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Chapter 4 - Spatial processes R packages andsoftware
Lecture notes
Odd Kolbjørnsen and Geir Storvik
February 13, 2017
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Last time
General set up for ”Kriging type problems”
Introductory exampleKriging caseGeneral case
Change of support
Spatial moving average models
ConstructionCorrelation function
Non gaussian observations
Monte CarloLaplace approximationINLA
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Today
To day computations
SoftwareComputer
RINLA
Next timel lattice models
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Software
There is allways an existing software that ”does your job”
The challenge is to figure out how this works
Even if the software does not do excatly what you want maybe it isgood enough
Reasons not to make your own computer code
existing code is tested (less bugs)existing code is optimized (speed)often related to publicationseasier to get others to ”accept it”
Reasons to make your own computer code
understand the methodology betterimprove/develop exsiting methodologycombining with other techniquescompete with existing computer codebecause you have too much spare time
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Software options
Wiki List of spatial analysis software
.../wiki/List_of_spatial_analysis_software/
LuciadLightspeed,Open GeoDa, CrimeStatc, SaTScan, SAGA,IDRISI, Biodiverse, ERDAS IMAGINE, TerraLens ++
ArcGIS: Geoprocessing, visualization(GIS = Geographic information system)
GeoBUGS: Bayesian analysis (WinBugs)
GeoDa: Explantory Spatial Anaysis +
STARS: Space-time
SAS/STAT: spatial analysis (limited)
SPLUS: SpatialStats
Matlab: Spatial-statistics toolbox (fast but limited)
GEOEAS, SGeMS, GSLIB ,COHIBA, CRAVA ++
R
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Spatial analysis in R
CRAN (The Comprehensive R Archive Network)
Classes for spatial data: sp, spacetimeHandling spatial data: geosphereReading and writing spatial data: rgdalVisualisationPoint pattern analysisGeostatistics: geoR, gstat, spatialDisease mapping and areal data analysis: INLASpatial regression: nlmeEcological analysis
install.packages(...)
update.packages(...)
library(...)
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Hierarcical and hierarcical Bayesian approach
Hierarchical model
Variable Densities Notation in bookData model: Z p(Z|Y,θ) [Z|Y,θ]Process model: Y p(Y|θ) [Y|θ]Parameter: θ
Simultaneous model: p(y, z|θ) = p(z|y,θ)p(y|θ)Marginal model: L(θ) = p(z|θ) =
∫y p(z, y|θ)dy
Inference: θ̂ = argmaxθL(θ)
Bayesian approach: Include model on θ
Variable Densities Notation in bookData model: Z p(Z|Y,θ) [Z|Y,θ]Process model: Y p(Y|θ) [Y|θ]Parameter model: θ p(θ) [θ]
Simultaneous model: p(y, z,θ)Marginal model: p(z) =
∫θ∫
y p(z, y|θ)dydθ
Inference: θ̂ =∫θ θp(θ|z)dθ
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INLA
INLA = Integrated nested Laplace approximation (software)C-code, R-interfaceCan be installed by
source("http://www.math.ntnu.no/inla/givemeINLA.R")
Both empirical Bayes and Bayes
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Full presentation on the course web page
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Example - simulated data
Assume
Y (s) =β0 + β1x(s) + σδδ(s), {δ(s)} matern correlation
Z (si ) =Y (si ) + σεε(si ), {ε(si )} independent
Simulations:
Simulation on 20× 30 grid
{x(s)} sinus curve in horizontal direction
Parametervalues (β0, β1) = (2, 0.3), σδ = 1, σε = 0.3 and(θ1, θ2) = (2, 3)
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Latent models in INLA
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Matern model - setup
Code on webpage
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Matern model - setup
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Process model mean and covariance
Mean: 20× 30 = 600, Covariance: 600× 600 = 360000
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Matern model - parametrization
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Matern model - parametrization
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Output INLA
Empirical Bayes vs Bayes
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Output INLA
Empirical Bayes
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Matern model - parametrization
Model Range par Smoothness parBook ∝ {||h||/θ1}θ2Kθ2(||h||/θ1) θ1 θ2geoR ∝ {||h||/φ}κKκ(||h||/φ) φ κINLA ∝ {||h|| ∗ κ}νKν(||h|| ∗ κ) 1/κ ν
Note: INLA reports an estimate of another “range”, defined as
r =
√8
κ
corresponding to a distance where the covariance function isapproximately zero. An estimate of the range parameter can then beobtained by
1
κ=
r√8
Note: INLA works with precisions τy = σ−2y , τz = σ−2z .
