A Poisson process reparameterisation for Bayesian inference for extremes

Transcription

1 A Poisson process reparaeterisation for Bayesian inference for extrees Paul Sharkey and Jonathan A. Tawn Eail: STOR-i Centre for Doctoral Training, Departent of Matheatics and Statistics, Lancaster University, Lancaster, LA1 4YF, U.K. Abstract A coon approach to odelling extree values is to consider the excesses above a high threshold as realisations of a non-hoogeneous Poisson process. While this ethod offers the advantage of odelling using threshold-invariant extree value paraeters, the dependence between these paraeters akes estiation ore difficult. We present a novel approach for Bayesian estiation of the Poisson process odel paraeters by reparaeterising in ters of a tuning paraeter. This paper presents a ethod for choosing the optial value of that near-orthogonalises the paraeters, which is achieved by iniising the correlation between the asyptotic posterior distribution of the paraeters. This choice of ensures ore rapid convergence and efficient sapling fro the joint posterior distribution using Markov Chain Monte Carlo ethods. Saples fro the paraeterisation of interest are then obtained by a siple transfor. Results are presented in the cases of identically and non-identically distributed odels for extree rainfall in Cubria, UK. Keywords: Covariate odelling. Poisson processes, Extree value theory, Bayesian inference, Reparaeterisation, 1 A Poisson Process odel for Extrees The ai of extree value analysis is to odel rare occurrences of an observed process to extrapolate to give estiates of the probabilities of unobserved levels. In this way, one can ake predictions of future extree behaviour by estiating the behaviour of the process using an asyptotically justified liit odel. Let X 1, X 2,..., X n be a series of independent and identically distributed (iid) rando variables with coon distribution function F. Defining M n = ax{x 1, X 2,..., X n }, if there exists sequences of noralising constants a n > 0 and b n such that: { } Mn b n Pr x G(x) as n, (1) a n 1

2 where G is non-degenerate, then G follows a generalised extree value (GEV) distribution, with distribution function { [ ( x µ G(x) = exp 1 σ )] 1/ }, (2) where x = ax(x, 0), σ > 0 and µ, R. Here, µ, σ and are location, scale and shape paraeters respectively. Using a series of block axia fro X 1,..., X n, typically with blocks corresponding to years, the standard inference approach to give estiates of (µ, σ, ) is the axiu likelihood technique, which requires nuerical optiisation ethods. In these probles, particularly when covariates are involved, such ethods ay converge to local optia, with the consequence that paraeter estiates are largely influenced by the choice of starting values. The standard asyptotic properties of the axiu likelihood estiators are subject to certain regularity conditions outlined in Sith (1985), but can give a poor representation of true uncertainty. In addition, flat likelihood surfaces can cause identifiability issues (Sith, 1987a). For these reasons, we choose to work in a Bayesian setting. Bayesian approaches have been used to ake inferences about θ = (µ, σ, ) using standard Markov Chain Monte Carlo (MCMC) techniques. They have the advantage of being able to incorporate prior inforation when little is known about the extrees of interest, while also better accounting for paraeter uncertainty when estiating functions of θ, such as return levels (Coles and Tawn, 1996). For a recent review, see Stephenson (2016). An approach to inference that is considered to be ore efficient than using block axia is to consider a odel for threshold excesses, which is superior in the sense that it reduces uncertainty due to utilising ore extree data (Sith, 1987b). Given a high threshold u, the conditional distribution of excesses above u can be approxiated by a generalised Pareto (GP) distribution (Pickands, 1975) such that Pr(X u > x X > u) = ( 1 x ) 1/, x > 0, ψ u where ψ u > 0 and R denote the scale and shape paraeters respectively, with ψ u dependent on the threshold u, while is identical to the shape paraeter of the GEV distribution. This odel conditions on an exceedance, but a third paraeter λ u, denoting the rate of exceedance of X above the threshold u, ust also be estiated. Both of these extree value approaches are special cases of a unifying liiting Poisson process characterisation of extrees (Sith, 1989; Coles, 2001). Let P n be a sequence of point processes 2

3 such that P n = {( i n 1, X ) } i b n : i = 1,..., n, a n where a n > 0 and b n are the noralising constants in liit (1). The liit process is non-degenerate since the liit distribution of (M n b n )/a n is non-degenerate. Sall points are noralised to the sae value b L = li n (x L b n )/a n, where x L is the lower endpoint of the distribution F. Large points are retained in the liit process. It follows that P n converges to a non-hoogeneous Poisson process P on regions of the for A y = (0, 1) [y, ), for y > b L. The liit process P has an intensity easure on A y given by Λ(A y ) = [ 1 ( y µ σ )] 1/. (3) It is typical to assue that the liit process is a reasonable approxiation to the behaviour of P n, without noralisation of the {X i }, on A u = (0, 1) [u, ), where u is a sufficiently high threshold and a n, b n are absorbed into the location and scale paraeters of the intensity (3). It is often convenient to rescale the intensity by a factor, where > 0 is free, so that the n observations consist of blocks of size n/ with the axiu M of each block following a GEV(µ, σ, ) distribution, with invariant to the choice of. The Poisson process likelihood can be expressed as { [ ( u µ L(θ ) = exp 1 σ )] 1/ } r 1 σ [ 1 ( xj µ σ )] 1/ 1, (4) where θ = (µ, σ, ) denotes the rescaled paraeters, r denotes the nuber of excesses above the threshold u and x j > u, j = 1,..., r, denote the exceedances. It is possible to ove between paraeterisations associated with different nubers of blocks. If for k blocks the block axiu is denoted by M k and follows a GEV distribution with the paraeters θ k = (µ k, σ k, ), then for all x Pr(M k < x) = Pr (M < x) k/. As M k is GEV(µ k, σ k, ) and M is GEV(µ, σ, ) it follows that ( µ k = µ σ ( ) ) k 1 σ k = σ ( k ). (5) In this paper, we present a ethod to iprove inference for θ k, the paraeterisation of interest. For an optial choice of we first undertake inference for θ before transforing our results to give inference for θ k using the apping in expression (5). 3

