Extreme Value Analysis and Spatial Extremes

Transcription

1 Extreme Value Analysis and Department of Statistics Purdue University 11/07/2013

2 Outline Motivation 1 Motivation 2 Extreme Value Theorem and 3 Bayesian Hierarchical Models Copula Models Max-stable Models

3 What are (Spatial) Extremes? Extreme events are defined to be rare and unexpected 100 year flood the level of flood water expected to be equaled or exceeded every 100 years on average Many extreme events of interest are spatio-temporal in natural

4 What are (Spatial) Extremes? Extreme events are defined to be rare and unexpected 100 year flood the level of flood water expected to be equaled or exceeded every 100 years on average Many extreme events of interest are spatio-temporal in natural

5 What are (Spatial) Extremes? Extreme events are defined to be rare and unexpected 100 year flood the level of flood water expected to be equaled or exceeded every 100 years on average Many extreme events of interest are spatio-temporal in natural

6 What are (Spatial) Extremes? Extreme events are defined to be rare and unexpected 100 year flood the level of flood water expected to be equaled or exceeded every 100 years on average Many extreme events of interest are spatio-temporal in natural

7 Why study extremes? Although infrequent, extremes have large human impact European heat wave example. Around 70,000 were killed!

8 Why study extreme? "There is always going to be an element of doubt, as one is extrapolating into into areas one does not know about. But what EVT is doing is making the best use of whatever data you have about extreme phenomena." Richard Smith

9 Usual vs Extremes Relies on asymptotic theory to provide models for the tail Uses only the "extreme" observations to fit the model

10 History Motivation 1920 s: Foundations of asymptotic argument developed by Fisher and Tippett 1940 s: Asymptotic theory unified and extended by Gnedenko and von Mises 1950 s: Use of asymptotic distributions for statistical modelling by Gumbel and Jenkinson 1970 s: Classic limit laws generalized by Pickands

11 History Motivation 1980 s: Leadbetter (and others) extend theory to stationary processes 1990 s: Multivariate and other techniques explored as a means to improve inference 2000 s: Interest in spatial and spatio-temporal applications, and in finance

12 Probability Framework Let X 1,, X n iid F and define Mn = max{x 1,, X n } Then the distribution function of M n is P(M n x) = P(X 1 x,, X n x) = P(X 1 x) P(X n x) = F n (x) Remark F n (x) n = { 0 if F(x) < 1 1 if F(x) = 1

13 Classical Limit Laws Motivation Recall the Central Limit Theorem: X n µ σ n d N(0, 1) rescaling is the key to obtain a non-degenerate distribution Question: Can we get the limiting distribution of M n b n a n for suitable sequence {a n } > 0 and {b n }?

14 Classical Limit Laws Motivation Recall the Central Limit Theorem: X n µ σ n d N(0, 1) rescaling is the key to obtain a non-degenerate distribution Question: Can we get the limiting distribution of M n b n a n for suitable sequence {a n } > 0 and {b n }?

15 Theorem (Fisher Tippett Gnedenko theorem) If there exist sequences of constants a n > 0 and b n such that, as n P( M n b n x) d G(x) a n for some non-degenerate distribution G, then G belongs to either the Gumbel, the Fréchet or the Weibull family Gumbel : G(x) = exp(exp( x)) < x < ; { 0 x 0, Fréchet : G(x) = exp( x α ) x > 0, α > 0; { exp( ( x) Weibull : G(x) = α ) x < 0, α > 0, 1 x 0;

16 Example: Exponential Maxima X Exp(1) F n (x + log n) = (1 exp( x log n)) n = (1 1 n exp( x))n n exp{ exp( x)}

17 Generalized Extreme Value Distribution (GEV) This family encompasses all three extreme value limit families: where x + = max(x, 0) { [ G(x) = exp 1 + ξ( x µ σ ] 1 } ) ξ + µ and σ are location and scale parameters ξ is a shape parameter determining the rate of tail decay, with ξ > 0 giving the heavy-tailed (Fréchet) case ξ = 0 giving the light-tailed (Gumbel) case ξ < 0 giving the bounded-tailed (Weibull) case

