Max-stable processes: Theory and Inference

Transcription

1 1/39 Max-stable processes: Theory and Inference Mathieu Ribatet Institute of Mathematics, EPFL Laboratory of Environmental Fluid Mechanics, EPFL joint work with Anthony Davison and Simone Padoan

2 2/39 Outline Introduction Max-stable processes Spatial dependence of max-stable processes Inference for max-stable processes Model Selection Application References

3 3/39 Outline Introduction Max-stable processes Spatial dependence of max-stable processes Inference for max-stable processes Model Selection Application References

4 4/39 EVT has been applied to many environmental processes - rainfall, temperature, river flow, wind, fire sizes,... However many of these processes are spatial Univariate/Multivariate analysis of extremes are not enough Problem Let Y 1,...,Y n be i.i.d. replicates of a stochastic process Y defined on K R d. We might be interested in estimating Pr[Y (x) > f (x),x K], f some deterministic function Pr[ K 1 Y (x)dx > y], K 1 K Pr[Y (x 2 ) > y 2 Y (x 1 ) > y 1 ], (x 1,x 2 ) K K

5 5/39 Outline Introduction Max-stable processes Spatial dependence of max-stable processes Inference for max-stable processes Model Selection Application References

6 Let X 1,X 2,... be independent replications of a stochastic processes on R d with continuous sample path We are interested in the limiting process { max i=1,...,n X i (x) b n (x) a n (x) } x K d {Y (x)} x R d (1) for some non-degenerate process Y, a n (x) > 0 and b n (x) continuous functions. de Haan shows that the class of limit process in (1) coincides with the class of max-stable processes. Definition A stochastic process Y with non-degenerate marginals is called max-stable if there are continuous function a n (x) > 0 and b n (x) such that if Y 1,Y 2,... are i.i.d. copies of Y then n = 1,2,... (2) Y i b n max i=1,...,n a n d = Y, 6/39

7 7/39 It is often more convenient to work with specified margins Most authors work with unit Fréchet margins i.e. Pr[Y (x) z] = exp( 1/z) Definition (Special case) A stochastic process Z is said max-stable with unit Fréchet margins if max i=1,...,n n 1 d Z i = Z, n = 1,2,... (3) Theorem (Spectral representation) Let {ξ i } i 1 be the points of a Poisson process on R + with intensity dλ(ξ) = ξ 2 dξ, and {Y i ( )} i 1 be i.i.d. replicates of a process on R d such that E[max{0,Y (x)}] = 1. Then Z(x) = max i 1 ξ i max{0,y i (x)} is a max-stable process with unit Fréchet margins.

8 8/39 Using the spectral representation, it is possible get parametric max-stable models through the definition of the process Y In the next slides, I will introduce some models

9 9/39 Smith s model [Smith, 1990] Let Y i (x) = ϕ(x X i ) where {X i } i 1 are the points of a homogeneous Poisson process and ϕ is the zero mean multivariate normal density with covariance matrix Σ, both on R d. Remark This model is often referred to as the storm profile model Lets focus on rainfall events. You can think about ξ i as the intensity of the i-th storm, {X i } as the storm centres ϕ as a deterministic function that drives how the amount of rain decreases from the storm centre. ξ iϕ(x Xi) X 3 X 1 X 2

10 10/39 Schlather s model [Schlather, 2002] Later Schlather proposed to take Y i (x) = 2πY,i (x) where Y,i are independent replications of a standard Gaussian process with correlation ρ Remark This model has no deterministic shape for the storms......but random ones Storm structure is driven by the correlation function ρ(h) But it has some drawbacks - see later ξ imax{0, Yi(x)}

11 11/39 Other models One can define various models in the spirit of the Schlather one The most popular are Geometric gaussian model Y i (x) = exp where ε is a standard Gaussian process. } {σε(x) σ2 2 Brown-Resnick processes { } Y i (x) = exp ε(x) σ2 (x) 2 where ε is a Gaussian process with stationary increments, σ 2 (x) = Var[ε(x)]. We ll see later what are the assets and drawbacks of these models

12 12/39 y y x x y y x Not yet implemented! x Figure: Top left: Smith. Top right: Schlather. Bottom left: Geometric. Bottom right: Brown Resnick. Gumbel margins. Simulations were performed using the SpatialExtremes package.

