Asymptotics for the Spatial Extremogram

1. Richard A. Davis* Columbia University Asymptotics for the Spatial Extremogram Collaborators: Yongbum Cho, Columbia University Souvik Ghosh, LinkedIn Claudia Klüppelberg, Technical University of Munich Christina Steinkohl, Technical University of Munich

2. Plan • Warm-up • Desiderata for spatial extreme models • The Brown-Resnick Model • Limits of Local Gaussian processes • Extremogram • Estimation—pairwise likelihood • Simulation examples • Estimating the extremogram • Asymptotics in random location case • Composite likelihood • Simulation examples • Two examples

3. Extremogram in Space Setup: Let be a stationary (isotropic?) spatial process defined on (or on a regular lattice ).

4. Extremal Dependence in Space and Time

5. Illustration with French Precipitation Data Data from Naveau et al. (2009). Precipitation in Bourgogne of France; 51 year maxima of daily precipitation. Data has been adjusted for seasonality and orographic effects.

6. Warm-up Desiderata for Spatial Models in Extremes: {Z(s), s  D} Model is specified on a continuous domain D (not on a lattice). Process is max-stable: assumption may be physically meaningful, but is it necessary? (i.e., max daily rainfall at various sites.) Model parameters are interpretable: desirable for any model! Estimation procedures are available: don’t have to be most efficient. Finite dimensional distributions (beyond second order) are computable: this allows for computation of various measures of extremaldependence as well as predictive distributions at unobserved locations. Model is flexibleand capable of modeling a wide range of extremal dependencies.

7. Building a Max-Stable Model in Space • Building blocks: Let be a stationary Gaussian process on with mean 0 and variance 1. • Transform the processes viawhereFis the standard normal cdf. Then has unit Frechetmarginals, i.e., • Note: For any (nondegenerate) Gaussian process we have • and hence • In other words, • observations at distinct locations are asymptotically independent. • not good news for modeling spatial extremes!

8. Building a Max-Stable Model in Space-Time Now assume is isotropic with covariance function where > 0is the range parameter and (0,2]isthe shape parameter. Note that and hence where . It follows that

9. Then, • onHere the are IID replicates of the GP andis a Brown-Resnick max-stable process. • Representation for : • where • of a Poisson process with intensity measure • (are iid Gaussian processes with mean zero and variogram Building a Max-Stable Model in Space-Time

10. Then, • on • Bivariate distribution function: • where • and Note • V(x,y;) → x-1 + y-1 as →∞ (asymptotic indep) • V(x,y;) →1/min(x,y) as → 0 (asymptotic dep) Building a Max-Stable Model in Space-Time

11. Scaled and transformed Gaussian random fields (fixed time point)

12. Scaled and transformed Gaussian random fields (fixed time point)

13. Lattice vscont space Setup: Let be a RV stationary (isotropic?) spatial process defined on (or on a regular lattice ). Consider the former—latter is more straightforward. regular grid random pattern

14. France Precipitation Data regular or random? Regular or random

15. Regular grid regular grid h = 1; # of pairs = 4 h = sqrt(10); # of pairs = 8 h = sqrt(18); # of pairs = 4 h = sqrt(2); # of pairs = 4 h = 2; # of pairs = 4

16. Random pattern random pattern h = 1; # of pairs = 0 h = 1 .25

17. Random pattern Zoom-in buffer Estimate of extremogram at lag h = 1 for red point: weight “indicators of points” in the buffer. Bandwidth: half the width of the buffer.

18. Random pattern random pattern • Note: • Expanding domain asymptotics: domain is getting bigger. • Not infill asymptotics: insert more points in fixed domain.

19. Estimating the extremogram--random pattern Setup: Suppose we have observations, at locations of some Poisson process with rate in a domain . Here, number of Poisson points in Weight function : Let be a bounded pdf and set where the bandwidth

20. Estimating extremogram--random pattern Kernel estimate of : Note: )is product measure off the diagonal.

21. Limit Theory Theorem: Under suitable conditions on (i.e., regularly varying, mixing, local uniform negligibility, etc.), then where is the pre-asymptotic extremogram, ( is the quantile of ). Remark: The formulation of this estimate and its proof follow the ideas of Karr (1986) and Li, Genton, and Sherman (2008).

