#TITLE SLIDE -And thanks to everyone for coming along, I know that many of us have lots of marking to do at the minute, so it's appreciated! -The work I'll present today has been pieced together from various bits of work that I've done by myself, work I've done jointly with former MMathStat students (Amy and Matthew), and more recently with a collaborator in Mexico City. -Should say that Amy Green is now a PhD student supervised jointly between Stats and Engineering, and Matthew Capewell works for Technology consultancy firm ROSEN, but they allow him time out every week to continue developing the work he started during his MMathStat #STRUCTURE SLIDE -There'll be three main sections to the talk: (1) Kick off with an extreme value primer, for those of you who've not heard me speak before, with a particular focus on return level estimation and some work - in my mind - that has been left hanging for a good few years -Why was it left hanging - there's been a huge focus over the last 10/15 years on developing theory, and models, aimed at analysing extremes that occur spatially - and univariate stuff has been largely left behind -I'll given an example of a spatial analysis of wind speeds, and through this attempt to justify a return to Univariate considerations -Then in the last part of the talk I'll give some pointers on what we've been doing to attend to this unfinished business, focussing mainly on dealing with extremal dependence. I'll also discuss some potential impact of this work #PICTURES SLIDE 1 -I often like to ease people into this sort of stuff with some nice pictures - I guess that depends on your idea of "nice" is - if "death and destruction", then you might like them -primarily aimed at giving a practical motivation for this work -The top two pictures show the effects of extreme precipitation in central/eastern Europe in May/June 2013: in many cities flood barriers were overwhelmed when rivers burst their banks, resulting in devastating floods in towns and cities in the basins of the Elbe and Danube rivers -We might be interested in being better-prepared against wind speed extremes - the pics along the bottom show the effects of extreme wind speeds after Hurricane Michael made landfall in Alabama, US, and the effects of tornadoes in Birmingham (UK!) in 2005 #PICTURES SLIDE 2 -We are often interested in the effects of sea surge extremes - an underwater earthquake off the SE coast of Japan led to the Fukushima nuclear disaster in 2011, when the resulting surge resulted in flooded generators, loss of cooling and three nuclear meltdowns -There was a near miss at the Blayais Nuclear Facility in SW France in 1999, again die to flooidng of the site - the floods this time occurring as a result of the combined effects of very heavy rainfall and hurricane-strength wind speeds driving sea-surge inland -A similar near miss also occurred at the Fort Calhoun nuclear facility in Nebraska in 2011 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ #THEORY SLIDE 1 -So now a very quick EV primer... -It's easy to see that, in the case of independence, the distribution function for M_n is just F to n... -In their 1988 book, Leadbetter and Rootzen showed that to account for dependence we further power by theta... extremal index... -small errors in the estimation of theta... #THEORY SLIDE 2 -In the 1920s through to the 1950s, those working in the field sought to describe the limiting behaviour of M_n - again, assuming initially independent r.v.s, the GEV distribution with parameters... -Raising this to the power theta to account for dependence results in another GEV distribution, with the extremal index iteslf being absorbed into the location and scale parameters of the model... #THEORY SLIDE 3 -On our whistlestop tour of the development of EVT we're now in the 1970s, and an alternative representation of extremes was suggested by Pickands, giving rise to the GPD as the limiting model for excesses over some high threshold u... -Again, this result is valid for (in the limit) for independent threshold excesses... in practice, and for environmental variables, consecutive excesses are NOT independent: but, as we know, we can power by theta to account for dependence -However, doing so here does not give such a convenient result as it did in the block maxima case... #THEORY SLIDE 4 -The ultimate aim of any extreme value analysis is usually the estimation of "r-year return levels" - levels of the variable we can expect to be over-topped (on average) once every r years -We do this be referring to high quantiles of the fitted distribution -For threshold excesses... -Another piece of unfinished business, I think, is how we deal with seasonal variation. One of the benefits of a threshold-based analysis over a block maxima analysis is the inclusion of more extremes from which to estimate return levels. In the spirit of... -For seasonally-varying data, it might make sense to include extremes from all seasons - especially if it's the same type of weather system that generates extremes throughout the year, and so the extremes observed within each season can be seen as informative of the process for generating extremes more generally... #EXAMPLES SLIDE -Many organisations responsible for protecting human life use return level estimates as design requirements for things like buildings, bridges and flood defence systems... -Just to illustrate this uncertainty, I've included some estimated confidence intervals on return levels from the literature - including the infamous NERC report of 1995, reproting massive uncertainty on the estimated 100 year river flow in the River Nidd; huge uncertainty in estimated wave heights in s hurricane-prone region in the US; some of our own work, showing in particular large variability in the upper endpoint of the 95\% CI for sea-surges at Newlyn, in SWE, depending on the method chose to account for extremal dependence; and large ucnertainty in the estimated return period for an extreme solar flare event +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ #SPATIAL! SLIDE -OK, so we'll put those bits of "unfinished business" on the backburner for now, and think for a bit about spatial extremes -Every other year there's an annual meeting of the Extremes community - I've attended three of these events (2005, 2007 and 2009), and I remember being hugely disappointed at the last one I attended - only around haLf of talks at the conference focused an a real-life application -All EVA programmes for conference since 2009 have been placed on a general conference webpage, and a couple of weeks ago I spent some time trawling (SKIMMING) through these programmes -My very rough and ready calculations show (for me) an annoying trend! Further reductions in the number of applications talks! We can also see a steady increase in the