This document discusses statistical methods for analyzing climate extremes using extreme value theory. It introduces the generalized extreme value distribution (GEV) for fitting block maxima and the generalized Pareto distribution (GPD) for fitting values above a threshold. These distributions allow inference of return levels to communicate the risk of extreme events like 100-year rainfall. Diagnostic tools help select the appropriate distribution and threshold.

1. Extreme Value Analysis for Climate Research
Whitney Huang
Statistical and Applied Mathematical Sciences Institute (SAMSI)
University of North Carolina at Chapel Hill
whuang@samsi.info
CLIM Undergraduate Workshop, Oct. 23, 2017

2. Climate extremes
Figure: Source: St¨ockli, Simmon and Herring, NASA Earth Observatory (Left).
http://abcnews.go.com/US/tropical-storm-harvey-expect-historic-storm/story?id=49435050(Right).

3. Scientific questions
How large is the magnitude of extreme events (e.g. 100-year
rainfall)?
How extremes vary in space and time?
How extremes may change in future climate conditions?

4. Probability distribution and statistical modeling
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0.000.050.100.150.20
Probability distribution of X
x
Density
Histogram of data
data
Density 0 2 4 6 8 10 12
0.000.050.100.150.20
How can we infer the underlying probability distribution from data?

5. Normal distribution for sample means
x
Density
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0.000.050.100.150.200.25
x100
Density 3.5 4.0 4.5 5.0 5.5 6.0 6.5
0.00.20.40.60.81.0

6. Which distribution to use for sample maxima?
x
Density
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0.000.050.100.150.200.25
m100
Density 10 20 30 40 50
0.000.020.040.060.08

7. Some theorems
Central limit theorem:
Normal for sample means
Extremal types theorem:
Generalized extreme value distribution for
sample maxima

8. Fort Collins daily precipitation, 1900 ∼ 1999
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1900 1920 1940 1960 1980 2000
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Time (Year)
Dailyprecipitation(in)

9. Seasonal variation for daily precipitation
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1999
Dailyprecipitation(in)
Jan
Feb
Mar
Apr
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Jun
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Aug
Sep
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Dec
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1900 ~ 1999
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec

10. Summer vs. winter daily precipitation
Summer (JJA) prec (in)
Density
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02468
Winter (DJF) prec (in)
0 1 2 3 4
02468

11. Block maxima method
1900 1920 1940 1960 1980 2000
Summer Maximum
Fort Collins
Year
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4
Precipitation(in)
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Density
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12. GEV fit diagnostics
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012345
Quantile Plot
Model
Empirical

13. Inference for 100-year event
95% CI for 100−yr RL
summer max (in)
Density
0 2 4 6 8
0.00.20.40.60.81.01.2

14. Recall the block maxima method
1900 1920 1940 1960 1980 2000
Summer Maximum
Fort Collins
Year
0
1
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3
4
Precipitation(in)
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Density
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Can we use data more efficiently?

15. Threshold exceedances method
Theorem: GEV for block maxima ⇒ GPD for excesses over high
threshold
1900 1920 1940 1960 1980 2000
Year
DailyPrecip(inches)
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3
4
Precipitation(in)
Density
2
1
0

16. How to choose the threshold?
0 1 2 3
0.00.51.01.5
Mean Residual Life
Threshold (in)
MeanExcess

17. GPD fit diagnostics
0 1 2 3 4 5 6
0123456
Quantile Plot
Model
Empirical

18. Inference for 100-year event
95% CI for 100−yr RL
Threshold excess (in)
Density
2 4 6 8 10
01234

19. Summary
Climate extremes can have large impacts on both human
society and environmental systems
Extreme value theory provides a framework to model extreme
values
GEV for fitting block maxima
GPD for fitting threshold exceedances
Return level for communicating risk