Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
Non asymptotic bounds for the distribution of
the maximum of Random fields
12 Janvier 2009
Jean-Marc A ZA ÏS
Institut de Mathématiques, Université de Toulouse
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
1 / 21
Examples
The record method
The maxima method
1
Examples
2
The record method
3
The maxima method
Second order
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
2 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
Signal + noise model
Spatial Statistics often uses “ signal + noise model”, for example :
precision agriculture
neuro-sciences
sea-waves modelling
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
3 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
Signal + noise model
Spatial Statistics often uses “ signal + noise model”, for example :
precision agriculture
neuro-sciences
sea-waves modelling
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
3 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
Signal + noise model
Spatial Statistics often uses “ signal + noise model”, for example :
precision agriculture
neuro-sciences
sea-waves modelling
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
3 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
Precision agriculture
Representation of the yield per unit by GPS harvester .
Is there only noise or some region with higher fertility ? ?
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
4 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
Neuroscience
The activity of the brain is recorded under some particular action and
the same question is asked
source : Maureen CLERC
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
5 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
Sea-waves spectrum
Locally in time and frequency the spectrum of waves is registered.
We want to localize transition periods.
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
6 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
In all these situations a good statistics consists in observing the
maximum of the (absolute value) of the random field for deciding if it
is typically large (Noise) of not (signal).
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
7 / 21
Examples
The record method
The maxima method
1
Examples
2
The record method
3
The maxima method
Second order
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
Non asymptotic bounds for the distribution of the maximum of Random fields
The record method
8 / 21
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
The record method
The maxima method
The record method
Hypothesis
S is a regular set of R2 (compact, simply connected + piecewise C 1
parametrization of the boundary by arc length) X is such that
∂X
-the bivariate process Z = (X, ∂t
) has C 1 sample paths and non
2
degenerated Gaussian distribution.
∂X ∂ 2 X
- the distribution of (X(t), X 0 (t)), X(t), ∂t
, ∂t2 do not degenerate
1
2
Then Roughly speaking the event
{M > u}
is almost equal to the events
”The level curve at levelu is not empty”
”The point at the southern extremity of the level curve exists”
”There exists a point on the level curve :
∂X
∂X
= 0; X01 (t) = ∂t
>0
X(t) = u; X10 (t) = ∂t
1
2
X20 (t) =
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
∂2X
∂t12
< 0”
9 / 21
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
The record method
The maxima method
The record method
The computation of a probability is bounded by the expectation of the
number of roots of the process Z(t) − (u, 0) that can be computed by
means of a Rice formula. Reintroducing the boundary to get an exact
statement, we get
Z
P{M > u} ≤ P{Y(0) > u} +
L
E(|Y 0 (`)|)Y(`) = u)pY(`) (u)d`
0
Z
+
E(| det(Z 0 (t) 1IX20 (t)<0 1IX01 (t)>0 |X(t) = u, X01 (t) = 0)pX(t),X01 (t) (u, 0)dt,
S
Where Y(`) is the process X on the boundary and ` = 0 corresponds
to the southern extremity. The difficulty lies in the computation of the
expectation of the determinant
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
10 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
The record method
The key point is that under the condition {X(t) = u, X01 (t)} = 0, the
quantity
X10 X01 0
| det(Z (t)| = X11 X02 is simply equal to |X10 X02 | . Taking into account conditions, we get the
following expression for the second integral
Z
+ E(|X20 (t)− X01 (t)+ X(t) = u, X01 (t) = 0)pX(t),X01 (t) (u, 0)dt.
S
Moreover under stationarity or some more general hypotheses, these
two random variables are independent.
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
11 / 21
Examples
The record method
The maxima method
1
Examples
2
The record method
3
The maxima method
Second order
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
Non asymptotic bounds for the distribution of the maximum of Random fields
The maxima method
12 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
The maxima method
Consider a realization with M > u, then necessarily there exist a local
maxima or a border maxima above U
Border maxima : local maxima in relative topology
If the consider sets with polyedral shape union of manifold of
dimension 1 to d
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
13 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
The maxima method
In fact result are simpler (and stronger) in term of the density pM (x) of
the maximum. Bound for the distribution are obtained by integration.
Theorem
1
[p (x) + pEC
M (x)]with
2 M
Z
pM (x) := E | det(X”(t))|/X(t) = x, X 0 (t) = 0 pX(t),Xj0 (t) (x, 0)dt+boundary term
pM (x) ≤ b
pM (x) :=
S
and
pEC
M (x)
d
Z
:= (−1)
S
E det(X”(t))/X(t) = x, X 0 (t) = 0 pX(t),Xj0 (t) (x, 0)dt+boundary
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
14 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
The maxima method
Quantity pEC
M (x) is easy to compute using the work by Adler and
properties of symmetry of the order 4 tensor of variance of X” ( under
the conditional distribution) )
Lemma
E det(X”(t))/X(t) = x, X 0 (t) = 0 = det(Λ)Hd (x)
where Hd (x) is the dth Hermite polynomial and Λ := Var(X 0 (t))
main advantage of Euler characteristic method lies in this result.
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
15 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
The maxima method
computation of pm
The key point is the following
If X is stationary and isotropic with covariance ρ(ktk2 ) normalized by
Var(X(t)) = 1 et Var(X 0 (t)) = Id
Then under the condition X(t) = x, X 0 (t) = 0
p
p
X”(t) = 8ρ”G + ξ ρ” − ρ02 Id + xId
Where G is a GOE matrix (Gaussian Orthogonal Ensemble), and ξ a
standard normal independent variable. We use recent result on the
the characteristic polynomials of the GOE. Fyodorov(2004)
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
16 / 21
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
The record method
The maxima method
The maxima method
Theorem
Assume that the random field X is centered, Gaussian, stationary
and isotrpic and is “regular” Let S have polyhedral shape. Then,


