Geometrically frustrated spin systems Ising model with ...

Modified planar rotator model for efficient spatial prediction Milan Žukovič Institute of Physics P.J. Šafárik University in Košice

Dionisios Hristopulos Geostatistics research unit Technical University of Crete

Modified planar rotator

IC-MSQUARE, Mykonos, Jun 5-8, 2015

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Outline

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Spatial prediction problem  “Interaction” based approach 

Modified planar rotator model

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Results

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Spatial correlations

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Prediction of missing data

Summary



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Spatial prediction problem  -

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Typical problem in geostatistics: Available sample data ZS={zi} at GS={(xi, yi)} Task: to predict values ZP at prediction points GP Classical geostatistical methods (e.g. kriging) Computationally intensive for large data Usually assume multivariate normal distribution Require human (subjective) input



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“Interaction” based approach 

Short-range interaction based approach [1]

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Computationally efficient Applicable to gridded and scattered data Requires minimal user input Limited to Gaussian data

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Spin model based on “energy matching” approach [2]

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Heuristic assumption of the existence of relevant correlations Discrete predictions even for continuous data

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[1] D.T. Hristopulos, Gibbs random field models for geostatistical applications, SIAM J. Scient. Comput., 24(6):2125-2162, (2003). [2] M. Žukovič and D T. Hristopulos, Phys. Rev. E 80 011116 (2009); Journal of Statistical Mechanics: Theory and Experiment P02023 (2009).



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Planar rotator model 

Hamiltonian H = – J ∑si.sj = – J ∑cos(i- j)

where J – exchange interaction constant si – spin at site i (2d unit vector) i – turn angle of si

Types of vortices

Vortex pair (dipole)

A. Kaser at al., 2D XY model (web project) Modified planar rotator


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Mapping Zi ↔ i max

Zmax


Spin angles

Gesostatistical data Zmin

2π

Energy H

π

HΔ=0 = HΔ=2π Degeneracy!

0

min


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Modified planar rotator model 

Hamiltonian

where q ≤ 1/2

H* = – J ∑cos[q(i- j)]

(a) q = 1


(b) q = 1/2


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Modeling spatial correlations Matérn correlation function where

Variogram function where

[3] B. Minasny, A.B. McBratney, Geoderma 128 192 (2005)



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Results Spatial correlations Variation in time for temp. T = 0.01 and (a) τ1 = 10, (b) τ2 = 102,

(c) τ3 = 103, (d) τ4 = 104



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Results Spatial correlations Variation in time for temp. T = 0.0001 and (a) τ1 = 10, (b) τ2 = 102,

(c) τ3 = 103, (d) τ4 = 104



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Results Prediction 1. 2.

conditional MC simulations in equilibrium at T estimated from samples predictions obtained as average values from Nc eqilibrium realizations

(a) Lidar elevation data (83% missing) Modified planar rotator

(b) Prediction map (T=10-5, Nc=100)


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Results Prediction performance Synthetic data with Matérn covariance (ξ = 5, ν = 2.5) and p% missing values p=30%

p=90%

L=1024

L=256

L=64

Original

Reconstructed p=60%



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Results Prediction performance L=64 p = 30%

p = 60%

p = 90%

AAE

0.1850

0.3525

1.2711

ARE [%]

-0.02

-0.10

0.01

AARE [%]

0.39

0.76

2.68

RASE

0.2543

0.5015

1.7325

L=1024

p = 30%

p = 60%

p = 90%

AAE

0.1757

0.3140

1.0973

ARE [%]

-0.05

-0.09

-0.39

AARE [%]

0.37

0.67

2.34

RASE

0.2302

0.4232

1.4886



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Results Computational performance Relaxation process

In Matlab® on desktop computer with 1.93 GB of RAM and Intel® Core ™2 CPU 6320 @ 1.86 GHz 1.86 GHz

(a) Number of MC sweeps


(b) CPU time


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Summary Modified planar rotator model 

Spatial correlations  

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Short-range nature – typical for many geophysical and environmental applications Great variability in the parameter space of temperature and relaxation time

Spatial prediction by conditional MC simulations    

Universal - no assumptions about data distribution Automatic – no user imput required Efficient – suitable for large (remote sensing) data Accuracy – ??? more testing needed



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Thank you for your attention!



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