Fixed-size Minimax for Committee Elections: Approximation and Local Search Heuristics
Stochastic Local Interaction Models and Space-Time ......The Errors of the Karhunen-Loeve Expansion...
Transcript of Stochastic Local Interaction Models and Space-Time ......The Errors of the Karhunen-Loeve Expansion...
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Stochastic Local Interaction Models andSpace-Time Covariance Functions based on
Linear Response Theory
Dionissios T. Hristopulos
Geostatistics Laboratory, School of Mineral Resources EngineeringTechnical University of Crete, Chania, Greece
Stochastic Weather Generators Workshop, May 17-20, 2016University of Bretagne Sud, Vannes, France
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Motivation for Local Random Field Models
I Spatial dependence can be encoded using local interactions inGibbs random fields
I Locality⇒ Sparse precision matrix⇒ Computational efficiency
I Locality leads to rational spectral densities
I Rational spectral density⇒ Connection with SPDEs (similar toMatern random fields)
I Linear response theory⇒ Extension to space-time SPDEs
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A Brief Roadmap of the Presentation
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Statistical Field Theories are Local and can be usedas Geostatistical Models
I In geostatistics, we write the joint pdf of spatial data as
fX ∝ exp
− N∑i,j=1
xi C−1x ; i,j xj
I In statistical field theories, the joint pdf of random fields is
determined from spatial interactions, i.e.,
fX ∝ exp(−∫
d~s φ[x(~s)])
I A local framework: Geostatistical models based on localinteractions: Spartan Spatial Random Fields (SSRFs)
I Motivation:1. Flexible parametrization of spatial dependence2. Efficient parameter estimation3. Interpolation and simulation of large datasets
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Spartan Spatial Random Fields (SSRFs) are GibbsRFs with Local Structure
I Gibbs probability density function (PDF)
f [x(~s)] = Z−1 e−H[x(~s)], H[x(~s)] : energy functional,
I Z: partition function Z =∫Dx(~s) e−H[x(~s)]
I Hfgc[x(~s)] =∫ d~s
2η0ξd
{[x(~s)
]2+ η1 ξ
2[∇x(~s)
]2+ ξ4
[∇2x(~s)
]2}I Properties: Gaussian, zero-mean, stationary, isotropic SRF
FGC-SSRF Coefficientsη0 : scale, η1 : rigidity, ξ: characteristic length; kc : spectral cutoff
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SSRF Covariance for Time Series and Drill-hole Data
G(h) =η0
4e−hβ2
[cos(hβ1)
β2+
sin(hβ1)
β1
], |η1| < 2
G(h) = η0(1 + h)
4 eh, η1 = 2
G(h) =η0
2 ∆
( e−hω1
ω1−
e−hω2
ω2
), η1 > 2
NotationI h = |r |/ξ : normalized lag
I β1,2 =(|2∓η1|
4
)1/2
I ω1,2 =(|η1∓∆|
2
)1/2
I ∆ = |η21 − 4|
12
d=1
0 1 2 3 4 5−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Distance lag
Cor
rela
tion
η
1=−1.9
η1=−1
η1=1
η1=16
Hristopulos and Elogne (2007), IEEETransactions on Information Theory,
53(12), 4667-4679
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Classical Damped Harmonic Oscillator in a Heat Bathis a one-dimensional SSRF
Langevin equation: x(t) + Γ x(t) + ω20 x(t) = ε(t), G(h) = E[x(t + h) x(h)]
G(h) =η0
4e−hβ2
[cos(hβ1)
β2+
sin(hβ1)
β1
], |η1| < 2, �Underdamping
G(h) = η0(1 + h)
4 eh, η1 = 2, �Critical damping
G(h) =η0
2 ∆
( e−hω1
ω1−
e−hω2
ω2
), η1 > 2, �Overdamping
0 1 2 3 4 5 6 7 8h
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ρxx
(h;θ
)
η1=-0.5
η1=2
