An Introduction to Extreme Value Statistics

1 An Introduction to Extreme Value StatisticsMarielle Pinheiro and Richard GrotjahniiThis tutorial is a basic Introduction to Extreme Value analysis and the R package,extRemes. Extreme valueanalysis has application in a number of different disciplines ranging from finance to hydrology, but here theexamples will be presented in the form of climate will begin with a brief background on Extreme Value analysis , presenting the two main methods andthen proceeding to show examples of each method. Readers interested in a more detailed explanation of themath should refer to texts such as Coles 2001 [1], which is cited frequently in the Gilleland and [2] detailing the various tools provided in theextRemespackage. Also, theextRemesdocumentation,which is useful for functions syntax, can be found guidance on the R syntax and R scripting, many resources are available online. New users might want tobegin with the Cookbook for R ( Quick-R ( )Contents1 Extreme Value Theory.)

2 Generalized Extreme Value (GEV) versus Generalized Pareto (GP) .. Stationarity versus non-stationarity ..32 ExtRemes example: Using Davis station data from Explanation of thefevdinput and output .. used in this example .. Working with the data: Generalized Extreme Value (GEV) distribution fit .. Working with the data: Generalized Pareto (GP) distribution fit .. Using the model fit: probabilities and return periods .. relationship between probability and return period .. case: 2013 and 2014 records .. model probabilities and return periods .. empirical probabilities and return periods .. : comparing empirical versus model return periods .. Discussion: GEV vs. GP .. well does each method capture the data distribution? .. do the results from each method compare to one another? .. 14 Appendix A .. Otherfevdoptions .. 18 Appendix B GP Threshold selection.

3 Declustering data .. Nonstationary threshold calculation .. Sine function .. Harmonics function .. 22 Appendix C Some R syntax Installing and running R .. Installing R .. Running R .. Installing Packages .. Formatting text files .. Reading and subsetting data in R .. Some user-defined functions .. Plots .. 29iiiivCONTENTSC hapter Extreme Value TheoryIn general terms, the chance that an event will occur can be described in the form of a probability. Think of acoin toss; in an ideal scenario, there is a 50% chance that the coin will land either heads or tails up in a singletrial, and as multiple tosses are made, we gather additional information about the probability of landing onheads versus tails. With this knowledge, we can make predictions about the outcomes of future coin toss scenario is an example of a simple binomialprobability distribution(frequency of headsversus frequency of tails), but the fundamental concept can be expanded to encompass more complex scenarios,described by other probability distributions.

4 Here, we are interested in formulating a mathematical represen-tation ofextremes, or events with a low probability of occurrence. The definition of extremes varies by fieldand methodology; in the context of climate, we will talk about Extreme temperatures, such as the higher-than-average temperatures experienced over the course of a heat wave. An Extreme weather event is an occurrencethat deviates substantially from typical weather at a specific location and time of year. Specific definitionsvary depending on the distribution of local weather patterns and method of categorization. analysis of extremeweather is made more difficult by the fact that Extreme events are, by definition, rare, and therefore reliabledata is Value theory deals with the stochasticity of natural variability by describing Extreme events withrespect to a probability of occurrence. The frequency of occurrence for events with varying magnitudes can bedescribed as a series of identically distributed random variablesF=X1,X2,X3.

5 XN( )whereFis some function that approximates the relationship between the magnitude of the event (variableXN) and the probability of its it is possible to do analysis with the overall distribution of temperature magnitudes, we are focusing onjust the Extreme temperatures, which can also be described in terms of a probability distribution function. Wecan use the information from the resultant distribution to analyze trends and the likelihood that catastrophicevents will occur. Here are just a few of the possibilities: Predict how often catastrophic events are likely to occur (return level) Extreme temperatures (heat waves, cold air outbreaks) precipitation levels, flooding and droughts hurricane frequency and magnitude Perform simulations utilizing the distributions, and use the results to anticipate future concerns How does the occurrence of current temperatures match the calculated probability of occurrence? Ina changing climate, what can we expect to change in terms of temperature trends?

