CLIM: Transition Workshop - On the Probability Distributions of the Intensity and Duration of Heatwaves - Sohini Raha, May 15, 2018
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Heatwaves are affecting the human mortality and the global climate adversely. There are many existing ad hoc definitions of heatwaves in literature. Here, we are identifying a notion of intensity and duration of the possible heatwaves and finding a proper definition using their distributions. We are modelling these two factors and attempting to find the predictive ability of our model. We have also obtained some preliminary results on obtaining the probability distribution of duration and intensity for a sequence of stationary random variables. We are exploring these concepts on a heat data of Atlanta from the year 1991-2012. The next goals are to relax the stationary assumption and obtaining more general results.
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CLIM: Transition Workshop - On the Probability Distributions of the Intensity and Duration of Heatwaves - Sohini Raha, May 15, 2018
- 1. On the probability distributions of the duration of heatwaves Sohini Raha & Dr. Sujit Ghosh In collaboration with Dr. Howard Chang & Dr.Jesse Bell Department of Statistics North Carolina State University May 15, 2018 1 / 30 Heatwave © 2018 by Raha & Ghosh
- 2. Outline Many Facets of Heatwaves A Probabilistic Framework Case Studies Fixed location: Atlanta Data Varying locations: USCRN Data Future Directions 2 / 30 Heatwave © 2018 by Raha & Ghosh
- 3. Section I: Many Facets of Heatwaves Definitions What is a Heatwave? " A period of abnormally and uncomfortably hot and unusually humid weather. Typically a heat wave lasts two or more days" (NOAA) 3 / 30 Heatwave © 2018 by Raha & Ghosh
- 4. Section I: Many Facets of Heatwaves Definitions No Universally accepted Definition of Heatwave "A universal quantitative definition of what constitutes an extreme heat wave is unlikely to ever be agreed upon "(Wehner et al. (2018)) Location specific definitions. Xu et al.(2018) evaluates 29 different definitions of Heatwave Vaidyanathan et al.(2016) had 92 different definitions 4 / 30 Heatwave © 2018 by Raha & Ghosh
- 5. Section I:many Facets of Heatwaves Definitions Examples of Definitions of Heatwave Definition 1:(Xu et al. 2018) Temperature (Min, Max, Ambient, Apparent etc) greater than T° for at least d days Definition 2 : (Frich et al. 2002)HWDI:At least 5 consecutive days, the maximum temperature exceeds the normal temperature by 5°C (9°F) based on a period of 1961-1990 . 5 / 30 Heatwave © 2018 by Raha & Ghosh
- 6. Section I:many Facets of Heatwaves Definitions Definition of Heatwave (Wehner et al.(2018)) Heat Index (HI) based on a complicated nonlinear function of Relative Humidity and Temperature. Caution if HI is 80-90°F Extreme Caution if HI is 90-103°F Danger if HI is 103-125°F Extreme Danger if HI over 125°F 6 / 30 Heatwave © 2018 by Raha & Ghosh
- 7. Section I:many Facets of Heatwaves Definitions Definition of Heatwave (Peng et al.(2011)) T1 = 97.5 percentile T2 = 81 percentile Heatwave is longest period of consecutive days if daily max temp ≥ T1 for ≥ 3 days daily max temp ≥ T2 for everyday average of daily max temp ≥ T1 7 / 30 Heatwave © 2018 by Raha & Ghosh
- 8. Section II: Probabilistic Framework Duration and Intensity Duration and Intensity 8 / 30 Heatwave © 2018 by Raha & Ghosh
