Statistical Models of Spatial Processes Based on Local-Interaction - - PowerPoint PPT Presentation

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Statistical Models of Spatial Processes Based

• n Local-Interaction Energy Functionals

Dionisis T. Hristopulos

Technical University of Crete, Greece

Presented at the Uncertainty Quantification Workshop ICERM, Brown University October 9-13, 2012

• Dionisis Hristopulos: dionisi@mred.tuc.gr

ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 1/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Overview

1

Introduction

2

Classical Geostatistical Approach

3

Spartan Spatial Random Fields (SSRF)

4

Parameter Inference

5

Interpolation of Scattered Data

6

Time-Series Application

7

Simulations

8

A Method Comparison

9

Anisotropy Analysis

10 Non-Gaussian Models 11 Conclusions 12 Some References

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 2/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Applications - Natural Resources Estimation

Groundwater Level Estimation Estimate water level in aquifers Sparse spatial network Quantify uncertainties Resolve temporal patterns Prediction forward in time Extreme low level predictions

Figure: Estimated map of Mires basin (Messara valley, Crete) groundwater level using regression kriging combined with Thiem’s multiple well equation as the trend.

Varouchakis and Hristopulos, Adv. Water Resour., http://dx.doi.org/10.1016/j.advwatres. 2012.08.002 (2012).

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 3/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Applications - Natural Resources Estimation

Groundwater Level Estimation Estimate water level in aquifers Sparse spatial network Quantify uncertainties Resolve temporal patterns Prediction forward in time Extreme low level predictions

Figure: Wet versus dry period variability. From E.

Varouchakis, “Geostatistical Analysis and Space-Time Models of Aquifer Levels,” PhD Dissertation, Technical University of Crete (2012).

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 3/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Applications - Energy Resources

Coal reserves & quality Estimate total reserves and uncertainty Quantify local variability (homogenization) Estimates of lignite quality parameters (e.g. ash, calorific value)

900 1000 1100 1200 1300 1400 1500 1600 1700 1800 −3 −2 −1 1 2 −2 −1 1 2 1000 1200 1400 1600 1800 Lower Calorific Value

Figure: Top: Locations of drill holes and

estimated ash content (%) South Field Mine. Bottom: Estimated lower calorific value (kcal/kg) Amyndeo Mine.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 4/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Applications - Environmental Monitoring

Radioactivity monitoring

Efficient generation of accurate space-time maps Automatic data processing, model selection & emergency detection

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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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• [0,3]

(3,3.5] (3.5,4] (4,4.5] (4.5,5] (5,5.5] (5.5,6] (6,6.5] (6.5,7] (7,7.5] (7.5,8] (8,8.5] (8.5,9] (9,9.5] (9.5,10] (10,10.5] (10.5,11] (11,11.5] (11.5,12] OneDomainIso OneDomainAniso Isotropic Anisotropic 2 4 6 8 10 12

Radioactivity dose rate (nSv/h) using background measurements and simulated accident (Bundesamt f¨ ur Strahlenschutz). Spiliopoulos et al., Computers & Geosciences, 37, 320-330 (2011).

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 5/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Uncertainty Quantification in Spatial Processes

Directions of Further Research Non-Gaussian, non-stationary models of spatial variability Flexible and efficient spatiotemporal statistical models Incorporation of physical laws (PDEs, SPDEs) in statistical models Developing connections between spatial modelling and statistical physics

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 6/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Data Reconstruction Problem

From a sample (X1, . . . , XN) of a process X(s), s ∈ Rd at points (s1, . . . , sN): Estimate missing data X(zp), p = 1, . . . , P, i.e., ˆ X(zp) = Φ(X1, . . . , XN)

Sampling: Missing data

−2 2 4 x 10

5

1 2 3 4 5 6 x 10

5

X (m) Y (m) Training set GDR (nSv/h) 60 80 100 120 140

Interpolated maps

Right panel: (a) NNI training (b) NNI prediction (c) SSRF SpartFit (d) SSRF Spart

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 7/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Data Reconstruction Problem

From a sample (X1, . . . , XN) of a process X(s), s ∈ Rd at points (s1, . . . , sN): Estimate missing data X(zp), p = 1, . . . , P, i.e., ˆ X(zp) = Φ(X1, . . . , XN)

Sampling: Missing data

x

100 200 300 50 100 150

200 250 300 350 400 450 500 550

Daily ozone (June 2007, NASA Earth Science Data) measurements

Interpolated maps

x

100 200 300 50 100 150

200 250 300 350 400 450 500 550

DGC interpolation based on 10 simulations (by M. ˇ Zukoviˇ c)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 7/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Preliminaries

X(s, ω) ∈ R, s ∈ Rd is a random function or spatial random field Wide-sense (second-order) Stationarity:

1 Constant mean: E [X(s)] = mx 2 Two-point covariance: G(r) = E [X(s) X(s + r)] − m2

x

Statistical Isotropy: G(r) = G(r) Intrinsic Stationarity:

1 Constant mean: E [X(s + r) − X(s)] = 0 2 Semivariogram (structure) function:

γ(r) = 1

2 E

• [X(s) − X(s + r)]2

Second-order stationarity: G(r) = σ2

x − γ(r)

Yaglom, Correlation Theory of Stationary and Related Random Functions, Springer, 1987.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Preliminaries

X(s, ω) ∈ R, s ∈ Rd is a random function or spatial random field Wide-sense (second-order) Stationarity:

1 Constant mean: E [X(s)] = mx 2 Two-point covariance: G(r) = E [X(s) X(s + r)] − m2

x

Statistical Isotropy: G(r) = G(r) Intrinsic Stationarity:

1 Constant mean: E [X(s + r) − X(s)] = 0 2 Semivariogram (structure) function:

γ(r) = 1

2 E

• [X(s) − X(s + r)]2

Second-order stationarity: G(r) = σ2

x − γ(r)

Yaglom, Correlation Theory of Stationary and Related Random Functions, Springer, 1987.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Preliminaries

X(s, ω) ∈ R, s ∈ Rd is a random function or spatial random field Wide-sense (second-order) Stationarity:

1 Constant mean: E [X(s)] = mx 2 Two-point covariance: G(r) = E [X(s) X(s + r)] − m2

x

Statistical Isotropy: G(r) = G(r) Intrinsic Stationarity:

1 Constant mean: E [X(s + r) − X(s)] = 0 2 Semivariogram (structure) function:

γ(r) = 1

2 E

• [X(s) − X(s + r)]2

Second-order stationarity: G(r) = σ2

x − γ(r)

Yaglom, Correlation Theory of Stationary and Related Random Functions, Springer, 1987.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Preliminaries

X(s, ω) ∈ R, s ∈ Rd is a random function or spatial random field Wide-sense (second-order) Stationarity:

1 Constant mean: E [X(s)] = mx 2 Two-point covariance: G(r) = E [X(s) X(s + r)] − m2

x

Statistical Isotropy: G(r) = G(r) Intrinsic Stationarity:

1 Constant mean: E [X(s + r) − X(s)] = 0 2 Semivariogram (structure) function:

γ(r) = 1

2 E

• [X(s) − X(s + r)]2

Second-order stationarity: G(r) = σ2

x − γ(r)

Yaglom, Correlation Theory of Stationary and Related Random Functions, Springer, 1987.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Preliminaries

X(s, ω) ∈ R, s ∈ Rd is a random function or spatial random field Wide-sense (second-order) Stationarity:

