928
1936
1944
1952
1960
1968
-1
-1.5
-2
 N

Yi , N 
 X, Y, N     Xi , max
1i  N
 i 1

year
Negative events
TETLG( ,p)
NEW stochastic model for the joint distribution of duration, magnitude,
and maximum of the events.
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NEVADA – A STATE OF EXTREMES
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Thank You