- 9. Section II: Probabilistic Framework Duration and Intensity X1, X2, ... : A Stationary sequence of RVs M : Threshold (location specific) T1, T3, ... : times of the upcrossings T2, T4, ... : times of the downcrossings Duration : T2k − T2k−1, k= 1,2,... Intensity : XT2k−1 + ... + XT2k −1 − M(T2k − T2k−1) 9 / 30 Heatwave © 2018 by Raha & Ghosh
- 10. Section II: Probabilistic Framework Duration and Intensity Connection to existing Definitions (Frich et al.(2002)) T2k − T2k−1 ≥ 5 Xt = Mt − Avg(Mt) where Mt = Max temp on day t Average over temperature from 1961-1990 (Wehner et al.(2018)) Xt = HIt 10 / 30 Heatwave © 2018 by Raha & Ghosh
- 11. Section II: Probabilistic Framework Theorem Theorem Let Bt = I(Xt > M). Under a set of regularity conditions, the distributions of T2m − T2m−1 and T2m+1 − T2m can be approximated by T2m − T2m−1 ∼ Geo(e−P(B1=1) ) m = 1, 2, ... T2m+1 − T2m ∼ Geo(e−P(B1=0) ) m = 0, 1, 2, , ... 11 / 30 Heatwave © 2018 by Raha & Ghosh
- 12. Section II: Probabilistic Framework Theorem Lemma (Chen et al.(2013)) {Xα : α ∈ J} be Bernoulli Random variables with success probabilities pα, α ∈ J. Let W = α∈J Xα and λ = EW = α∈J pα. Then, for any collection of sets Bα ⊂ J, α ∈ J dTV (L(W), Poi(λ)) ≤ (1 ∧ 1 λ )(b1 + b2) + (1 + 1.4√ λ ) |P[W = 0] − e−λ | ≤ (1 ∧ 1 λ )(b1 + b2 + b3), where b1 = α∈J β∈Bα pαpβ, b2 = α∈J β∈Bα{α} E(XαXβ), b3 = α∈J |E(Xα|Xβ, β ∈ Bα) − pα| Let Bt = I(Xt > M) Apply lemma on Bt to prove Theorem 12 / 30 Heatwave © 2018 by Raha & Ghosh
- 13. Section II: Probabilistic Framework Model Fitting Hierarchical Model M = Threshold or q-th quantile of Xn dij(M) : i-th duration (days) within year j based on M dij(M) ∼ Geo(pj(M)) (from previous theorem) pj(M) ∼ Beta(α(M)τ(M), (1 − α(M))τ(M)) (to be verified) Definition:Expected length of duration for threshold M is defined to be E(dij(M)) = E(E(dij(M))|pj(M)) = τ(M)−1 (1−α(M))τ(M)−1 13 / 30 Heatwave © 2018 by Raha & Ghosh
- 14. Section III :Case Studies Atlanta Data Atlanta Data (fixed location) 22-year Meteorology Data in Atlanta 1991-2012 "daily" and "hourly" Ambient, Dew-point, Apparent temperature, Heat index, etc.. Thanks to Dr. Howard Chang for the data We are using daily Maximum ambient temperature. 14 / 30 Heatwave © 2018 by Raha & Ghosh
- 15. Section III :Case Studies Atlanta Data Duration and Intensity in 1991 (M= 81 percentile of daily max temp) Durations: 2,1,7,1,3,1,1,6,2,6,7,1,2,9,1,2,5,3,9 150 200 250 65707580859095 days temperature 1991 15 / 30 Heatwave © 2018 by Raha & Ghosh
- 16. Section III :Case Studies Atlanta Data Fitted vs Observed Duration (1991-1994) 0123456 1991 0123456 1992 01234 1993 02468 1994 16 / 30 Heatwave © 2018 by Raha & Ghosh
- 17. Section III :Case Studies Atlanta Data CVM test for Geometric Distributions 1995 2000 2005 2010 0.00.20.40.60.81.0 year p_value p−values of CVM test 17 / 30 Heatwave © 2018 by Raha & Ghosh
- 18. Section III :Case Studies Atlanta Data Fitting the Beta distribution Estimate the parameters (α, τ) of Beta distribution using MLE of the marginal log-likelihood. KS test: p-value = 0.76 AD test: p-value = 0.68 0.0 0.2 0.4 0.6 0.8 1.0 0.00.20.40.60.81.0 Fitting Beta value CDF 18 / 30 Heatwave © 2018 by Raha & Ghosh
- 19. Section III :Case Studies Atlanta Data Expected Length of Duration for quantiles 0.70 to 0.95 2 4 6 8 0.70 0.75 0.80 0.85 0.90 0.95 threshold Expectedlength 19 / 30 Heatwave © 2018 by Raha & Ghosh
- 20. Section III :Case Studies Atlanta Data Next we try to see whether there is a linear relationship, or quadratic or log-linear relationship between the expected length and the quantiles. The quadratic is the best fit. Definition: For each quantile, if the duration of any up-crossings exceeds the predicted expected length of duration for that quantile, we will call it a Heatwave. 20 / 30 Heatwave © 2018 by Raha & Ghosh