1 Constant mean: E [X(s)] = mx 2 Two-point covariance: G(r) = E [X(s) X(s + r)] − m2

x

Statistical Isotropy: G(r) = G(r) Intrinsic Stationarity:

1 Constant mean: E [X(s + r) − X(s)] = 0 2 Semivariogram (structure) function:

γ(r) = 1

2 E

• [X(s) − X(s + r)]2

Second-order stationarity: G(r) = σ2

x − γ(r)

Yaglom, Correlation Theory of Stationary and Related Random Functions, Springer, 1987.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Preliminaries

X(s, ω) ∈ R, s ∈ Rd is a random function or spatial random field Wide-sense (second-order) Stationarity:

1 Constant mean: E [X(s)] = mx 2 Two-point covariance: G(r) = E [X(s) X(s + r)] − m2

x

Statistical Isotropy: G(r) = G(r) Intrinsic Stationarity:

1 Constant mean: E [X(s + r) − X(s)] = 0 2 Semivariogram (structure) function:

γ(r) = 1

2 E

• [X(s) − X(s + r)]2

Second-order stationarity: G(r) = σ2

x − γ(r)

Yaglom, Correlation Theory of Stationary and Related Random Functions, Springer, 1987.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Preliminaries

X(s, ω) ∈ R, s ∈ Rd is a random function or spatial random field Wide-sense (second-order) Stationarity:

1 Constant mean: E [X(s)] = mx 2 Two-point covariance: G(r) = E [X(s) X(s + r)] − m2

x

Statistical Isotropy: G(r) = G(r) Intrinsic Stationarity:

1 Constant mean: E [X(s + r) − X(s)] = 0 2 Semivariogram (structure) function:

γ(r) = 1

2 E

• [X(s) − X(s + r)]2

Second-order stationarity: G(r) = σ2

x − γ(r)

Yaglom, Correlation Theory of Stationary and Related Random Functions, Springer, 1987.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Preliminaries

X(s, ω) ∈ R, s ∈ Rd is a random function or spatial random field Wide-sense (second-order) Stationarity:

1 Constant mean: E [X(s)] = mx 2 Two-point covariance: G(r) = E [X(s) X(s + r)] − m2

x

Statistical Isotropy: G(r) = G(r) Intrinsic Stationarity:

1 Constant mean: E [X(s + r) − X(s)] = 0 2 Semivariogram (structure) function:

γ(r) = 1

2 E

• [X(s) − X(s + r)]2

Second-order stationarity: G(r) = σ2

x − γ(r)

Yaglom, Correlation Theory of Stationary and Related Random Functions, Springer, 1987.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Preliminaries

X(s, ω) ∈ R, s ∈ Rd is a random function or spatial random field Wide-sense (second-order) Stationarity:

1 Constant mean: E [X(s)] = mx 2 Two-point covariance: G(r) = E [X(s) X(s + r)] − m2

x

Statistical Isotropy: G(r) = G(r) Intrinsic Stationarity:

1 Constant mean: E [X(s + r) − X(s)] = 0 2 Semivariogram (structure) function:

γ(r) = 1

2 E

• [X(s) − X(s + r)]2

Second-order stationarity: G(r) = σ2

x − γ(r)

Yaglom, Correlation Theory of Stationary and Related Random Functions, Springer, 1987.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 8/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Classical Geostatistical Approach

The semivariogram function Semivariogram structure function

γ(r) = 1 2 E

• {X(s) − X(s + r)}2

Empirical semivariogram (MoM)

ˆ γ(rk) = 1 2 n(rk)

N

• i,j=1
• X(si) − X(sj)

2 ϑij(rk), (k = 1, . . . , Nc) ϑij(rk) =

• 1, si − sj ∈ B(rk)

0, si − sj ∈ B(rk) .

Scattered data (SIC 2004)

• G. Matheron, Trait´

e de g´ eostatistique applique´ e, Editions Technip, France, 1962-63

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Classical Geostatistical Approach

The semivariogram function Semivariogram structure function

γ(r) = 1 2 E

• {X(s) − X(s + r)}2

Empirical semivariogram (MoM)

ˆ γ(rk) = 1 2 n(rk)

N

• i,j=1
• X(si) − X(sj)

2 ϑij(rk), (k = 1, . . . , Nc) ϑij(rk) =

• 1, si − sj ∈ B(rk)

0, si − sj ∈ B(rk) .

Scattered data (SIC 2004)

−2 2 4 x 10

5

1 2 3 4 5 6 x 10

5

X (m) Y (m) Training set GDR (nSv/h) 60 80 100 120 140

• G. Matheron, Trait´

e de g´ eostatistique applique´ e, Editions Technip, France, 1962-63

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Classical Geostatistical Approach

Numerical complexity of ˆ γ(rk) estimation is O(N2)

1 2 3 4 5 x 10

5

50 100 150 200 250 300 350 400 450 lag distance (m) Empirical semivariogram

WLS fit to theoretical variogram (spherical model)

• G. Matheron, Trait´

e de g´ eostatistique applique´ e, Editions Technip, France, 1962-63

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Classical Geostatistical Approach

Numerical complexity of ˆ γ(rk) estimation is O(N2)

1 2 3 4 5 x 10

5

50 100 150 200 250 300 350 400 450 lag distance (m) Empirical semivariogram

WLS fit to theoretical variogram (spherical model)

1 2 3 4 5 x 10

5

50 100 150 200 250 300 350 400 450 lag distance (m) Semivariograms Empirical Model

• G. Matheron, Trait´

e de g´ eostatistique applique´ e, Editions Technip, France, 1962-63

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Classical Geostatistical Approach

Ordinary Kriging is a Best Linear Unbiased Estimator (BLUE)

ˆ X(s) =

m

• j=1

λjX(sj), sj ∈ B(s; rc), B : search neighborhood

Best: minimum (ensemble) mean square estimation error

{λ1, λ2, . . . , λm} = arg min

λ1,λ2,...,λm

• E
• ˆ

X(s) − X(s) 2

• λ1 + λ2 + . . . λm = 1
• Spatial weights follow from the linear system:

         γ(s1 − s1) γ(s1 − s2) . . . γ(s1 − sm) 1 γ(s2 − s1) γ(s2 − s2) . . . γ(s2 − sm) 1 . . . . . . . . . . . . . . . γ(sm − s1) γ(sm − s2) . . . γ(sm − sm) 1 1 1 . . . 1                 λ1 λ2 . . . λm µ        =        γ(s1 − s) γ(s2 − s) . . . γ(sm − s) 1       

• G. Matheron, Trait´

e de g´ eostatistique applique´ e, Editions Technip, France, 1962-63

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Classical Geostatistical Approach

OK Problems Computational cost of inverting covariance matrix ∝ O(N3) Empirical solution: use

• f kriging search radius

Implicit assumption: covariance truncation Optimality conditions: Gaussian data and knowledge of true semivariogram function Kriging radius

−2 2 4 x 10

5

1 2 3 4 5 6 x 10

5

X (m) Y (m)

• G. Matheron, Trait´

e de g´ eostatistique applique´ e, Editions Technip, France, 1962-63

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Classical Geostatistical Approach

OK Problems Computational cost of inverting covariance matrix ∝ O(N3) Empirical solution: use

• f kriging search radius

Implicit assumption: covariance truncation Optimality conditions: Gaussian data and knowledge of true semivariogram function Kriging radius

• G. Matheron, Trait´

e de g´ eostatistique applique´ e, Editions Technip, France, 1962-63

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 9/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Motivation for SSRF