13 13/39 Outline Introduction Max-stable processes Spatial dependence of max-stable processes Inference for max-stable processes Model Selection Application References

14 14/39 Motivation The modelling of univariate extreme events is limited if the process is spatial by nature The knowledge of dependence between extreme events occuring at several locations is essential In this section, we will introduce some notions describing the spatial structure of extreme events

15 15/39 Any multivariate extreme value distribution has the form Pr[Z(x 1 ) z 1,...,Z(x k ) z k ] = exp { V (z 1,...,z k )} (4) where V has some homogeneity properties depending on the margins. If we suppose unit Fréchet margins, V is homogeneous of order 1 so that log (Pr[Z(x 1 ) z,...,z(x k ) z]) = V (z,...,z) hom. = θ k z where θ k = V (1,...,1) is called the extremal coefficient. θ k is a measure of dependence between extremes and satisfies 1 θ k k θ k = 1 complete dependence θ k = k independence For max-stable processes, an important measure of dependence is the extremal coefficient function θ(x 1 x 2 ) = z log Pr[Z(x 1 ) z,z(x 2 ) z] (5)

16 16/39 It is easy to see that the extremal coefficient functions for the models introduced are ( ) (x1 x Smith θ(x 1 x 2 ) = 2Φ 2 ) T Σ 1 (x 1 x 2 ) 2 1 ρ(x 1 x 2 ) Schlather θ(x 1 x 2 ) = 1 + ( 2 Geometric Gaussian θ(x 1 x 2 ) = 2Φ 2 ( ) γ(x1 x Brown Resnick θ(x 1 x 2 ) = 2Φ 2 ) σ 2 (1 ρ(x 1 x 2 )) The Schlather model has an upper bound of in R 2 and in R 3. In addition, as most of the correlation functions take only positive values, the upper bound is 1 + 1/2. Independence never reached! The geometric gaussian model has an upper bound of 2Φ(σ/ 2). Hence, if σ is large enough, independence is reached - at least numerically. 2 )

17 17/39 Variogram-based approaches In conventional geostatistics, the variogram determines the degree of spatial dependence γ(x 1 x 2 ) = 1 2 Var [Y (x 1) Y (x 2 )] = σ 2 {1 ρ(x 1 x 2 )} But for max-stable processes having unit Fréchet margins, E[Z(x)] = Var[Z(x)] = + Hence we need variogram-based approaches for max-stable random fields (6)

18 18/39 Let Z be a stationary max-stable random fields with unit Fréchet margins Let F(x) = exp( 1/z) The F-madogram [Cooley et al., 2006] is defined as Then we have ν F (x 1 x 2 ) = 1 2 E [ F {Z(x 1)} F {Z(x 2 )} ] (7) 2ν F (x 1 x 2 ) = θ(x 1 x 2 ) 1 θ(x 1 x 2 ) + 1 (8) or conversely θ(x 1 x 2 ) = 1 + 2ν F(x 1 x 2 ) 1 2ν F (x 1 x 2 ) (9) For the proof just note that x y = 2max{x,y} x y and you re set.

19 19/39 Outline Introduction Max-stable processes Spatial dependence of max-stable processes Inference for max-stable processes Model Selection Application References

20 20/39 A first statement It is almost impossible to get analytical CDF for Z( ) Indeed as Z(x) = max i ξ i Y + i (x), Y + i (x) = max{0,y i (x)} Hence to get the k-variate CDF we need to compute F(z 1,...,z k ) = Pr[max ξ i Y + i i (x j ) z j,j = 1,...,k] [ = Pr ξ i z ] j Y +, i,j = 1,...,k i (x j ) [ ( ) z j = exp I ξ > min R d R j y j + ξ 2 dξdp Y (y 1,...,y k ) [ = exp max R d j y + j z j dp Y (y 1,...,y k ) The number of possible cases becomes quickly intractable when k gets large Only the bivariate densities can easily be derived analytically ]

21 21/39 Bivariate CDF Smith [1990] j Pr[Z(x 1 ) z 1, Z(x 2 ) z 2 ] = exp 1 a Φ z a log z «2 1 a Φ z 1 z a log z «ff 1 z 2 where Φ is the standard normal CDF and a 2 = x T Σ 1 x. Schlather [2002] ( Pr[Z(x 1 ) z 1, Z(x 2 ) z 2 ] = exp «s!) z 1 z {ρ(h) + 1} 2 z 1 z 2 (z 1 + z 2 ) 2 where h = x 1 x 2. Geometric Gaussian and Brown Resnick processses The bivariate CDFs are the same as for the Smith model where Geom a 2 = 2σ 2 {1 ρ(h)} B. R. a 2 = γ(x 2 x 1 ), γ( ) is the variogram of ǫ( ) If γ(h) h 2, h 0 then B. R. Smith (isotropic).