η1=4 ⇐ Oscillator displacement correlation function
Nørrelykke, S.F., Flyvbjerg, H.: Harmonic oscillator in heat bath:
Exact simulation of time-lapse-recorded data and exact
analytical benchmark statistics. Phys. Rev. E 83, 041103 (2011)
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An Optimal Basis for the 1D SSRF model is given bythe Karhunen-Loeve Expansion
Karhunen-Loeve Expansion of SSRFs in a NutshellI Determining the K-L basis requires solving a fourth-order ODE
with a 4× 4 system of boundary conditions
I There are two eigenfunction branches: the first involves onlyharmonic functions, while the second involves a superposition ofharmonic and hyperbolic eigenfunctions
I Each branch contains eigenfunctions with even, f (s) = f (−s),and odd, f (s) = −f (−s), symmetry
I The eigenvalues are determined by solving (numerically)transcendental equations
For more details see: Tsantili and Hristopulos (2016), Probabilistic EngineeringMechanics, 43, 132-147
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Convergence of the Variance of the Karhunen-LoeveExpansion for d = 1 SSRFs
0 10 20 30 40n
-5
-4
-3
-2
-1
0
1
2
3
Log(
Eig
enva
lues
)
6
6$
6
6$
0 10 20 30 40n
0
20
40
60
80
100
En=E
tot(%
)(Left) Eigenvalues λ, λ∗, λ, λ∗ obtained for SSRF with η0 = 2, η1 = −1 and ξ = 5.(Right) Cumulative energy En =
∑n′<n En′ by superposition of largest n eigenvalues of K-L basis
as percentage of σ2 =∑∞
n′=1 En′ . Open circles: First-branch, even-sector eigenvalues.
Squares: First-branch, odd-sector eigenvalues. Diamonds: Second-branch, even-sector
eigenvalues. Filled circles: Second-branch, odd-sector eigenvalues.
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The Errors of the Karhunen-Loeve Expansion can beControlled
K-L Expansion - Local Approximation Errors
Local approximation errors involved for SSRF correlation function with η0 = 2, η1 = −1and ξ = 5. (Left) K-L approximation with four terms and domain length L ≈ 13.6.(Right) K-L approximation with twenty four terms and domain length L = 100.
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SSRF Covariance Functions for Processes in thePlane
G(h) =η0 = [K0(h z+)]
π√
4− η21
, |η1| < 2
G(h) =
(η0 h4π
)K−1(h), η1 = 2
G(h) =η0 [K0(h z+)− K0(h z−)]
2π√η2
1 − 4, η1 > 2
NotationI =: Imaginary partI z± =
√−t∗±
I t∗± =
(−η1 ±
√η2
1 − 4)/2
I Kν(z): modified Bessel functionof the second kind and order ν
d=2
0 1 2 3 4 5 6 7 8−0.05
0
0.05
0.1
0.15
0.2
0.25
h
G(h
)
η1=−1.2
η1=−0.5
η1=0.5
η1=2
η1=4
Hristopulos (2015), Stoch. Environ. Res. RiskAssesss., 29(3), 739–754.
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SSRF Covariance Functions for Spatial Processes in3D Space (in R3 or in R2 × T )
G(h) = η0e−hβ2
2π∆
[sin (hβ1)
h
], |η1| < 2
G(h) =η0
8πe−h, η1 = 2
G(h) =1
4π∆
[e−hω1 − e−hω2
h (ω2 − ω1)
], η1 > 2
NotationI h = ‖~r‖/ξ
I β1,2 =(|2∓η1|
4
)1/2
I ω1,2 =(|η1∓∆|
2
)1/2
I ∆ = |η21 − 4|
12
d=3
0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Distance lag
Cor
rela
tion
η1=−1
η1=2
η1=8
η1=16
Hristopulos and Elogne, IEEE Trans. Inform.Theor., 2007
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Applications - Groundwater Level Estimation
Estimated map of Mires basin (Messara valley,Crete) groundwater level using regression krigingcombined with Thiem’s multiple well equation as
trend.
Varouchakis and Hristopulos, Advances in Water
Resources, (2012)
Wet versus dry period variability.
