Gaussian processes for spatial modelling in environmental health: parameterizing for flexibility vs. computational efficiency

Transcription

1 Gaussian processes for spatial modelling in environmental health: parameterizing for flexibility vs. computational efficiency Chris Paciorek March 11, 2005 Department of Biostatistics Harvard School of Public Health

2 Increased attention to spatial analysis in public health data availability: geocoding and GPS for assigning point locations to individuals and monitors GIS software: easy data management and manipulation graphical presentation spatially-varying covariate generation interest amongst researchers: strong applied interest in kriging and related smoothing methods opportunities for more sophisticated spatio-temporal modelling, particularly Bayesian hierarchical modelling 1

3 Petrochemical exposure in Kaohsiung, Taiwan leukemia brain cancer northing (km) northing (km) easting (km) easting (km) n = 495 n = 433 n 1 = 141 n 1 = 121 2

4 Possible approaches for health analysis Explicitly estimate pollutant exposure - difficult retrospectively Use distance to exposure source as covariate Use a moving window/multiple testing to detect clusters of cases default approach - software available Include space as a covariate to provide a map of risk H i Ber(p(x i, s i )) logit(p(x i, s i )) = x T i β + g θ (s i ) 3

5 Particulate matter exposure in the Nurses Health Study estimate individual exposure, EPA monitoring for large-scale spatio-temporal heterogeneity spatially-varying covariates for local heterogenity distance to roads, climate variables, local land use,... generated using GIS basic additive exposure model: log E i N(f(x i, s i ), η 2 ) f(x i, s i ) = p h p (x i ) + g θ (s i ) geocoding of individual residences every two years relate estimated exposure to health outcomes (chronic heart disease) 4

6 geocoding and GIS make this possible; spatial statistics provides a rigorous framework 5

7 Health outcomes by postcode in NSW, Australia methodological challenges areal (postcode) units vary drastically in size data misalignment relate areal data to a latent smooth process, g θ ( ) (Kelsall & Wakefield, Rathouz) computational challenges: 650 units, 5 years daily data, 2 sexes, 9 age groups 6

8 Outline Motivating examples Introduction to Gaussian processes (GPs) Fast Gaussian process modelling Flexible Gaussian process modelling Bayes and overfitting The future: flexibility + efficiency + hierarchical modelling 7

9 Kriging as a GP model Y i N(g(s i ), η 2 ) g( ) GP(µ, C( ; θ)) Bayesian model specifies prior distributions for θ (Bayesian kriging) Empirical Bayes/marginal likelihood (i.e., kriging) integrate g train = (g(s 1 ),..., g(s n )) out of model estimate θ maximizing marginal posterior maximizing marginal likelihood fitting variogram model for C( ; θ) point estimate for spatial process: E(g test Y, θ) based conditional normal calculations 8

10 GAUSSIAN PROCESS DISTRIBUTION Infinite-dimensional joint distribution for g(x), x X : Example: g( ) a spatial process, X = R 2 g( ) GP(µ( ), C(, )) Finite-dimensional marginals are normal Types of covariance functions, C(x i, x j ): stationary, isotropic stationary, anisotropic nonstationary X x j x j x i x i 9

11 GAUSSIAN PROCESS DISTRIBUTION Infinite-dimensional joint distribution for g(x), x X : Example: g( ) a spatial process, X = R 2 g( ) GP(µ( ), C(, )) Finite-dimensional marginals are normal Types of covariance functions, C(x i, x j ): stationary, isotropic stationary, anisotropic nonstationary X x j x i x j x i 10

12 GAUSSIAN PROCESS DISTRIBUTION Infinite-dimensional joint distribution for g(x), x X : Example: g( ) a spatial process, X = R 2 g( ) GP(µ( ), C(, )) Finite-dimensional marginals are normal Types of covariance functions, C(x i, x j ): stationary, isotropic stationary, anisotropic nonstationary X x j x j x i x i 11

13 Stationary Correlation Functions Correlation Correlation fxns ρ = 0.2 ρ = Distance Z(x) Sample fxns x Matérn form ( ) ν ( ) 1 R(τ) = 2 ντ 2 ντ Γ(ν)2 ν 1 ρ Kν ρ ; ν > 0, ρ > 0 Differentiability controlled by ν, asymptotic advantages (Stein) Familiar exponential (ν = 0.5) and squared exponential (Gaussian) (ν ) correlations as special and limiting cases 12