22 22/39 Definition Composite likelihoods Let {f (y;θ),y Y,θ Θ} a parametric statistical model, where Y R K, Θ R p, K 1 and p 1. Consider a set of (marginal or conditional) events {A i : A i F,i I }, where I N and F is a σ-algebra on Y. A log-composite likelihood is defined as l c (θ;y) = i I w i log f (y A i ;θ) where f (y A i ;θ) = f ({y j Y : y j A i } ;θ), y = (y 1,...,y n ) and {w i,i I } is a set of suitable weights. In a nutshell, log-composite likelihood are just a linear combination of (smaller) log-likelihood entities

23 23/39 Why does it works? First, note that the full likelihood is a special case of composite likelihood For i being fixed, log f (y A i ;θ) is a valid log-likelihood Thus leading to an unbiased estimating equation log f (y A i ;θ) = 0 Finally l c (θ;y) = i I w i log f (y A i ;θ) = 0 is unbiased - as a linear combination of unbiased estimating equations For max-stable processes, as only the bivariate densities are known we will consider the pairwise likelihood l p (y;θ) = n log f (y (i) k,y(j) k ;θ) k=1 i<j

24 24/39 Asymptotics Instead of having n{ H(θ)} 1/2 d (ˆθ θ) N(0,Id p ), n + where H(θ) = E[ 2 l(θ;y)], (M 1/2 ) T M 1/2 = M 1 When we work under misspecification - which is the case when using composite likelihoods, we now have n{h(θ)j(θ) 1 H(θ)} 1/2 d (ˆθ θ) N(0,Id p ), n + where J(θ) = Var[ l(θ; Y)] Note that when the model is correctly specified, the so-called second Bartlett idendity holds i.e. E[ 2 l(θ;y)] + Var[ l(θ;y)] = 0 (10) In other words, we have H(θ) = J(θ) and H(θ)J(θ) 1 H(θ) = H(θ) 1 Beware I ll use this abuse of notation several times for shortness

25 25/39 Pairwise likelihood The log-pairwise likelihood is given by l p (θ;y) = n K 1 K k=1 i=1 j=i+1 log f (y (i) k,y(j) k ;θ) (11) Under the usual regularity conditions for the MLE and that θ is identifiable from the densities in (11), n{h(θ)j(θ) 1 H(θ)} 1/2 (ˆθ p θ) d N(0,Id p ), n + H(θ) and J(θ) can be consistently estimated by Ĥ(ˆθ p ) = Ĵ(ˆθ p ) = 1 n 1 n n 2 l p (ˆθ p ;y i ) i=1 n { l c (ˆθ p ;y i ) l c (ˆθ p ;y i ) T} i=1

26 From unit Fréchet margins to ordinary GEV ones Most of the max-stable processes are given with unit Fréchet margins Usually the margins will be GEV - not necessary identical Lets consider the following mapping ( t : y 1 + ξ y µ ) 1/ξ (12) σ If Y GEV(µ,σ,ξ), then Z = t(y ) Fréchet(1) Hence if Y ( ) is a max-stable process such that Y (x i ) GEV(µ i,σ i,ξ i ), the pairwise likelihood for max-stable processes defined with unit Fréchet margins becomes l p (θ;y) = where z (i) k n k=1 i<j { log f (z (i) k,z(j) k } ;θ) + log J{t(y(i) k )}J{t(y(j) k )} = t(y (i) k ) = {1 + ξ i(y (i) k µ i )/σ i } 1/ξ i and (13) J{t(y (i) k )} is the Jacobian of the transformation t 26/39

