asymptotics for the spatial extremogram
DESCRIPTION
Richard A. Davis* Columbia University. Asymptotics for the Spatial Extremogram. Collaborators : Yongbum Cho, Columbia University Souvik Ghosh, LinkedIn Claudia Klüppelberg , Technical University of Munich Christina Steinkohl, Technical University of Munich. Plan. Warm-up - PowerPoint PPT PresentationTRANSCRIPT
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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68
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
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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
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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
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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
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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
Building a Max-Stable Model in Space-Time
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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)
Building a Max-Stable Model in Space-Time
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Scaled and transformed Gaussian random fields (fixed time point)
11
−1log ¿¿
−1log ¿¿
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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 )))
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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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x
y
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regular grid random pattern
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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?
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Regular grid
regular grid
x
y
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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
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Random pattern
random pattern
h = 1; # of pairs = 0h = 1 .25
x
y
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x
y
0 1 2 3 4 5
45
67
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.
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Random pattern
random pattern
Note: • Expanding domain asymptotics: domain is getting bigger.• Not infill asymptotics: insert more points in fixed domain.
x
y
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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
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Estimating extremogram--random pattern
Kernel estimate of :
Note: ) is product measure off the diagonal.
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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 ).
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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
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Limit Theory
=
=
where Making the change of variables and the expected value is
Numerator:
=|
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Limit Theory
Remark: We used the following in this proof.
• and .
•
Need a condition for the latter.
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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, .
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Limit Theory
Outline of argument:
• Under LUNC already shown asymptotic unbiasedness of numerator
and denominator.
with .
Strategy: Show joint asymptotic normality of and
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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.
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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.
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Limit Theory
Subdivide into big blocks and small blocks.
where and size
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y
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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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y
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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
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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).
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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.
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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
37
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
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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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Simulations of spatial 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),(
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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),(
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Data Example: extreme rainfall in Florida
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Data Example: extreme rainfall in Florida
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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.
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Data Example: extreme rainfall in Florida
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Hourly accumulated rainfall time series in the wet season 2002 at 2 locations.
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Data Example: extreme rainfall in Florida
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Hourly accumulated rainfall fields for four time points.
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Data Example: extreme rainfall in Florida
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Empirical extremogram in space (left) and time (right)
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Data Example: extreme rainfall in Florida
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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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Data Example: extreme rainfall in Florida
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Empirical extremogram in space (left) and time (right)
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Data Example: extreme rainfall in Florida
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Computing conditional return maps.Estimate such that
where satisfies is pre-assigned.A straightforward calculation shows that must solve,
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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.
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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
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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:
Estimation—composite likelihood approach
N
jii
N
jjiji sysyfPL
1 1, ));(),((log),( dbf
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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
Estimation—composite likelihood approach
N
i Lssjjiji
ij
sysyfPL1 |:|
, ));(),((log),( dbf
N
i Dssjjiji
iij
sysyfPL1 |:|
, ));(),((log),( dbf
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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
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Process: Limit process d = |s-t|, 1 , .2; 9 nearest neighbors
Simulation Examples
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Simulation Examples
Process: Limit process d = |s-t|, 1 , .2; 25 nearest neighbors
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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.
Illustration with French Precipitation Data
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Estimated spatial correlation function before transformation to unit Frechet. Correlation function
Illustration with French Precipitation Data
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
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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
Illustration with French Precipitation Data
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
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Plot of r(l; |s-t|) = limn→∞ P(Yn(s) > n(1-l) | Yn(t) > nl)
Illustration with French Precipitation Data
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Plot of r(l; |s-t|) = limn→∞ P(Yn(s) > n(1-l) | Yn(t) > nl)
Illustration with French Precipitation Data
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Illustration with French Precipitation Data
Results from random permutations of data—just for fun. Since random permutations have no spatial dependence, r should be 0 for |s-t| >0.