4 In any practical probles, k is taken to be n y, the nuber of years of observation, so that the annual axiu has a GEV distribution with paraeters θ ny = (µ ny, σ ny, ). Although inference is for the annual axiu distribution paraeters θ ny, the Poisson process odel akes use of all data that are extree, so inferences are ore precise than estiates based on a direct fit of the GEV distribution to the annual axiu data as noted above. To help see how the choice of affects inference, consider the case when = r, the nuber of excesses above the threshold u. If a likelihood inference was being used with theis choice of, the axiu likelihood estiators (ˆµ r, ˆσ r, ˆ) = (u, ˆψ u, ˆ), see Appendix A for ore details. Therefore, Bayesian inference for the paraeterisation of the Poisson process odel when = r is equivalent to Bayesian inference for the GP odel. Although inference for the Poisson process and GP odels is essentially the sae approach when = r, they differ in paraeterisation, and hence inference, when r. The GP odel is advantageous in that λ u is globally orthogonal to ψ u and. Chavez-Deoulin and Davison (2005) achieved local orthogonalisation of the GP odel at the axiu likelihood estiates by reparaeterising the scale paraeter as ν u = ψ u (1 ). This ensures all the GP tail odel paraeters are orthogonal locally at the likelihood ode. However, the scale paraeter is still dependent on the choice of threshold. Unlike the GP, the paraeters of the Poisson process odel are invariant to choice of threshold, which akes it ore suitable for covariate odelling and hence suggests that it ay be the better paraeterisation to use. In contrast, it has been found that the paraeters are highly dependent, aking estiation ore difficult. As we are working in the Bayesian fraework, strongly dependent paraeters lead to poor ixing in our MCMC procedure (Hills and Sith, 1992). A coon way of overcoing this is to explore the paraeter space using a dependent proposal rando walk Metropolis-Hastings algorith, though this requires a knowledge of the paraeter dependence structure a priori. Even in this case, the dependence structure potentially varies in different regions of the paraeter space, which ay require different paraeterisations of the proposal to be applied. The alternative approach is to consider a reparaeterisation to give orthogonal paraeters. However, Cox and Reid (1987) show that global orthogonalisation cannot be achieved in general. This paper illustrates an approach to iproving Bayesian inference and efficiency for the Poisson process odel. Our ethod exploits the scaling factor as a eans of creating a near-orthogonal 4

5 representation of the paraeter space. While it is not possible in our case to find a value of that diagonalises the Fisher inforation atrix, we focus on iniising the off-diagonal coponents of the covariance atrix. We present a ethod for choosing the best value of such that nearorthogonality of the odel paraeters is achieved, and thus iproves the convergence of MCMC and sapling fro the joint posterior distribution. Our focus is on Bayesian inference but the reparaeterisations we find can be used to iprove likelihood inference as well, siply by ignoring the prior ter. The structure of the paper is as follows. Section 2 exaines the idea of reparaeterising in ters of the scaling factor and how this can be ipleented in a Bayesian fraework. Section 3 discusses the choice of to optiise the sapling fro the joint posterior distribution in the case where X 1,..., X n are iid. Section 4 explores this choice when allowing for non-identically distributed variables through covariates in the odel paraeters. Section 5 describes an application of our ethodology to extree rainfall in Cubria, UK, which experienced ajor flooding events in Noveber 2009 and Deceber Bayesian Inference Bayesian estiation of the Poisson process odel paraeters involves the specification of a prior distribution π(θ ). Then using Bayes Theore, the posterior distribution of θ can be expressed as π(θ x) π(θ )L(θ ), where L(θ ) is the likelihood as defined in (4) and x denotes the excesses of the threshold u. We saple fro the posterior distribution using a rando walk Metropolis-Hastings schee. Proposal values of each paraeter are drawn sequentially fro a univariate Noral distribution and accepted with a probability defined as the posterior ratio of the proposed state relative to the current state of the Markov chain. In all cases throughout the paper, each individual paraeter chain is tuned to give the acceptance rate in the range of 20% 25% to satisfy the optiality criterion of Roberts et al. (2001). For illustration purposes, results in Sections 2 and 3 are fro the analysis of siulated iid data. A total of 300 exceedances above a threshold u = 30 are siulated fro a Poisson process odel with θ 1 = (80, 15, 0.05). Figure 1 shows individual paraeter chains for θ k fro a rando walk Metropolis schee run for iterations with a burn-in of 1000 reoved, where k = 1 and a chosen = 1. This figure shows the clear poor ixing of each coponent of θ 1, indicating non-convergence and strong dependence in the posterior sapling. 5

6 µ 1 σ Figure 1: Rando-walk Metropolis chains run for each coponent of θ 1. We explore how reparaeterising the odel in ters of can iprove sapling perforance. For a general prior on the paraeterisation of interest θ k, denoted by π(θ k ), Appendix B derives that the prior on the transfored paraeter space θ is ( ) π(θ ) = π(θk ). (6) k In this exaple, independent Unifor priors are placed on µ 1, log σ 1 and, which gives π(θ 1 ) 1 σ 1 ; µ 1 R, σ 1 > 0, R. (7) This choice of prior results in a proper posterior distribution, provided there are at least 4 threshold excesses (Northrop and Attalides, 2016). By finding a value of that near-orthogonalises the paraeters of the posterior distribution π(θ x), we can run an efficient MCMC schee on θ before transforing the saples to θ k. It is noted in Wadsworth et al. (2010) that setting to be the nuber of exceedances above the threshold, i.e. = r, iproves the ixing properties of the chain, as is illustrated in Figure 2. This is approxiately equivalent to inference using a GP odel, as discussed in Section 1. Given this choice of, the MCMC schee is run for θ before transforing to estiate the posterior of θ 1 using the apping in (5), where k = 1 in this case. Figure 3 shows contour plots of estiated joint posterior densities of θ 1. It copares the saples fro directly estiating the posterior of θ 1 with that fro transforing fro the MCMC saples of the posterior of θ to give 6