In spatial statistics the “key players” are the data and the covariance model In statistical field theories the key players are spatial interactions and their coupling strengths (Gaussian, Landau - Ginzburg, Ising models) Idea: To use interaction-based models of spatial dependence inspired from statistical physics, i.e., Spartan Spatial Random Fields (SSRFs) Motivation: Flexible parametrization of spatial dependence & Efficient interpolation/simulation

1 No variograms? 2 Fast spatial prediction algorithms (no kriging system?)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 10/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Motivation for SSRF

In spatial statistics the “key players” are the data and the covariance model In statistical field theories the key players are spatial interactions and their coupling strengths (Gaussian, Landau - Ginzburg, Ising models) Idea: To use interaction-based models of spatial dependence inspired from statistical physics, i.e., Spartan Spatial Random Fields (SSRFs) Motivation: Flexible parametrization of spatial dependence & Efficient interpolation/simulation

1 No variograms? 2 Fast spatial prediction algorithms (no kriging system?)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 10/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Motivation for SSRF

In spatial statistics the “key players” are the data and the covariance model In statistical field theories the key players are spatial interactions and their coupling strengths (Gaussian, Landau - Ginzburg, Ising models) Idea: To use interaction-based models of spatial dependence inspired from statistical physics, i.e., Spartan Spatial Random Fields (SSRFs) Motivation: Flexible parametrization of spatial dependence & Efficient interpolation/simulation

1 No variograms? 2 Fast spatial prediction algorithms (no kriging system?)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 10/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Motivation for SSRF

In spatial statistics the “key players” are the data and the covariance model In statistical field theories the key players are spatial interactions and their coupling strengths (Gaussian, Landau - Ginzburg, Ising models) Idea: To use interaction-based models of spatial dependence inspired from statistical physics, i.e., Spartan Spatial Random Fields (SSRFs) Motivation: Flexible parametrization of spatial dependence & Efficient interpolation/simulation

1 No variograms? 2 Fast spatial prediction algorithms (no kriging system?)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 10/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Fluctuation-Gradient-Curvature (FGC) SSRF

Gibbs probability density function (PDF) f[X(s)] = e−H[X(s)] Z , H[X(s)] : energy functional, Z: partition function Z =

• DX(s)e−H[X(s)]

FGC energy functional - for simplicity assume E [X(s)] = 0 Hfgc[X(s)] = 1 2η0ξd

• D

ds

• [X(s)]2 + η1 ξ2 [∇X(s)]2 + ξ4

∇2X(s) 2 Properties: Gaussian, zero-mean, stationary, isotropic SRF

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 11/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Fluctuation-Gradient-Curvature (FGC) SSRF

Gibbs probability density function (PDF) f[X(s)] = e−H[X(s)] Z , H[X(s)] : energy functional, Z: partition function Z =

• DX(s)e−H[X(s)]

FGC energy functional - for simplicity assume E [X(s)] = 0 Hfgc[X(s)] = 1 2η0ξd

• D

ds

• [X(s)]2 + η1 ξ2 [∇X(s)]2 + ξ4

∇2X(s) 2 Properties: Gaussian, zero-mean, stationary, isotropic SRF

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 11/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Fluctuation-Gradient-Curvature (FGC) SSRF

Gibbs probability density function (PDF) f[X(s)] = e−H[X(s)] Z , H[X(s)] : energy functional, Z: partition function Z =

• DX(s)e−H[X(s)]

FGC energy functional - for simplicity assume E [X(s)] = 0 Hfgc[X(s)] = 1 2η0ξd

• D

ds

• [X(s)]2 + η1 ξ2 [∇X(s)]2 + ξ4

∇2X(s) 2 Properties: Gaussian, zero-mean, stationary, isotropic SRF

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 11/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Fluctuation-Gradient-Curvature (FGC) SSRF

Gibbs probability density function (PDF) f[X(s)] = e−H[X(s)] Z , H[X(s)] : energy functional, Z: partition function Z =

• DX(s)e−H[X(s)]

FGC energy functional - for simplicity assume E [X(s)] = 0 Hfgc[X(s)] = 1 2η0ξd

• D

ds

• [X(s)]2 + η1 ξ2 [∇X(s)]2 + ξ4

∇2X(s) 2 Properties: Gaussian, zero-mean, stationary, isotropic SRF FGC-SSRF Coefficients η0 : scale, η1 : stiffness, ξ: characteristic length; kc : spectral cutoff

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 11/43

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FGC-SSRF Energy Functional on Hypercubic Lattice

Grid structure = ⇒ Gauss-Markov random fields

Hfgc[X(s)] ∝ 1 2η0ξd

N

• n=1
• X 2(sn) + η1ξ2

d

• i=1

X(sn + ai ˆ ei) − X(sn) ai 2 +ξ4

d

• i=1

X(sn + ai ˆ ei) − 2X(sn) + X(sn − ai ˆ ei) a2

i

2 ˆ ei, i = 1, . . . , d : unit vectors in lattice directions ai : lattice steps

2 4 6 8 10 1 2 3 4 5 6 7 8 9 10

Hfgc[X(s)] = α0S0 + α1SG + α2Sc

Rue and Held, Gaussian Markov Random Fields: Theory and Applications, Chapman and Hall/CRC, 2005

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Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Covariance Function & Spectral Density

Covariance: G(r) = E [X(s) X(s + r)]. Fourier transform pair: ˜ G(k) =

• dr e−k·r G(r),

G(r) = 1 (2 π)d

• dk ek·r ˜

G(k). Covariance spectral density: ˜ G(k) = ✶kc≥κ(κ) η0ξd 1 + η1κ2ξ2 + κ4ξ4 , κ = k, ✶B(·) : indicator function, Permissibility conditions (Bochner’s theorem): For any kc: η0 > 0, ξ > 0, η1 > −2 For finite kc: η1 < −2, if kcξ <

• |η1|−∆

2

∆ =

• η2

1 − 4

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 13/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Covariance Function & Spectral Density

Covariance: G(r) = E [X(s) X(s + r)]. Fourier transform pair: ˜ G(k) =

• dr e−k·r G(r),

G(r) = 1 (2 π)d

• dk ek·r ˜

G(k). Covariance spectral density: ˜ G(k) = ✶kc≥κ(κ) η0ξd 1 + η1κ2ξ2 + κ4ξ4 , κ = k, ✶B(·) : indicator function, Permissibility conditions (Bochner’s theorem): For any kc: η0 > 0, ξ > 0, η1 > −2 For finite kc: η1 < −2, if kcξ <

• |η1|−∆

2

∆ =

• η2

1 − 4

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 13/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Covariance Function & Spectral Density

Covariance: G(r) = E [X(s) X(s + r)]. Fourier transform pair: ˜ G(k) =

• dr e−k·r G(r),

G(r) = 1 (2 π)d

• dk ek·r ˜

G(k). Covariance spectral density: ˜ G(k) = ✶kc≥κ(κ) η0ξd 1 + η1κ2ξ2 + κ4ξ4 , κ = k, ✶B(·) : indicator function, Permissibility conditions (Bochner’s theorem): For any kc: η0 > 0, ξ > 0, η1 > −2 For finite kc: η1 < −2, if kcξ <

• |η1|−∆

2

∆ =

• η2

1 − 4

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 13/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Covariance Function & Spectral Density

SPD: Positive stiffness SPD: Negative stiffness

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 13/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Covariance Function & Spectral Density

Covariance (d = 2): Positive stiffness Covariance (d = 2): Negative stiffness

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 13/43

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Spectral Representation of the Covariance Function