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Result - INLA - empirical Bayes
Parameter True value Estimate Run 1β1 2.0 1.7991β2 0.3 0.3385σz 0.3 1√
13.213= 0.275
σy 1.0 1√1.398
= 0.846
φ1 2.0 4.901√8
= 1.7328
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Spatial prediction Run 1
Prior mean ”True” intensity Gaussian data
Empirical Bayes and Bayesian prediction
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Result - INLA - empirical Bayes
Parameter True value Estimate Run 1 Estimate Run 2β1 2.0 1.799 1.9719β2 0.3 0.3385 0.4741σz 0.3 1√
13.213= 0.275 1√
13.6917= 0.2756
σy 1.0 1√1.398
= 0.846 1√0.0014
= 26.72*
φ1 2.0 4.901√8
= 1.7328 17.44√8
=6.16*
* Commonly seen variance vs range tradeoff.Large variance and long range vs
”on scale” variance and ”on scale” rangeAllways compare your estimates with data variance.
Here: V̂ar(Z ) = 1.03, theortically (Var(Z ) = 1.09 = 1.0 + 0.32 )This is an argument for a Bayesian approach with an informative prior
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Spatial prediction Run 2
Prior mean ”True” intensity Gaussian data
Empirical Bayes and Bayesian prediction
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Example - simulated data
Assume
Y (s) =β0 + β1x(s) + σδδ(s), {δ(s)} matern correlation
Z (si ) ∼Poisson(exp(Y (si ))
Simulations:
Simulation on 20× 30 grid
{x(s)} sinus curve in horizontal direction
Parametervalues (β0, β1) = (1, 0.3), σδ = 1, σε = 0.3 and(θ1, θ2) = (2, 3)
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Spatial prediction, Code Poisson
Empirical Bayes (note constant term changed from 2 to 1, hence y-1 )
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Parameter prediction, Poisson
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Spatial prediction, Poisson
”True” intensity Poisson data (counts)
Estimated intensity
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Inla engine: GMRFLib
Principle in GRMF = Gaussian Markov Random field is to invert therelation in ”smoothing of independent Gaussian variables”
Y = Lε, ε ∼ N(0, I)⇒ Y ∼ N(0,Σ = LLT )
Select the smooting kernel to be the cholesky factor: Y = Lε
Then flip the equation and use the pressision matrix Q = Σ−1:
L−1Y = ε, ε ∼ N(0, I)⇒ Y ∼ N(0,Q = L−1L−T )
The formulations are equivalent but L and L−1 differs greately. This hasmajor impact on computations.
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Inla engine: GMRFLib
Principle in GRMF = Gaussian Markov Random field is to invert therelation in ”smoothing of independent Gaussian variables”
Y = Lε, ε ∼ N(0, I)⇒ Y ∼ N(0,Σ = LLT )
Select the smooting kernel to be the cholesky factor: Y = Lε
Then flip the equation and use the pressision matrix Q = Σ−1:
L−1Y = ε, ε ∼ N(0, I)⇒ Y ∼ N(0,Q = L−1L−T )
The formulations are equivalent but L and L−1 differs greately. This hasmajor impact on computations.
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Choleskey versus inverse choleskey in AR(1)
Y1 = σ1ε1, Yi = αYi−1 + σεi
”Inverse formulation”:
ε1 =1
σ1Y1, εi =
1
σYi −
α
σYi−1, i > 1
”Direct formulation”:
Yi = αi−1σ1ε1 + σ
i∑j=2
αi−jεj
The ”Inverse formulation” can model long range dependency with asparse matrix (few non-zero elements).
Complexity inverse formulation, – solving L−1Y = ε: ∼ n
Complexity direct formulation, – multiply Y = Lε: ∼ n2
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