Asymptotics for the Spatial Extremogram

Zagreb June 5, 2014

Asymptotics for the Spatial Extremogram

Richard A. Davis*Columbia University

Collaborators:Yongbum Cho, Columbia UniversitySouvik Ghosh, LinkedInClaudia Klüppelberg, Technical University of MunichChristina Steinkohl, Technical University of Munich

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Zagreb June 5, 2014

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

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Setup: Let be a stationary (isotropic?) spatial process defined on (or on a

regular lattice ).

Extremogram in Space

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68

1012

14

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1012

1412345

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Extremal Dependence in Space and Time

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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.

Illustration with French Precipitation Data


Desiderata for Spatial Models in Extremes: {Z(s), s D}

1. Model is specified on a continuous domain D (not on a lattice).

2. Process is max-stable: assumption may be physically meaningful,

but is it necessary? (i.e., max daily rainfall at various sites.)

3. Model parameters are interpretable: desirable for any model!

4. Estimation procedures are available: don’t have to be most efficient.

5. Finite dimensional distributions (beyond second order) are

computable: this allows for computation of various measures of

extremal dependence as well as predictive distributions at

unobserved locations.

6. Model is flexible and capable of modeling a wide range of extremal

dependencies.

Warm-up


Building blocks: Let be a stationary Gaussian process on with mean 0

and variance 1.

• Transform the processes via

where F is the standard normal cdf. Then has unit Frechet marginals,

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!

Building a Max-Stable Model in Space


Now assume is isotropic with covariance function

where > 0 is the range parameter and (0,2] is the shape parameter.

Note that

and hence

where .

It follows that

Building a Max-Stable Model in Space-Time


Then,

on Here the are IID replicates of the GP and is 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



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)


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Scaled and transformed Gaussian random fields (fixed time point)

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−1log ¿¿

−1log ¿¿

Zagreb June 5, 2014

Scaled and transformed Gaussian random fields (fixed time point)

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𝜂𝑛 (𝑠)≔ 1𝑛 ¿ 𝑗=1¿𝑛− 1

log (Φ (𝑍 𝑗 (𝑠𝑛𝑠 )))


Setup: Let be a RV stationary (isotropic?) spatial process defined on (or

on a regular lattice ). Consider the former—latter is more straightforward.

Lattice vs cont space

x

y

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regular grid random pattern


Regular or random

France Precipitation Data

0.5 1.0 1.5

0.5

1.0

1.5

distance

L(t)

K-Function

5500 6000 6500 7000 7500 8000

1850

019

000

1950

020

000

x

y

Site Locations

regular or random?


Regular grid

regular grid

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h = 1; # of pairs = 4h = sqrt(2); # of pairs = 4h = 2; # of pairs = 4

h = sqrt(10); # of pairs = 8 h = sqrt(18); # of pairs = 4


Random pattern

random pattern

h = 1; # of pairs = 0h = 1 .25

x

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x

y

0 1 2 3 4 5

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89

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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.


Random pattern

random pattern

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

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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


Estimating extremogram--random pattern

Kernel estimate of :

Note: ) is product measure off the diagonal.


Limit Theory

Theorem: Under suitable conditions on (i.e., regularly varying, mixing,

local uniform negligibility, etc.), then

where is the pre-asymptotic extremogram,

(|𝑆𝑛|𝜆𝑛2

𝑚𝑛)

12 ( �̂� 𝐴 ,𝐵 (h )−𝜌𝐴 ,𝐵 ,𝑚 (h ) )→𝑁 ( 0 , Σ ) ,

Remark: The formulation of this estimate and its proof follow the ideas

of Karr (1986) and Li, Genton, and Sherman (2008).

( is the quantile of ).


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


Limit Theory

=

=

where Making the change of variables and the expected value is

Numerator:

=|


Limit Theory

Remark: We used the following in this proof.

• and .

•

Need a condition for the latter.


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, .


Limit Theory

Outline of argument:

• Under LUNC already shown asymptotic unbiasedness of numerator

and denominator.

with .

Strategy: Show joint asymptotic normality of and


Limit Theory

Proof of (i): Sum of variances + sum of covariances

→𝜇 (𝐴)𝜈 +∫

ℝ 2

❑

𝜏 𝐴 ,𝐴 (𝑦 ) 𝑑𝑦

Step 1: Compute asymptotic variances and covariances.

i.

ii.


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.

