A Model for Extremeson a Regular Spatial Lattice

Dan CooleyGrant Weller

Department of StatisticsColorado State University

Support obtained from NSF grant DMS-0905315

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Outline

1. Motivating Data

2. Multivariate Extremes and the Angular Measure

3. A Big (i.e., High Dimensional) Extremes Model from SmallExtremes Models

4. Inference via a Decomposition

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Motivating Data

−130 −125 −120 −115 −110 −105

3035

4045

5055

longitude

latit

ude

010

2030

4050

60

Daily precipitation amounts from a regional climate modelfor August 31, 2012.

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Motivating Data

Characteristics of this data:• Data occurs on a regular lattice.

• Data is very high dimensional (∼ 2500 locations).

• We wish to work with daily data, and not use block (an-nual) maxima.

• Extremes occur on a small subregion of the spatial domain.

−130 −125 −120 −115 −110 −105

3035

4045

5055

longitude

latit

ude

010

2030

4050

60

Goal: Construct a threshold exceedance model which cap-tures the spatial tail dependence in this data.

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Max-stable Processes

Spatial extremes work has centered around max-stable pro-cesses.

Q: Are max-stable processes the right answer?A: Yes. Limiting distribution of pointwise maxima.

Max-stable process notes:

• process models (i.e., infinite dimension)

• have been used to model block-maximum data

• fitting: composite likelihoods (size limitations?)

5

Proposed Approach

• not a process model. Instead, a high-dimensional multi-variate model.

• designed to handle threshold exceedances.

• constructed from the perspective of multivariate regular-variation.

• describes tail dependence via the angular (spectral) mea-sure.

• overall spatial model is composed of smaller models.

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Multivariate Regular Variation

Idea: Joint tail behavior like a power function.

So What? Because it is defined in terms of tail behavior, itprovides a framework for describing the joint tail.

One definition: Let R = ‖Z‖ and W = ‖Z‖−1Z. Z is regu-lar varying if there exists a normalizing sequence {bn} whereP (b−1

n ‖Z‖ > r) ∼ 1/n, such that

nP(b−1n R > r,W ∈ B

)v→ r−αH(B)

where d is the dimension of Z, and where H is some proba-bility measure on the unit ‘ball’ Sd = {z ∈ Rd | ‖z‖ = 1}.

• left says “as points get big”

• measure on right is a product measure.

• angular measure H describes distribution of directions –completely describes dependence.

• definition requires a common tail behavior.

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Multivariate Regular Variation, in Practice

• it is standard practice to convert marginals to a commonand convenient distribution, similar to copula methods.

• as points get large, it is natural to think in terms of pseudopolar coordinates.

• goal is to model the angular (or spectral) measure H.

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Center-of-Mass Condition

We assume α = 1 and common marginals.

If we choose to use the L1 norm: ‖Z‖ = Z1 + . . .+ Zd, theseassumptions imply ∫

Sd

wiH(dw) = 1/d;

the “center-of-mass” condition.

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More about the Simplex Sd

Sd = {w ∈ [0,1]d | ‖w‖ = 1}

• actually of dimension (d− 1), often denoted Sd−1.

One can speak of lower-dimensional simplices. FollowingColes and Tawn (1991), let l ∈ {1, . . . , d} denote the (lower)dimension, and c = {v1, . . . , vl} be an index variable of sub-spaces of size l.

Then Sl,c = {w ∈ Sd | wi = 0, i /∈ c;wj > 0, j ∈ c}.

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A Big Model from Small Models

Toy Example:

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

Let i = 1, . . . , d denote the locations on a lattice.

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Defining Subregions

Using a ‘template’, define overlapping subregions Rj = {i :cell i ∈ subregion j} to cover spatial domain. Denote by Rthe set of subregions {R1, . . . , RJ}.

Template: ‘cross’

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1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

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1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

R1 = {1,2,6} R2 = {1,2,3,7} R13 = {8,12,13,14,18}

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Building a Big EV Model from Small EV Models

Fundamental Model AssumptionWe assume that H(Sl,c) = 0 if c /∈ R.

• That is, we assume that cells are asymptotically dependentonly if there is a subregion which contains them.

• Made for modeling purposes.

• Corresponds to the idea that extreme events have a limitedspatial extent and occur only on small subregions of thestudy region.

We call Sj the simplex that corresponds to Rj.

Furthermore, let us assume that H is differentiable on Sj, anddenote by hj the angular density on Sj .

Idea: build models for hj, combine to get an overall model.