27 Thus one can define trend surfaces for the evolution of the GEV parameters in space i.e. and similarly for σ and ξ µ = X µ β µ (14) so that the parameters for our max-stable process have the form ψ = (θ,β µ,β σ,β ξ ), where θ is the parameter for the simple max-stable process - i.e. with unit Fréchet margins. The definition of trend surfaces allows 1. predictions of the GEV parameters, quantiles x K 2. correct estimates of the standard errors 2 2 the two step procedure i.e. first margins, then dependence leads to std. err. underestimations 27/39

28 28/39 Outline Introduction Max-stable processes Spatial dependence of max-stable processes Inference for max-stable processes Model Selection Application References

29 29/39 Information criteria When several models M0, M1,...are fitted to our data, one would prefer the one minimising AIC = 2l(ˆθ;y) + 2p where p is the number of parameters to be estimated. Under misspecification, one should use TIC = 2l(ˆθ;y) 2tr {Ĵ(ˆθ)Ĥ(ˆθ) 1} as the 2nd Bartlett idendity is not true anymore Note that if the model is correctly specified J(θ) = H(θ), E [ 2 l(θ;y) ] + Var [ l(θ;y)] = 0 so that TIC = 2l(ˆθ;y) + 2tr {I p } = AIC

30 30/39 Likelihood ratio When nested models are considered, it is always preferable to use the likelihood ratio statistic - cf. Neyman Pearson Let (ˆκ T, ˆφ T ) T be the MLE and (κ T 0, ˆφ T κ 0 ) T the MLE under the restriction κ = κ 0. Then { } W (κ 0 ) = 2 l(ˆκ, ˆφ) l(κ 0, ˆφ κ0 ) χ 2 p, n + (15) where p is the dimension of κ 0. Under mis-specification, this result differs { W (κ 0 ) = 2 l(ˆκ, ˆφ) l(κ 0, ˆφ } κ0 ) p λ i X i, n + i=1 where X i iid χ 2 1, λ i are the eigenvalues of (H 1 JH 1 ) κ { (H 1 ) κ } 1 and M κ is the submatrix of M corresponding to the elements of κ. (16)

31 31/39 The problem with (16) is that the limiting distribution is generally unknown. As an approximation, Rotnitski and Jewell suggest to treat pw (κ 0 )/ p i=1 λ i as a χ 2 p ] E Var [ pw (κ0 ) p i=1 λ i [ ] pw (κ0 ) p i=1 λ i p, n + 2p, n + Results based on quadratic forms for Normal random variables might be more accurate Another approach consists in adjusting the likelihood so that the usual convergence to a χ 2 p is met

32 32/39 Lets consider the following adjustment of the log-likelihood l A (θ) = l(θ ), θ = ˆθ + C(θ ˆθ) (17) The following adjustment has to interesting properties 1. ˆθ maximises l A 2. At ˆθ, l A has curvature C T H(θ 0 )C To have the usual χ 2 p convergence, we need that l A has curvature H(θ 0 )J(θ 0 ) 1 H(θ 0 ) at ˆθ If we choose C = {H(θ 0 ) 1/2 } 1 {H(θ 0 )J(θ 0 ) 1 H(θ 0 )} 1/2 (18) then C T H(θ 0 )C = H(θ 0 )J(θ 0 ) 1 H(θ 0 ) and W A (κ 0 ) = 2 Note the presence of ˆφ κ0,a! { } l A (ˆκ, ˆφ) l A (κ 0, ˆφ κ0,a) χ 2 p, n +

33 33/39 Outline Introduction Max-stable processes Spatial dependence of max-stable processes Inference for max-stable processes Model Selection Application References

34 34/39 Extreme precipitations around Zurich Elevation (meters) 35 stations (47 obs./stations) kept for validation κ = R 2 10 km Zurich Smith Covariance σ 11 σ 12 σ 22 h h + DoF l p TIC Isotropic 259 (45) 0 ( ) σ 22 = σ Anisotropic 251 (46) 64 (13) 290 (50) Schlather Covariance λ κ h h + DoF l p TIC Whittle 38.1 (10.7) 0.45 ( ) Cauchy 8.0 ( 2.2) 0.34 (0.16) Stable 34.8 (11.5) 0.95 (0.16) Exponential 34.1 ( 9.0) Bessel 0.95 (0.11) 150 ( ) Geometric Covariance σ 2 λ κ h h + DoF l p TIC Whittle 5.77 (8.64) (456.3) 0.40 ( ) Cauchy 36.0 (673) 5.19 (1.54) 9e 3 (0.16) Stable 5.74 (14.9) 237 (961) 0.79 (0.17) Exponential 2.42 (0.93) 53.2 (18.4) Bessel 1.28 (0.30) 0.61 (0.03) 150 ( ) 7.7 never h α Brown Resnick: γ(h) = λ Covariance λ α h h + DoF l p TIC (3.43) 0.74 (0.07)