7 µ 300 σ Figure 2: Rando-walk Metropolis chains run for paraeters θ r, where r = 300 is the nuber of exceedances in the siulated data. a posterior saple for θ 1. Figure 3 indicates that θ 1 are highly correlated, with the result that we only saple fro a sall proportion of the paraeter space when exploring using independent rando walks for each paraeter. This explains the poor ixing if we were to run the MCMC without a transforation. It is clear that, after back-transforing to θ 1, the reparaeterisation enables a ore thorough exploration of the paraeter space. However, while this transforation is a useful tool in enabling an efficient Bayesian inference procedure, further investigation is necessary in the choice of to achieve near-orthogonality of the paraeter space and thus axiising the efficiency of the MCMC procedure. 3 Choosing optially As illustrated in Section 2, the choice of in the Poisson process likelihood can iprove the perforance of the MCMC required to estiate the posterior density of odel paraeters θ k. We desire a value of such that near-orthogonality of θ is achieved, before using the expressions in (5) to transfor to the paraeterisation of interest, e.g. θ 1 or θ ny. As a easure of dependence, we use the asyptotic expected correlation atrix of the posterior distribution of θ x. In particular, we explore how the off-diagonal coponents of the atrix, that is, the correlation between paraeters, changes with. The covariance atrix associated with θ x can be derived analytically by inverting the Fisher inforation atrix of the Poisson process log-likelihood (see Appendix C). The correlation atrix is then obtained by noralising so that the atrix has a unit diagonal. 7

8 σ µ µ σ 1 σ µ µ σ 1 Figure 3: Contour plots of the estiated joint posterior of θ 1 for 4000 iterations (top) and iterations (botto) created fro the transfored saples drawn fro the MCMC procedure for θ (in black) and saples of θ 1 drawn directly (in red). Other choices for the easure of the dependence of the posterior could have been used, such as the inverse of the Hessian atrix (or the expected Hessian atrix) of the log-posterior, evaluated at the posterior ode. For inference probles with strong inforation fro the data relative to the prior there will be liited differences in the approach and siilar values for the optial will be found. In contrast, if the prior is strongly inforative and the nuber of threshold exceedances is sall then the choice of fro using our approach could be far fro optial. Also the use of the 8

9 observed, rather than expected, Hessian ay better represent the actual posterior distribution of θ and deliver a choice of that better achieves orthogonalisation, see Efron and Hinkley (1978) and Tawn (1987) respectively. We prefer our choice of easure of dependence as for iid probles it gives closed for results for which can be used without the coputational work required for other approaches, and this gives valuable insight into the choice of. Furtherore, inforative priors rarely arise in extree value probles, and so inforation in the data typically doinates inforation in the prior, particularly around the posterior ode. It should be pointed out however, that the prior is used in the MCMC so there is no loss of prior inforation in our approach. Also standard MCMC diagnostics should be used even after the selection of an optial, so if the asyptotic posterior correlations differ uch fro the posterior correlations, aking our choice of poor, this will be obvious and a ore coplete but coputationally burdensoe analysis can be conducted using the ethods described above. In this section, we use the data introduced in Section 2. For all integers [1, 500], axiu likelihood estiates ˆθ are coputed and pairwise correlations calculated by substituting ˆθ into the expressions for the Fisher inforation atrix, in Appendix C, and taking the inverse. Figure 4 shows how paraeter correlations change with the choice of, illustrating that the asyptotic posterior distributions of µ and are orthogonal when = r, the nuber of excesses above a threshold, which explains the findings of Wadsworth et al. (2010). Figure 4: Left: Estiated paraeter correlations changing with : ρ µ,σ (black), ρ µ, (red), ρ σ, (blue). Right: Expanded region of the graph showing ρ µ, = 0 for close to r where r = 300 is the nuber of excesses above the threshold, while ρ µ,σ = 0 when

10 It is proposed that MCMC ixing can be further iproved by iniising the overall correlation in the asyptotic posterior distribution of θ. Therefore, we would like to find the value of such that ρ(θ ) is iniised, where ρ(θ ) is defined as ρ(θ ) = ρ µ,σ ρ µ, ρ σ,, (8) where ρ µ,σ denotes the asyptotic posterior correlation between µ and σ for exaple. We also Figure 5: How ρ(θ ) changes with (top left) and how correlations in each individual estiated paraeter, as easured by ρ µ, ρ σ and ρ, change with. look at the su of the asyptotic posterior correlation ters involving each individual paraeter estiate. For exaple, we define ρ µ, the asyptotic posterior correlation associated with the estiate of µ, to be: ρ µ = ρ µ,σ ρ µ,. (9) Figure 5 shows how the asyptotic posterior correlation associated with each paraeter varies with. Fro Figure 5 we see that while ρ µ is iniised at the value of for which ρ µ,σ = 0 (see Figure 4), ρ σ and ρ have inia at the value of for which ρ σ, = 0. We denote the latter 10