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Building a Big EV Model from Small EV Models

Assume that for each subregion Rj, we have gj(w), a valid|Rj|-dimensional angular measure; that is, it meets the ‘center-of-mass’ conditions.

Assume hj = ajgj. Can we find weights aj, which yield a valid(overall) angular measure?

We need ∫Sd

wiH(dw) = 1/d.

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Building a Valid Overall Angular Measure

∫Sd

wiH(dw) =J∑

j=1

∫Sj

wihj(w)dw

=J∑

j=1

aj

∫Sj

wigj(w)dw

=J∑

j=1

ajI{i∈Rj}

∫Sj

wigj(w)dw

=J∑

j=1

ajI{i∈Rj}1

|Rj|.

Setting this equation equal to 1/d for all i yields a linearsystem

Γa = (1/d)1,

where Γ is the d×J matrix whose {i, j}th element is I{i∈Rj}1|Rj|

.

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Solving for Weights

Model: ‘cross’

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1 2 3 4 5

6 7 8 9 10

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16 17 18 19 20

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

Say we define subregions by centering the template at eachcell, creating 20 subregions.

Γ =

1/3 1/4 0 0 0 1/3 0 . . .

1/3 1/4 1/4 0 0 0 1/5 . . .

0 1/4 1/4 1/4 0 0 0 . . ....

Solving Γa = (1/d)1 for a yields strange results.

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Solving for Weights

Problem is due to edge effects. Specifically, it is that cells atcorners and edges appear in too few subregions.

Fix: Expand the study region and account for missing subre-gions...

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1 2 3 4 5

6 7 8 9 10

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Solving for Weights

Problem is due to edge effects. Specifically, it is that cells atcorners and edges appear in too few subregions.

Fix: Expand the study region and account for missing sub-regions...and add columns to Γ which correspond to the newsubregions (of dimension 1 in this model).

Γ =

1/3 1/4 0 0 0 1/3 0 . . . 1 1 0 . . .

1/3 1/4 1/4 0 0 0 1/5 . . . 0 0 1 . . .

0 1/4 1/4 1/4 0 0 0 . . . 0 0 0 1...

Now system is underconstrained. Γ is d× J.

A solution: Impose additional constraints such as: subregionswith equal coverage get equal weights. Procedure: use SVDto find null space of Γ . . .

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Solving for Weights

SVD procedure not really necessary, as setting aj =|Rj|Dd

whereD is the dimension of the template is a solution the linearsystem Γa = (1/d)1 yielding equal weights.

Further interpretation of aj:Let ARj = {z ∈ Rd | ‖z‖ > r, z‖z‖−1 ∈ Rj}, andAr = {z ∈ Rd | ‖z‖ > r}. Then,

P(b−1n Z ∈ ARj | b

−1n Z ∈ Ar

)=

nP(b−1n Z ∈ ARj

)nP (b−1

n Z ∈ Ar)

→r−1H(Sj)

r−1

= aj.

That is, probability a large point occurs in region Rj is aj.

It is easily shown that∑Jj=1 aj = 1.

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So Far ....

We can take angular measure models on the lower-dimensionalsimplices which correspond to our pre-defined subregions andcombine them to obtain a valid overall angular measure model.

How do you fit the model?

Tasks:

1. How do you define a threshold exceedance and extractthem?

2. Estimating model parameters of gj. We will make a sta-tionary assumption; assume dependence parameters donot change with location.

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Model Fitting

Angular measure models have been fit via point-process basedlikelihood methods applied to threshold exceedances.

• Dirichlet of Coles and Tawn (1991)

• Pairwise Beta of Cooley et al. (2010)

• Models of Ballani and Schlather (2011)

Problem: Model assumes that limiting angular measure onlyhas mass on the lower-dimensional simplices–that is, angularcomponents have many elements that are exactly zero. Ourobserved data, when ‘angularized’ (i.e., ‖Z‖−1Z) will not havecomponents that are exactly zero.

That is, for finite sample sizes, the higher-order terms haven’tcompletely disappeared.

Proposed Solution: Account/penalize for the higher orderterms.

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Decomposition of Z

Theorem: (Weller and Cooley, 2012) Assume Z is regularlyvarying on cone C with limiting measure ν with correspondingangular measure H. Further assume, ∃ C0 ⊂ C s.t. ν(C0) = 0and Z exhibits hidden regular variation (Maulik and Resnick, 2004) onC0. Then there exists independent random vectors Y and Es.t. Z is tail equivalent to Y +E on both C and C0.