35 35/39 Schlather Geometric Brown Resnick θ(h) Whittle Cauchy θ(h) h Whittle Cauchy h θ(h) Brown Resnick Smith h Figure: Comparison between the empirical extremal coefficients and the fitted extremal coefficient functions. Grey points : fitting data set. Black points : Validation data set. Although there seems to be a slight underestimation of the dependence, the fits look reasonably well The empirical estimates suggest the use of a nugget effect that has no theoretical meaning...

36 36/39 Comments on the Geometric Gaussian model θ(h) λ = 200 λ = 400 λ = 600 λ = 800 λ = h σ Figure: Impact of fixing the scale parameter λ in the Geometric gaussian model. The scale λ and variance σ 2 are far from being orthogonal - identifiability problem? Remember that σ 2 drives the upper bound of θ(h). Our application study does not cover the full range of dependence, difficulties to estimate both λ and σ 2 The safest strategy seems to fix λ and estimate σ 2 and hence the upper bound of θ(h). λ

37 Observed Observed Observed Model Model Model Observed Observed Observed Model Model Model Observed Observed Observed Model Model Model Figure: Model checking for the geometric models having a Whittle Matérn correlation function. Top row compares pairwise maxima estimated from the model and the observed ones for pairs of stations separated by 7 km (left), 45 km (middle) and 83 km. The middle row compares the observed and predicted minima (left), mean (middle) and maxima (right) for a group of 5 stations chosen randomly. The bottom row compares the observed and predicted minima (left), mean (middle) and maxima (right) for all the stations kept for model validation. The 95% confidence envelopes are also reported. The plot is transformed to the unit Gumbel scale using the empirical distribution function for the observations and the integral transform for the model. 37/39

38 Smith Schlather Geometric Precipitation (cm) Zurich Zurich Zurich Figure: One realisation of the best Smith, Schlather and geometric gaussian max-stable models. Smith s model leads unrealistic realisation while the Schlather and Geometric ones seem more plausible. Note how the Schlather model leads to larger range effects - as θ(h) 1 + 1/2 Simulations from fitted models are available (but difficult) and the modelling of spatial extremes in a subset of our region is therefore possible 38/39

39 39/39 Outline Introduction Max-stable processes Spatial dependence of max-stable processes Inference for max-stable processes Model Selection Application References

40 39/39 Chandler, R. E. and Bate, S. (2007). Inference for clustered data using the independence loglikelihood. Biometrika, 94(1): Cooley, D., Naveau, P., and Poncet, P. (2006). Variograms for spatial max-stable random fields. In Springer, editor, Dependence in Probability and Statistics, volume 187, pages Springer, New York, lecture notes in statistics edition. Davison, A. (2003). Statistical Models. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press. de Haan, L. (1984). A spectral representation for max-stable processes. The Annals of Probability, 12(4): de Haan, L. and Pereira, T. (2006). Spatial extremes: Models for the stationary case. The Annals of Statistics, 34: Kent, J. (1982). Robust properties of likelihood ratio tests. Biometrika, 69: Naveau, P., Guillou, A., and Cooley, D. (2009). Modelling pairwise dependence of maxima in space. Biometrika, 96(1):1 17.

41 39/39 Padoan, S., Ribatet, M., and Sisson, S. (2009). Likelihood-based inference for max-stable processes. To appear in the Journal of the American Statistical Association (Theory & Methods). Ribatet, M. (2009). SpatialExtremes: Modelling Spatial Extremes. R package version Rotnitzky, A. and Jewell, N. (1990). Hypothesis testing of regression parameters in semiparametric generalized linear models for cluster correlated data. Biometrika, 77: Schlather, M. (2002). Models for stationary max-stable random fields. Extremes, 5(1): Schlather, M. and Tawn, J. (2003). A dependence measure for multivariate and spatial extremes: Properties and inference. Biometrika, 90(1): Smith, R. (1990). Max-stable processes and spatial extreme. Unpublished manuscript.