11 iniu by 1 and the forer by 2. In ters of the covariance function, this can be written as: ACov(σ 1, x) = ACov(µ 2, σ 2 x) = 0, (10) where ACov denotes the asyptotic covariance. Figure 5 shows that 2 also iniises the total asyptotic posterior correlation in the odel. One would expect that the values of for which ρ(θ ) is iniised would correspond to the MCMC chain of θ with good ixing properties. We exaine the effective saple size (ESS) as a way of evaluating this objectively. ESS is a easure of the equivalent nuber of independent iterations that the chain represents (Robert and Casella, 2009). MCMC saples are often positively autocorrelated, and thus are less precise in representing the posterior than if the chain was independent. The ESS of a paraeter chain φ is defined as ESS φ = n 1 2 i=1 ν, (11) i where n is the length of the chain and ν i denotes the autocorrelation in the sapled chain of φ at lag i. In practice, the su of the autocorrelations is truncated when ν i drops beneath a certain level. Figure 6 shows how ESS varies with for each paraeter in θ. For these data the ESS follow a pattern we found to typically occur. We see that ESS µ is axiised at = 2 due to the near-orthogonality of µ 2 with σ 2 and. We find that ESS σ is axiised for 1 < < 2, as σ 1 reains substantially postively correlated with µ 1 and σ 2 is negatively correlated with. Siilarly, ESS is axiised at a value of close to 1, but is negatively correlated with µ 1, which explains the slight distortion. Fro these results, we postulate that a selection of in the interval ( 1, 2 ) would ensure the ost rapid convergence of the MCMC chain of θ, thus enabling an effective sapling procedure fro the joint posterior. Figure 6 shows clearly the benefits of the proposed approach. For exaple, ESS µ310 = and ESS µ1 = 24.44, illustrating that the forer paraeterisation is over 310 ties ore efficient than the latter. In addition, by introducing the interval ( 1, 2 ), this approach gives a degree of flexibility to the choice of and giving a balance of ixing quality across the odel paraeters. The quantities 1 and 2 can be found by nuerical solution of the equations ACov(σ, x) = 0 and ACov(µ, σ x) = 0 respectively, using the asyptotic covariance atrix of the posterior of θ, which is given by the inverse of the Fisher inforation (see Appendix C). Approxiate analytical expressions for 1 and 2 can be derived using Halley s ethod for root-finding (Gander, 11

12 µ σ ESS ESS ESS Figure 6: How ESS varies with for each paraeter in θ. The blue dashed lines represent = 1 (left) and = 2 (right) in the siulated data exaple, where 1 and 2 are defined by property (10). In the calculations, the su of the autocorrelations were truncated when the autocorrelations in the chain drop below ) applied to equations (10). This ethod yields the following approxiations of 1 and 2 : ( [ ]) (2 1) 1 2 ( 1) log ˆ 1 = r ( ]) (12) (2 1) 3 2 ( 1) log [ ˆ 2 = r (13) In practice, the values of ˆ 1 and ˆ 2 are estiated by using an estiate of, such as the axiu likelihood estiate. Figure 7 shows how ˆ 1 and ˆ 2 change relative to r for a range of. This illustrates that for negative estiates of the shape paraeter, r is not a suitable candidate to be the optial value of as it is not in the range ( 1, 2 ). In the siulated data used in this section, although a selection of = r is reasonable, Figure 6 shows that this ay not be wise if one was priarily concerned about sapling well fro, for exaple. In this case, ˆ 2 is relatively close to r, but Figure 7 shows that this is not the case for odels with a larger positive estiate of. A siulation study was carried out to assess the suitability of expressions ˆ 1 and ˆ 2 as approxiations to 1 and 2 respectively. A total of 1000 Poisson processes were siulated with different values of θ. The approxiations were calculated and copared with the true values of 1 and 2, which were obtained exactly by nuerical ethods. It was found that ˆ i i < 0.1 for 12

13 Figure 7: How ˆ 1 and ˆ 2 change as a ultiple of r with respect to ˆ: ˆ 1 /r (black), ˆ 2 /r (red). i = 1, 2 always, while ˆ i i < 0.01 for 78% and 88.2% of the tie for i = 1, 2 respectively. Both quantities were copared to the perforance of other approxiations derived using Newton s ethod, which unlike Halley s ethod does not account for the curvature in a function. Siulations show that the root ean square errors are significantly saller for estiates of i using Halley s ethod (0.2% and 5% saller than Newton s ethod for i = 1, 2 respectively). A suary of the reparaeterisation ethod is given in Algorith 1. Algorith 1: Sapling fro the posterior distribution of the Poisson process odel paraeters after reparaeterising Data: Threshold excesses x Result: Saples fro the posterior distribution π(θ k x) 1 Choose paraeterisation of interest θ k ; 2 if iid then 3 Obtain estiate of shape paraeter using axiu likelihood, for exaple; 4 Copute ˆ 1 and ˆ 2 as defined in (12) and (13); 5 Choose in range ( ˆ 1, ˆ 2 ); 6 else 7 Choose to be the value of that nuerically solves ρ µ (0),σ = 0; 8 Obtain MCMC saples for posterior distribution π(θ x); 9 Transfor to obtain saples fro π(θ k x) 13

