EMD9/3/2009
1
Extreme wind speed estimation
Outline• Definition
• Data extraction
• Choice of asymptote
• Convergence to asymptote (bias I)
• Plotting positions (bias II)
• Bias I & Bias II – A Monte Carlo experiment
• The plot
• Choice of fit?
• Conclusion/Recommendations
Lasse Svenningsen, EMD
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EMD9/3/2009
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Extreme wind speed estimation
Outline• Definition
• Data extraction
• Choice of asymptote
• Convergence to asymptote (bias I)
• Plotting positions (bias II)
• Bias I & Bias II – A Monte Carlo experiment
• The plot
• Choice of fit?
• Conclusion/Recommendations
A matter of choices!
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Definition of “50 year” wind?
Annual risk of exceedance R=2%
Or T = 1/R = 50y
The real task is:
• Estimate the CDF for annual extremes, FA(u)
• Given FA(u) find: u | FA(u) =1-R=98%
• Or reduced variate: u | y=-ln(-ln(0.98))=3.9
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Data extraction
“Annual Max” or “Peak-over-Threshold”?
• AM:
• Only use the max of each year
• POT:
• Use all independent storms above uthreshold
• Stormrate: λ
• Parent:
• For known N & Weibull CDF: FA(u)= WCDF(A,k)N
• ….
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LS2
Slide 4
LS2 Largest / n'th order statitsticLasse Svenningsen, 01/09/2009
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Data extraction
“Annual Max”
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0 2000 4000 6000 8000 10000 120000
5
10
15
20
25
30
days from 1st sample
u[m
/s]
1800 2000 2200 2400 2600 2800 3000 3200 3400
16
18
20
22
24
26
28
days from 1st sample
u[m
/s]
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Data extraction
“Peak-over-Threshold”
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0 2000 4000 6000 8000 10000 120000
5
10
15
20
25
30
days from 1st sample
u[m
/s]
1800 2000 2200 2400 2600 2800 3000 3200
18
20
22
24
26
28
days from 1st sample
u[m
/s]
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Choice of asymptote
GEV distribution type 1, 2 or 3??
Type 1: “Gumbel”
• Parent PDF: exponential tail
• E.g.: Weibull, Gauss, …
Type 2: “Fréchet”
• Parent PDF: power-law tail (heavy tailed)
• E.g.:Log-normal, Pareto
Type 3: “Reverse Weibull”
• Parent PDF: bounded above
• E.g.: Beta
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Choice of asymptote
GEV distribution type 1, 2 or 3??
From Palutikof (1999)Ty
pe2
Type 1
Type 3
u
y
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Choice of asymptote
GEV distribution type 1, 2 or 3??
If Weibull…..
….Then Gumbel!!
Common observations:
- Type 3 tends to fit better?
- Type 1 too conservative?
Why?
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Convergence to asymptote
Extreme distributions are only asymptotic!
• Assumes: extremes of N=∞ independent samples
• In reality N<<∞
• Nyear=52560 x 10min but…
• Only ~ 2300 are independent (Bergström,1992)
• The consequence?
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BiasI
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Convergence to asymptote
Rayleigh type Weibulls (k=2) converge slowly
From Cook (1982)
Type
3
u
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BiasI
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Convergence to asymptote
Exponential Weibulls (k=1) converge extremely fast
From Cook (1982)
u
y
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BiasI
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Convergence to asymptote
The fix is “preconditioning”:• Transform z(u) to make WCDF(z) exponential
• Transform result back to u after the analysis
• Optimum transform: z=uk
•Weibull CDF:
−−= k
k
CDF AuAkuW exp1),,(
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BiasI
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Plotting positions
Probability plotting positions, Fest.?
• The choices:
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BiasII
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Bias I & Bias II….Extr
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A Monte Carlo experiment - setup:
• 1000 random realizations generated for:
Weibull parent with:Nsamples = 2300 �(Bergström, 1992)A = 10 m/sk = 1.7
⇒ V50, exact= 42.4 m/s (~ IEC class I/II limit)
Nyears = 1….20
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Bias I & Bias II….Extr
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A Monte Carlo experiment - setup:
AM: Pick annual maxima
POT: Select storm threshold / fixed number of storms(independence assured)
Fit: LSQ
Bias estimate: V50, ensemble mean / V50, exact
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Bias I & Bias II….Extr
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A Monte Carlo experiment – Bias II:
• Choice of plotting position: AM
2 4 6 8 10 12 14 16 18 200.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
Number of years
Bia
s: m
ean/
exac
t
WeibullHazenGringorton
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Bias I & Bias II….Extr
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A Monte Carlo experiment – Bias II:
• Effect of k-factor preconditioning: AM
2 4 6 8 10 12 14 16 18 200.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
Number of years
Bia
s: m
ean/
exac
t
WeibullHazenGringorton
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Bias I & Bias II….Extr
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A Monte Carlo experiment – Bias II:
• Choice of plotting position: POT (fixed uthreshold = 24m/s)
0 5 10 15 200.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
Number of years
Bia
s: m
ean/
exac
t
WeibullHazenGringorton
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Bias I & Bias II….Extr
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A Monte Carlo experiment – Bias II:
• Choice of plotting position: POT (fixed NS=20)
0 5 10 15 200.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
Number of years
Bia
s: m
ean/
exac
t
Weibull
Hazen
Gringorton
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Bias I & Bias II….Extr
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A Monte Carlo experiment – Bias II:
• Effect of k-factor preconditioning: POT (fixed NS=20)
0 5 10 15 200.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
Number of years
Bia
s: m
ean/
exac
t
WeibullHazenGringorton
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Bias I & Bias II….Extr
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A Monte Carlo experiment – conclusion:
• Total bias < 1% if…
• AM:
• k-factor preconditioning
• Hazen plotting positions
• POT:
• Fixed “storm” number ~ 20
• k-factor preconditioning
• Weibull plotting positions
(some exceptions!)