E. Varouchakis, “Geostatistical Analysis and
Space-Time Models of Aquifer Levels,” PhD
Dissertation, Technical University of Crete (2012)
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Applications - Lignite Mining
−2.2 −2.1 −2 −1.9 −1.8 −1.7
x 104
2.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65x 10
4
West−East (m)
So
uth
−N
ort
h (
m)
Mavropigi mine
Estimation of area density of lignite energy content (Gcal/m2) for Mavropigi mine (WesternMacedonia, Greece) using regression kriging with Spartan variogram Pavlides et al. (2015),Energy, 93(Part 2), 1906–1917.
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An SSRF on a Regular Grid is a Gauss-Markov RFwhose Energy involves the Squares of theFluctuations, their Gradient and their Curvature
H[x(~s)] =λN∑
n=1
{[x(~sn)− µX
]2+ c1
d∑i=1
[x(~sn + ai ei)− x(~sn)
ai
]2
+c2
d∑i=1
[x(~sn + ai ei)− 2x(~sn) + x(~sn − ai ei)
a2i
]2}
ei , i = 1, . . . , d : unit vectors in lattice directions
ai : lattice steps
0 2 4 6 8 100
1
2
3
4
5
6
7
8
9
10
Hfgc [X (~s)] = λS0 + c1 λSG + c2 λSc
Rue and Held, Gaussian Markov Random Fields: Theory
and Applications, Chapman and Hall/CRC, 2005
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Reconstruction of Walker Lake Data (260× 300 grid)Anisotropic model based on 39 000 sampling points (50%)
0
200
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600
800
1000
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1400
1600
0
200
400
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800
1000
1200
1400
1600
Left top: Full data setLeft bottom: Sample
Below: Scatter plot of samplevs SLI predictions
Right top: SLI-based mapRight bottom: SLI spatial error
SLI Prediction0 500 1000 1500
Sam
ple
0
200
400
600
800
1000
1200
1400
1600 0
200
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-800
-600
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-200
0
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800
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Reconstruction of Walker Lake Data (260× 300 grid)Anisotropic model based on 7 800 sampling points (10%)
0
200
400
600
800
1000
1200
1400
1600
0
200
400
600
800
1000
1200
1400
1600
Left top: Full data setLeft bottom: Sample
Below: Scatter plot of samplevs SSRF predictions
Right top: SSRF-based mapRight bottom: SSRF spatial
error
SLI Prediction0 500 1000 1500
Sam
ple
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200
400
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800
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Stochastic Local Interaction Model for Scattered DataCombining SSRF ideas with kernel functions the following energy
functional can be used for scattered data xS ≡ (x1, . . . , xN)T
H(xS;θ) =1
2λ
[S0(xS) + α1 S1(xS;~h1) + α2 S2(xS;~h2)
],
where ~hi = (hi,1, . . . ,hi,N)T , i = 1,2, is a vector of local bandwidths.
