Lecture

2023-11-01

Today

Extreme values

Case studies

Theoretical Frameworks

Terminology

Challenges

- Likelihood of extreme events
- Extrapolate
- Univariate (for now)

- Engineering design
- Emergency management
- Regulation
- Insurance
- Managing financial assets

- Streamflow
- Precipitation rates or totals
- Wind speed
- Temperatures

- There are simple and clever methods for estimating the likelihood of rare events
- Fundamentally, extrapolating is hard
- Key sources of uncertainty:
- Estimation uncertainty
- Model structure uncertainty
- Sampling uncertainty

Today

Extreme values

Case studies

Theoretical Frameworks

Terminology

Challenges

I was asked to calculate what would happen under some specific assumptions (no suggestion of unbiasedness!)

- These were plausible assumptions, not necessarily the “right” or “best” assumptions (other side had many reasonable objections!)
- Large differences between estimates made under different assumptions underscore the challenge of “objective” esimates
- A hard problem (interacting drivers of nonstationarity, short records, etc)

- TWDB project
- Atlas 14 (Perica et al., 2018) does not account for climate change
- Use more stations and account for climate change
- Joint TWDB/TAMU/Rice project
- Variable studied: precipitation at multiple durations across the entire state

- What should we plan for?
- Was it an “unprecedented” event?
- Variable studied:
- temperature at grid cells
- aggregated “population-weighted” index

Today

Extreme values

Case studies

Theoretical Frameworks

Terminology

Challenges

- Define a threshold
- Model the distribution of events above the threshold
- Model the probability of seeing an event over the threshold
- Advantage: focus on meaningful events, even if they’re rare
- Disadvantage: threshold is arbitrary, modeling arrival turns out to be tricky

- Define a block size (e.g., 1 year – how you define “a year” matters)
- Model the distribution of the extreme in each block
- Advantage: easier to communicate and implementations are more flexible
- Disadvantages: timing of extremes; two extremes in one year; sometimes your min/max is not special

Today

Extreme values

Case studies

Theoretical Frameworks

Terminology

Challenges

- Exceedance probability (often AEP): \(p\)
- Return period or recurrence interval (\(T\)): \(\frac{1}{p}\)
- Return level: the value that will be exceeded with probability \(\frac{1}{T}\), ie the quantile

Two ideas:

- Ranks: if you have \(N\) events, the largest is rank 1, the second largest rank 2, etc.
- Use the points that you have as
*return levels*and estimate the associated*return periods* - Common estimator is Weibull plotting position \(p = \frac{m}{N+1}\) where \(m\) is the rank
- Lots of bickering in the literature about right choice
- When you see return period plots, if the observations are shown they are using a plotting position of some sort

- Different variables have different properties
- USGS likes this distribution for streamflow

- Model block extremes
- As we will see, has strong theoretical justification
- Three parameters: location, scale, shape
- Shape can be tricky to estimate \(\rightarrow\) large parameter uncertainty

- Similar: location, scale, and shape parameter
- Requires a separate model for “arrivals”

Today

Extreme values

Case studies

Theoretical Frameworks

Terminology

Challenges

- Many parameter values can be consistent with the data
- But lead to very different conclusions about the likelihood of extreme events
- Hard to pin down
- Relatively easier to resolve through clever approaches we will see

- Different assumptions lead to different inferences
- GEV / GPD are theoretically justified
- As we add clever approaches, we add model structure uncertainty

- We have a finite sample of data and are trying to estimate parameters that tell us about rare events
- If Harvey had never occurred, we would likely have a very different estimate of the 100-year rainfall!

- These models don’t explicitly account for climate change
- Interannual variability / correlations

- Coles (2001): canonical extreme value textbook

- Exams
- Working on grading exams 1 and 2

- Class
- Friday 11/3: lab
- Monday 11/6: POT and Block Maxima Models
- Wednesday 11/8: GEV models and estimators

Coles, S. (2001). *An introduction to statistical modeling of extreme values*. London ; Springer.

Perica, S., Pavlovic, S., St. Laurent, M., Trypaluk, C., Unruh, D., & Wilhite, O. (2018). *NOAA Atlas 14* (No. Volume 11 Version 2.0: Texas) (p. 283). Silver Spring, MD: National Weather Service, National Oceanic and Atmospheric Administration, U.S. Department of Commerce. Retrieved from https://www.weather.gov/media/owp/oh/hdsc/docs/Atlas14_Volume11.pdf