18 Quantiles and Return Levels In terms of quantiles, take 0 < p < 1 and define x p such that: { [ G(x p ) = exp 1 + ξ( x p µ ] 1 } ξ ) = 1 p σ + x p = µ σ [ 1 { log(1 p) ξ }] ξ In the extreme value terminology, x p is the return level associated with the return period 1 p

19 Max-Stability Motivation Definition A distribution G is said to be max-stable if G k (a k x + b k ) = G(x), k N for some constants a k > 0 and b k Taking powers of G results only in a change of location and scale A distribution is max-stable it is a GEV distribution

20 Some Remarks Motivation There has been some work on the convergence rate of M n to the limiting regime, which depends on the underling distribution For statisticians, we use the GEV as an approximate distribution for sample maximal for "finite" n assess the fit empirically Direct use the GEV rather than three types separately

21 Block Maximum Approach Determine the block size and compute maximal for blocks usually annual maximal Fit the GEV to the maximal and assess fit usually via likelihood based techniques Perform inference for return levels, probabilities, etc

22 Diagnostics Motivation

23 Point Process Approach Motivation The block maximum method ignores much of the data which may also relevant to extreme we would like to use the data more efficient. Alternatives: peaks over thresholds r-largest order statistics Both are special cases of a point process representation, under which we approximate the exceedances over a threshold by a two-dimensional Poisson process

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25 Point Process Limit: Basic Idea iid Suppose X 1,, X n F and { M n b n a n } converges to GEV distribution Construct a sequence of point processes on R 2 by { i P n = ( n + 1, X } i b n ) : i = 1,, n a n P n n P, where P is a Poisson process

26 Formally Linking Extremes and Point Processes Given: P( Mn bn a n { } x) exp (1 + ξx) 1 ξ log P( M n b n a n log P n ( X b n a n x) (1 + ξx) 1 ξ x) (1 + ξx) 1 ξ n log(1 P( X b n a n > x)) (1 + ξx) 1 ξ np( X b n a n > x) (1 + ξx) 1 ξ

27 Extremes and Point Processes Cond t np( X b n a n > x) (1 + ξx) 1 ξ Therefore, creating a series of point processes { } Xi b n : i = 1,, n n a n these will converge to an inhomogeneous Poisson process with measure ν((x, )) = (1 + ξx) 1 ξ for sets bounded away from zero (exceedance example).

28 Generalized Pareto Distribution (GPD) for Exceedances P(X i > x + u X i > u) = np(x i > x + u) np(x i > u) ( 1 + ξ x+u b n ) 1 ξ a n 1 + ξ x bn a n = ( 1 + ) 1 ξx ξ a n + ξ(u b n ) Survival function of generalized Pareto distribution

29 Theorem (Pickands Balkema de Haan theorem) Let X 1, iid F, and let F u be their conditional excess distribution function. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions F, and large u, F u is well approximated by the generalized Pareto distribution GPD. That is: F u (y) GPD ξ,σ(u),u (y) u where GPD ξ,σ(u),u (y) = { 1 (1 + ξy σ(u) ) 1 ξ ξ 0, 1 exp( y σ(u) ) ξ = 0;

30 Fort Collins Data: Block Maximum vs. Threshold-exceedance GPD has lower standard error for ξ, lower estimate as well. GPD has narrower confidence interval. Have not yet discussed threshold selection procedure.

31 Threshold Selection Motivation Bias variance trade off: threshold too low bias because of the model asymptotics being invalid; threshold too high variance is large due to few data points Figure: Mean residual life plot(mrl): MRL is linear when GPD holds

32 Temporal Dependence Question: Is the GEV still the limiting distribution for block maxima of a stationary (but not independent) sequence {X i }? Answer: Yes, so long as mixing conditions hold. (Leadbetter et al., 1983) What does this mean for inference? Block maximum approach: GEV still correct for marginal. Since block maximum data likely have negligible dependence, proceed as usual Threshold exceedance approach: GPD is correct for the marginal. If extremes occur in clusters, estimation affected as likelihood assumes independence of threshold exceedances