proportion of talks that are spatial/high-dimensional in flavour -My experience of trying to do anything spatial in extremes is that it's HARD,... BUT LET'S GIVE IT A GO! -Problematic - there's a general classification of spatial models for extremes into those that are "max-stable" or not. Max-stable models give rise to asymptotic dependence between extremes - for example, the dependence between extremes at multiple locations persists no matter how far into the tails we move... as opposed to Gaussian processes which might display dependence above levels of practical interest but independence asymptotically... -there are graphical procedures available for determining the nature of the dependence present in your data, before you then go on to use a Brown-Resnick process, or a Gaussian process (for example). -However, some work that Dave and I have done recently has shown that these diagnostic plots are extremely difficult to interpret, and don't always work! #LV MODEL SLIDE 2 -We assume trend surfaces for our GEV parameters - at least for the location and scale; as is often the case in high-dimensional block maxima analyses, we keep the shape constant (across the region here); the location are allowed to vary with both longitude and latitude -S is a GRF with exponential covariance function with sill and range parameters alpha and lambda; h is the separation distance between sites -Conditional on the Gaussian processes at sites in D... we assume our random process is GEV descrived by those trend surfaces #LV MODEL SLIDE 3 -We need to specify a prior density for our parameter vector, which consists of the range and sill parameters in the covariance function for the GRF for each of the GEV parameters; and the collection of regression coefficients as given in the trend surfaces -We keep things cojugate where we can - we assume independence between the range, sill and regression parameters; we specify an inverse gamma prior for the sills; Normal priors for the regression coefficients; gamma priors for the ranges (for which we will have to use a Metropolis update owing to lack of conjugacy) -Values for the hyper-parameters are chosen to reflect a largely non-informative specification - we follow the prior specification as given in Anthony Davison's 2012 paper #MCMC SAMPLING SLIDE -Won't have time to go through the nuts and bolts of the MCMC in detail... -But step 1 involves updating the marginal GPD parameters at each site using a Metropolis update -In step 2 we update the regression parameters by drawing from the full conditionals -In step 3 we update the sill parameters of the covariance function by drawing from the full conditionals -In step 4 we use a Metropolis step tp update the range parameters +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ #####BACK TO UNFINISHED BUSINESS #EXTREMAL DEPENDENCE SLIDE 1 -General aim might be to squeeze the life out of the data AND do anything else we can to increase the precision of return level estimates - a Bayesian analysis therefore has appeal, perhaps if we can specify informative priors; (1) doesn't lend itself well to Bayesian inference as we estimate theta empirically from the arrival times of the threshold excesses; (2) is quite restrictive - we need to assume a particular form of temporal dependence... back to the question of asymptotic dependence/independence again (although in a time series context now). Also, which model within these classes? What model order? (We assume first order here for simplicity...) #EXTREMAL DEPENDENCE SLIDE 2 -Thomai, in Mexico, reminded me of a Biometrika paper published by Antony Ledford and Jon Tawn in 1997 -They specify the joint tail distribution for consecutive pairs in our process as the sum of the marginals F, plus the bivariate function F_bar, where F_bar is given by the product of a slowly varying function L and the log of the marginals raised to 1/2eta -Marginal are assumed GPD with scale, shape and threshold excess parameters sigma, xi and lambda; eta is known as the coefficient of tail dependence and can identify the `type' of extremal dependence -Bortot proposes a particular form for the slowly varying function that allows the incorporation of extremal dependence according to a class of bivariate extreme value models with parameter beta -a>0 is a parameter constrained so that the function evaluates to something positive #EXTREMAL DEPENDENCE SLIDE 3 -As I said, eta is the coefficient of tail dependence - if eta<1: asymptotic independence, eta=1: asymptotic dependence -Bortot(1998) then links the (threshold-dependent) extremal index to this through Eq.(3), meaning that although it's not explicitly one of the model parameters, it is determined by the other parameters -Thomai has coded this all up and we've performed a little simulation study to test this estimator out. We're using profile log-likelihood at the minute, but we'd like to think about how to incorporate this more completely in a fully Bayesian analysis, leading to return level estimates that have been informed by the extremal dependence in the process under study #SIMSTUDY RESULTS 1 -I've produced plots like these before for previous talks; now we have the addition of an extra line (colour) for this "new" estimator for theta -Have 10,000 replications of a first-order MC of length 10,000, constructed using three different bivariate extreme value distributions to describe the evolution of the process -First column: shows sampling distribution means for five different estimators of theta (ours is shown in red) across a range of strengths of extremal dependence as provided by these three models -Second column: subsequent return level estimation using these extremal index estimators, with vertical lines superimposed at the estimated values for one of the French wind speed sites -Third column: Fixed dependence; return level estimates across a range of return periods, with POT estimators superimposed for comparison #SIMSTUDY RESULTS 2 -Results on the last slide were for asymptotically dependent chains; we've looked at a range of processes, including those giving asymptotic independence -In each case, you would expect the "Markov chain" estimator to do well: We *know* the exact form of dependence, as we have constructed the chains, so the only source of error here will come from the estimation procedure -In reality, we *don't* know the form of dependence/structure of the chain, so this is where the "new" estimator comes in handy - it's completely "automatic" and it does pretty well -The arrival times estimator is also "automatic", but as I said it's non-parametric and does not lend itself to Bayesian inference (which is where we want to go) -The cluster and blocks methods, based on identifying clusters of extremes and commonly used for their simplicity, do not perform very well at all