- 21. Section III :Case Studies Atlanta Data Quadratic fit to the Expected length Expected length = 23.48 -31.96 q + 10.87 q2 Adjusted R2 =0.96 0.70 0.75 0.80 0.85 0.90 0.95 3456 Quadratic fit quantiles expected_length 21 / 30 Heatwave © 2018 by Raha & Ghosh
- 22. Section III :Case Studies Atlanta Data Heatwave Comparison properties Our defini- tion with 90 percentile Our defini- tion with 80 percentile Frich et al. Defintition Number of Heatwaves 59 91 142 Avg Duration 7.59 days 11.35 days 16.52 days Max Duration 34 days 36 days 118 days 22 / 30 Heatwave © 2018 by Raha & Ghosh
- 23. Section III :Case Studies USCRN data USCRN data (varying locations) 226 weather stations in USA and 6 stations outside 2000-2017 years of data daily average temperature and hourly temperature Thanks to Dr. Jesse Bell for the data We are only considering 126 stations where we have at least 10 years of data and we will just consider the daily temperature first. 23 / 30 Heatwave © 2018 by Raha & Ghosh
- 24. Section III :Case Studies USCRN data USCRN stations map 24 / 30 Heatwave © 2018 by Raha & Ghosh
- 25. Section III :Case Studies USCRN data Expected lengths for quantiles from 0.7 to 0.95 for each stations 25 / 30 Heatwave © 2018 by Raha & Ghosh
- 26. Section III :Case Studies USCRN data Quadratic fit to the expected lengths 26 / 30 Heatwave © 2018 by Raha & Ghosh
- 27. Section III :Case Studies USCRN data Variation in quadratic plots across the station 0.70 0.75 0.80 0.85 0.90 0.95 05101520 Quadratic plots over the stations quantiles expectedlength 27 / 30 Heatwave © 2018 by Raha & Ghosh
- 28. Section IV: Future Directions Future Directions Explore the spatial variations of the quadratic equation. Explore the spatial variations of the expected lengths of durations for a fixed quantile. Develop similar distributional properties for intensities (as a function of M and locations) 28 / 30 Heatwave © 2018 by Raha & Ghosh
- 29. Section IV: Future Directions References [1] Xu, Z., Cheng, J., Hu, W., & Tong, S. (2018). Heatwave and health events: A systematic evaluation of different temperature indicators, heatwave intensities and durations. Science of The Total Environment, 630, 679-689. [2] Frich, P., et al. "Observed coherent changes in climatic extremes during the second half of the twentieth century." Climate research19.3 (2002): 193-212. [3] Wehner, M., Castillo, F., & Stone, D. (2017-04-26). The Impact of Moisture and Temperature on Human Health in Heat Waves.Oxford Research Encyclopedia of Natural Hazard Science. Retrieved 8 May.2018 [4] Chen, L. H., & Rollin, A. (2013). Approximating dependent rare events. Bernoulli, 19(4), 1243-1267. [5] Vaidyanathan, A., Kegler, S. R., Saha, S. S., & Mulholland, J. A. (2016). A statistical framework to evaluate extreme weather definitions from a health perspective: a demonstration based on extreme heat events. Bulletin of the American Meteorological Society, 97(10), 1817-1830. [6] Peng, R. D., Bobb, J. F., Tebaldi, C., McDaniel, L., Bell, M. L., & Dominici, F. (2011). Toward a quantitative estimate of future heat wave mortality under global climate change. Environmental health perspectives, 119(5), 701. 29 / 30 Heatwave © 2018 by Raha & Ghosh
- 30. Section IV: Future Directions Thank You ! 30 / 30 Heatwave © 2018 by Raha & Ghosh