22. Limit theory Asymptotic “unbiasedness”: is a ratio of two terms; will show that both are asymptotically unbiased. Denominator: By RV, stationarity, and independence of and

23. Limit Theory Numerator: = = where Making the change of variables and the expected value is =|

24. Limit Theory • Remark: We used the following in this proof. • and . • Need a condition for the latter.

25. Limit theory Local uniform negligibility condition (LUNC): For any , there exists a such that Proposition: If is a strictly stationary regularly varying random field satisfying LUNC, then for This result generalizes to space points, .

26. Limit Theory • Outline of argument: • Under LUNCalready shown asymptotic unbiasedness of numerator and denominator. • with . • Strategy: Show joint asymptotic normality of and

27. Limit Theory Step 1: Compute asymptotic variances and covariances. Proof of (i):Sum of variances + sum of covariances

28. Limit Theory Step 2: Show joint CLT for and using a blocking argument. Idea: Focus on Set and put We will show is asymptotically normal.

29. Limit Theory Subdivide into big blocks and small blocks. where and size

30. Limit Theory Insert block inside with dimension : Subdivide into big blocks and small blocks. where and size

31. Limit Theory Recall that is a (mean-corrected) double integral over

32. Limit Theory Remaining steps: Show Let be an iid sequence with whose sum has characteristic function Show Intuition. The sets are small by proper choice of Use a Lynapounov CLT (have a triangular array). Use a Bernstein argument (see next page).

33. Limit Theory Useful identity: (by indep of ) whereis a strong mixing bounding function that is based on the separationhbetween two sets UandVwith cardinality rands.

34. Strong mixing coefficients Strong mixing coefficients: Let be a stationary random field on Then the mixing coefficients are defined by where the sup is taking over all sets with and Proposition (Li, Genton, Sherman (2008), Ibragimov and Linnik (1971)): Let and be closed and connected sets such that and and are rvs measurable wrtand respectively, and bded by 1, then .

35. Limit Theory Mixing condition:for some Returning to calculations:

36. Simulations of spatial extremogram Extremogram for one realization of B-R process (function of level) Note:black dots = true; blue bands are permutation bounds Lattice Non-Lattice

37. Simulations of spatial extremogram Max-moving average (MMA) process: Let be an iid sequence of Frechetrvs and set where MMA(1): Extremogram:

38. Simulations of spatial extremogram Box-plots based on 1000 (100) replications of MMA(1) (left) and BR (right) Lattice Non-lattice; (left) (right)

39. Space-Time Extremogram Extremogram. For the special case, , For the Brown-Resnick process described earlierwe find that )and )

40. Inference for Brown-Resnick Process Semi-parametric: Use nonparametric estimates of the extremogram and then regress function of extremogram on the lag. Regress: , The intercepts and slopes become the respective estimates of and Asymptotic properties of spatial extremogram derived by Cho et al. (2013). Pairwise likelihood: Maximize the pairwise likelihood given by

41. Inference for Brown-Resnick Process Semi-parametric: Use nonparametric estimates of the extremogram and then regress function of extremogram on the lag. Regress:) ) , The intercepts and slopes become the respective estimates of and Asymptotic properties of spatial extremogram derived by Cho et al. (2013). Pairwise likelihood: Maximize the pairwise likelihood given by

42. Data Example: extreme rainfall in Florida

43. Data Example: extreme rainfall in Florida Radar data: Rainfall in inches measured in 15-minutes intervals at points of a spatial 2x2km grid. Region: 120x120km, results in 60x60=3600 measurement points in space. Take only wet season (June-September). Block maxima in space: Subdivide in 10x10km squares, take maxima of rainfall over 25 locations in each square. This results in 12x12=144 spatial maxima. Temporal domain: Analyze daily maxima and hourly accumulated rainfall observations. Fit our extremal space-time model to daily/hourly maxima.

44. Data Example: extreme rainfall in Florida Hourly accumulated rainfall time series in the wet season 2002 at 2 locations.

45. Data Example: extreme rainfall in Florida Hourly accumulated rainfall fields for four time points.

46. Data Example: extreme rainfall in Florida Empirical extremogram in space (left) and time (right)

47. Data Example: extreme rainfall in Florida Empirical extremogram in space (left) and time (right): spatial indep for lags > 4; temporal indep for lags > 6.

48. Data Example: extreme rainfall in Florida Empirical extremogram in space (left) and time (right)

49. Data Example: extreme rainfall in Florida Computing conditional return maps. Estimate such that wheresatisfiesis pre-assigned. A straightforward calculation shows that must solve,

50. 100-hour return maps (): time lags = 0,2,4,6 hours (left to right on top and then right to left on bottom), quantiles in inches.