Zagreb June 5, 2014

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Limit Theory

Subdivide into big blocks and small blocks.

where and size

Zagreb June 5, 2014

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Limit Theory

Subdivide into big blocks and small blocks.

where and size

𝐵

Insert block inside with dimension :

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Limit Theory

𝐷𝑖𝐵𝑖

Recall that is a (mean-corrected) double integral over ~𝐴𝑛= ∫𝑆𝑛×𝑆𝑛

𝑤𝑛 (h+𝑠1−𝑠2 )𝐻 (𝑠1 , 𝑠2 )𝑁 (2 ) (𝑑 𝑠1 ,𝑑 𝑠2 )

¿∑𝑖=1

𝑘𝑛

∫𝐷𝑖×𝐷𝑖

𝑤𝑛(h+𝑠1−𝑠2 )𝐻 (𝑠1 ,𝑠2 )𝑁 ( 2) (𝑑𝑠1 ,𝑑 𝑠2)¿∑𝑖=1

𝑘𝑛

∫𝐵𝑖×𝐵 𝑖

𝑤𝑛 (h+𝑠1−𝑠2 )𝐻 (𝑠1 ,𝑠2 )𝑁 (2 ) (𝑑 𝑠1 ,𝑑 𝑠2 )

¿∑𝑖=1

𝑘𝑛~𝑎𝑛𝑖+

~𝐴𝑛−∑𝑖=1

𝑘𝑛

~𝑎𝑛𝑖

Zagreb June 5, 2014

Limit Theory

Remaining steps:

~𝑎𝑛𝑖= ∫𝐵𝑖×𝐵𝑖

𝑤𝑛 (h+𝑠1−𝑠2)𝐻 (𝑠1 ,𝑠2 )𝑁 ( 2 ) (𝑑 𝑠1 ,𝑑 𝑠2 )i. Show

ii. Let be an iid sequence with whose sum has characteristic function Show

iii.

Intuition.

(i) The sets are small by proper choice of

(ii) Use a Lynapounov CLT (have a triangular array).

(iii) Use a Bernstein argument (see next page).

Zagreb June 5, 2014

Limit Theory

Useful identity:

¿𝜙𝑛 (𝑡 )−𝜙𝑛𝑐 (𝑡)∨¿∨𝐸∏

𝑖=1𝑒𝑖𝑡

~𝑎𝑛𝑖❑−𝐸∏

𝑖=1

𝑘

𝑒𝑖𝑡~𝑐𝑛𝑖∨¿¿

¿∨𝐸∑𝑖=1

𝑘𝑛

∏𝑗=1

𝑖−1

𝑒𝑖𝑡~𝑎𝑛 𝑗(𝑒𝑖𝑡~𝑎𝑛𝑖−𝑒𝑖𝑡~𝑐𝑛𝑖) ∏𝑗=𝑖+1

𝑘𝑛

𝑒𝑖𝑡~𝑐𝑛𝑖∨¿¿

(by indep of )

where is a strong mixing bounding function that is based on the

separation h between two sets U and V with cardinality r and s.

Zagreb June 5, 2014

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 wrt and respectively, and bded by 1, then

.

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Limit Theory

Mixing condition: for some

Returning to calculations:

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Simulations of spatial extremogram

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Extremogram for one realization of B-R process (function of level)

Note: black dots = true; blue bands are permutation bounds

Lattice Non-Lattice

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Max-moving average (MMA) process:

Let be an iid sequence of Frechet rvs and set

where

MMA(1):

Extremogram:

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Box-plots based on 1000 (100) replications of MMA(1) (left) and BR (right)

Lattice

Non-lattice; (left) (right)

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Space-Time Extremogram

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Extremogram.

For the special case, ,

For the Brown-Resnick process described earlier we find that) and

)

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Inference for Brown-Resnick Process

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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

ilk iijDttlk Dssji

ljki tstsfPL|:|),( |:|),(

),);,(),,((log),(

Zagreb June 5, 2014

Inference for Brown-Resnick Process

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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

ilk iijDttlk Dssji

ljki tstsfPL|:|),( |:|),(

),);,(),,((log),(

Zagreb June 5, 2014

Data Example: extreme rainfall in Florida

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Zagreb June 5, 2014


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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.

Zagreb June 5, 2014


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Hourly accumulated rainfall time series in the wet season 2002 at 2 locations.

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Hourly accumulated rainfall fields for four time points.

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Empirical extremogram in space (left) and time (right)

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Empirical extremogram in space (left) and time (right): spatial indep for lags > 4; temporal indep for lags > 6.