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Bias I & Bias II….Extr
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A Monte Carlo experiment – conclusion:
• Scatter around mean of the 1000 realisations??
0 5 10 15 20 250.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Number of years
Ens
embl
e C
OV
- 1
000
real
isat
ions
POT-20AM
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Bias I & Bias II….Extr
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A Monte Carlo experiment – conclusion:
• Correlation with max? Nyears = 1
POT r = 0.95 AM
25 30 35 40 45 50 5530
35
40
45
50
55
60
65
Max u each period (N years) [m/s]
Res
ultin
g V
50 f
or p
erio
d [m
/s]
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Bias I & Bias II….Extr
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A Monte Carlo experiment – conclusion:
• Correlation with max? Nyears = 5
POT r = 0.95 AM r = 0.95
32 34 36 38 40 42 44 46 48 50 5230
35
40
45
50
55
60
65
70
Max u each period (N years) [m/s]
Res
ultin
g V
50 f
or p
erio
d [m
/s]
30 35 40 45 50 55 6030
35
40
45
50
55
60
65
70
Max u each period (N years) [m/s]
Res
ultin
g V
50 f
or p
erio
d [m
/s]
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Bias I & Bias II….Extr
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A Monte Carlo experiment – conclusion:
• Correlation with max? Nyears = 10
POT r = 0.95 AM r = 0.95
30 35 40 45 50 5535
40
45
50
55
60
65
Max u each period (N years) [m/s]
Res
ultin
g V
50 f
or p
erio
d [m
/s]
30 35 40 45 50 55 6035
40
45
50
55
60
65
Max u each period (N years) [m/s]
Res
ultin
g V
50 f
or p
erio
d [m
/s]
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The Plot
Order of the axes?
Which has the greater uncertainty, u or y?
• Hopefully u is measured accurately!
• y uncertainty (FA) is large especially for highest u!
Or
u
y
y
u
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The Plot
Order of the axes?
Which has the greater uncertainty, u or y?
• Hopefully u is measured accurately!
• y uncertainty (FA) is large especially for highest u!
u
y
y
u
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The Plot
Making AM and POT plots directly comparable
• AM:
• Plot: (uk,y)
• Extrapolate to: y = -ln(-ln(1-1/T)) = 3.9
• POT:
• Plot: (uk,y)
• Extrapolate to: y = -ln(-ln( 1-1/λT )) = ?Extr
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Differen
t y-axes!
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The Plot
The problem:
• AM:
• Estimates CDF of annual extremes directly: FA(u)
• Extrapolate to risk: R = 0.02
• POT:
• Estimates CDF of the extracted λ storms/year: FS(u)
• Extrapolate to risk: R = 0.02/λ
The solution:
• Get FA from FS (Cook, 82): FA(u) = FS(u)λ
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The Plot
Making AM and POT plots comparable
• AM:
• Plot: (uk,y)
• Extrapolate to: y = -ln(-ln(1-1/T)) = 3.9
• POT:
• Plot: (uk, y0-ln(λ) )
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Same y-axes!
!!!!
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Choice of fit
Common types of fit used:
• LSQ (L2)
• PWM (probability weighted moments)
• W-LSQ (weighted L2)
• Which weights to use? Harris’? Gumbels?
• LAD (L1)?
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Conclusion/suggestionsExtr
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att
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ices • Choose POT for real life applications (always too few data!)
• Use the GEV Type 1 asymptote (i.e. Gumbel)
• Use k-factor preconditioning (z=uk)
• AM: Use “Hazen” plotting positions
• POT: Use “Weibull” plotting positions (y0), and…
• Use fixed “storm” number ~ 20, not fixed threshold
• Plot: z=uk versus y=y0-ln(λ), extrapolate to y=3.9
• Fit: ???