Spectral Representation (Inverse Hankel transform For isotropic covariance functions the following holds: G(r) = η0 ξd r (2πr)d/2 kc dκ κd/2Jd/2−1(κr) 1 + η1(κξ)2 + (κξ)4 Jd/2−1(r): Bessel function of the first kind of order d/2 − 1 For kc → ∞ the integral exists for d ≤ 3

• I. J. Schoenberg, “Metric spaces and completely monotone functions,” Ann. Math.,

39(4), 811841, 1938

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

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Spectral Representation of the Covariance Function

Unlimited band covariance function d = 1, kc → ∞ G(h) = η0 4 e−hβ2 cos(hβ1) β2 + sin(hβ1) β1

• ,

|η1| < 2 G(h) = η0 (1 + h) 4 eh , η1 = 2 G(h) = η0 2 ∆ e−h ω1 ω1 − e−h ω2 ω2

• ,

η1 > 2 h = |r|/ξ : normalized lag, β1,2 =

• |2∓η1|

4

1/2 , ω1,2 =

• |η1∓∆|

2

1/2 , ∆ = |η2

1 − 4|

1 2

Hristopulos and Elogne, IEEE Trans. Information Theory, 53(12), 4667–4679, 2007

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

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Spectral Representation of the Covariance Function

1 2 3 4 5 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

Distance lag Correlation

η1=−1.9 η1=−1 η1=1 η1=16

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Spectral Representation of the Covariance Function

Unlimited band covariance function d = 3, kc → ∞ G(h) = η0 e−hβ2 ∆ sin (hβ1) h

• ,

|η1| < 2 G(h) = η0 4 e−h, η1 = 2 G(h) = 1 2 ∆ e−hω1 − e−hω2 h

• ,

η1 > 2 h = r/ξ, β1,2 =

• |2∓η1|

4

1/2 , ω1,2 =

• |η1∓∆|

2

1/2 , ∆ = |η2

1 − 4|

1 2

Hristopulos and Elogne, IEEE Trans. Information Theory, 53(12), 4667–4679, 2007

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Spectral Representation of the Covariance Function

0.2 0.4 0.6 0.8 1 −0.2 0.2 0.4 0.6 0.8 1 1.2

Distance lag Correlation

η1=−1 η1=2 η1=8 η1=16

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 14/43

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FGC-SSRF Realizations d = 1

η1 = −1.999 η1 = −1

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FGC-SSRF Realizations d = 1

η1 = 0.15 η1 = 15

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 15/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Length Scales

Definitions

1 Integral range:

ℓc . =

• dr G(r)

G(0) 1/d =Adσ−2/d ˜ G(0) 1/d

2 Correlation length:

rc . =

• dr r 2 G(r)
• dr G(r)

1/2 =

• d2 ˜

G(k)/dk2 2˜ G(k)

• k=0
• =
• |η1| ξ

2-D Integral range FGC-SSRF Integral Range

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 5 10 15 20 25 30 35 kc ξ A2(η1,kc ξ)

η1=−1.8 η1=1 η1=2 η1=4

Hristopulos and ˇ Zukoviˇ c, Stoch. Env.

• Res. Risk Assess., 25, 2511 (2011)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 16/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Length Scales

Definitions

1 Integral range:

ℓc . =

• dr G(r)

G(0) 1/d =Adσ−2/d ˜ G(0) 1/d

2 Correlation length:

rc . =

• dr r 2 G(r)
• dr G(r)

1/2 =

• d2 ˜

G(k)/dk2 2˜ G(k)

• k=0
• =
• |η1| ξ

2-D Integral range FGC-SSRF Integral Range

2 4 6 8 10 12 14 16 18 20 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Ratio of Integral Range to Characteristic Length Shape parameter η1 A2(η1) d=2

Hristopulos and ˇ Zukoviˇ c, Stoch. Env.

• Res. Risk Assess., 25, 2511 (2011)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 16/43

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FGC-SSRF Parameter Inference

The FGC-SSRF joint PDF belongs to the exponential family, i.e., f[X(s)] ∝ e−H[X(s)], where H[X(s)] ∝ S0 + α1SG + α2SC We use a modified method of moments: the PDF is determined by means of ˆ S0, ˆ SG, ˆ SC which are sample-based estimators of E [S0] , E [SG] , E [SC] Lattice estimators are based on sample averages, e.g., ˆ SG = 1 N

N

• n=1

d

• i=1

X(sn + ai ˆ ei) − X(sn) ai 2 For irregularly sampled data we adapt kernel averaging methods (Watson-Nadaraya estimators)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 17/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Parameter Inference

The FGC-SSRF joint PDF belongs to the exponential family, i.e., f[X(s)] ∝ e−H[X(s)], where H[X(s)] ∝ S0 + α1SG + α2SC We use a modified method of moments: the PDF is determined by means of ˆ S0, ˆ SG, ˆ SC which are sample-based estimators of E [S0] , E [SG] , E [SC] Lattice estimators are based on sample averages, e.g., ˆ SG = 1 N

N

• n=1

d

• i=1

X(sn + ai ˆ ei) − X(sn) ai 2 For irregularly sampled data we adapt kernel averaging methods (Watson-Nadaraya estimators)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 17/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Parameter Inference

The FGC-SSRF joint PDF belongs to the exponential family, i.e., f[X(s)] ∝ e−H[X(s)], where H[X(s)] ∝ S0 + α1SG + α2SC We use a modified method of moments: the PDF is determined by means of ˆ S0, ˆ SG, ˆ SC which are sample-based estimators of E [S0] , E [SG] , E [SC] Lattice estimators are based on sample averages, e.g., ˆ SG = 1 N

N

• n=1

d

• i=1

X(sn + ai ˆ ei) − X(sn) ai 2 For irregularly sampled data we adapt kernel averaging methods (Watson-Nadaraya estimators)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 17/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Parameter Inference

The FGC-SSRF joint PDF belongs to the exponential family, i.e., f[X(s)] ∝ e−H[X(s)], where H[X(s)] ∝ S0 + α1SG + α2SC We use a modified method of moments: the PDF is determined by means of ˆ S0, ˆ SG, ˆ SC which are sample-based estimators of E [S0] , E [SG] , E [SC] Lattice estimators are based on sample averages, e.g., ˆ SG = 1 N

N

• n=1

d

• i=1

X(sn + ai ˆ ei) − X(sn) ai 2 For irregularly sampled data we adapt kernel averaging methods (Watson-Nadaraya estimators)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 17/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

The Nadaraya-Watson Kernel Average

Definition We use kernel functions K(r), r ∈ Rd such that: (i) K(r) ∈ R and K(r) ≥ 0, (ii)

• dr K(r) = 1 and (iii) K(r) = K(−r).