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Empirical extremogram in space (left) and time (right)

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Computing conditional return maps.Estimate such that

where satisfies is pre-assigned.A straightforward calculation shows that must solve,


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.


For dependent data, it is often infeasible to compute the exact likelihood

based on some model. An alternative is to combine likelihoods based on

subsets of the data.

To fix ideas, consider the following data/model setup:

(Here we have already assumed that the data has been transformed to a

stationary process with unit Frechet marginals.)

Data: Y(s1), …, Y(sN) (field sampled at locations s1, …, sN )

Model: max-stable model defined via the limit process

maxj=1,…,nYn(j)(s) →d X(s),

• Yn(s) = Y(s /(log n)1/b)) = 1/log(F(Z(s/(log n)1/b))

• Z(s) is a GP with correlation function r(|s-t|) = exp{-|s-t|b/f}

Estimation—composite likelihood approach


Bivariate likelihood: For two locations si and sj, denote the pairwise

likelihood by

f(y(si), y(sj); di,j ) = ∂2/(∂x∂y) F(Y(si) ≤ x, Y (sj) ≤ x)

where F is the CDF

F(Y(si) ≤ x, Y (sj) ≤ x)

= exp{-(x-1F(log(y/x)/(2d) + d) + y-1F(log(x/y)/(2d) + d))},

and di,j |si – sj|b/f is a function of the parameters b and f.

Pairwise log-likelihood:


N

jii

N

jjiji sysyfPL

1 1, ));(),((log),( dbf


Potential drawbacks in using all pairs:

• Still may be computationally intense with N2 terms in sum.

• Lack of consistency (especially if the process has long memory)

• Can experience huge loss in efficiency.

Remedy? Use only nearest neighbors in computing the pairwise

likelihood terms.

1. Use only neighbors within distance L. The new criterion becomes

2. Use only nearest K neighbors. If Di denotes the distance of kth

nearest neighbor to si , then criterion is


N

i Lssjjiji

ij

sysyfPL1 |:|

, ));(),((log),( dbf

N

i Dssjjiji

iij

sysyfPL1 |:|

, ));(),((log),( dbf


Simulation setup:• Simulate 1600 points of a spatial (max-stable) process Y(s) on a

grid of 40x40 in the plane.• Choose a distance L or number of neighbors K (usually 9, 15, 25)• Maximize

with respect to f and b• Calculate summary dependence statistics: r(l; |s-t|) = l-1(1 – l(1l)V(l, 1l; d)), where V(l,1-l; d) = l-1F(log ((1-l)/l) /(2d)+d) + (1-l)-1F(log (l/(1-l))/(2d)+d) d = |s-t|b/f

Simulation Examples

N

i Dssjjiji

iij

sysyfPL1 |:|

, ));(),((log),( dbf


Process: Limit process d = |s-t|, 1 , .2; 9 nearest neighbors

Simulation Examples


Simulation Examples

Process: Limit process d = |s-t|, 1 , .2; 25 nearest neighbors


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



Estimated spatial correlation function before transformation to unit Frechet. Correlation function


0 200 400 600 800

0.0

0.2

0.4

0.6

0.8

1.0

lag

cor

0 200 400 600 800

0.0

0.2

0.4

0.6

0.8

1.0

lag

cor


After transforming the data to unit Frechet marginals, we estimated and using pairwise likelihood (nearest neighbors with K = 25). 1.143; f 185.1 (if constrained to 1, then f 89.2)

Correlation function


0 200 400 600 800

0.0

0.2

0.4

0.6

0.8

1.0

lag

cor

0 200 400 600 800

0.0

0.2

0.4

0.6

0.8

1.0

lag

cor

0 200 400 600 800

0.0

0.2

0.4

0.6

0.8

1.0

lag

cor

Green: 1.143 ; f

185.1

Blue: 1 ; f 89.2

Black: 1 ; f 139.6

(MLE)

Red: 1.17 ; f 147.8

(Matern)

(MLE based on GP likelihood)

0 200 400 600 800

0.0

0.2

0.4

0.6

0.8

1.0

lag

cor


Plot of r(l; |s-t|) = limn→∞ P(Yn(s) > n(1-l) | Yn(t) > nl)



Plot of r(l; |s-t|) = limn→∞ P(Yn(s) > n(1-l) | Yn(t) > nl)




Results from random permutations of data—just for fun. Since random permutations have no spatial dependence, r should be 0 for |s-t| >0.

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Asymptotics for the Spatial Extremogram