14 4 Choosing in the presence of non-stationarity In any practical applications, processes exhibit trends or seasonal effects caused by underlying echaniss. The standard ethods for odelling extrees of non-identically distributed rando variables were introduced by Davison and Sith (1990) and Sith (1989), using a Poisson process and Generalised Pareto distribution respectively. Both approaches involve setting a constant threshold and odelling the paraeters as functions of covariates. In this way, we odel the non-stationarity through the conditional distribution of the process on the covariates. We follow the Poisson process odel of Sith (1989) as the paraeters are invariant to the choice of threshold if the odel is appropriate. We define the covariate-dependent paraeters θ (z) = (µ (z), σ (z), (z)), for covariates z. Often in practice, the shape paraeter is assued to be constant. A log-link is typically used to ensure positivity of σ (z). The process of choosing is coplicated when odelling in the presence of covariates. This is partially caused by a odification of the integrated intensity easure, which becoes [ ( )] u µ (z) 1/(z) Λ(A) = 1 (z) g(z)dz, (14) z σ (z) where g denotes the probability density function of the covariates, which is unknown and with covariate space z. The density ter g is required as the covariates associated with exceedances of the threshold u are rando. In addition, the extra paraeters introduced by odelling covariates increases the overall correlation in the odel paraeters. For siplicity, we restrict our attention to the case of odelling when the location paraeter is a linear function of a covariate, that is, µ (z) = µ (0) µ (1) z, σ (z) = σ, (z) =, where we centre the covariate z, as this leads to paraeters µ (0) and µ (1) being orthogonal. Note that the regression paraeter µ (1) is invariant to the choice of. A total of 233 excesses above a threshold of u = 15 are siulated fro a Poisson process odel with µ (0) 1 = 75, µ (1) 1 = 30, σ 1 = 15, = We choose g to follow an Exp(2) distribution, noting that one could also choose g to be the density of a covariate that is used in practice. We ipose an iproper Unifor prior on the regression paraeter µ (1) 1 and set up the MCMC schee in the sae anner as in Section 3. The objective reains to identify the value of that achieves near-orthogonality of the paraeters of the posterior distribution. Like before, we run an MCMC sapler on θ (z) and transfor the 14

15 saples back to the paraeterisation of interest θ k (z), which can be obtained as in (5) using the relations µ (0) k = µ (0) σ ( 1 ( ) ) k µ (1) k = µ (1) (15) σ k = σ ( k ). Figure 8: Contour plots of estiated posterior densities of θ 1 (z) having sapled fro the joint posterior directly (red) and having transfored using (15) after reparaeterising fro θ 85 (z) (black). The coplication of the integral ter in the likelihood for non-identically distributed variables eans that it is no longer feasible to gain an analytical approxiation for the optial value of. A referee has suggested a possible route to obtaining such expressions for in the non-stationary case, is by building on results in Attalides (2015) and using a non-constant threshold as in Northrop and Jonathan (2011), but as this oves away fro our constant threshold case we do not pursue this. We therefore choose a value of that iniises the asyptotic posterior correlation in the odel. Analogous to Section 3, the optial coincides with the value of such that ρ µ (0),σ = 0. Using nuerical ethods, we identify that this corresponds to a value of = 85 for the siulated data exaple. Figure 8 shows contour plots of estiated posterior densities of θ 1 (z), coparing the sapling fro directly estiating the posterior θ 1 (z) with that fro transforing the saples fro the estiated posterior of θ (z) to give a saple fro the posterior of θ 1 (z). Fro this figure, we see that the reparaeterisation iproves the sapling fro the posterior θ 1 (z). 15

16 µ (0) µ (1) ESS ESS σ ESS ESS Figure 9: Effective saple size of each paraeter chain of the MCMC procedure. We again inspect the effective saple size for each paraeter as a way of coparing the efficiency of the MCMC under different paraeterisations. Figure 9 shows how the effective saple size varies with for each paraeter. This figure shows how the quality of ixing is approxiately axiised in µ (0) for µ (1) for the value of that iniises the asyptotic posterior correlation. Mixing is consistent across all values of. Interestingly, ixing in increases as the value of increases. Without a foral easure for the quality of ixing across the paraeters, it is found that, when averaging the effective saple size over the nuber of paraeters, the ESS is stable with respect to in the interval spanning fro the value of such that ρ (0) µ,σ (0) = 0 and the value of such that ρ (0) σ = 0, like in Section 3. For a suary of how the reparaeterisation, ethod can be used in the presence of non-stationarity, see Algorith 1. 16

17 5 Case study: Cubria rainfall In this section, we present a study as an exaple of how this reparaeterisation ethod can be used in practice. In particular, we analyse data taken fro the Met Office UKCP09 project, which contains daily baseline averages of surface rainfall observations, easured in illietres, in 25k 25k grid cells across the United Kingdo in the period In this analysis, we focus on a grid cell in Cubria, which has been affected by nuerous flood events in recent years, ost notably in 2007, 2009 and In particular, the Deceber 2015 event resulted in an estiated 5 billion worth of daage, with rain gauges reaching unprecedented levels. Many explanations have been postulated for the seeingly increased rate of flooding in the North West of England, including cliate change, natural cliate variability or a cobination of both. The baseline average data for the flood events in Deceber 2015 are not yet available, but this event is widely regarded as being ore extree than the event in Noveber 2009, the levels of which were reported at the tie to correspond to return periods of greater than 100 years. We focus our analysis on the 2009 event, looking in particular at how a phase of cliate variability, in the for of the North Atlantic Oscillation (NAO) index, can have a significant ipact on the probability of an extree event occurring in any given year. Rainfall datasets on a daily scale are coonly known to exhibit a degree of serial correlation. Analysis of autocorrelation and partial autocorrelation plots indicates that rainfall on day t is dependent on the rainfall of the previous five days. In addition, the data ay exhibit seasonal effects. However, while serial dependence affects the effective saple size of a dataset, it does not affect correlations between paraeters, and is thus unlikely to influence the choice of. For the purposes of illustrating our ethod, we ake the assuption for now that the rainfall observations are iid and proceed with the ethod outlined in Section 3. We wish to obtain inforation about the paraeters corresponding to the distribution of annual axia, i.e. θ 55. Standard threshold diagnostics (Coles, 2001) indicate a threshold of u = 15 is appropriate, which corresponds to the 95.6% quantile of the data. There are r = 880 excesses above u (see Figure 10). We obtain bounds 1 and 2, then choose a value 1 < < 2 that will achieve near-orthogonality of the Poisson process odel paraeters to iprove MCMC sapling fro the joint posterior distribution. We obtain ˆ = using axiu likelihood when = r, which we use to obtain approxiations for 1 and 2 as in (12) and (13). Fro this, we obtain ˆ and ˆ We checked that ˆ 1 and ˆ 2 represent good approxiations by solving equations (10) to obtain 1 = and 2 = Since r = 880 is contained in the interval ( 1, 2 ), we choose = r. We run an MCMC chain for θ 880 for iterations, discarding the first 1000 saples as burn-in. We 17