If K(r) is a kernel function, then Kh = h−d K(r/h) is also a kernel function. Kernel averages of field values and two-point distances ui,j(h) = Kh(si − sj) N

i=1

N

j=1,j=i Kh(si − sj)

Φ(Xi, Xj) = N

i=1

N

j=1,j=i Φ(Xi, Xj) ui,j(h)

si − sjβ = N

i=1

N

j=1,j=i si − sjβ ui,j(h)

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 18/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Typical” Step and Kernel Bandwidth Selection

1 Find the Delaunay triangulation of the sampling network:

computational cost is O(N logN)

2 Estimate ˆ

α using the ℓ2−norm of the near-neighbor distances (Delaunay triangle edges) of the sampling points ˆ α =

• 1

N0

N0

i=1 ∆2 i

1/2

3 Determine a uniform bandwidth by solving consistency relations,

e.g., for curvature constraint si − sj4h = ˆ α4

4 Adaptive bandwidth estimation is also possible

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 19/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Typical” Step and Kernel Bandwidth Selection

1 Find the Delaunay triangulation of the sampling network:

computational cost is O(N logN)

2 Estimate ˆ

α using the ℓ2−norm of the near-neighbor distances (Delaunay triangle edges) of the sampling points ˆ α =

• 1

N0

N0

i=1 ∆2 i

1/2

3 Determine a uniform bandwidth by solving consistency relations,

e.g., for curvature constraint si − sj4h = ˆ α4

4 Adaptive bandwidth estimation is also possible

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 19/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Typical” Step and Kernel Bandwidth Selection

1 Find the Delaunay triangulation of the sampling network:

computational cost is O(N logN)

2 Estimate ˆ

α using the ℓ2−norm of the near-neighbor distances (Delaunay triangle edges) of the sampling points ˆ α =

• 1

N0

N0

i=1 ∆2 i

1/2

3 Determine a uniform bandwidth by solving consistency relations,

e.g., for curvature constraint si − sj4h = ˆ α4

4 Adaptive bandwidth estimation is also possible

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 19/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Typical” Step and Kernel Bandwidth Selection

1 Find the Delaunay triangulation of the sampling network:

computational cost is O(N logN)

2 Estimate ˆ

α using the ℓ2−norm of the near-neighbor distances (Delaunay triangle edges) of the sampling points ˆ α =

• 1

N0

N0

i=1 ∆2 i

1/2

3 Determine a uniform bandwidth by solving consistency relations,

e.g., for curvature constraint si − sj4h = ˆ α4

4 Adaptive bandwidth estimation is also possible

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 19/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Typical” Step and Kernel Bandwidth Selection

Example 1 - Sampling pattern

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Uniform step calculation ˆ α =

• 1

N0

N0

i=1 ∆2 i

1/2

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 19/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Typical” Step and Kernel Bandwidth Selection

Example 1 - Sampling pattern

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Uniform step calculation

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

ˆ α =

• 1

N0

N0

i=1 ∆2 i

1/2

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 19/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

FGC-SSRF Interpolation

FGC Spartan predictor Given the sample (X1, . . . , XN), where Xi = X(si), estimate ˆ X(zp) at point zp ∋ (s1, . . . , sN) Local interactions ⇒ Efficient prediction: Estimate ˆ X(zp) maximizes the joint PDF, i.e., minimizes the FGC energy functional over D ∪ zp An explicit expression follows for the predictor ˆ X(zp) =

N

• i=1

φi(zp) X(si) The linear weights φi(zp) depend on network geometry and the kernel function

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 20/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Spartan” Interpolators

Use of simplifying assumptions Various interpolation methods can be formulated based on the FGC-SSRF energy function. We use the following terms: Spartan or FGC-SSRF: An interpolator based on the FGC energy functional General Uniform Bandwidth Spartan (GUBS): Explicit summations over the sampling points and uniform kernel bandwidth Asymptotic General Uniform Bandwidth Spartan (AGUBS): Uniform kernel bandwidth and discrete summations replaced by integrals Asymptotic Locally Adaptive Spartan (ALAS): Locally tuned kernel bandwidth that adapts to sampling density variations

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 21/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Spartan” Interpolators

Use of simplifying assumptions Various interpolation methods can be formulated based on the FGC-SSRF energy function. We use the following terms: Spartan or FGC-SSRF: An interpolator based on the FGC energy functional General Uniform Bandwidth Spartan (GUBS): Explicit summations over the sampling points and uniform kernel bandwidth Asymptotic General Uniform Bandwidth Spartan (AGUBS): Uniform kernel bandwidth and discrete summations replaced by integrals Asymptotic Locally Adaptive Spartan (ALAS): Locally tuned kernel bandwidth that adapts to sampling density variations

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 21/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Spartan” Interpolators

Use of simplifying assumptions Various interpolation methods can be formulated based on the FGC-SSRF energy function. We use the following terms: Spartan or FGC-SSRF: An interpolator based on the FGC energy functional General Uniform Bandwidth Spartan (GUBS): Explicit summations over the sampling points and uniform kernel bandwidth Asymptotic General Uniform Bandwidth Spartan (AGUBS): Uniform kernel bandwidth and discrete summations replaced by integrals Asymptotic Locally Adaptive Spartan (ALAS): Locally tuned kernel bandwidth that adapts to sampling density variations

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 21/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

“Spartan” Interpolators

Use of simplifying assumptions Various interpolation methods can be formulated based on the FGC-SSRF energy function. We use the following terms: Spartan or FGC-SSRF: An interpolator based on the FGC energy functional General Uniform Bandwidth Spartan (GUBS): Explicit summations over the sampling points and uniform kernel bandwidth Asymptotic General Uniform Bandwidth Spartan (AGUBS): Uniform kernel bandwidth and discrete summations replaced by integrals Asymptotic Locally Adaptive Spartan (ALAS): Locally tuned kernel bandwidth that adapts to sampling density variations

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 21/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Spartan (FGC-SSRF) Interpolation

Equations of FGC General Uniform Bandwidth Spartan predictor

Linear coefficients: φi(zp) = ν(si, zp) N

i=1 ν(si, zp)

Network weights: ν(si, zp) = 4

q=1 bqui,p(hq) + (N + 1)−2

Normalized kernel weights: ui,j(hq) =

Khq (si −sj ) N

i=1

N

j=1,j=i Khq (si −sj )

FGC - SSRF coeffs: b1 = βd η1

ξ2 ˆ α2 , b2 = δd g1 ξ4 ˆ α4 , b3 = −ζd g2 ξ4 ˆ α4 , b4 = −βd ξ4 ˆ α4

Curvature bias corrections: g1, g2 = 1 + o(1) Step - bandwidth relations: ˆ α ≈ h1 √ B2, ˆ α ≈ h2

4

√ B4, h3 = √ 2 h2, h4 = 2 h2 Kernel moment ratio: Bp = R

0 ds K(s) sd+p−1 R 0 ds K(s) sd−1

Hristopulos and Elogne (2009), IEEE TSP, 57(9), 3475.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 22/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Numerical Complexity of FGC-SSRF interpolation

Numerical complexity of Ordinary kriging (OK) vs Spartan

1 OK: ∝ O(N3 + P N2) or (using search radius) ∝ O(P M

3)

2 FGC (GUBS): ∝ O(N2 + P N) or (asymptotic GUBS) ∝ O(P N)

Comparison of CPU times: ALAS vs OK

Table: Gaussian RF is sampled at N + P randomly selected points inside a square domain. P = 1000 points are removed and used for prediction.1 Method N = 500 N = 700 N = 1000 N = 2000 N = 3000 Spartan 1.5 sec 2.1 sec 2.8sec 5.5 sec 7.3sec Kriging 398.7 sec 821.1 sec 1737.2 sec 7588.3 sec 18250.1 sec 6.64 min 13.68 min 28.95 min 2.10 hr 5.06 hr

1In Matlabenvironment running under Windows XP

, on a laptop with AMD Turion processor, clock speed 1.6 GHz, 960 MB RAM

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 23/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Time-series Interpolation