6 What do current precipitation levels mean for reservoir levels and overall water usage? What changes can we expect for the intensity and frequency of hurricanes?12 CHAPTER 1. Generalized Extreme Value (GEV) versus Generalized Pareto(GP)We will focus on two methods of Extreme Value analysis . The first approach, GEV, looks at distribution ofblock maxima (a block being defined as a set time period such as a year); depending on the shape parameter,a Gumbel, Fr echet, or Weibull1distribution will be produced. The second method, GP, looks at values thatexceed a defined threshold2; depending on the shape parameter, an Exponential, Pareto, or Beta distributionwill be two methods are summarized below; a demonstration of each method follows in the next function of standardized max-ima (or minima) block maxima/minimaapproachprobability of exceeding pre-determinedthreshold peaks over threshold approachParameterslocation : position of the GEV meanthresholdu: Reference Value for whichGP excesses are calculatedscale : multiplier that scales functionshape : Parameter that describes the relative distribution of the function (CDF)for Extreme valuez,G(z) = exp[ {1 + (z )} 1/ +]for threshold excessx,H(x) = 1 [1 + (x u u)] 1/ +Limit as 0 Gumbel:Exponential:G(z) = exp[ exp{ (z )}]H(x) = 1 exp( x u ) >0Fr echetPareto <0 WeibullBetaInterpretation of resultsReturn level: valuezpthat is expected tobe exceeded on average once every 1/ppe-riods, where 1 pis the probability associ-ated with the quantile.

7 Findzpsuch thatG(zp) = 1 pReturn level: valuexmthat is exceededeverymtimes. Begin by estimating u,the probability of exceeding the ,xmisxm={u+ u [(m u) 1] 6= 0u+ uln(m u) = 0 Table : Description of the two basic types of Extreme Value distributionsProbability density functions (PDFs) and cumulative distribution functions (CDFs)Theprobability density function(as shown in Figure ), plots the relative likelihood (on the yaxis) that a variable will have valueX(on the x axis). Contrast this with thecumulative distributionfunction(as shown in Figure ), in which the probability ofXis defined by integrating the PDF overthe range in which the variable X. Note that the equations in Table are CDFs, not that the Weibull distribution has a finite right endpoint; Gumbel and Fr echet have infinite right endpoints2 The GP function can be approximated as the tail of a GEV; the scale parameter uis a function of the threshold and isequivalent to g+ (u ), where g, and are all parameters of a corresponding GEV STATIONARITY VERSUS NON-STATIONARITY3 Figure : Probability density functions for (left) GEV and (right) GP.}

8 For each plot,xand =1, = For , blue= , black=0, and red= : Cumulative density functions for (left) GEV and (right) GP. For each plot,xand =1, = For , blue= , black=0, and red= Stationarity versus non-stationarityIn calculating the model fit, it is useful to determine whether the model distribution remains the same as timeprogresses. Figure shows how the probability of the extremes might change in the future under possibleproposed climate scenarios. Note that the model that was fit to an Extreme temperature distribution from the20th century might not work for 21st century : Temperature magnitudes and probability of occurrence in the context of a warmer climate. Plotsobtained from IPCC Summary for Policymakers (2012), Figure change can be incorporated into the model as a change with respect to time; ,x, , and can berepresented as some sort of function of time. This is known as anon-stationary model, in contrast to astationary modelin which the model parameters are fixed example, if we anticipated that there would be a greater proportion of Extreme high temperatures in thefuture, the shape of a GEV function fitted to the data might increase towards a more heavy-tailed distribution,as seen in Figure 1.

9 BACKGROUNDF igure : Altering in a GEV distribution as a function of time: (t) = Black:t= 0 Green:t= 15 Purple:t= 50. Other parameters remain is most often seen with GP distributions where the threshold is defined as a function. Ifwe wanted to increase the GP threshold linearly, as might be the case in an increasingly warmer climate, wecould define the threshold asx(t) =x0+x1t( )wherex0would be the initial threshold Value andx1would increment the threshold Value over time. Thesame can be done for the other parameters as well; for example,The parameters could also be tailored to incorporate some sort of seasonal cycle; Figure demonstratesthat there is a definite trend to the temperature values throughout the JJAS months, so we tailor the thresholdaccordingly by writing the threshold as a sine function (see Appendix for equation). Note the differencein the number of points that exceed the non-stationary threshold (red) as opposed to the stationary threshold(blue).

10 We gain some points in June and September, and lose some points in July and method, which will be used in this study (see section for a demonstration), utilizes the Value ofthe 90th percentile for each day in the season to provide an initial estimate for a threshold Value , and then fitsa sum of sines and cosines to those estimates; this is known as a sum of harmonics. The 90th percentile valuesare plotted on as the green line, with the corresponding harmonics sum plotted in black. At manyspots throughout the season, the black line is reasonably close to the sine function, but there are notable dipsin mid-August and demonstrates how the distribution of the excesses changes with each of the methods. Altering theshape of the threshold also changes the magnitude of the excesses, and this leads to a different GP distribution,highlighting the importance of choosing a proper of non-stationary threshold in extRemesCertain methods in theextRemespackage will be unavailable when using a non-stationary threshold;for example, , which is used to determine an optimal threshold Value , will not workwith a non-constant threshold.