14 Computational challenges of GPs even marginal likelihood in normal error model is intensive: g( ) GP(µ( ), C θ (, )) Y N(µ, C θ + η 2 I) O(n 3 ) fitting: C θ + η 2 I and (C θ + η 2 I) 1 (Y µ1) non-gaussian spatial models particularly difficult spatial process can t be integrated out MCMC mixing is very slow because of high-level structure correlation amongst process values and between process values and process hyperparameters value ACF iteration lag/10 13

15 Petrochemical exposure in Kaohsiung, Taiwan leukemia brain cancer northing (km) northing (km) easting (km) easting (km) n = 495 n = 433 n 1 = 141 n 1 =

16 Modelling Framework H i Ber(p(x i, s i )) logit(p(x i, s i )) = x i T β + g θ (s i ) basic spatial model for g s θ = (g θ (s 1 ),..., g θ (s n )) GAM: g θ ( ) is a two-dimensional smooth term basis representation g s θ = Zu Gaussian process representation: g( ) GP(µ( ), C θ (, )) g s θ N(µ, C θ ) GLMM: g s θ = Zu correlated random effects, u N(0, Σ) 15

17 Approaches Bayesian spectral basis model fit by MCMC (Wikle, 2002) [B-SB] penalized likelihood based on mixed model (radial basis functions) with REML smoothing (Kammann and Wand, 2003; Ngo and Wand, 2004) [PL-PQL] penalized likelihood with GCV smoothing (Wood, 2001, 2003, 2004) [PL-GCV] Bayesian mixed model/radial basis functions fit by MCMC (Zhao and Wand 2004) [B-Geo] 16

18 Bayesian spectral basis function model computationally efficient basis function construction (Wikle 2002) g # = Zu and g s = σp g # piecewise constant gridded surface on k by k grid P maps observation locations to nearest grid point Z is the Fourier (spectral) basis and Zu is the inverse FFT Zu is approximately a Gaussian process (GP) when... u N(0, diag(π θ (ω))) for Fourier frequencies, ω spectral density, π θ ( ), of GP covariance function defines V(u) 17

19 Bayesian spectral basis functions ω 2 = 0 ω 2 = 1 ω 2 = 2 ω 2 = 3 ω 1 = 0 ω 1 = 1 ω 1 = 2 ω 1 = 3 18

20 Comparison with usual GP specification spectral basis uses FFT O ( (k 2 ) log(k 2 ) ) additional observations are essentially free for fixed grid fast computation and prediction of surface given coefficients a priori independent coefficients give fast computation of prior and help with mixing 19

21 Penalized likelihood using GLMM framework with REML [PL-PQL] g s = Zu, Z = Ψ nk Ω 1 2 kk, u N(0, σ2 u) - variance component provides complexity penalty Ω contains pairwise spatial covariances between k knot locations and Ψ between n data locations and k knot locations potential covariance functions: thin plate spline generalized covariance function, ( C(τ) = τ 2 log τ ) ν ( ) 1 Matérn correlation function, R(τ) = 2 ντ 2 ντ Γ(ν)2 ν 1 ρ Kν ρ, with ρ and ν fixed computationally efficient approximation of a Gaussian process representation for g s PQL approach - IWLS fitting of (β, u) with REML estimation of σ 2 u within the iterations using MM software 20

22 GLMM basis functions radial basis functions centered at the knots 4 of 64 functions displayed: TPS Matérn 21

23 Penalized likelihood using GCV [PL-GCV] thin plate spline basis for g( ) truncated eigendecomposition of basis matrix increases computational efficiency IWLS fitting of (β, u) with GCV estimation of penalty easy implementation using the R mgcv library gam() 22

24 Bayesian geoadditive model [B-Geo] Bayesian version of GLMM framework already described g s = Zu, Z = Ψ nk Ω 1 2 kk, u N(0, σ2 u) natural Bayesian complexity penalty through prior on u thin plate spline covariance or Matérn correlation basis construction of Ψ and Ω MCMC implementation - ensuring mixing is not simple Metropolis-Hastings for u using conditional posterior mean and variance based on linearized observations joint proposals for σ 2 u and u to ensure that u remains compatible with its variance component 23