18 Rainfall accuulations () Rainfall accuulations() Day NAO Figure 10: (Left) Daily rainfall observations in the Cubria grid cell in the period The red line represents the extree value threshold of u = 15. (Right) Boxplots of rainfall against onthly NAO index. transfor the reaining saples using the apping in (5), where k = 55, to obtain saples fro the joint posterior of θ 55. The estiated posterior density for each paraeter is shown in Figure 11. To estiate probabilities of events beyond the range of the data, we can use the estiated paraeters to estiate extree quantiles of the annual axiu distribution. The quantity yn, satisfying: 1/N = 1 G(yN ), (16) is tered the N -year return level, where G is defined as in expression (2). The level yn is expected to be exceeded on average once every N years. By inverting (16) we get: µ σ55 [1 { log(1 1/N )} ] for 6= 0 55 yn = µ55 σ55 log{ log(1 1/N )} for = 0. (17) The posterior density of the 100-year return level in Figure 11 is estiated by inputting the MCMC saples of the odel paraeters into expression (17). We use the sae ethodology to explore the effect of the onthly NAO index on the probability of extree rainfall levels in Cubria. The NAO index describes the surface sea-level pressure difference between the Azores High and the Icelandic Low. The low frequency variability of the onthly scale is chosen to represent the large scale atospheric processes affecting the distribution of wind and rain. In the UK, a positive NAO index is associated with cool suers and wet winters, while a negative NAO index typically corresponds to cold winters, pushing the North Atlantic stor track 18

19 Figure 11: Estiated posterior densities of µ 55, σ 55, and the 100-year return level. further south to the Mediterranean region (Hurrell et al., 2003). In this analysis, we incorporated the effect of NAO by introducing it as a covariate in the location paraeter. The threshold of u = 15 was retained for this analysis. To obtain the value of that iniises the overall correlation in the odel, we solve nuerically the equation ρ µ (0),σ = 0. We obtain a kernel density estiate of the NAO covariate, which represents g as defined in expression (14). We use this to obtain axiu likelihood estiates ˆθ r (x). These quantities are substituted into the Fisher inforation atrix. The atrix is then inverted nuerically to estiate = 920. This represents a slight deviation fro ˆ 2 estiated during the iid analysis. We would expect this as the covariate effect is sall, as shown in Figure 12. This exaple illustrates the benefit of nuerically solving for when odelling non-stationarity, as the range ( 1, 2 ) estiated analytically during the iid analysis no longer contain the optial value of. 19

20 We run an MCMC chain for θ 920 (x) for iterations before discarding the first 1000 saples as burn-in. We transfor the reaining MCMC saples to the annual axiu scale using the apping in (15) where k = 55. Figure 12 indicates that NAO has a significantly positive effect on the location paraeter. Density Density (0) µ (1) µ 55 Density Density σ Figure 12: Estiated posterior densities of µ (0) 55, µ(1) 55, σ 55 and. We wish to estiate return levels relating to the Noveber 2009 flood event, which is represented by a value of in the dataset. Return levels corresponding to the distribution of Noveber axia are shown in Figure 13. We can also use the predictive distribution in order to account for both paraeter uncertainty and randoness in future observations (Coles and Tawn, 1996). On the basis of threshold excesses x = (x 1,..., x n ), the predictive distribution of a future Noveber axiu M is: Pr{M y x} = Pr{M y θ 55 }π(θ 55 x)dθ 55, (18) θ 55 20

21 where { exp Pr{M y θ 55 } = exp [ ( 1 z [ 1 )] 1/ y (µ (0) 55 µ(1) 55 z) σ 55 ( y (µ (0) 55 µ(1) σ 55 } )] 1/ 55 z) g N (z)dz where z is known where z is unknown, (19) where g N is the density of NAO in Noveber and the integral is evaluated nuerically using adaptive quadrature ethods. The integral in (18) can be approxiated using a Monte Carlo suation over the saples fro the joint posterior of θ 55. Fro this, we estiate the predictive probability of an event exceeding in a typical Noveber is , with a 95% credible interval of (0.0063, ), which corresponds to an 89-year event, (54, 158). For Noveber 2009, when an NAO index of 0.02 was easured, the probability of such an event was , (0.0062, ), corresponding to a 90-year event, (54, 161). For the axiu observed value of NAO in Noveber, with NAO = 3.04, the predictive probability of such an event is , (0.0073, ), which corresponds to a 75-year flood event, (47, 136). This illustrates the ipact that different phases of cliate variability can have on the probabilities of extree events. Figure 13: Return levels corresponding to Noveber axia. The full line represents the posterior ean and the two dashed lines representing 95% credible intervals. 21

