# Extreme Value Theory

Topics: Generalized extreme value distribution, Weibull distribution, Extreme value theory Pages: 8 (1156 words) Published: June 19, 2013
Extreme Value Theory and Distribution: Application to earthquake, flood and wind

ESE158/C1

MIURA, Airei R
ESE-3/2010101255

Extreme Value Distribution
← To assess the risk
← Deals with rare events
• Extreme floods and snow falls
• High wind speeds
• Extreme temperature
• Large fluctuation in exchange rates
• Market crashes

Extreme Value Theory
← Distributions for maxima or minima
← Deals with extreme events
← 3 types of distribution
• Gumbel distribution
• Frechet distribution
• Weibull distribution
• Extreme Value Theory
← Ordinary Statistics
• try to describe a distribution of data
← Extreme Value Theory
• try to characterize the tail of the distribution

Extreme Value Theory
← Importance
• to quantify the magnitude of a worst-case or really-bad-case scenario ← Application areas:
• hydrology (stream/river fows)
• climate variables: precipitation, wind, heat waves • insurance/reinsurance
• engineering (structural design, failure)

Gumbel Distribution
← Extreme Value Type I distribution

← Focuses on engineering problems

• Annual flood flows

[pic]
← Where
• z = (x-µ)/

• µ - location of parameter

• σ - is the distribution scale (σ >0)

← Largest extreme value (maxima)

[pic]

The graph shows the PDF for σ=1 and µ=0

Frechet Distribution
← Extreme Value Type II
← devised one possible limiting distribution for a sequence of maxima ← Used in
• Finance
• Modeling of market-returns
← Largest extreme value (maxima)
[pic]

← Where
• α is the shape of parameter (α>0)
• β is the scale parameter (β>0)
← Heavy up tail graph
← Bounded on the lower side (x>0)
← Largest extreme value (maxima)
[pic]
PDF for β=1 and various values of α

Weibull Distribution
← Extreme Value Type III
← Strength of materials and fatigue analysis
← used in the problems of material science
← 2 parameter of the density function
[pic]
← Is defined for
• X>0
• Both distribution parameters (α – shape, β – scale) are positive

← Generalized formula by adding the location (shift) parameter γ [pic]
← Where
• γ – location paramter
– any real number
• x>γ
← When
• α=1, distribution reduces to exponential model • α=2, mimics the Reyleigh distribution (mainly used in telecommunication) • α=3.5, normal dostribution
← Smallest extreme value (minima)
← Commonly used in practice

Generalized Extreme Value Distribution
← GEV combines the Gumbel, Frchet, and weibull
← Apply Fisher-Tippett Theorem
[pic]
← PDF:
[pic]
← Where
• Z=(x-μ)/σ
• k – shape parameters
• σ – scale parameters
• μ – location parameters

← Another formula combined as one
[pic]

← The range of the GEV distribution is depends on k
[pic]
← GEV has 3 types depending on shape parameter, k
• Gumbel – k=0
• Frechet – k=1/(α>0)
• Weibull – k= -1/(α>0)
[pic]

APPLICATION

APPLICATION OF EXTREME FLOOD
← Application: Extreme Flood
← Heavy ﬂoods and destructive landslides in northern Taiwan (1997) • Economic losses
• 40 death
← used daily stream ﬂow data from the Pachang River (southern Taiwan) ← Got 39 yearly stream flow record ftom1961 to 1999
← Daily stream flow exceeding a given threshold
← 4 properties: characterize ﬂood events
• ﬂood peak - maximum daily ﬂow during the ﬂood period • ﬂood volume - cumulative ﬂow volume...