Transcript Extreme Value Theory with Applications to Aviation Safety

Extreme Value Theory (EVT):
Application to Runway Safety
Wang Yao
Department of Statistics
Rutgers University
[email protected]
Mentor: Professor Regina Y. Liu
DIMACS -- July 17, 2008
Motivation
Task: allow multiple runway usage
to ease air traffic congestion!
Cut-off point: Require all landings to be
completed before the cut-off
point with certain “guarantee”
*
X
s
Q: How to determine s such that:
P(X> s)  .0000001=
(Extremely small!)
Difficulty (why Extreme Value Theory)
• Extremely small tail probability
e.g. p= 0.0000001
• Few or no occurrences (observations) in reality
e.g. Even with sample size=2000
2000  0.0000001  0.0002  1
Difficulty: No observations!
Possible Solution: Extreme Value Theory (EVT)
Overview of EVT
X1 , X 2 ,
X1:n  X 2:n 
, X n : Random sample from unknown distr. fun. F
 X n:n : Order statistics
 X n:n  bn

 x  
 G( x),
 {bn ; n  1} and {an  0; n  1}, s.t. P 
( n  )
 an


1
G ( x)  G ( x)  exp( (1   x)  ),   ,1   x  0
(1   x)

1

 e x for   0
 : Tail index ↔ Characterizes tail thickness of F
heavy tail
  0 : Fréchet distribution 

in between, e.g. normal dist.
  0 : Gumbel distribution 
  0 : Weibull distribution
finite end point, e.g. uniform dist.


Extreme Quantile
For F  D(G ) want to find
x p s.t. 1  F ( xp )  p
n
F
(an x  bn )  G( x)
F  D(G) 
1

 t (1  F (at x  bt ))   log G ( x)  (1   x) 

Y

b
Y

b
1
1
t
t
Y  at x  bt1  F (Y )   log G (
)  (1  
) 
t
at
t
at
1
Let
Take
n
t  with k
k
n, then
ˆ

 k 
 np   1

xp  
a n  bn

k
k
Estimated by
 k 
 np   1
xˆ p   
aˆ n  bˆn
ˆ
k
k
Learning from Some Known Distributions
e.g. Normal, Exponential, Chi-square,…
• Generate random samples n  100
• For p= 0.001, estimate the p-th upper quantile
P=0.001 Distribution
Estimated
xp
True
xp
Error
Case A
N(0,1)
4.37951
3.09023
1.28928
Case B
Exp(1)
5.99578
6.907755
0.911975
14.1312
16.2662
2.135
Case C
Chi-square(3)
• Analysis: Small sample size Method of moments
• Bootstrap Method:
a resampling technique for obtaining
limiting distribution of any estimator
P
Real Data
e.g. Landing distance: underling distribution/model unknown!
xp
* ********* * *

 *   
k2
k1
Task: Applying Bootstrap Method to find a proper k
(Bootstrap method: completely nonparametric approach and
does not need to know the underlying distribution)
X
Yet to be completed
• Analyze landing data collected from airport runways
• Apply bootstrap method with proper choice of k
• Determine the suitable cut-off point -estimate the tail index  , and extreme quantile x p
ˆ
 k 
 np   1

xˆ p  
aˆ n  bˆn
ˆ
k
k
Remarks:
Important project with real application.
Well motivated and requires new interesting statistical methodology
I learned some interesting new subjects, e.g. EVT, bootstrap method.
Statistics is a practical field and theoretically challenging.
Questions?
Acknowledgment: Thanks to DIMACS REU!