Main IdeasI Use kernel functions to express the energy terms Si(xS)
I Adjust the kernel bandwidths adaptively according to localsampling densityhi = µDi,[k ](SN ), where Di,[k ](SN ) is the distance between ~si and its k -nearestneighbor in SN , and µ > 1 is a data-dependent parameter
I Estimate parameters using leave-one-out cross validation (faster)or maximum likelihood (slower)
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Thanks to SLI Precision Matrix Formulation, Predictiondoes not Require Covariance Matrix Inversion
I Energy functional (Hristopulos (2015), Computers & Geosciences, 85, 26–37)
H(xS;θ) =12
(xS − µX)T J(θ) (xS − µX)
I Precision matrix (sparse, explicit)
J(θ) =1λ
{INN
+ α1 c1 J1(~h1) + α2
[c2,1 J2(~h2)− c2,2 J3(
√2~h2)− c2,3 J4(2~h2)
]}I Gradient and Curvature Precision sub-matrices
[Jq(~hq)]i,j = −ui,j (hq;i )− ui,j (hq;j ) + [IN ]i,j
N∑l=1
[ui,l (hq;i ) + ul,i (hq;l )
],
I Kernel Weights
ui,j (hq;i ) =K((~si −~sj )/hq,i
)∑Ni=1∑N
j=1 K((~si −~sj )/hq,i
) .Technical University of Crete Geostatistics Laboratory: http://www.geostatistics.tuc.gr/4908.html
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SLI Mode Predictor has O(Nα) Numerical Complexity,where 1 ≤ α ≤ 2 for Scattered Data and α = 1 onRegular Grids
I Mode predictor: xp = arg minxp H(xS, xp;θ∗) , where xp is the
value at the prediction point
I Prediction equation: xp = µX − 1Jp,p(θ∗)
∑Ni=1 Jp,i(θ
∗) (xi − µX)
I Property 1: Unbiased estimator, i.e., E [xp] = µX
I Property 2: Not necessarily an exact interpolator
I Property 3: Interpolation “limited” within the convex hull of thedata
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Study of Groundwater Level Data (Small Data Set forIllustration Purposes)
Scattered data with modest size (N = 250) are used to compare withOrdinary Kriging (based on the Spartan variogram)
Plot of sample values
5 5.05 5.1 5.15
x 105
4.54
4.545
4.55
4.555
x 106
Easting (m)
No
rth
ing
(m
)
50
100
150
200
250
300
350
400
450
Frequency histogram
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SLI Interpolation Analysis: Bandwidth & PrecisionMatrixBi-triangular kernel, k = 2, Estimated parameters: α1 ≈ 5.38, α2 ≈ 0.53, µ ≈ 2.93
Relative size of localbandwidths per site
0 20 40 60 80 1000
20
40
60
80
100
X
Y
h1
5
10
15
20
25
30
35
Precision matrix
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SLI Interpolation Analysis: Bandwidth & PrecisionMatrixBi-triangular kernel, k = 2, Estimated parameters: α1 ≈ 5.38, α2 ≈ 0.53, µ ≈ 2.93
Relative size of localbandwidths per site
0 20 40 60 80 1000
20
40
60
80
100
X
Y
h1
5
10
15
20
25
30
35
Precision matrixPrecision Matrix
50 100 150 200 250
50
100
150
200
250 −0.5
0
0.5
1
1.5
2
2.5
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SLI Interpolation Analysis: Bandwidth & PrecisionMatrixBi-triangular kernel, k = 2, Estimated parameters: α1 ≈ 5.38, α2 ≈ 0.53, µ ≈ 2.93
Relative size of localbandwidths per site
0 20 40 60 80 1000
20
40
60
80
100
X
Y
h1
5
10
15
20
25
30
35
Precision matrixPrecision Matrix
50 100 150 200 250
50
100
150
200
250
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SLI Interpolation Analysis - MapsBi-triangular kernel, k = 2, Estimated parameters: α1 ≈ 5.38, α2 ≈ 0.53, µ ≈ 2.93
Ordinary Kriging Map SLI Generated Map
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Cross Validation Analysis - A difference is a differenceonly if it makes a difference
Table: Leave-one-out Cross Validation results of SLI interpolation(Bi-triangular kernel) and OK (with Spartan variogram).
