Quantifying Uncertainty in the Rarity of Extreme Multivariate Weather and Climate Events

Introduction Simulation Results: Step 1 Potential Uses References Proposed Methodology Background Methodological Next Steps ● Extreme weather and climate events are becoming more commonplace. However, what makes these events so hazardous is not necessarily an extreme event in a single variable but rather an extreme combination of variables that compound to produce dangerous conditions. ○ Examples: hot, dry, and windy conditions conducive to wildfires, or hot, humid, and stagnant conditions with increased human mortality risk ● Fifth National Climate Assessment (NCA5): studying compound events will be crucial to studying impacts of climate change and climate resiliency ● An important tool to assess the risk of extreme bivariate events is a p-isoline, a bivariate analog of a univariate probability distribution’s quantile ● Building on existing estimation methods, we propose a procedure to draw confidence tubes as a means to assess uncertainty in estimating p-isolines Quantifying Uncertainty in the Rarity of Extreme Multivariate Weather and Climate Events James Butler1, Jon McAuliffe1,2, Michael Wehner3 1Department of Statistics, UC Berkeley; 2The Voleon Group; 3Applied Mathematics and Computational Research Division, LBNL Acknowledgements 1. Construct estimate of base p-isoline ○ Choose p not too small, not too large ○ Obtain the p-level set of estimated bivariate survival function (empirical, Gaussian KDE, etc.) ● Current issue: projected regions not covering their extreme q-isolines. Why? ○ Transformation T doesn’t properly convert data to a space in which scaling/projection relation holds? (relies on estimates of marginal distribution functions..) ○ Even if we knew exact marginal distribution functions, projection doesn’t convert p-isolines to q-isolines exactly, only asymptotically well as you go further out in the tail. ● To make progress, studying statistical extreme value theory to: ○ Determine under what conditions transformed distribution admits projection property ○ Build inexactness of transformation and projection into the construction of confidence tubes, adjusting width accordingly JB and MFW were supported by the Director, Office of Science, Office of Biological and Environmental Research of the U.S. Department of Energy under Contract No. DE340AC02-05CH11231 under the Regional and Global Model Analysis (RGMA) program. Special thanks to Daniel Cooley (CSU) and Wolfgang Polonik (UC Davis) for sharing insight into their research results on which this project heavily relies! Cooley, D., Thibaud, E., Castillo, F., Wehner, M. F. (2019). A nonparametric method for producing isolines of bivariate exceedence probabilities. Extremes, 22, 373-390. doi: https://doi.org/10.1007/s10687-019-00348-0 Mammen, E., Polonik, W. (2013). Confidence regions for level sets. Journal of Multivariate Statistical Analysis, 122, 202-214. doi: https://doi.org/10.1016/j.jmva.2013.07.017 Observe data 2. Project estimate into tail to obtain extreme q-isoline estimate ○ Choose desired q, extremely small probability of interest ○ If distribution asymptotically dependent, do: Transform Scale/Project Back-Transform ○ If distribution asymptotically independent, do: Scale/ProjectTransform Back-Transform We propose a similar two-step procedure, but for confidence tubes Existing Method to Estimate Extreme Isolines (Cooley et al., 2019) Proposed Two-Step Method to Estimate Extreme Confidence Tubes Observe data 1. Construct 1-𝛼 confidence tube for base p-isoline ○ Choose p not too small not too large ○ Use Mammen and Polonik (2013) to draw base tube Data-dependent thresholds, chosen to have 1-𝛼 coverage probability 2. Scale base tube using same projection scheme to obtain confidence tube for extreme q-isoline, with q an extremely small probability of interest Simulation Results: Step 1 Pick a known distribution and a sample size n. For each of 500 simulations: ● Generate n random draws from distribution ● Generate 1-𝛼 confidence tube for base p-isoline ● Check if tube contains or covers known distribution’s true p-isoline Proportion of covers should be approx. 1-𝛼 for each p Data from Cooley et al. (2019), Karachi, Pakistan Known distributions: Bivariate t, Bivariate Gaussian, and Beta KDE on Cooley et al. (2019) Karachi Temp/Humidity Data Simulation Results: Step 2 Method appears to achieve asymptotic coverage for base regions for p = 0.1 and p = 0.05; p = 0.01 exhibits overcoverage that decreases with increasing sample size (higher thresholds due to less data that far into tail) Similar results for 𝛼 = 0.1 and 𝛼 = 0.01 Pursuing Step 2 is Still Worthwhile Similar results for Bivariate Gaussian and Karachi Beta KDE Severe miscoverage, even as n increases! Step 1 by itself (MP Only) leads to overly conservative (wide) confidence tubes Step 1 + Step 2 (MP + Cooley) gives much more narrow, informative confidence tubes. If undercoverage issue is fixed, this will be powerful! 1. Risk assessment for extreme combinations of climate events and their return times 2. Detecting distributional shift: are bivariate climate events of a fixed return period becoming more extreme? ○ Duality of confidence intervals and hypothesis testing ○ Can use large simulated climate ensembles to simulate draws from future distributions, compare confidence tube of q-isoline with that built on past/present data i. Can also see what kinds of combinations are becoming more severe (maybe not all combinations are more severe!) becomes Ex: 95% confidence tubes for 0.005-isolines, with 10000 draws from bivariate t distributions ● Future: df = 3 (heavier tailed) ● Past: df = 4 p-isoline: set of bivariate thresholds jointly exceeded with probability p Traces out a curve in the real plane Knowledge of a distribution’s p-isoline allows one to understand bivariate tail event risk Successful Cover Failed Cover Detects “future” distribution is heavier tailed than “past”!