Excursion sets over high levels of non-Gaussianinfinitely divisible random fields: extreme values,

algebra, and geometry

Robert Adler, Gennady Samorodnitsky and Jonathan Taylor

June 2013

Let(X (t), t ∈ T

)be a random field indexed by some parameter

space T .

In many applications, including statistical hypothesis testing, one isinterested in, whether or not, the random field crosses a level u,often a high level u > 0.

Specifically: suppose we use a test statistic supt∈T X (t).

Let u > 0 be so large that, under the null hypothesis on theprobability law of the random field

(X (t), t ∈ T

), the level

crossing probability P(supt∈T X (t) > u

)is small.

Decision rule: reject the null hypothesis if supt∈T X (t) > u.

The excursion set

The excursion set of the random field(X (t), t ∈ T

)over the level

u is the random set

Au ={

t ∈ T : X (t) > u}.

The statisical decision rule supt∈T X (t) > u can be reformulated:

reject the null hypothesis if Au is non-empty.

More powerful statistical tests can be potentially constructedif one uses more delicate algebraic and/or geometricproperties of the excursion set Au other than it being emptyor not.

One of the very useful geometric characteristics of a set is its Eulercharacteristic.

For a “nice” set A the Euler characteristic χ(A) depends only thehomotopy type of the set A:

for a “nice” one-dimensional set A, χ(A) = number ofconnected components (intervals) in A;

for a “nice” two-dimensonal set A, χ(A) = number ofconnected components in A minus the number of “holes”;

for a three-dimensonal set A, χ(A) = number of connectedcomponents in A minus the number of “handles” plus thenumber of “holes’, etc.

Smooth Gaussian random fields

Let T be a “nice” manifold, and let(X (t), t ∈ T

)be a smooth

zero mean constant variance σ2 Gaussian random field on T .

Smoothness assumption includes existence of two continuouspartial derivatives plus other verifiable technical assumptions.

Consider the excursion set

Au ={

t ∈ T : X (t) ≥ u}.

Then an explicit non-asymptotic expression for the expected Eulercharacteristic of the excursion set exists.

For u > 0,

E(χ(Au)

)=

N∑j=0

Lj(T )ρj(u/σ)

(Adler and Taylor (2007)), where N is the dimension of themanifold, and for j ≥ 1,

ρj(u) = (2π)−(j+1)/2Hj−1(u)e−u2/2

with Hk being the kth Hermite polynomial.

Further,

ρ0(u) = Ψ(u) =

∫ ∞u

(2π)−1/2e−x2/2 dx

is the standard normal tail.

The coefficients Lj(T ), j = 0, . . . ,N are the so-calledLipschitz-Killing curvatures of T . They are independent of u.

In general, the coefficients(Lj(T )

)depend on the covariance

function of the random field. However: if the random field isisotropic, the coefficients depend only on the manifold T and thesecond spectral moment of the field.

Much effort is being put into computation of the Lipschitz-Killingcurvatures for various manifolds.

The expected Euler characteristic of the excursion set of a smoothGaussian randon field provides an excellent approximation to thelevel crossing probability P

(supt∈T X (t) > u

).

Let T be a “nice” manifold, and let(X (t), t ∈ T

)be a smooth

zero mean constant variance σ2 Gaussian random field on T . Thenthere is a number a > 0 such that for u > 0 large enough,∣∣∣P(sup

t∈TX (t) > u

)− E

(χ(Au)

)∣∣∣ ≤ e−(1+a)u2/2σ2

(Taylor, Takemura, Adler (2005)).

That is, the approximation is superexponentially good.

The reason the approximation of the level crossing probability bythe expected Euler characteristic of the excursion set is so good isthat, for smooth Gaussian random fields, exceedance of a highlevel u occurs locally:

the maximum of the field, supt∈T X (t), is achieved at aunique point, with probability 1;

if supt∈T X (t) > u, the excursion set Au is very likely to be a“nice ball-like” neighbourhood of the point where themaximum is achieved;

away from a neighborhood of that point the values of theprocess are very likely to be below u.

Therefore, the excursion set Au over a high level u, if non-empty, isvery likely to have a single connected component, without anyholes or other algebraic/geometric features, and its Eulercharacteristic is very likely to be equal to 1. In particular,

E(χ(Au)

)= E

(χ(Au)1

(Au 6= ∅

))≈ P

(Au 6= ∅

)= P

(supt∈T

X (t) > u),

and the approximation ≈ is very good.