d0
X
X
|ρ0 | j/2

Hj (x) + Rj (x) gj
σ
b0 (t) +
p(x) = ϕ(x)
(1)


π
t∈S0
j=1
R
gj = Sj θbj (t)σj (dt), σ
bj (t) is the normalized solid angle of the cone
of the extended outward directions at t in the normal space with
the convention σd (t) = 1.
For convex or other usual polyhedra σ
bj (t) is constant for t ∈ Sj ,
Hj is the j th(probabilistic) Hermite polynomial.
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
17 / 21
Non asymptotic bounds for the distribution of the maximum of Random fields
Examples
The record method
The maxima method
The maxima method
Theorem (continued)
Rj (x) =
2ρ00
π|ρ0 |
2j Γ((j+1)/2 R +∞
−∞
π
Tj (v) exp −
v := −(2)−1/2 (1 − γ 2 )1/2 y − γx
Tj (v) :=
X
j−1
k=0
In (v) = 2e
−v2 /2
y2
2
dy
with γ := |ρ0 |(ρ00 )−1/2 ,
Hj (v)
Hk2 (v) −v2 /2
e
− j
Ij−1 (v),
k
2 k!
2 (j − 1)!
(2)
(3)
[ n−1
2 ]
(n − 1)!!
Hn−1−2k (v)
(n − 1 − 2k)!!
k=0
√
n
+ 1I{n even} 2 2 (n − 1)!! 2π(1 − Φ(x))
X
2k
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
(4)
18 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
The maxima method
Second order
Second order study
Using an exact implicit formula
Theorem
Under conditions above + Var(X(t)) ≡ 1 Then
limx→+∞ −
2
log b
pM (x) − pM (x)
2
x
σt2
≥
1 + inf
t∈S
σt2
1
+ λ(t)κ2t
Var X(s)/X(t), X 0 (t)
:= sup
(1 − r(s, t))2
s∈S\{t}
and κt is some geometrical characteristic et Λt = GEV(Λ(t))
The right hand side is finite and > 1
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
19 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
The maxima method
Level Sets and Extrema
of Random Processes
and Fields
References
Jean-Marc Azaïs and Mario Wschebor
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
20 / 21
Examples
The record method
The maxima method
Non asymptotic bounds for the distribution of the maximum of Random fields
The maxima method
References
Adler R.J. and Taylor J. E. Random fields and
geometry. Springer.
Mercadier, C. (2006), Numerical Bounds for the
Distribution of the Maximum of Some One- and
Two-Parameter Gaussian Processes, Adv. in Appl.
Probab. 38, pp. 149–170.
Jean-Marc A ZA ÏS ( Institut de Mathématiques, Université de Toulouse )
21 / 21