Extreme Value Analysis
What is extreme value
analysis?
 Different statistical distributions
that are used to more accurately
describe the extremes of a
distribution
 Normal distributions don’t give
suitable information in the tails of
the distribution
 Extreme value analysis is
primarily concerned with
modeling the low probability, high
impact events well
Extreme Value Analysis
Fit
Extreme Value Analysis-Why is
it Important to Model the
Extremes Correctly?
 Imagine a shift in the
mean, from A to B
 In the new scenario (B)
most of the data is pretty
similar to A
 However, in the
extremes of the
distribution we see
changes > 200%!
Extreme Value Analysis

Changes in the mean, variance
and/or both create the most
significant changes in the
extremes

Risk communication is critical
“Man can believe the impossible, but
man can never believe the improbable”
--Oscar Wilde (Intentions, 1891)
Extreme Value Analysis - Uses

Climatology
 Hurricanes, heat waves, floods

Reinsurance Industry
 Assessing risk of extreme events

Wall Street
 Market extremes and threshold
exceedence potentials

Hydrology
 Floods, dam design

Water Demand!
Two Approaches To EVA
Block Maxima
Points over Threshold
 location parameter µ
 scale parameter σ
 shape parameter k

Used…
 …in instances where
maximums are plentiful
 …when user would like to
know the magnitude of an
extreme event
 shape parameter k
 scale parameter σ
 threshold parameter θ

Used…
 …in instances where data is
limited
 …when user would like to
know with what frequency
extreme events will occur
Case Study Introduction
 Water demand data
from Aurora, CO
 Used for
NOAA/AWWA
study on the
potential impacts of
climate change on
water demand
Generalized Extreme Value
Distribution: Block Maxima
Approach
‘Block’ or Summer Seasonal
Maxima in Aurora, CO
Issues

For water demand data ‘blocks’
could be annual or seasonal

However, this leaves us with a
very limited amount of data to fit
the GEV with for Aurora

This is not an appropriate
method to use because of the
limited data
GEV: Block Maxima Approach
Aurora, CO Seasonal
Monthly Maximums
Compromise

Not a true maxima

However, it allows GEV
modeling on smaller data sets

An acceptable approach for GEV
modeling
GPD: Points Over Threshold
Approach
Approach
Daily Water Demand; Aurora, CO

Choose some high threshold

Fit the data above the threshold
to a GPD to get intensity of
exceedence

Fit the same data to Point
Process to get frequency of
exceedence
GPD: Points Over Threshold
Approach
Capacity of Points Over Threshold Process

Uses more data than GEV

Can answer questions like ‘what’s the
probability of exceeding a certain threshold
in a given time frame?’ or ‘How many
exceedences do we anticipate?’

We can also see how return levels will change
under given IPCC climate projections

This will give an idea about the impact of
climate on water demand
Points Over Threshold
Use

The point process fit is a Poisson
distribution that indicates
whether or not an exceedence
will occur at a given location

The point process fit couples with
the GPD fit will be used to
model the data
Non-Stationary EVA
Benefits

Allows flexible, varying models

Improved forecasting capacity

Trends in models apparent
   0  1 x

Potential covariates
 Precipitation
 Temperatures
 Spell statistics
 Population
 Economic forecasts
 etc
4e-04
Stationary GEV
2e-04
1e-04
0e+00
PDF
3e-04
Unconditional GEV
0
2000
4000
6000
Maximum Streamflow (cfs)
8000
4e-04
3e-04
2e-04
1e-04
0e+00
PDF
2e-04
1e-04
0e+00
PDF
3e-04
4e-04
Conditional GEV Shifts with Climate
Covariates
0
0
20002000
40004000
60006000
Maximum
Streamflow
(cfs) (cfs)
Maximum
Streamflow
80008000
(Towler et al., 2010)
4e-04
4e-04
Conditional GEV Shifts with Climate
Covariates
2e-04
2e-04
P[S>Q90Uncond] ??
40%
1e-04
1e-04
10%
3%
0e+00
0e+00
PDF
PDF
3e-04
3e-04
Q90
0
0
2000
2000
4000
4000
6000
6000
Maximum Streamflow
Streamflow (cfs)
(cfs)
Maximum
8000
8000
(Towler et al., 2010)
Non-Stationary Case

We can allow the extreme value
parameters to vary with respect to a
variety of covariates

Covariates will be the climate
indicators we have been building
(temp, precip, PDSI, spells, etc)

Forecasting these covariates with
IPCC climate models will give the best
forecast of water demand

Climate is non-stationary, water
demand fluctuations with respect to
climate will also not be stationary
Generalized Parateo
Distribution