1 Hfgc[X; θ] =

1 2η0ξ

∞

−∞ dt

• [X(t)]2 + η1 ξ2[ ˙

X(t)]2 + ξ4[¨ X(t)]2

2 Hfgc [X; θ] = 1

2 X(ti) Jx(ti, tj; θ) X(tj)

3 Jx(ti, tj; θ) =

1 η0ξ

• J0(ti, tj) + η1

ξ2 α2 J1(ti, tj) + ξ4 α4 J2(ti, tj)

• Precision matrix

J1=        1 −1 0 · · · −1 2 −1 · · · ... ... ... 0 · · · −1 2 −1 0 · · · 0 −1 1        , J2=          1 −2 1 0 · · · −2 5 −4 1 · · · 1 −4 6 −4 1 ... ... ... ... 0 · · · 1 −4 5 −2 0 · · · 1 −2 1         

Interpolation

50 100 150 200 250 300 350 20 40 60 80 100 120

time [α = 40 min.] Aerosol concentration [ µg/m3]

×: Validation •: Training

• : Estimates

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 24/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Time-series Interpolation

ˆ X(zp) = −

• tl ∈V(zp)

Jx(θ; tl, zp) Jx(θ; zp, zp) X(tl), p = 1, ..., P

Spartan vs. Kriging Comparison

Spartan-Kriging performance comparison on aerosol concentration data. Model parameters are inferred by MLE, using training set of 121 points. Statistics are calculated on validation set of 232

• points. Category (i, j) includes points in the training set with i nearest and j next-nearest neighbors

(2,2) (1,2) (0,2) (0,1) (0,0) (1,1) (1,0) (2,0) (2,1) MAE SP 1.75 2.67 3.85 5.30 8.15 2.88 3.24 2.05 2.03 OK 1.80 2.68 3.87 5.32 8.10 2.90 3.25 2.06 2.06 MARE [%] SP 4.3 6.6 9.3 12.5 19.1 6.8 7.5 4.5 4.9 OK 4.4 6.7 9.4 12.6 19.0 6.9 7.6 4.5 5.0 MRE [%] SP −0.4 −1.1 −1.4 −2.6 −5.2 −0.9 −0.9 −0.2 −0.8 OK −0.4 −0.9 −1.3 −2.5 −5.0 −0.8 −0.8 −0.2 −0.6 RMSE SP 2.15 3.72 5.14 7.44 11.04 4.23 4.65 2.92 2.93 OK 2.19 3.73 5.12 7.46 10.97 4.25 4.65 2.92 2.98

ˇ Zukoviˇ c & Hristopulos, Atmospheric Environment, 42 (2008) 7669-7678

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 24/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Unconditional Simulations

Spectral Methods

Regular lattice: X(sn) = IFFT

• u(ki)
• ˜

G(ki) 1/2 , u(ki) ∼ N(0, 1) Unstructured lattice: X(sn) ≈ σ

• 2

NM

NM

• p=1

cos (kp · sn + ϕp), φp ∼ U(0, 2π), PDF(k) ∝ ˜ G(k)

η0 = 1, η1 = 0.2, ξ = 5 η0 = 3, η1 = −0.2, ξ = 50

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 25/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Unconditional Simulations

3D Simulations η0 = 1, η1 = 0.2, ξ = 5 η0 = 3, η1 = −0.2, ξ = 20

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 25/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Conditional Simulations

Spectral simulation + SSRF interpolation: Z c(s) = Z u(s) +

• ˆ

Z(s) − ˆ Z u(s)

• .

Conditional simulations 300 × 300 grid

20 40 60 80 100 120 140 160 100 200 200 400 600 100 200 100 200 200 400 600 100 200 100 200 200 400 600 100 200 100 200 200 400 600 100 200 100 200 200 400 600 100 200

Mean & StD of 1000 states; Tcpu ≈ 20s per state

−50 50 100 150 200 250 100 200 300 400 500 600 60 70 80 90 100 110 120 130 140 150 −50 50 100 150 200 250 100 200 300 400 500 600 2 4 6 8 10 12 14 16 18 20 22

Mean based on 1000 Simulations Standard Deviation based on 1000 Simulations

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 26/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Karhunen-Lo` eve Expansions of SSRFs

Karhunen-Lo` eve Theorem

1 A second-order X(s) with continuous covariance covariance G(s, s′)

can be expanded on a closed and bounded domain D as: X(s) = mx(s) +

∞

• m=1

√ λm cm ψm(s). The convergence is uniform on D.

2 The λm and ψm(s) are respectively, eigenvalues and eigenfunctions of

the covariance operator, that satisfy the Fredholm integral equation

• D

ds′G(s, s′) ψm(s′) = λm ψm(s′).

3 The cm are zero-mean, uncorrelated random variables, i.e, E [cm] = 0

and E [cm cn] = δn,m, ∀n, m ∈ N.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 27/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Karhunen-Lo` eve Expansions of SSRFs

η0 = 2, η1 = −1.5, ξ = 5 η0 = 2, η1 = 0, ξ = 5 η0 = 2, η1 = 1.5, ξ = 5 η0 = 2, η1 = 15, ξ = 5

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 27/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Karhunen-Lo` eve Expansions of SSRFs

SRRF Variance Evolution versus number of ordered eigenvalues

10 20 30 40 50 60 70 80 90 100 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 Mode number σ η1=−1.5 η1=0 η1=1.5 η1=−15

Figure: SSRF standard deviation of K-L simulation at s0 = (25, 25) on 100 × 100 square domain

with pinned boundaries. SSRF parameters: η0 = 2, ξ = 5 and η1 = (−1.5, 0, 1.5, 15)T .

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 27/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data

1 Two data sets of radioactivity gamma dose rates (GDR) over

• Germany. GDR is measured in nanoSievert per hour (nSv/h)

2 First data set: “Routine” measurements of background radiation 3 Second data set: Simulated radioactivity release; a few hot spots 4 Data are available at: http://www.ai-geostats.org/bin/

view/AI_GEOSTATS/AI_GEOSTATSData

Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data

1 Two data sets of radioactivity gamma dose rates (GDR) over

• Germany. GDR is measured in nanoSievert per hour (nSv/h)

2 First data set: “Routine” measurements of background radiation 3 Second data set: Simulated radioactivity release; a few hot spots 4 Data are available at: http://www.ai-geostats.org/bin/

view/AI_GEOSTATS/AI_GEOSTATSData

Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data

1 Two data sets of radioactivity gamma dose rates (GDR) over

• Germany. GDR is measured in nanoSievert per hour (nSv/h)

2 First data set: “Routine” measurements of background radiation 3 Second data set: Simulated radioactivity release; a few hot spots 4 Data are available at: http://www.ai-geostats.org/bin/

view/AI_GEOSTATS/AI_GEOSTATSData

Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data

1 Two data sets of radioactivity gamma dose rates (GDR) over

• Germany. GDR is measured in nanoSievert per hour (nSv/h)

2 First data set: “Routine” measurements of background radiation 3 Second data set: Simulated radioactivity release; a few hot spots 4 Data are available at: http://www.ai-geostats.org/bin/

view/AI_GEOSTATS/AI_GEOSTATSData

Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data

Locations of training (+) and prediction (dots) sets

100 200 100 200 300 400 500 600 x (km) y (km)

Histogram of routine training data and best-fit normal PDF

Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data

Locations of training (+) and prediction (dots) sets

100 200 100 200 300 400 500 600 x (km) y (km)

Histogram of routine training data and best-fit normal PDF

Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Brief Description of Spatial Interpolation Comparison (SIC) 2004 Data

Locations of training (+) and prediction (dots) sets

100 200 100 200 300 400 500 600 x (km) y (km)