33 Temporal Dependence Question: Is the GEV still the limiting distribution for block maxima of a stationary (but not independent) sequence {X i }? Answer: Yes, so long as mixing conditions hold. (Leadbetter et al., 1983) What does this mean for inference? Block maximum approach: GEV still correct for marginal. Since block maximum data likely have negligible dependence, proceed as usual Threshold exceedance approach: GPD is correct for the marginal. If extremes occur in clusters, estimation affected as likelihood assumes independence of threshold exceedances

34 Temporal Dependence Question: Is the GEV still the limiting distribution for block maxima of a stationary (but not independent) sequence {X i }? Answer: Yes, so long as mixing conditions hold. (Leadbetter et al., 1983) What does this mean for inference? Block maximum approach: GEV still correct for marginal. Since block maximum data likely have negligible dependence, proceed as usual Threshold exceedance approach: GPD is correct for the marginal. If extremes occur in clusters, estimation affected as likelihood assumes independence of threshold exceedances

35 Temporal Dependence Question: Is the GEV still the limiting distribution for block maxima of a stationary (but not independent) sequence {X i }? Answer: Yes, so long as mixing conditions hold. (Leadbetter et al., 1983) What does this mean for inference? Block maximum approach: GEV still correct for marginal. Since block maximum data likely have negligible dependence, proceed as usual Threshold exceedance approach: GPD is correct for the marginal. If extremes occur in clusters, estimation affected as likelihood assumes independence of threshold exceedances

36 Remarks on To estimate the tail, EVT uses only extreme observations Tail parameter ξ is extremely important but hard to estimate GPD is the limiting distribution for threshold exceedances Threshold exceedance approaches allow the user to retain more data than block-maximum approaches, thereby reducing the uncertainty with parameter estimates Temporal dependence in the data is more of an issue in threshold exceedance models. One can either decluster, or alternatively, adjust inference

37 Remarks on To estimate the tail, EVT uses only extreme observations Tail parameter ξ is extremely important but hard to estimate GPD is the limiting distribution for threshold exceedances Threshold exceedance approaches allow the user to retain more data than block-maximum approaches, thereby reducing the uncertainty with parameter estimates Temporal dependence in the data is more of an issue in threshold exceedance models. One can either decluster, or alternatively, adjust inference

38 Remarks on To estimate the tail, EVT uses only extreme observations Tail parameter ξ is extremely important but hard to estimate GPD is the limiting distribution for threshold exceedances Threshold exceedance approaches allow the user to retain more data than block-maximum approaches, thereby reducing the uncertainty with parameter estimates Temporal dependence in the data is more of an issue in threshold exceedance models. One can either decluster, or alternatively, adjust inference

39 Remarks on To estimate the tail, EVT uses only extreme observations Tail parameter ξ is extremely important but hard to estimate GPD is the limiting distribution for threshold exceedances Threshold exceedance approaches allow the user to retain more data than block-maximum approaches, thereby reducing the uncertainty with parameter estimates Temporal dependence in the data is more of an issue in threshold exceedance models. One can either decluster, or alternatively, adjust inference

40 Remarks on To estimate the tail, EVT uses only extreme observations Tail parameter ξ is extremely important but hard to estimate GPD is the limiting distribution for threshold exceedances Threshold exceedance approaches allow the user to retain more data than block-maximum approaches, thereby reducing the uncertainty with parameter estimates Temporal dependence in the data is more of an issue in threshold exceedance models. One can either decluster, or alternatively, adjust inference

41 Examples A central aim of multivariate extremes is trying to find an appropriate structure to describe tail dependence

42 What is a Multivariate Extreme? Let Z m = (Z m,1,, Z m,d ) T, m N be an iid sequence of random vectors. Want to extract a subset of data considered "extreme" Block-maximum: Mn = ( n m=1 Z m,1,, n Leads to modeling with multivariate max-stable distributions m=1 Z m,d) T Marginal-exceedance: For each marginal i = 1,, d, find an appropriate threshold u i and retain data where Z m,i > u i. Leads to multivariate generalized Pareto distribution Norm-exceedance: For a given norm retain data where Z m > z. Leads to description by multivariate regular variation