22 Appendix A Proof: ˆµ r = u when = r We can write the full likelihood for paraeters θ r given a series of excesses {x i } above a threshold u as: L(θ r ) = L 1 L 2, where L 1 is the Poisson probability of r exceedances of u and L 2 is the joint density of these r exceedances, so that: { L 1 = 1 [ ( u µr r 1 r! σ r r [ ( 1 xi µ r L 2 = 1 σ i 1 r [ ( By defining Λ = 1 u µr σ r in ters of θ = (Λ, ψ u, ) to give: )] 1/ σ r L(θ ) Λ r exp { rλ} )] 1/ )] 1/ 1 } r { [ ( u µr exp r 1 [ 1 ( u µr σ r σ r )] 1/. )] 1/ and using identity (A), we can reparaeterise the likelihood r 1 ψ i=1 u [ 1 ( xi u Taking the log-likelihood and axiising with respect to Λ, we get: ψ u )] 1/ 1 ( ) r [ ( )] l(θ ) := log L(θ ) = r log ˆΛ r ˆΛ 1 r log ψ u 1 xi u log 1 l Λ = rˆλ r = 0, which gives ˆΛ = 1. Then, by the invariance property of axiu likelihood estiators, ˆµ r = u. Using the identity ψ u = σ (u µ ), which is invariant to choice of, we get ˆσ r = ˆψ u. i=1 } ψ u Because the -dependent ter in the loglikelihood is identical to that in a GP log-likelihood, the axiu likelihood estiators of the two odels coincide., B Derivation of prior for inference on θ We define a joint prior on the paraeterisation of interest θ k. However, as we are aking inference for the optial paraeterisation θ, we ust derive the prior for θ. We can calculate the prior 22

23 density of θ by using the density ethod for one-to-one bivariate transforations. Inverting (5) to get expressions for µ and σ, i.e. µ = µ k σ ( ( k ) ) 1 = g 1 (µ k, σ k ) k ( ) σ = σ k = g2 (µ k, σ k ), k we can use this transforation to calculate the prior for θ. π(θ ) = π(µ, σ, ) = π(µ k, σ k, ) det J µk =g 1 1 (µ,σ),σ k=g 1 2 (µ,σ),=, where det J = = µ µ µ k σ k σ σ µ k σ k µ k σ k µ µ µ k σ k σ 0 σ k 0 0 = σ σ k ( = k ). µ σ µ σ Therefore, π(θ ) = ( ) π(θk k ). C Fisher inforation atrix calculations for iid rando variables The log-likelihood of the Poisson process odel with paraeterisation θ = (µ, σ, ) can be expressed as [ l(θ ) = 1 ( u µ σ )] 1/ ( ) r [ 1 r log σ 1 log 1 ( )] xj µ where r is the nuber of exceedances of X above the threshold u. For siplicity, we drop the [ ] subscript in subsequent calculations. In order to produce analytic expressions for the asyptotic covariance atrix, we ust evaluate the observed inforation atrix Î(θ ). For siplicity, we define v = u µ σ and z j, = x j µ σ. σ, 23

24 2 l µ 2 2 l σ 2 = ( 1) σ 2 [1 v ] 1/ 2 ( 1) σ 2 = 2 σ 2 [1 v ] 1/ 1 v r ( 1) σ 2 2 l 2 = [1 v ] 1/ 2 2 [1 v ] 1 v 2 3 z 2 j, [1 z j, ] 2, r log [1 z j, ] l µ σ = σ 2 [1 v ] 1/ 1 1 r σ 2 [1 z j, ] 1 2 l µ 2 l σ r [1 z j, ] 2, ( 1) σ 2 [1 v ] 1/ 2 v 2 r 2( 1) σ 2 σ 2 [ 1 v2 [1 v ] log [1 v ] ( 1 2 log [1 v ] 1 [1 v ] 1 v r [1 z j, ] 1 z j, 1 ( 1) σ 2 [1 v ] 1/ 2 v r ( 1) σ 2 [1 z j, ] 2 z j,, ) 2 ] r [1 z j, ] 1 z j, r [1 z j, ] 2 zj,, 2 = [ 1 σ 2 [1 v ] 1/ 1 log [1 v ] 1 ] [1 v ] 1/ 2 v 1 σ 1 σ r [1 z j, ] 2 z j,, = [ 1 v σ 2 [1 v ] 1/ 1 log [1 v ] 1 ] [1 v ] 1/ 2 v 1 σ 1 σ r [1 z j, ] 2 zj, 2 r [1 z j, ] 1 To obtain the Fisher inforation atrix, we take the expected value of each ter in the observed inforation with respect to the probability density of points of a Poisson process. Let Z = X µ σ, and R be a rando variable denoting the nuber of excesses of X above u. The density of points in the set A u can de defined by f(x) = λ(x) [1 z] 1/ 1 = Λ(A u ) [1 v ] 1/, r [1 z j, ] 1 z j, 24

25 where λ is a function denoting the rate of exceedance. Then, for exaple, R R E Z,R [1 z j, ] 2 = E RE Z R [1 z j, ] 2 { { = E R RE Z [1 Z] 2}} { } = E R R[1 v ] 1/ [1 z] 1/ 3 dz = v 2 1 [1 v ] 1/ 2 Following this process, we can write the Fisher inforation atrix I(θ ) as: { } E 2 l µ 2 { } E 2 l σ 2 { } E 2 l 2 { } E 2 l µ σ { } E 2 l µ { } E 2 l σ = ( 1) σ 2 [1 v ] 1/ 2 ( 1) (2 1)σ 2 [1 v ] 1/ 2, = 2 [1 v ] 1/ 1 v σ 2 2 σ 2 [1 v ] 1/ 1 [1 ( 1)v ] ( 1) σ 2 [1 v ] 1/ 2 v 2 r σ 2 (2 1)σ 2 [1 v ] 1/ 2 [ ( )v 2 (4 2)v 2 ], [ = [1 v ] 1/ 1 v2 [1 v ] log [1 v ] ( 2 2 [1 v ] 1 1 v 2 log [1 v ] 1 ) ] 2 [1 v ] [1 v ] 1/ 2 [ log [1 v ]] ( 1) 2 [1 v ] 1/ 1 [1 ( 1)v ] (2 1) [1 v ] 1/ 2 [ ( )v 2 (4 2)v 2 ], ( 1) = σ 2 [1 v ] 1/ 2 v (2 1)σ 2 [1 v ] 1/ 2 [1 (2 1)v ], = [ 1 σ 2 [1 v ] 1/ 1 log [1 v ] 1 ] [1 v ] 1/ 2 v σ ( 1) [1 v ] 1/ 1 σ (2 1) [1 v ] 1/ 2 [1 (2 1)v ], = [ 1 v σ 2 [1 v ] 1/ 1 log [1 v ] 1 ] [1 v ] 1/ 2 v σ ( 1) [1 v ] 1/ 1 [1 ( 1)v ] σ (2 1) [1 v ] 1/ 2 [ ( )v 2 (4 2)v 2 ]. By inverting the Fisher inforation atrix using a technical coputing tool like Wolfra Matheatica, aking the substitution r = [1 v ] 1/, the expected nuber of exceedances, and 25