Bubble plot of emergency training data

100 200 100 200 300 400 500 600 x (km) y (km) 200 400 600 800 1000 1200 1400

Elogne, Hristopulos, Varouchakis, Stoch. Environ. Res. Risk Assess. (2008) 22:633-646

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 28/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Comparison of “Spartan” & Ordinary Kriging

Cross validation - SIC 2004 Background radiation data OK versus True

60 80 100 120 140 160 180 60 80 100 120 140 160 180 OK True values

Spartan - adaptive vs True

60 80 100 120 140 160 180 60 80 100 120 140 160 180 ALAS True values

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 29/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Comparison of “Spartan” & Ordinary Kriging

Ordinary kriging

x (km) y (km) OK 100 200 100 200 300 400 500 600 70 80 90 100 110 120 130 140

Spartan - adaptive

x (km) y (km) ALAS 100 200 100 200 300 400 500 600 70 80 90 100 110 120 130 140

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 29/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Comparison of “Spartan” & Ordinary Kriging

Cross validation - SIC 2004 emergency radiation data OK versus True

500 1000 1500 200 400 600 800 1000 1200 1400 OK True values

Spartan - adaptive vs True

500 1000 1500 200 400 600 800 1000 1200 1400 ALAS True values

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 30/43

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Comparison of “Spartan” & Ordinary Kriging

OK contour plots

x (km) y (km) OK 100 200 100 200 300 400 500 600 200 400 600 800 1000 1200

ALAS contour plots

x (km) y (km) ALAS 100 200 100 200 300 400 500 600 100 200 300 400 500 600 700 800 900 1000

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 30/43

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Comparison of “Spartan” & Ordinary Kriging

OK contour plots - zoom on spreading plume ALAS contour plots - zoom on spreading plume

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 30/43

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Cross validation - SIC 2004 data set

Table: ME: Mean error; MAE: Mean absolute error; RMSE: Root mean square error; RMSRE: Root mean square relative error; RS: Spearman rank correlation coefficient. δxmin=ˆ xmin − xmin; δxmax=ˆ xmax − xmax

. ME MAE MARE RMSE RMSRE RS δxmin δxmax SIC 2004 Background data set OK

• 1.30

9.08 0.09 12.42 0.12 0.77 12.55

• 54.61

ALAS2

• 1.38

9.35 0.09 12.74 0.12 0.75 1.20

• 44.79

SIC 2004 Emergency data set OK 0.80 21.91 0.16 77.83 0.43 0.73 1.20

• 900.77

ALAS 3 0.34 20.74 0.15 74.43 0.33 0.78 1.20

• 1045.41

2with quadratic kernel 3with tricubic kernel

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Brief Introduction to Anisotropy

Simulated anisotropic spatial random field Schematic of isotropy restoring transformation

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The Covariance Tensor Identity (CTI) Method

The covariance tensor identity (Swerling, 1962) Assume an anisotropic, differentiable, second-order stationary spatial random field (SRF) X(s) with a covariance function G(r). Then: Qi,j ≡ E ∂X(s) ∂si ∂X(s) ∂sj

• = −∂2G(r)

∂ri ∂rj

• r=(0,0)

Expression for the covariance Hessian matrix (CHM) in 2D Q11 Q12 Q11 Q12

• = σ2

xζ2

ξ2

1

• cos2 θ + R2 sin2 θ

sin θ cos θ(1 − R2) sin θ cos θ(1 − R2) R2 cos2 θ + sin2 θ

• Hristopulos (2002). Stoch. Environ. Res. Risk Assess., 16(1), 43-62

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Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Explicit CTI Solution

Relation between CHM and (R, θ) qdiag := Q22 Q11 = R2 + tan2 θ 1 + R2 tan2 θ qoff := Q21 Q11 = tan θ(1 − R2) 1 + R2 tan2 θ Solution of nonlinear equations θ = 1 2 arctan

• 2ˆ

qoff 1 − ˆ qdiag

• ,

θ ∈

• −π

4, π 4

• R2 = 1 +

1 − ˆ qdiag ˆ qdiag − (1 + ˆ qdiag) cos2 θ, R ∈ [0, ∞).

Chorti and Hristopulos, IEEE Trans. Signal Proc. 56(10), 4738-4751, 2008

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Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Explicit CTI Solution

Relation between CHM and (R, θ) qdiag := Q22 Q11 = R2 + tan2 θ 1 + R2 tan2 θ qoff := Q21 Q11 = tan θ(1 − R2) 1 + R2 tan2 θ Solution of nonlinear equations θ = 1 2 arctan

• 2ˆ

qoff 1 − ˆ qdiag

• ,

θ ∈

• −π

4, π 4

• R2 = 1 +

1 − ˆ qdiag ˆ qdiag − (1 + ˆ qdiag) cos2 θ, R ∈ [0, ∞).

Chorti and Hristopulos, IEEE Trans. Signal Proc. 56(10), 4738-4751, 2008

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 34/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Sample-based Estimation of CHM

On regular grids the CHM is estimated using discretization of the first-order partial derivatives, e.g. on square grids of step a: ˆ Qij = 1 N

N

• k=1

X(sk + a ei) − X(sk) a X(sk + a ej) − X(sk) a where ei, ej are the unit vectors in the respective directions For scattered data, estimation of partial derivatives is based on :

1 Interpolation of scattered data on background grid followed by finite

differencing

2 or, estimation of derivatives using Savitzky-Golay (SG) polynomial

filters

3 or, SG filtering is applied directly to the scattered data

Interpolation on background grid is performed using the bilinear, bicubic and biharmonic spline methods

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 35/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Sample-based Estimation of CHM

On regular grids the CHM is estimated using discretization of the first-order partial derivatives, e.g. on square grids of step a: ˆ Qij = 1 N

N

• k=1

X(sk + a ei) − X(sk) a X(sk + a ej) − X(sk) a where ei, ej are the unit vectors in the respective directions For scattered data, estimation of partial derivatives is based on :

1 Interpolation of scattered data on background grid followed by finite

differencing

2 or, estimation of derivatives using Savitzky-Golay (SG) polynomial

filters

3 or, SG filtering is applied directly to the scattered data

Interpolation on background grid is performed using the bilinear, bicubic and biharmonic spline methods

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 35/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Outline of Steps Involved in Joint PDF Approximation

1 Approximate the JPDF fˆ Q11,ˆ Q12,ˆ Q22 with multivariate Gaussian

(Central Limit Theorem + short-range correlations).