This points to certain drawbacks of smooth Gaussian random fieldsas models in the situations where high level crossings areimportant.

The excursion set is nearly always of the same shape,algebraically the same as a ball.

Very precise information is available about the expected Eulercharacteristic of the excurson set, but not other features ofthe Euler characteristic (e.g. its variance).

The level crossing probabilities are very small for high levels u.

Non-Gaussian infinitely divisible random fields

In order to obtain a tractable class of models with a richer structureof high level excursion sets, we will look at non-Gaussian infinitelydivisible random fields. For simplicity of notation, in this talk wewill look at random fields on a unit cube,

(X (t), t ∈ [−1, 1]d

)and

study their excursion sets over a high level.

For technical reasons, we would like to have a random field definedon a neigborhood of the unit cube, and have “nice” propertiesthere.

Let G be an open set in Rd such that [−1, 1]d ⊂ G . We consideran infinitely divisible random field of the form

X (t) =

∫Sf (s; t)M(ds), t ∈ G ,

where

M is an infinitely divisible random measure on a measurablespace (S ,S);

f (·; t), t ∈ G is a family of nonrandom measurable functionson S satisfying certain integrability assumptions.

An infinitely divisible random measure is characterized by its Levymeasure. This is a σ-finite measure on S ×

(R \ {0}

)of the form

F (A) =

∫Sρ(s;As

)m(ds)

for a measurable A ⊂ S ×(R \ {0}

), where

As ={x ∈ R \ {0} : (s, x) ∈ A

}is the s-section of the set A;

m is a σ-finite measure on (S ,S), the so-called controlmeasure of the random measure M;(ρ(s; ·)

)is a measurable family of Levy measures on R, the

so-called local Levy measures.

We will assume that the local Levy measures of the infinitelydivisible random measure M possess the following regular variationproperty:

there exists a function H : (0,∞)→ (0,∞) that is regularlyvarying at infinity with exponent −α, α > 0, and nonnegativemeasurable functions w+ and w− on S such that

limu→∞

ρ(s; (u,∞)

)H(u)

= w+(s), limu→∞

ρ(s; (−∞,−u)

)H(u)

= w−(s).

In combination with certain integrability assumptions on the kernelf (·; t), t ∈ G , this means that for every t ∈ G ,

limu→∞

P(X (t) > u

)H(u)

= c+(t), limu→∞

P(X (t) < −u

)H(u)

= c−(t)

for some c+, c− ≥ 0 and, moreover,

limu→∞

P(supt∈[−1,1]d X (t) > u

)H(u)

= cmax,

with

cmax =

∫S

[w+(s) sup

t∈[−1,1]df (s, t)α++w−(s) sup

t∈[−1,1]df (s, t)α−

]m(ds).

Critical points

For a smooth non-Gaussian infinitely divisible random field we willstudy the Euler characterstic of its excursion Au by first studyingthe critical points of the random field over the level u.

A critical point of a smooth function f on Rd is a point wherethe gradient of f vanishes.

Critical points can be of different types, depending on theHessian matrix at the critical point.

Numbers of critical points of a function above a level arerelated to the Euler characteristic of the excursion set abovethe level through the Morse theory.

We need to take into account that the notion of a type of a criticalpoint differs from face to face of the cube [−1, 1]d .

Let Jk be the collection of faces of dimension k = 0, 1, . . . , d ofthe cube [−1, 1]d . For each face J ∈ Jk there is a correspondingset σ(J) ⊆ {1, 2, . . . , d} of cardinality k and a sequenceε(J) ∈ {−1, 1}{1,2,...,d}\σ(J) such that

J ={

t = (t1, . . . , td) ∈ [−1, 1]d : tj = εj for j /∈ σ(J)

and 0 < tj < 1 for j ∈ σ(J)}.

Let g be a “nice” C 2 function on an open set G containing thecube [−1, 1]d . For J ∈ Jk and i = 0, 1, . . . , k let Cg (J; i) be theset of points t ∈ J satisfying the following conditions.

∂g

∂tj= 0 for each j ∈ σ(J) ,

εj∂g

∂tj> 0 for each j /∈ σ(J) ,

the matrix

(∂2g

∂tm∂tn

)m,n∈σ(J)

is non-degenerate

and has i negative eigenvalues.

