9 th International Extreme Value Analysis Conference Ann Arbor, Michigan. 15 June 2015

Transcription

1 Severe Storm Environments and Extreme Value Analysis Eric Gilleland Research Applications Laboratory Weather and Climate Impacts Assessment Science Project 9 th International Extreme Value Analysis Conference Ann Arbor, Michigan. 15 June 2015

2 Background

3 Background Fort Collins Estes Park Boulder Golden DIA Denver NCAR/NCEP reanalysis CCSM3 Global Climate Model

4 Background CAPE X Shear J kg -2 ms -1 W max X Shear (WmSh, m 2 s -2 ) Convective Available Potential Energy (CAPE, J kg -2 ) 0 6 km vertical wind shear (Shear, ms -1 ) Non severe Severe Significant Non tornadic Significant Tornadic 0e+00 2e+05 4e W max = (2 CAPE) 0.5 (ms -1 )

5 The end at the beginning The End Spring WmSh (m 2 s -2 ) G. et al. (2013, Environmetrics, 24 (6), , DOI: /env.2234)

6 Measure the energy in the spatial field at each point in time (e.g., mean over space, upper quartile over space, etc.) in order to obtain a univariate time series that measures the amount of spatial energy. Here, we choose the upper quartile, and call it q75.

7 Distribution, of interest, is: WmSh,...,WmSh q75 > u 1 n With this distribution, can look at many different quantities. For example, the mean at each grid point, or the 95 th percentile, etc.

8 Distribution, of interest, is: WmSh,...,WmSh q75 > u 1 n Empirically, the process would be to take all of the time points where q75 > u, and then take the average (or 95 th percentile, or ) at each grid point. From the conditional EV model (Heffernan and Tawn, 2004, JRSS B, 66 (3), ) simulate numerous realizations from the distribution (at each grid point) and then take the average (or 95 th percentile, or ) at each grid point.

9 WmSh 1,...,WmSh n q75 > u The mean of WmSh at each grid point conditioned on high q Spring WmSh (m 2 s -2 )

10 WmSh 1,...,WmSh n q75 > u Summer WmSh (m 2 s -2 )

11 WmSh 1,...,WmSh n q75 > u Fall WmSh (m 2 s -2 )

12 WmSh 1,...,WmSh n q75 > u Winter WmSh (m 2 s -2 )

13 WmSh 1,...,WmSh n q75 > u Difference of second period minus the first Sig. Diff. only Winter WmSh (m 2 s -2 )

14 WmSh 1,...,WmSh n q75 > u Difference of second period minus the first Sig. Diff. only Fall WmSh (m 2 s -2 )

15 Conditioning WmSh on stream flow. Red River at Port Royal, TN Port Royal Kingston Springs Lobelville

16 Conditioning WmSh on stream flow. WmSh (m 2 /s 2 ) conditioned on high river flow (cfs) Mean Red River at Port Royal, Tennessee Lower 5-th percentile Upper 95-th percentile Median

17 Conditioning WmSh on stream flow. WmSh (m 2 /s 2 ) conditioned on lower river flow (cfs) Red River at Port Royal, Tennessee

18 Advances in estimation strategy Keef et al (2013, Environmetrics, 24, 13 21) introduce a fast method to estimate α that does not involve Z. Keef et al (2013, JMVA) impose a joint constraint on (α, β). Cheng et al. (2014, Stat, 3 (1) , DOI: /sta4.71) introduce an empirical Bayes estimation strategy.

19 Key tricks Empirical Bayes Strategy Suppose that X and Z are not deterministically equivalent, but dependent random variables so that X Y >u = α y + y β Z + ε Use empirical Bayes using the first two moments of Z in order to handle the lack of a distribution for the unknown G. To obtain good prior information, can use fast estimate for α, then use the relation X Y >u = αy + Y β Z Y >u to get ln( X Y >u ˆα y) = β ln y + ln Z Y >u and find an initial estimate for β using usual least squares where ln Z is the residual term.

20 Empirical Bayes Strategy α,β,θ Z,X,Y > u Y Z,X,Y α,β,θ,y > u Y = X,Y Z,α,β,θ,Y > u Y Z α,β,y > u Y α,β,θ η [ η ] = X Y,Z,α,β,θ Y Z,α,β,θ,Y > u Y Z α,β,y > u Y [ ] α,β,θ η η [ ] α,β,θ η η marginal distribution for X marginal distribution for Y G(z) using empirical Bayes as surrogate for not knowing the distribution

21 Conclusions The Heffernan and Tawn conditional EVA model is useful for modeling patterns of severe storm environments in the presence of high field energy. Provides a link between usual climate analyses and something far more sophisticated and useful. That is, beyond the mean. To be truly useful, need to resolve issues with the estimation strategy. Empirical Bayes approach is promising. Use extension of the HT model to allow for covariates in the parameter estimates, or is it better/sufficient to stratify out by time and season?

22 Thank You R package extremes (univariate analysis) and texmex (Heffernan and Tawn model). Reference list for spatial and spatiotemporal EVA papers at