2 Evaluate the sequence of transformations:

fˆ

Q11,ˆ Q12,ˆ Q22 → fˆ qdiag,ˆ qoff → fˆ θ,ˆ R 3 Neglecting covariance-specific corrections in CovQ a model

independent approximation of the JPDF is obtained

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Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Detection of Anisotropy Change in SIC2004 Data

GDR Histogram: routine (background) data

Histogram of cluster 1

field Frequency 60 80 100 120 140 160 10 20 30 40

GDR Histogram: emergency (simulated release) data

Histogram of cluster 1

field Frequency 500 1000 1500 50 100 150 200

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Anisotropy Estimates of GDR data - SIC 2004

Anisotropy statistics for “background” and “emergency” cases CTI estimates Anisotropy statistics ˆ R ˆ θ Background radioactivity 1.2 −36◦ Simulated accidental release 0.68 −39◦ Is the difference significant? Comparison of confidence regions for ˆ R, ˆ θ can provide an answer

Petrakis and Hristopulos, arxiv: 1203.5010v1, 2012

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Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Anisotropy Estimates of GDR data - SIC 2004

Anisotropy statistics for “background” and “emergency” cases CTI estimates Anisotropy statistics ˆ R ˆ θ Background radioactivity 1.2 −36◦ Simulated accidental release 0.68 −39◦ Is the difference significant? Comparison of confidence regions for ˆ R, ˆ θ can provide an answer

Petrakis and Hristopulos, arxiv: 1203.5010v1, 2012

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Anisotropy Estimates of GDR data - SIC 2004

Figure: Comparison of the JPDFs for the normal and emergency data sets

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Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Interpolation of missing lattice data: Definitions

Define a series of discretization thresholds tq, q = 1, . . . , Nc Threshold-dependent sample S(q) ⊂ D & prediction domains P(q) ⊂ D Spin assignments: s(q)

i

= 1 × sign(Xi − tq) Sample correlation energy E(q)(S(q)) = s(q)

i

s(q)

j

i,j, i = 1, . . . , N Total correlation energy E(q,m)(D) = s(q,m)

i

s(q,m)

j

i,j, i = 1, . . . , N + P Define cost functional (m: state index) U(q,m) =

• 1 − E(q,m)(D)/E(q)(S(q))

2 , E(q)(S(q)) = 0 U(q,m) = E(q,m)(D)2, E(q)(S(q)) = 0.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 39/43

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Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Schematic of spin state assignment

Figure: Majority rule assignment of initial spin values using adaptive

• stencil. NaN: sites of missing data

Schematic of interacting pairs

Figure: Sample sites (solid circles) and nearest-neighbor pairs (solid circles linked with bold lines)

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Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Simplified “spin” algorithm for missing lattice data

1 Set q → 1; initialize discrete index 2 Discretize sample data w.r. to tq : s(q)

i

→ ±1, i = 1, . . . , N

3 Calculate sample correlation energy E(q)(S(q)) 4 Set m → 1; initialize missing spins ˆ

s(q,m)

i

, i = N + 1, . . . , N + P

5 Minimize U(q,m) using greedy Monte Carlo

(ˆ sq

1, . . . ˆ

sq

p) = arg min sq

1 ,...sq p

U(sq

1, . . . sq p|Sq)

6 Assign spins ˆ

s(q,mf )

i

= −1 the label q and include in S(q+1)

7 q → q + 1; While q ≤ Nc goto 2

ˇ Zukoviˇ c & Hristopulos, Phys. Rev. E 80, 011116; J. Stat. Mech., P02023 (2009)

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Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Digital image: 512 × 512 grid, 256 levels, 33% missing pixels Restored image using “spin-interaction” model

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 39/43

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Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Digital image: 512 × 512 grid, 256 levels, 90% missing pixels Restored image using GC method

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 39/43

Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Digital image: 256 × 256 grid, 256 levels

50 100 150 200 250 50 100 150 200 250

50 100 150 200

Detail of the moon’s surface (see rectangle on the left)

5 10 15 20 25 30 5 10 15 20 25 30

50 100 150 200

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 39/43

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Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Image thinned by 66%

50 100 150 200 250 50 100 150 200 250

Detail thinned by 66%

5 10 15 20 25 30 5 10 15 20 25 30

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Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Moon reconstruction using GC method

50 100 150 200 250 50 100 150 200 250

Detail reconstruction using GC method

5 10 15 20 25 30 5 10 15 20 25 30

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Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Rainfall map (Indonesia)

x y

10 20 30 40 50 10 20 30 40 50

100 200 300 400 500 600 700 800

Map with missing block

x y

10 20 30 40 50 10 20 30 40 50

100 200 300 400 500 600 700 800 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 39/43

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Non-Gaussian distributions: Discrete “Spin” & Gradient-Curvature Models

Reconstruction using kriging

x y

10 20 30 40 50 10 20 30 40 50

100 200 300 400 500 600 700 800

Reconstruction using GC method

x y

10 20 30 40 50 10 20 30 40 50

100 200 300 400 500 600 700 800 Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 39/43

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Summary

Advantages of SSRF interpolation and simulation of spatial processes

Flexible spatial model able to incorporate physical or intuitive constraints in the joint PDF Explicit Spartan interpolation schemes (FGC: GUBS, AGUBS, ALAS) Computational efficiency (fast parameter estimation & interpolation) Reduced requirements for user input (kernel functions)

Disadvantages

Spartan predictor needs locally adaptive bandwidth to become exact interpolator Prediction variance is not guaranteed to be optimal; bandwidth tuning? Negative rigidity coefficient may lead to non-permissible functional for irregular sampling patterns

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Intro Geostats SSRF Infer Interpolation 1-D Simula Compar Aniso Spin Conclu Biblio

Summary

Advantages of SSRF interpolation and simulation of spatial processes

Flexible spatial model able to incorporate physical or intuitive constraints in the joint PDF Explicit Spartan interpolation schemes (FGC: GUBS, AGUBS, ALAS) Computational efficiency (fast parameter estimation & interpolation) Reduced requirements for user input (kernel functions)

Disadvantages

Spartan predictor needs locally adaptive bandwidth to become exact interpolator Prediction variance is not guaranteed to be optimal; bandwidth tuning? Negative rigidity coefficient may lead to non-permissible functional for irregular sampling patterns

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 40/43

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Future Directions

1 Development of constrained simulation in 2D − 3D 2 Extension to non-stationary covariance functions 3 Impact of kernel function on prediction and prediction variance 4 Extension to non-Gaussian Spartan PDFs 5 Extension to space-time models 6 Links with Gaussian processes & machine learning (Bayesian

predictive framework)

7 Development of high-dimensional FGC kernel functions

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Thank you for your attention!

Collaborations with Dr. S. Elogne, Dr. M. ˇ Zukoviˇ c, Dr. A. Chorti, Mr.

• A. Pavidis, Mr. M. Petrakis, Mr. I. Spiliopoulos and Mr. E. Varouchakis

are acknowledged

Past research partly funded by the European projects SPATSTAT (Marie Curie) and Intamap. Current research funded by project SPARTA implemented under the ”ARISTEIA” Action of the

• perational programme ”Education and Lifelong 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 New

Class 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 random

fields,” in geoENV VI - Geostatistics for Environmental Applications, pp. 477-488 (ed. by A. Soares et al.) 512p.

• A. Chorti and D. T. Hristopulos (2008). “Non-parametric Identification of Anisotropic (Elliptic)

Correlations in Spatially Distributed Data Sets” IEEE Trans. Signal Proc., 56(10), 4738-4751.

• S. Elogne, D. T. Hristopulos, M. Varouchakis (2008). “An application of Spartan spatial random

fields in environmental mapping: focus on automatic mapping capabilities,” Stoch. Envir. Res. Risk A., 22(5), 633-646.

• M. ˇ

Zukoviˇ c, and D. T. Hristopulos (2009a). “The method of normalized correlations: a fast parameter estimation method for random processes and isotropic random fields that focuses on short-range dependence,” Technometrics, 15(2), 173-185.

• D. T. Hristopulos and S. N. Elogne (2009). “Computationally efficient spatial interpolators based on

Spartan spatial random fields”, IEEE Trans. Signal Proc., 57(9), 3475-3487.

• M. ˇ

Zukoviˇ c, and D. T. Hristopulos (2009b). “Classification of missing values in spatial data using spin models,” Phys. Rev. E, 80(1), 011116.

Dionisis Hristopulos: dionisi@mred.tuc.gr ICERM Uncertainty Quantification Workshop Spatial Processes Based on Local-Interaction Energy Functionals 43/43