For u ∈ R, k = 0, 1, . . . , d , J ∈ Jk and i = 0, 1, . . . , k let

µg (J; i : u) = Card(Cg (J; i) ∩ {t : g(t) > u}

)be the numbers of the critical points of different types and ondifferent faces of the cube, of function g over the level u.

LetAu(g) =

{t ∈ [−1, 1]d : g(t) ≥ u

}be the excursion set of the function g over the level u. Then

χ(Au(g)

)=

d∑k=0

∑J∈Jk

k∑i=0

(−1)iµg (J; i : u) .

The goal: obtain the limiting conditional joint distribution of thenumbers of the critical points of different types and on differentfaces of the cube of a non-Gaussian infinitely divisible random fieldover a high level u given that the random field does exceed thatlevel.

This will allow us to compute, for example, the limiting conditionaldistribution of the Euler characteristic of the excursion set of thelevel u, given that the level is exceeded.

Knowing the full limiting conditional distribution we cancompute the conditional mean of the Euler characteristic,conditional variance, etc.

The approach: the assumptions on the random field, particularlythe assumption of regular variation, imply that, at high levels, theexcursions of the random field

X (t) =

∫Sf (s; t)M(ds), t ∈ [−1, 1]d

are similar to the excursions of a much simpler random field,

Z (t) = Xf (W , t), t ∈ [−1, 1]d ,

where (W ,X ) ∈ S × (R \ {0}) is a random pair whose law is thefinite restriction of the Levy measure F to the set{

(s, x) ∈ S × (R \ {0}) : supt∈G|xf (s; t)| > 1

},

normalized to be a probability measure on that set.

Main theorem

For k = 0, 1, . . . , d , a face J ∈ Jk and i = 0, 1, . . . , k letci (J; s) = µf (s;·)(J; i) be the number of the critical points ofthe s-section of f of the appropriate type.

Let(tl(J; i ; s), l = 1, . . . , ci (J; s)

)be an enumeration of these

critical points.

Let(f(J;i :+)[m] (s), m = 1, 2, . . .

)be the mth largest of the

positive parts(f (s; tl(J; i ; s))

)+, l = 1, . . . , ci (J; s);

let(f(J;i :−)[m] (s), m = 1, 2, . . .

)be the mth largest of the

negative parts.

Theorem Assume that an infinitely divisible random field(X (t), t ∈ [−1, 1]d

)is smooth, satisfies the assumption of regular

variation and additional technical assumtions. Then for anynumbersn(J; i) = 0, 1, 2, . . . , J ∈ Jk , k = 0, 1, . . . , d , i = 0, 1, . . . , k

limu→∞

P(µX (J; i : u) ≥ n(J; i), J ∈ Jk ,

k = 0, 1, . . . , d , i = 0, 1, . . . , k∣∣∣ sup

t∈[−1,1]dX (t) > u

)

=

∫S

[w+(s)

(minJ,i f

(J;i :+)[n(J;i)] (s)

)α+ w−(s)

(minJ,i f

(J;i :−)[n(J;i)] (s)

)α]m(ds)∫

S

[w+(s) supt∈[−1,1]d f (s, t)α+ + w−(s) supt∈[−1,1]d f (s, t)α−

]m(ds)

Moving average random fields

A moving average random field is the random field

X (t) =

∫Rd

g(s + t)M(ds), t ∈ G ,

where the control measure of the infinitely divisible randommeasure M is the d-dimensional Lebesgue measure, and the localLevy measure ρ(s, ·) = ρ(·) is independent of s ∈ Rd .

The measure ρ(·) is assumed to be regularly varying.

Choosing the kernel g of different shapes, one can obtain verydifferent algebraic/geometrical properties of the high levelexcursion sets.

Example 1. Let g(t) = exp{−‖t‖2/2}, t ∈ Rd . Because of the

rotational invariance of this kernel and its radial monotonicity, thehigh level excursion set Au, if not empty, looks, algebraically, “likea ball” and, hence, Euler characteristic equal to 1, as in the case ofsmooth Gassian random fields.

Example 2. Let us take d = 1 and

g(t) =(1 + cos γt

)e−t

2/2, t ∈ R .

In this case the high level excursion set will have a random numberof “holes”, and the Euler characteristic a non-degenerateconditional distribution.