The Extremogram in Space (and Time):
Richard A. Davis*
Columbia University
Collaborators:
Yongbum Cho, Columbia University
Souvik Ghosh, LinkedIn
Claudia Klüppelberg, Technical University of Munich
Christina Steinkohl, Technical University of Munich
* Support of the Villum Kann Rasmussen Foundation is gratefully
acknowledged.
Copenhagen May 27-30, 2013
1
Plan
 Extremogram in space
 lattice vs continuous space
 Estimating extremogram—random pattern
 Limit theory for empirical extremogram
 Simulation examples
Copenhagen May 27-30, 2013
2
1
Extremal Dependence in Space and Time
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Extremogram in Space
Setup: Let
∈
be a stationary (isotropic?) spatial process defined on
∈
(or on a regular lattice
).
5
4
3
2
1
14
12
10
10
8
14
8
6
6
4
4
2
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2
4
2
Lattice vs cont space
be a RV stationary (isotropic?) spatial process defined on
Setup: Let
∈
(or on a regular lattice
∈
). Consider the former—latter is more
straightforward.
lim
,

regular grid
,
∈
0
0
2
2
4
4
y
y
6
6
8
8
10
10
random pattern
0
2
4
6
8
10
0
2
4
x
6
8
10
x
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Regular grid
0
2
4
y
6
8
10
regular grid
0
2
4
h = 1; # of pairs = 4
6
x
h = sqrt(2); # of pairs = 4
8
10
h = sqrt(10); # of pairs = 8
h = sqrt(18); # of pairs = 4
h = 2; # of pairs = 4
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3
Random pattern
0
2
4
y
6
8
10
random pattern
0
2
4
h = 1; # of pairs = 0
h=1
6
8
10
x
.25
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Random pattern
8
9
Zoom-in
4
5
6
y
7
buffer
0
1
2
3
4
5
x
Estimate of extremogram at lag h = 1 for red point: weight
“indicators of points” in the buffer.
Bandwidth: half the width of the buffer.
Copenhagen May 27-30, 2013
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4
Random pattern
0
2
4
y
6
8
10
random pattern
0
2
4
6
8
10
x
Note:
• Expanding domain asymptotics: domain is getting bigger.
• Not infill asymptotics: insert more points in fixed domain.
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Estimating extremogram--random pattern
,…,
Setup: Suppose we have observations,
,…,
with rate
of some Poisson process
Here,
number of Poisson points in
Weight function
: Let
Copenhagen May 27-30, 2013
in a domain
,
~
↑
.
.
⋅ be a bounded pdf and set
1
where the bandwidth
at locations
→ 0and
,
→ ∞.
10
5
Estimating extremogram--random pattern
lim
,

,
/
,
∈
Kernel estimate of :
|
,
|
∑
∈
|
|
∑
∈
∈
,
∈
|
,
Note:
∈
,
∈
|
)
is product measure off the
diagonal.
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11
Limit Theory
, (i.e., regularly varying,
Theorem: Under suitable conditions on
mixing, local uniform negligibility, etc.), then
,
where
, ,
is the 1
0, Σ ,
is the pre-asymptotic extremogram,
, ,
(
→
, ,
1/
quantile of
,
/
,
∈
,
).
Remark: The formulation of this estimate and its proof follow the ideas
of Karr (1986) and Li, Genton, and Sherman (2008).
Copenhagen May 27-30, 2013
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6
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
0 ∈
0 ∈
→
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Limit Theory
Numerator:
∈
∈
0 ∈
=
,
,
∈
=
0 ∈
where
variables
and
,
∈
.Making the change of
, the expected value is
∩
=
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|
∩
|/|S |
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7
Limit Theory
∩
|
|
S
→
,
.
,
Remark: We used the following in this proof.
•
|
∩
|
→ 1 and
→
0 ∈
•
.
,
∈
→
.
,
Need a condition for the latter.
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Limit theory
Local uniform negligibility condition (LUNC): For any ,
0, there
exists a ′ such that
limsup
Proposition: If
sup |
|
is a strictly stationary regularly varying random
field satisfying LUNC, then for
0
∈ ,
→ 0,
∈
This result generalizes to space points, 0,
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.
→
,
,…,
.
16
8
Limit Theory
Outline of argument:
• Under LUNC already shown asymptotic unbiasedness of numerator
and denominator.
with
•
̂
,
→
•
̂
, ,
→
,
,
,
/
.
Strategy: Show joint asymptotic normality of ̂
var ̂
→
,
⟹ ̂
,
,
and ̂
, ,
→
,
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Limit Theory
Step 1: Compute asymptotic variances and covariances.
var ̂
i.
→
,
var ̂
ii.
,
→
,
Proof of (i): Sum of variances + sum of covariances
̂
,
|
→
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∈
|
|
∈
|
,
∈
,
,
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9
Limit Theory
Step 2: Show joint CLT for ̂
Idea: Focus on ̂
,
and ̂
using a blocking argument.
, ,
. Set
, ,
∈
. We will show
and put
∈
,
is asymptotically normal.
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Limit Theory
Subdivide
into big blocks and small blocks.
where
hasdiminesions
and size
0
2
4
y
6
8
10
∪
0,
0
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2
4
6
8
10
20
10
Limit Theory
Insert block
Subdivide
| |
where
hasdiminesions
:
and size
0
2
4
y
6
8
10
∪
inside
dimension
0,
intowith
big blocks
and small blocks.
0
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2
4
6
8
10
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Limit Theory
Recall that
, . .,
is a (mean-corrected) double integral over
,
,
,
,
,
10
,
4
0
2
y
6
8
0
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2
4
6
8
10
x
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Limit Theory
,
Remaining steps:
i.
Show var
ii.
Let
̃
∑
be an iid sequence with ̃
→ 0.
whose sum has
. Show
characteristic function
iii.
,
→ exp
.
→ 0.
Intuition.
(i)
The sets
∖
are small by proper choice of and .
(ii) Use a Lynapounov CLT (have a triangular array).
(iii) Use a Bernstein argument (see next page).
Copenhagen May 27-30, 2013
Limit Theory
Useful identity: ∏
|
∏
|
∑
⋯
⋯
̃
|
̃
|
where ,
|
∑
|cov ∏
∑
|
∑
4
,
cov ∏
∪
̃
|
| (by indep of ̃ )
,
|
,
is a strong mixing bounding function that is based on the
separation h between two sets U and V with cardinality r and s.
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Strong mixing coefficients
Strong mixing coefficients: Let
be a stationary random field on
.
Then the mixing coefficients are defined by
sup
,
∩
,
∈
where the sup is taking over all sets
,
, and
,
,
,
∈
, with
.
Proposition (Li, Genton, Sherman (2008), Ibragimov and Linnik (1971)):
Let
and
,
,
be closed and connected sets such that
. If and are rvs measurable wrt
and
,
and
respectively, and bded by 1, then
,
cov
16
,
, 4
,
.
Copenhagen May 27-30, 2013
Limit Theory
/
Mixing condition:sup
2.
for some
Returning to calculations:
|
|
16
16
|cov
∪
,
|
,
∪
16
→ 0 4
2
0 .
Copenhagen May 27-30, 2013
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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
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Simulations of spatial extremogram
Box-plots based on 1000 (100) replications of MMA(1) (left) and BR (right)
Lattice
Non-lattice;
1/ log (left)
5/ log (right)
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