26 using the apping in (5), we can get expressions for asyptotic posterior covariances. ACov(µ, ) = 1 ( r ) ( ( r ) ( ( r ( r ) )) 2 r ( 1)σ ( 1) log (2 1) 1 ) ACov(µ, σ ) = 1 ( ( r ( r ) ( ( r ( ( r ) 2 r σ2 ( 1) log ( 1) log 3 1 ) ) ) ) ( ( r ) (( 2) 3) 1 ( 1)(2 1) log 1 ) ) ACov(σ, ) = 1 r ( 1)σ ( ( r ) ( 1) log 1 ) When = r, ACov(µ, ) = 0. In addition, the for which ACov(µ, σ ) = 0 coincides with the value of that iniises ρ θ as defined in (8). This root can easily be found nuerically, but an analytical approxiation can be calculated using a one-step Halley s ethod. By using = r as the initial seed, and using the forula: x n1 = x n f(x n ) f (x n ) f(xn)f (x n) 2f (x n) we get the expression (13) for ˆ 2 after one step. The quantity for ˆ 1, given by expression (12) requires two iterations of this ethod. Acknowledgeents We gratefully acknowledge the support of the EPSRC funded EP/H023151/1 STOR-i Centre for Doctoral Training, the Met Office and EDF Energy. We extend our thanks to Jenny Wadsworth of Lancaster University and Sion Brown of the Met Office for helpful coents. We also thank the Met Office for the rainfall data. References Attalides, N. (2015). Threshold-based extree value odelling. PhD thesis, UCL (University College London). Chavez-Deoulin, V. and Davison, A. C. (2005). Generalized additive odelling of saple extrees. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(1): Coles, S. G. (2001). An Introduction to Statistical Modeling of Extree Values. Springer. Coles, S. G. and Tawn, J. A. (1996). A Bayesian analysis of extree rainfall data. Applied Statistics, 45:

27 Cox, D. R. and Reid, N. (1987). Paraeter orthogonality and approxiate conditional inference (with discussion). Journal of the Royal Statistical Society. Series B (Methodological), 49(1):1 39. Davison, A. C. and Sith, R. L. (1990). Models for exceedances over high thresholds (with discussion). Journal of the Royal Statistical Society. Series B (Methodological), 52(3): Efron, B. and Hinkley, D. V. (1978). Assessing the accuracy of the axiu likelihood estiator: Observed versus expected fisher inforation. Bioetrika, 65(3): Gander, W. (1985). On Halley s iteration ethod. Aerican Matheatical Monthly, 92(2): Hills, S. E. and Sith, A. F. (1992). Paraeterization issues in Bayesian inference. Bayesian Statistics, 4: Hurrell, J. W., Kushnir, Y., Ottersen, G., and Visbeck, M. (2003). An overview of the North Atlantic oscillation. Geophysical Monograph-Aerican Geophysical Union, 134:1 36. Northrop, P. J. and Attalides, N. (2016). Posterior propriety in Bayesian extree value analyses using reference priors. Statistica Sinica, 26(2): Northrop, P. J. and Jonathan, P. (2011). Threshold odelling of spatially dependent non-stationary extrees with application to hurricane-induced wave heights. Environetrics, 22(7): Pickands, J. (1975). Statistical inference using extree order statistics. The Annals of Statistics, 3(1): Robert, C. and Casella, G. (2009). Introducing Monte Carlo Methods with R. Springer Science & Business Media. Roberts, G. O., Rosenthal, J. S., et al. (2001). Optial scaling for various Metropolis-Hastings algoriths. Statistical Science, 16(4): Sith, R. L. (1985). Maxiu likelihood estiation in a class of nonregular cases. Bioetrika, 72(1): Sith, R. L. (1987a). Discussion of Paraeter orthogonality and approxiate conditional inference by D.R. Cox and N. Reid. Journal of the Royal Statistical Society. Series B (Methodological), 49(1): Sith, R. L. (1987b). A theoretical coparison of the annual axiu and threshold approaches to extree value analysis. Technical Report 53, University of Surrey. 27

28 Sith, R. L. (1989). Extree value analysis of environental tie series: an application to trend detection in ground-level ozone. Statistical Science, 4(4): Stephenson, A. (2016). Bayesian inference for extree value odelling. Extree Value Modeling and Risk Analysis: Methods and Applications, pages Tawn, J. A. (1987). Discussion of Paraeter orthogonality and approxiate conditional inference by D.R. Cox and N. Reid. Journal of the Royal Statistical Society. Series B (Methodological), 49(1): Wadsworth, J. L., Tawn, J. A., and Jonathan, P. (2010). Accounting for choice of easureent scale in extree value odeling. The Annals of Applied Statistics, 4(3):