Multivariate modelling and efficient estimation of Gaussian random fields with application to roller data

Transcription

1 Multivariate modelling and efficient estimation of Gaussian random fields with application to roller data Reinhard Furrer, UZH PASI, Búzios, NZZ.ch

2 Motivation Microarray data: construct alternative background correction 2

3 Motivation observations = fixed effects + spatial term + noise Chip 1 Chip 2 3

4 Motivation Microarray data Climate data synthesize projections 4

5 Motivation Microarray data Climate data Satellite data attribute spatial changes 5

6 Motivation (a) Observed change in vegetation activity ( ) y 10 S 20 S observations 0 (b) Change explained by climatic effects x T β (RT) 30 S 10 S 20 E 30 E 20 S trend 0 30 S 20 E 30 E (c) Structured change not explained by climatic effects h (GRF) 10 S 20 S spatial term 0 30 S 20 E 30 E (d) Residuals ε 10 S noise 0 20 S 30 S E 30 E 6

7 Motivation for roller data analysis Compaction for road construction: one vibrating drum (smooth drum or padfoot) rolling at 1m/s 20cm material per layer typical bed is 12 15m wide and m long sufficient compaction is manually tested after several layers of material (USA) 7

8 Motivation for roller data analysis Compaction for road construction: one vibrating drum (smooth drum or padfoot) rolling at 1m/s 20cm material per layer typical bed is 12 15m wide and m long sufficient compaction is manually tested after several layers of material (USA) Is an automatic quality assurance and intelligent compaction possible? 7

9 Motivation Current intelligent compaction: precise GPS positioning on-board visualization off-board processing Source: y (m) k s (MN m) x (m)

10 Motivation Relation between measurement and actual soil modulus is unknown. (Linear) relationship is determined with a second measurement device (lightweight deflection, nuclear density... 8

11 Spatial model Spatial, additive mixed effects model for roller measurement values: RMV = amplitude + roller type + driving direction trend(s) + spatial term(s) + error Y (s) = Xβ + α(s) + γ(s) + ε(s) s D R d, d 1 with Xβ: fixed effects and trend α(s): spline component (trend) γ(s): zero mean spatial Gaussian process ε(s): iid noise, orthogonal to γ(s) 9

12 Spatial model Spatial, additive mixed effects model for roller measurement values: RMV = amplitude + roller type + driving direction trend(s) + spatial term(s) + error Y (s) = Xβ + α(s) + γ(s) + ε(s) s D R d, d 1 with Xβ: α(s): γ(s): ε(s): fixed effects and trend coefficients β spline component (trend) basis function coefficients θ α ; smoothing parameter λ α zero mean spatial Gaussian process parameters θ γ describing the covariance function iid noise, orthogonal to γ(s) variance σ 2 10

13 Multivariate modeling: setting Spatial, additive mixed effects model: Y 1 (s) = X 1 β 1 + α 1 (s) + γ 1 (s) + ε 1 (s). Y p (s) = X p β p + α p (s) + γ p (s) + ε p (s) s D R d, d 1 11

14 Multivariate modeling: setting Spatial, additive mixed effects model: Y 1 (s) = X 1 β 1 + α 1 (s) + γ 1 (s) + ε 1 (s). Y p (s) = X p β p + α p (s) + γ p (s) + ε p (s) s D R d, d 1 Modeling the spatial processes themselves: Random field: GRF GMRF Dependency: cross-correlation model ➀ ➂ common process(es) ➁ ➃ 11

15 Random field modeling Dependency through a cross-correlation model ➀ : X N n (µ X, Σ X ) Y N n (µ Y, Σ Y ) ( ) X Y (( ) µx N 2n, µ Y Cov(X, Y) = Σ XY ( )) ΣX Σ XY Σ XY T (e.g., Gneiting, Kleiber, Schlather 2010, Apanasovich, Genton, Sun 2012,... ) Σ Y 12

16 Random field modeling Dependency through a cross-correlation model ➀ : X N n (µ X, Σ X ) Y N n (µ Y, Σ Y ) ( ) X Y (( ) µx N 2n, µ Y Cov(X, Y) = Σ XY ( )) ΣX Σ XY Σ XY T (e.g., Gneiting, Kleiber, Schlather 2010, Apanasovich, Genton, Sun 2012,... ) Σ Y Dependency through common process(es) ➁ : X N n (µ X, Σ X ) Y N n (µ Y, Σ Y ) ( X + Z Y + Z ) Z N n (0, Σ Z ) (( ) µx N 2n, µ Y ( ΣX + Σ Z Σ Z Σ Z Σ Y + Σ Z )) 12

