1. 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

2. 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%!

3. 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)

4. 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!

5. Two Approaches To EVA Block Maxima  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 Points over Threshold  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

6. Case Study Introduction  Water demand data from Aurora, CO  Used for NOAA/AWWA study on the potential impacts of climate change on water demand

7. 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

8. GEV: Block Maxima Approach Aurora, CO Seasonal Monthly MaximumsCompromise  Not a true maxima  However, it allows GEV modeling on smaller data sets  An acceptable approach for GEV modeling

9. GPD: Points Over Threshold Approach Daily Water Demand; Aurora, CO Approach  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

10. 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

11. 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

12. Non-Stationary EVA Benefits  Allows flexible, varying models  Improved forecasting capacity  Trends in models apparent  Potential covariates  Precipitation  Temperatures  Spell statistics  Population  Economic forecasts  etc

13. Stationary GEV Unconditional GEV

14. Conditional GEV Shifts with Climate Covariates (Towler et al., 2010)

15. P[S>Q90 Uncond ] ?? 10% 40% 3% Q90 Conditional GEV Shifts with Climate Covariates (Towler et al., 2010)

16. 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