Chapter 4 - Spatial processes R packages and software Lecture notes

Transcription

1 TK Intro 1 Chapter 4 - Spatial processes R packages and software Lecture notes Odd Kolbjørnsen and Geir Storvik February 13, 2017

2 STK Intro 2 Last time General set up for Kriging type problems Introductory example Kriging case General case Change of support Spatial moving average models Construction Correlation function Non gaussian observations Monte Carlo Laplace approximation INLA

3 STK Intro 3 Today To day computations Software Computer R INLA Next timel lattice models

4 STK Intro 4 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 is good enough Reasons not to make your own computer code existing code is tested (less bugs) existing code is optimized (speed) often related to publications easier to get others to accept it Reasons to make your own computer code understand the methodology better improve/develop exsiting methodology combining with other techniques compete with existing computer code because you have too much spare time

5 STK Intro 5 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

6 STK Intro 6 Spatial analysis in R CRAN (The Comprehensive R Archive Network) Classes for spatial data: sp, spacetime Handling spatial data: geosphere Reading and writing spatial data: rgdal Visualisation Point pattern analysis Geostatistics: geor, gstat, spatial Disease mapping and areal data analysis: INLA Spatial regression: nlme Ecological analysis install.packages(...) update.packages(...) library(...)

7 TK Intro 7 Hierarcical and hierarcical Bayesian approach Hierarchical model Variable Densities Notation in book Data 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) = p(z, y θ)dy y Inference: θ = argmax θ L(θ) Bayesian approach: Include model on θ Variable Densities Notation in book Data 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 p(z, y θ)dydθ y Inference: θ = θ θp(θ z)dθ

8 INLA INLA = Integrated nested Laplace approximation (software) C-code, R-interface Can be installed by source(" Both empirical Bayes and Bayes

9 Full presentation on the course web page STK Intro 9

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14 STK Intro 14 Example - simulated data Assume Y (s) =β 0 + β 1 x(s) + σ δ δ(s), Z(s i ) =Y (s i ) + σ ε ε(s i ), {δ(s)} matern correlation {ε(s i )} independent Simulations: Simulation on grid {x(s)} sinus curve in horizontal direction Parametervalues (β 0, β 1 ) = (2, 0.3), σ δ = 1, σ ε = 0.3 and (θ 1, θ 2 ) = (2, 3)

15 Latent models in INLA STK Intro 15

16 STK Intro 16 Matern model - setup Code on webpage

17 Matern model - setup STK Intro 17

18 TK Intro 18 Process model mean and covariance Mean: = 600, Covariance: =

19 Matern model - parametrization STK Intro 19

20 Matern model - parametrization STK Intro 20

21 STK Intro 21 Output INLA Empirical Bayes vs Bayes

22 STK Intro 22 Output INLA Empirical Bayes

23 STK Intro 23 Matern model - parametrization Model Range par Smoothness par Book { h /θ 1 } θ2 K θ2 ( h /θ 1 ) θ 1 θ 2 geor { h /φ} κ K κ ( h /φ) φ κ INLA { h κ} ν K ν ( h κ) 1/κ ν Note: INLA reports an estimate of another range, defined as 8 r = κ corresponding to a distance where the covariance function is approximately zero. An estimate of the range parameter can then be obtained by 1 κ = r 8 Note: INLA works with precisions τ y = σy 2, τ z = σz 2.

24 STK Intro 24 Result - INLA - empirical Bayes Parameter True value Estimate Run 1 β β σ z = σ y = φ =

25 STK Intro 25 Spatial prediction Run 1 Prior mean True intensity Gaussian data Empirical Bayes and Bayesian prediction

26 TK Intro 26 Result - INLA - empirical Bayes Parameter True value Estimate Run 1 Estimate Run 2 β β σ z = = σ y = = 26.72* φ = =6.16* * Commonly seen variance vs range tradeoff. Large variance and long range vs on scale variance and on scale range Allways compare your estimates with data variance. Here: Var(Z) = 1.03, theortically (Var(Z) = 1.09 = ) This is an argument for a Bayesian approach with an informative prior

27 STK Intro 27 Spatial prediction Run 2 Prior mean True intensity Gaussian data Empirical Bayes and Bayesian prediction

28 STK Intro 28 Example - simulated data Assume Y (s) =β 0 + β 1 x(s) + σ δ δ(s), Z(s i ) Poisson(exp(Y (s i )) {δ(s)} matern correlation Simulations: Simulation on grid {x(s)} sinus curve in horizontal direction Parametervalues (β 0, β 1 ) = (1, 0.3), σ δ = 1, σ ε = 0.3 and (θ 1, θ 2 ) = (2, 3)

29 STK Intro 29 Spatial prediction, Code Poisson Empirical Bayes (note constant term changed from 2 to 1, hence y-1 )

30 Parameter prediction, Poisson STK Intro 30

31 STK Intro 31 Spatial prediction, Poisson True intensity Poisson data (counts) Estimated intensity

32 STK Intro 32 Inla engine: GMRFLib Principle in GRMF = Gaussian Markov Random field is to invert the relation in smoothing of independent Gaussian variables Y = Lε, ε N(0, I) Y N(0, Σ = LL T )

33 STK Intro 33 Inla engine: GMRFLib Principle in GRMF = Gaussian Markov Random field is to invert the relation in smoothing of independent Gaussian variables Y = Lε, ε N(0, I) Y N(0, Σ = LL T ) Select the smooting kernel to be the cholesky factor: Y = Lε Then flip the equation and use the pressision matrix Q = Σ 1 : L 1 Y = ε, ε N(0, I) Y N(0, Q = L 1 L T ) The formulations are equivalent but L and L 1 differs greately. This has major impact on computations.

34 TK Intro 34 Choleskey versus inverse choleskey in AR(1) Inverse formulation : Direct formulation : Y 1 = σ 1 ε 1, Y i = αy i 1 + σε i ε 1 = 1 σ 1 Y 1, ε i = 1 σ Y i α σ Y i 1, i > 1 Y i = α i 1 σ 1 ε 1 + σ i α i j ε j The Inverse formulation can model long range dependency with a sparse matrix (few non-zero elements). j=2 Complexity inverse formulation, solving L 1 Y = ε: n Complexity direct formulation, multiply Y = Lε: n 2