43 What is a Multivariate Extreme?

44 Multivariate Models Motivation Let X 1, X 2, iid d-dimensional random vectors with distribution F Our interest is in the (non degenerate) limiting distribution { } Xi b i n P max x G(x) i=1,,n a n for some sequences a n > 0 and b n R d. G is called a multivariate extreme value distribution Transform to common marginals Z with unit Fréchet i.e., Z (x,,, ) = = Z (,,, x) = exp( 1 x ) to model the dependence structure

45 Multivariate Models cond t P(Z 1 { z 1,, Z d z} d ) = exp V (z 1,, z d ) z 1,, z d > 0 The exponent measure V (z 1,, z d ) satisfies V (tz 1,, tz d ) = t 1 V (z 1,, z d ) = V is homogeneous of order -1 V (z 1,, z d ) = { 1 z z D 1 min(z 1,,z D ) if Z 1,, Z d are independent if Z 1,, Z d are entirely dependent

46 Spectral Representations Pickands, 1981 V (z 1,, z d ) = S D max( w 1 z 1,, w D z D ) dm(w 1,, w D ) where M is a measure on the D-dimensional simplex S D wd dm(w 1,, w D ) = 1 for each d No simple parametric forms for V

47 measure of extremal dependence { P(Z 1 z,, Z D z) = exp V (1,,1) z } exp( θ D z ) z > 0 θ D is the extrmal coefficient θ D = 1 fully dependent θ D = D independent In bivariate case lim z P(Z 2 > z Z 1 > z) = 2 θ D When D = 2 madogram [Cooley, Naveau and Poncet, 2006] µ F = 1 2 E{ F(Z 1 F(Z 2 )) } θ 2 = 1 + 2ν F 1 2ν F provide a good estimator of θ 2

48 Motivation Developed originally to predict probability distributions of ore grades for mining operations is based largely on the theory of Gaussian random processes Y (x) = µ(x) + e(x) + ε(x), x D R 2 E(Y (x)) = µ(x), Cov(Y (x), Y (y)) = C( x y ) The main propose of geostatistics is interpolation

49 Figure: A realization of Gaussian random processes

50 Bayesian Hierarchical Models Copula Models Max-stable Models Three-Level Spatial Hierarchical Model Data level: Likelihood which characterizes the distribution of the observed data given the parameters at the process level Y i (x d ) {µ(x d ), σ(x d ), ξ(x d )} GEV {µ(x d ), σ(x d ), ξ(x d )} i = 1,, n, d = 1,, D Process level: Latent process captured by spatial model for the data level parameters Prior level: Prior distributions put on the parameters

51 Bayesian Hierarchical Models Copula Models Max-stable Models Three-Level Spatial Hierarchical Model Suppose the response variables {Y (x)} are independent conditionally on an unobserved latent process {S(x)} Assume the {S(x)} follows a Gaussian process and induce spatial dependence in {Y (x)} by integration over the latent process Common approach in geostatistics with non-normal response [Diggle et al. 1998,2007]. Usually performed in a Bayesian setting using Markov chain Monte Carlo (MCMC)

52 Pros and Cons Motivation Bayesian Hierarchical Models Copula Models Max-stable Models The quantile surfaces are realistic After averaging over S(x) the marginal distribution of {Y (x)} is NOT GEV The spatial dependence is ignored because of conditional independence

53 Pros and Cons Motivation Bayesian Hierarchical Models Copula Models Max-stable Models Figure: One realisation of the latent variable model, showing the lack of local spatial structure

54 Copula Motivation Bayesian Hierarchical Models Copula Models Max-stable Models The D dimensional joint distribution F of any random variable (Y 1,, Y D ) may be written as F (y 1,, y n ) = C{F 1 (y 1 ),, F D (y D )} Where F 1,, F D are univariate marginal distributions of Y 1,, Y D and C is a copula The copula is uniquely determined for distributions F with absolutely continuous marginal With continuous and strictly increasing marginals, the copula corresponds to F(y 1,, y n ) = C{F 1 (y 1 ),, F D (y D )} is as follows: { } C(u 1,, u D ) = F F 1 1 (u 1 ),, F 1 D (u D )