CV measures SLI (bi-tria) OK (SSRF)ME: Mean error (bias) 0.05 −0.02MAE: Mean absolute error 1.76 1.76MARE: Mean absolute relative error 0.44 0.43RMSE: Root mean square error 2.77 2.69RP: Pearson correlation coefficient 0.68 0.70SR: Spearman correlation coefficient 0.72 0.75Min Dif: min{z} −min{z} 0.79 0.76Max Dif: max{z} −max{z} −7.31 −8.5
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Fast SLI Interpolation: Campbell County Coal Data -Mapping of Coal ThicknessQuadratic kernel, k = 2, Ntrain = 8700, Estimated parameters: α1 ≈ 267.48, α2 ≈ 0.60, µ ≈ 2.23,Nval = 8700, 100× 100 grid
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Fast SLI Interpolation: Campbell County Coal Data -Mapping of Coal ThicknessQuadratic kernel, k = 2, Ntrain = 8700, Estimated parameters: α1 ≈ 267.48, α2 ≈ 0.60, µ ≈ 2.23,Nval = 8700, 100× 100 grid
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Linear Response of Local Interaction Gibbs Models
I Linear response theory describes the non-equilibrium responsedue to small deviation from equilibrium and leads to S-TLangevin equations Hohenberg and Halperin (1977), Reviews of ModernPhysics, 49(3), 435–479
∂x(~s, t)∂t
= −DδH[x(~s)]δx(~s)
∣∣∣∣x(~s)=x(~s,t)
+ ζ(~s, t) = V [x(~s, t)] + ζ(~s, t)
I Equilibrium-restoring velocity for the ST-SLI random field
V [x(~s, t)] = − 12ξd η0
(1− η1ξ
2∇2 + µ ξ4∇4) x(~s)∣∣∣∣x(~s)=x(~s,t)
I D is a diffusion coefficient
I ζ(~s, t) is the random velocity (e.g., Gaussian white noise)
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Equations of Motion for the ST-SLI Covariance can bederived using Linear Response Theory
EOM:∂Cx(r, τ)
∂τ= − sign(τ)
τc
(1− η1ξ
2∇2 + µ ξ4∇4) Cx(r, τ),
where τ−1c = D/(2ξd η0), and the initial condition is the SSRF spectral density
C(k , τ = 0) =η0 ξ
d
1 + η1(kξ)2 + µ(kξ)4.
Zero-µ solution in d = 1 dimension
C1(h,u) =η0 λ
4
[e−λ h erfc
(√u − λh
2√
u
)+ eλ h erfc
(√u +
λh2√
u
)],
λ = 1/√η1, h = r/ξ, u = |τ |/τc , and erfc(·) is the complementary error function.
C1(h, u) is equivalent (except for different parametrization) to a covariance derived froma parabolic SPDE in 1+1 dimensions by Heine (1955). Biometrika, 42(1-2), 170–178.
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Equations of Motion for the ST-SLI Covariance can bederived using Linear Response Theory
Zero-µ solution in d = 3 dimension
C3(h,u) =η0 ξ λ
2
8π r
[e−λ h erfc
(√u − λh
2√
u
)− eλ h erfc
(√u +
λh2√
u
)],
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The variance divergence can be tamed by adding thecurvature term
Small µ approximation
C(~r , τ) ≈2 e−D |τ |
(4π)d/2
2M∑m=0
(−µ)m
m!
Γ(d/2 + 2m)
Γ(d/2)Rm(r , τ)
Rm(r , τ) =
∫ ∞0
dκe−κβ0 vm
u4m+d 1F1
(2m +
d2,
d2
;−r2
4 u2
)where r = ‖~r‖, and
I v = ξ4(
D|τ |+ κβ0
)I u2 = η1ξ
2(
D |τ |+ κβ0
)I 1F1(a1, a2; z) is the confluent hypergeometric
function
Hristopulos & Tsantili (2015), Int. J. Modern Phys. B, 29,1541007
0 5 10 15-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
r
CHr,
0L
Η =-1.5Η =-1Η=-0.5
0 5 10 15-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
r
CHr,
3L
Η =-1.5Η =-1Η=-0.5
µ = 1, ξ = 3
Top: τ = 0. Bottom: τ = 3.
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Taming the variance divergence with space transformsI Space transforms are mathematical operations that can generate
higher-dimensional functions based on lower-dimensionalprojections, e.g., Ehrenpreis, L. (2003). The Universality of the RadonTransform, Oxford.