17 Backfitting Recall: Y 1 (s) = X 1 β 1 + α 1 (s) + γ 1 (s) + ε 1 (s). Y p (s) = X p β p + α p (s) + γ p (s) + ε p (s) 13

18 Backfitting Recall: Y 1 (s) = X 1 β 1 + α 1 (s) + γ 1 (s) + ε 1 (s). Y p (s) = X p β p + α p (s) + γ p (s) + ε p (s) Extending the classical backfitting approach to dependent data: repeat until convergence repeat until convergence estimate fixed effects for all stochastic effects estimate parameters predict smooth field See Furrer, Sain (2009) Heersink, Furrer (2012 3) 13

19 Sparse covariance matrices Calculate Σ:

20 Sparse covariance matrices Calculate Σ:

21 Sparse covariance matrices Calculate Σ: Distances : Correlation( Distances ) : 14

22 Sparse covariance matrices Sparseness is guaranteed when the covariance function has a compact support a compact support is (artificially) imposed tapering Covariance Matern ν = 0.5 spherical Distance, lag h 15

23 Sparse covariance matrices Sparseness is guaranteed when the covariance function has a compact support a compact support is (artificially) imposed tapering Covariance Matern ν = 0.5 spherical Matern * spherical Distance, lag h 16

24 Sparse covariance matrices Sparseness is guaranteed when the covariance function has a compact support a compact support is (artificially) imposed tapering Covariance Matern ν = 0.5 spherical Matern * spherical Distance, lag h spam for 16

25 Multivariate tapering Concept of Gaussian equivalent measures does not exist Domain increasing framework Σ Σ T 0 as n 17

26 Multivariate tapering Concept of Gaussian equivalent measures does not exist Domain increasing framework Σ Σ T 0 as n Covariance Matern ν = 0.5 spherical Matern * spherical Distance, lag h 17

27 Multivariate tapering Concept of Gaussian equivalent measures does not exist Domain increasing framework Σ Σ T 0 as n Difference Spherical Distance, lag h Covariance Matern ν = 0.5 spherical Matern * spherical Distance, lag h 17

28 Multivariate tapering Concept of Gaussian equivalent measures does not exist Domain increasing framework Σ Σ T 0 as n Difference Wendland Distance, lag h Covariance Matern ν = 1.5 Wendland Matern * Wendland Distance, lag h 18

29 Multivariate tapering Concept of Gaussian equivalent measures does not exist Domain increasing framework Σ Σ T 0 as n Difference Wendland Distance, lag h Covariance Matern ν = 1.5 Wendland Matern * Wendland Distance, lag h 19

30 Multivariate tapering Concept of Gaussian equivalent measures does not exist Domain increasing framework Σ Σ T 0 as n Difference Wendland Distance, lag h Covariance Matern ν = 1.5 Wendland Matern * Wendland Distance, lag h 19

31 Multivariate tapering Concept of Gaussian equivalent measures does not exist Domain increasing framework Σ Σ T 0 as n Difference Wendland Distance, lag h Covariance Matern ν = 1.5 Wendland Matern * Wendland Distance, lag h 19

32 Multivariate tapering Concept of Gaussian equivalent measures does not exist Domain increasing framework Σ Σ T 0 as n Difference Wendland Distance, lag h Covariance Matern ν = 1.5 Wendland Matern * Wendland Distance, lag h 19

33 Multivariate tapering Concept of Gaussian equivalent measures does not exist Domain increasing framework Σ Σ T 0 as n Tapering is purely pragmatic T = Q T i Q = ɛi + (1 ɛ)j 19

34 All models are wrong, but... Iterative approaches + Flexible, numerically feasible Uncertainties Maximum likelihood + Uncertainties, asymptotics Numerical issues Bayesian hierarchical models + Flexible, uncertainties MCMC SPDE models + flexible, scalable 20

35 Example: Backfitting Minnesota testbed: Cell 28 Cell 27 Subgrade, subbase, base (top to bottom). 21

36 Example: Backfitting Minnesota testbed: Model: Y 1 (s) = X 1 β 1 + γ 1 (s) + ε 1 (s) Y 2 (s) = X 2 β 2 + c 1 γ 1 (s) + γ 2 (s) + ε 2 (s) Y 3 (s) = X 3 β 3 + c 2 (c 1 γ 1 (s) + γ 2 (s)) + γ 3 (s) + ε 3 (s) 22