Idea: Assume Y has angular measure H, and that E is the‘error’ term which is regular varying (with lighter tail) andwhich accounts for the difference between the large observa-tions of a finite sample and the asymptotic limiting behavior.

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Example in two dimensions

0 500 1000 1500

0500

1000

1500

Y

Y1

Y2 +

0 500 1000 1500

0500

1000

1500

E

E1

E2 =

0 500 1000 1500

0500

1000

1500

Y +E

Y1 +E1

Y2+E2

No point has component exactly equal to zero.

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Inference for Z = Y +E

If Y and E were observed, we could use a point-processlikelihood approach applied to the large values of each.

Of course, we only observe Z.

Work in progress: A Monte-Carlo EM for obtaining MLE’sfor a likelihood which is valid only for the large values.

L(y, e; θ) =

J∏j=1

{1

mj!exp

[−aj(uY

bn

)−1]

×mj∏k=1

ajgj

(y(j,k)

‖y(j,k)‖; θ

)(‖y(j,k)‖bn

)−2}

×d∏

i=1

exp(−uE

cn

)mi!

mi∏k=1

(‖e(i,k)‖cn

)−3

Temporary solution: Assign Y to subregion with largest normand set these elements of E to zero.

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Example: Smith (1990) Fields

• 20× 20

• max-stable fields with unit Frechet marginals

• regular-varying fields with α = 1 and common marginals

5 10 15 20

510

1520

x

y

10

20

30

40

50

60

70

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Defining and Extracting Threshold Exceedances

5 10 15 20

510

1520

x

y

1000

2000

3000

4000

5 10 15 205

1015

20

x

y

1000

2000

3000

4000

5 10 15 20

510

1520

x

y

1000

2000

3000

4000

Original Field y e

‖z‖ = 46646 ‖y‖ = 17483 ‖e‖ = 29163

Note: Cross model likely doesn’t fit data well, as large valuesdo not appear to be contained by template.

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Model Fitting (Preliminary)

Cross Model (5 dimensional)

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In theory, we can choose any 5-dimensional angular measuremodel

• Dirichlet (Not spatial, Coles and Tawn (1991) )

• Pairwise Beta (Not spatial, Cooley et al. (2010))

• Smith (fit via an ad-hoc pairwise likelihood approach)

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Model Fitting (Preliminary)

Cross Model (5 dimensional)

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1.0

1.2

1.4

1.6

1.8

2.0

distance

thet

a

●

● ●

● ● ●

●

●

●

● ● ● ● ● ● ● ●

Dirichlet / Pairwise Beta / Smith

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Model Fitting (Preliminary)

Star Model (13-dimensional)

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9 10 11 12 13 14 15 16

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• Dirichlet

• Pairwise Beta

• Smith

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Model Fitting (Preliminary)

Star Model (13-dimensional)

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1.0

1.2

1.4

1.6

1.8

2.0

distance

theta

Dirichlet / Pairwise Beta / Smith

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Summary

• Proposed a finite-dimensional model for high-dimensionalspatial data.– Large extremes model constructed from small models.– Results in a valid angular-density model.– Makes an assumption of asymptotic independence.– Different approach from max-stable processes.– Designed to model spatial threshold exceedances.

• Model fitting/model selection.– Threshold exceedances via Y ’s radial component.– Models fit via a point-process likelihood.∗ Requires a flexible finite-dimensional angular measure

model at subregion level.∗ Possible EM algorithm for Y and E components.

– Model selection yet to be explored.∗ Possible likelihood-based with E serving as penalty?

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References

Ballani, F. and Schlather, M. (2011). A construction principle for multivariate extremevalue distributions. Biometrika, 98:633–645.

Coles, S. and Tawn, J. (1991). Modeling multivariate extreme events. Journal of theRoyal Statistical Society, Series B, 53:377–92.

Cooley, D., Davis, R. A., and Naveau, P. (2010). The pairwise beta: A flexible paramet-ric multivariate model for extremes. Journal of Multivariate Analysis, 101:2103–2117.

Maulik, K. and Resnick, S. (2004). Characterizations and examples of hidden regularvariation. Extremes, 7(1):31–67.

Smith, R. (1990). Max-stable processes and spatial extremes. Technical report, Uni-versity of North Carolina, Chapel Hill.

Weller, G. and Cooley, D. (2012). An alternative characterization of hidden regularvariation in joint tail modeling. Technical Report 2012/2, Colorado State University,Department of Statistics.

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