55 Copula Motivation Bayesian Hierarchical Models Copula Models Max-stable Models The D dimensional joint distribution F of any random variable (Y 1,, Y D ) may be written as F (y 1,, y n ) = C{F 1 (y 1 ),, F D (y D )} Where F 1,, F D are univariate marginal distributions of Y 1,, Y D and C is a copula The copula is uniquely determined for distributions F with absolutely continuous marginal With continuous and strictly increasing marginals, the copula corresponds to F(y 1,, y n ) = C{F 1 (y 1 ),, F D (y D )} is as follows: { } C(u 1,, u D ) = F F 1 1 (u 1 ),, F 1 D (u D )

56 Copula Motivation Bayesian Hierarchical Models Copula Models Max-stable Models The D dimensional joint distribution F of any random variable (Y 1,, Y D ) may be written as F (y 1,, y n ) = C{F 1 (y 1 ),, F D (y D )} Where F 1,, F D are univariate marginal distributions of Y 1,, Y D and C is a copula The copula is uniquely determined for distributions F with absolutely continuous marginal With continuous and strictly increasing marginals, the copula corresponds to F(y 1,, y n ) = C{F 1 (y 1 ),, F D (y D )} is as follows: { } C(u 1,, u D ) = F F 1 1 (u 1 ),, F 1 D (u D )

57 Example Motivation Bayesian Hierarchical Models Copula Models Max-stable Models Gaussian copula: Y 1,, Y D N d (0, Ω) { C(u 1,, u d ) = Φ Φ 1 (u 1 ),, Φ 1 (u d ) : Ω} Student t copula: { C(u 1,, u d ) = T ν Extremal Copula: Tν 1 (u 1 ),, Tν 1 (u D ) : Ω} Y 1,, Y D multivariate GEV By max-stability C(u m 1,, um D ) = Cm (u 1,, u D ), 0 < u 1,, u d < 1, m N

58 Pros and Cons Motivation Bayesian Hierarchical Models Copula Models Max-stable Models On the "copula scale : the dependence seems more or less OK But this is no longer true at the original, i.e., extremal scale. Figure: One simulation from the fitted Gaussian copula model

59 Max-stable Processes Bayesian Hierarchical Models Copula Models Max-stable Models Definition Let Y m (x), x D R 2, m = 1,, n be independent copies of Y (x), and let M n (x) = max Y m (x). Y (x) is termed max-stable if there exist {a n (x)} and {b n (x)} such that P( M n(x) b n (x) a n (x) y(x)) n P(G(x) y(x))

60 Spectral Representations Bayesian Hierarchical Models Copula Models Max-stable Models Theorem (Schlather, 2002) Z (x), x D R 2 is max-stable with unit Fréchet marginals There exist iid positive stochastic processes V 1 (x), V 2 (x), with E[V i (x)] = 1 x D and E[sup x D V (x)] < and an independent point process {s i } i=1 on R + with intensity measure r 2 dr such that Z (x) d = max i N s i V i

61 Some Models Motivation Bayesian Hierarchical Models Copula Models Max-stable Models Smith: Y i (x) = φ(x i U i )(), {U i } i 1 points of a homogeneous Poisson process on R 2 Schlather: Y i (x) = 2πε i (x), ε i ( ) standard Gaussian process Geometric: Y i (x) = exp{σε i (x) σ2 2 } Brown Res: Y i (x) = exp{ε i (x) γ(x)}, ε i ( ) intrinsically stationary Gaussian process with (semi) variogram γ Figure: One realization of max-stable processes

62 Inference Motivation Bayesian Hierarchical Models Copula Models Max-stable Models For the max-stable process models, only the bivariate distributions are known Composite likelihoods [Lindsay, 1988] are used to obtain estimation Since we have the bivariate distributions we will use the pair wise likelihood l p (θ, y) = n K 1 m=1 i=1 j=i+1 k log f (ym, i fm; j θ) Not a true likelihood Over uses the data each observation appears K-1 times

63 Pros and Cons Motivation Bayesian Hierarchical Models Copula Models Max-stable Models Justified by extreme value theory Able to describe asymptotic dependence Describe everything that is asymptotically independent as exactly independent Only suitable for observations that are annual maximal at this point

64 Motivation Although infrequent, extremes have large human impact Uses only the "extreme" observations to fit the model No role for normal distribution! Spatial extremes modeling is challenge both in theoretical and computational aspects

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