I For example [Mantoglou and Wilson (1982). Water Resources Research,18(5), 1379–1394]
C3(r , τ) =1r
∫ r
0dx C1(x , τ) =
1h
∫ h
0dy C1(y ,u)
I Applying the above to the d = 1 local-interaction covariancefunction we obtain
Space-transformed ST-SLI (linear response) covariance
C3(h, u) =η04h
[2 e−u erf
(λh
2√
u
)+ eλ h erfc
(√u + λh
2√
u
)− e−λ h erfc
(√u − λh
2√
u
)]Technical University of Crete Geostatistics Laboratory: http://www.geostatistics.tuc.gr/4908.html
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Visualization of the ST-SLI covariance obtained bymeans of the space transform
-1.5 -1 -0.5 0 0.5 1 1.5h (space lag)
-1.5
-1
-0.5
0
0.5
1
1.5u
(tim
e la
g)
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Visualization of the ST-SLI covariance obtained bymeans of the space transform
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Brief Review of Linear Response Theory ResultsI S-T local-interaction Langevin equation
∂x(~s, t)∂t
= −D
2ξd η0
(1− η1ξ
2∇2 + µ ξ4∇4)
x(~s, t) + ζ(~s, t)
I 1D+T SLI covariance
C1(h, u) =η0 λ
4
[e−λ h erfc
(√u −
λ h2√
u
)+ eλ h erfc
(√u +
λ h2√
u
)]
I 3D+T space-transformed SLI covariance
C3(h, u) =η0
4h
[2e−u erf
(λh
2√
u
)+ eλh erfc
(√u +
λh2√
u
)− e−λh erfc
(√u −
λh2√
u
)]
I The above covariances do not scale in u and h like the Gneiting(2002) class C(h,u) = 1
ψ(u2)d/2φ(
h2
ψ(u2)
), φ(·) : completely monotone,
ψ(·): Bernstein (positive definite with completely monotone derivative)
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Conclusions and Future Directions
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Thank you for your attention!
Collaborations: Dr. S. Elogne, Dr. M. Zukovic, Dr. A. Chorti, Dr. E. Varouchakis,
Mr. I. Spiliopoulos, Dr. Ivi Tsantili, Mr. A. Pavlides, Mrs. V. Agou, and Mr. M.Petrakis
Research funded by the project SPARTA implemented under the“ARISTEIA” Action of the operational programme “Education andLifelong Learning” co-funded by the European Social Fund (ESF)and National Resources
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For more information ...D. T. Hristopulos (2003). “Spartan Gibbs Random Field Models for Geostatistical Applications,”SIAM J. Sci. Comput., 24(6), 2125-2162.
D. T. Hristopulos and S. Elogne (2006). “Analytic Properties and Covariance Functions of a NewClass of Generalized Gibbs Random Fields,” IEEE Trans. Infor. Theory, 53(12), 4667 - 4679.
S. N. Elogne and D. T. Hristopulos (2008). “Geostatistical applications of Spartan spatial randomfields,” in geoENV VI - Geostatistics for Environmental Applications, pp. 477-488 (ed. by A. Soareset al.) 512p.
S. Elogne, D. T. Hristopulos, M. Varouchakis (2008). “An application of Spartan spatial randomfields in environmental mapping: focus on automatic mapping capabilities,” Stoch. Envir. Res. RiskA., 22(5), 633-646.
D. T. Hristopulos and S. N. Elogne (2009). “Computationally efficient spatial interpolators based onSpartan spatial random fields”, IEEE Trans. Signal Proc., 57(9), 3475–3487.
M. Zukovic, and D. T. Hristopulos (2009b). “Classification of missing values in spatial data usingspin models,” Phys. Rev. E, 80(1), 011116.
D. T. Hristopulos (2015). “Covariance functions motivated by spatial random field models with localinteractions,” Stoch. Envir. Res. Risk A., 29(3), 739–754.
D. T. Hristopulos (2015). “Stochastic Local Interaction (SLI) Model for Incomplete Data ind−Dimensional Metric Spaces” Computers & Geosciences, 85, 26–37.
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Absence of Space-Time Dimple Effect
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2h (spatial lag)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Cov
aria
nce
00.25 0.5 11.25 1.5 2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2u (time lag)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Cov
aria
nce
00.25 0.5 11.25 1.5 2
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