37 be uniformly blue. Inset: nonstationarity Figure 2: Data from the test bed in Florida, USA. RMVs collected from driving in the x-direction (left) and from driving in the y-direction (right) are depicted

38 be uniformly blue. Inset: nonstationarity Figure 2: Data from the test bed in Florida, USA. RMVs collected from driving in the x-direction (left) and from driving in the y-direction (right) are depicted lag dist lag dist 23 1

39 be uniformly blue. Inset: nonstationarity Figure 2: Data from the test bed in Florida, USA. RMVs collected from driving in the x-direction (left) and from driving in the y-direction (right) are depicted lag dist lag dist 23 1

40 be uniformly blue. Inset: nonstationarity Figure 2: Data from the test bed in Florida, USA. RMVs collected from driving in the Geometric anisotropy x-direction (left) and from driving in the y-direction (right) are depicted lag dist directionally colored noise lag dist 23 1

41 Example: Backfitting Minnesota testbed: Model: Y 1 (s) = X 1 β 1 + γ 1 (s) + ε 1 (s) Y 2 (s) = X 2 β 2 + c 1 γ 1 (s) + γ 2 (s) + ε 2 (s) Y 3 (s) = X 3 β 3 + c 2 (c 1 γ 1 (s) + γ 2 (s)) + γ 3 (s) + ε 3 (s) (add measurement operator... ) 24

42 Example: Backfitting Minnesota testbed: Model: Y 1 (s) = X 1 β 1 + γ 1 (s) + ε 1 (s) Y 2 (s) = X 2 β 2 + c 1 γ 1 (s) + γ 2 (s) + ε 2 (s) Y 3 (s) = X 3 β 3 + c 2 (c 1 γ 1 (s) + γ 2 (s)) + γ 3 (s) + ε 3 (s) (add measurement operator... ) Cell 27: Fitted smooths: 24

43 Multiresolution analysis Simultaneous inference for IC and QA which features are really there? Using Holmström et al. (2010), extensions from original SiZER Ready to use software Decomposition into different scales: 0 = λ 1 < λ 2 < < λ L = smoothing parameters γ = L 1 i=1 (S λi S λi+1 )γ + S λl γ z 1 + z z L 25

44 Multiresolution analysis Simultaneous inference for IC and QA which features are really there? Using Holmström et al. (2010), extensions from original SiZER Ready to use software Decomposition into different scales: 0 = λ 1 < λ 2 < < λ L = smoothing parameters γ = L 1 i=1 (S λi S λi+1 )γ + S λl γ z 1 + z z L 25

45 Example: multiresolution analysis Minnesota testbed: Cell 28 Cell 27 Subgrade, subbase, base (top to bottom). 26

46 Example: Multiresolution analysis Cell 27: Fitted smooths: Multiresolution analysis: subgrade subbase base 27

47 Afterthoughts/outlook Flexible setting... toolbox(es) Multivariate spatial (spatio-temporal) non-stationary data Bayesian framework Non-Gaussian data 28

48 Collaboration with: Daniel Heersink, now Research Scientist at CSIRO, Canberra Mike Mooney, CSM Roland Anderegg, FHNW... URPP Systems Biology/Functional Genomics & URPP Global Change and Biodiversity , ,

49 References (some) de Jong, Schaepman, Furrer, de Bruin, Verburget (2013) Spatial relationship between climatologies and changes in global vegetation activity, Global Change Biology, 19(6), Facas, Furrer, Mooney (2010) Anisotropy in the Spatial Distribution of Roller- Measured Soil Stiffness. International Journal of Geomechanics, 10(4), Furrer, Sain (2009) Spatial Model Fitting for Large Datasets with Applications to Climate and Microarray Problems, Statistics and Computing, 19(2), Heersink, Furrer (2013) Sequential Spatial Analysis of Large Datasets with Applications to Modern Soil Compaction Roller Measurement Values, Spatial Statistics, 6, Heersink, Furrer, Mooney (2013) Intelligent Compaction and Quality Assurance of Roller Measurement Values utilizing Backfitting and Multiresolution Scale Space Analysis, arxiv: Heersink, Furrer, Mooney (2013) Spatial Backfitting of Roller Measurement Values from a Florida Test Bed, arxiv: Heersink, Furrer (2012) Moore Penrose Inverses of Quasi-Kronecker Structured Matrices, Linear Algebra and its Applications, 436(3), Holmström, Pasanen, Furrer, Sain (2011) Scale Space Multiresolution Analysis of Random Signals. Computational Statistics and Data Analysis, 55,