25 Simulated datasets 3 case-control scenarios: n 0 = 1, 000; n 1 = 200; n test = 2500 on 50 by 50 grid 1 cohort scenario: n = 10, 000; n test = 2500 on 50 by 50 grid

26 Assessment on 50 simulated datasets flat isotropic MSE PL PQL PL GCV B Geo B SB MSE PL PQL PL GCV B Geo B SB anisotropic cohort MSE MSE PL PQL PL GCV B Geo B SB PL PQL PL GCV B SB null 25

27 Mixing and speed of Bayesian methods speed (1000 its) speed (1000 eff. its) log posterior trace σ trace ρ trace B-Geo 15 min. 104 hr B-SB 1.3 min. 6 hr

28 Taiwan revisited - assessment leukemia brain cancer Summed test deviance over 10-fold C-V sets leukemia brain cancer PL-GCV PL-PQL B-Geo B-SB null northing (km) northing (km) easting (km) easting (km) 27

29 Assessment on count simulations n = 225, n test = 2500 on 50 by 50 grid MSE MSE MSE PL PQL PL GCV B Geo B SB null PL PQL PL GCV B Geo B SB null PL PQL PL GCV B Geo B SB null 28

30 Penalization in the spectral approach GP representation zeroes out high-frequency coefficients as appropriate Spatial hyperparameter controls coefficient variances g( ) GP(µ( ), σ 2 R(, ; ρ, ν)) prior penalty sample functions log(variance) ρ = 0.1 ρ = 0.5 ρ = frequency variance ρ = 0.1 ρ = 0.5 ρ = frequency f(x) x 29

31 Heterogeneous penalties y true gam() spectral x spatially-varying penalties are one option (e.g., Lang & Brezger 2004; Crainiceanu et al. 2004) spatially-varying ρ in a GP context is another 30

32 Outline Motivating examples Introduction to Gaussian processes (GPs) Fast Gaussian process modelling Flexible Gaussian process modelling Why Bayes works for smoothing The future: flexibility + efficiency + hierarchical modelling 31

33 A nonstationary covariance Higdon, Swall, and Kern (1999) model: C NS (x i, x j ) = k xi (u)k xj (u)du R p guaranteed positive definite Gaussian kernels give closed form: k xi (u) exp ( (u x i ) T Σ 1 i (u x i ) ) ( R NS (x i, x j ) = c ij exp (x i x j ) T ( Σi + Σ j 2 ) 1 (x i x j )) g( ) GP(µ, σ 2 R NS (, ; Σ( ))) 32

34 Nonstationary GPs in 1-D Kernel standard deviation Some kernels f(x) Some sample functions f(x) x x x f(x) x f(x) x 33

35 Nonstationary GPs in 2-D Sample function Kernel structure Sample function image x 2 1 g(x[1], x[2]) x[2] x[1] x 2 x 1 x 1 34

36 Generalizing the kernel convolution approach Squared exponential form: exp ( ( τij ρ ) 2 ) c ij exp ( (x i x j ) T ( Σi + Σ j 2 ) 1 (x i x j )) infinitely-differentiable sample paths Distance measures : isotropy τij 2 = (x i x j ) T (x i x j ) anisotropy τij 2 = (x i x j ) T Σ 1 (x i x j ) ( ) nonstationarity Q ij = (x i x j ) T Σi +Σ 1 j 2 (xi x j ) Can we replace τ 2 ij with Q ij in other stationary correlation functions? 35

37 A class of nonstationary covariance functions Theorem 1: if R(τ) is positive definite for R P, P = 1, 2,..., then R NS (x i, x j ) = c ij R( Q ij ) is positive definite for R P, P = 1, 2,... Theorem 2: smoothness (differentiability) properties of the original stationary correlation retained Specific case of Matérn nonstationary covariance: 1 Γ(ν)2 ν 1 ( ) ν ( ) 2 ντ 2 ντ K ν ρ ρ 1 ( Γ(ν)2 ν 1 2 ) ν ( νq ij Kν 2 ) νq ij advantages: more flexible form, differentiability not constrained, possible asymptotic advantages 36

38 Exponential and Matérn sample functions (stationary) ν = 0.5 ν = 4 x x x 1 x 1 37

39 A basic Bayesian nonstationary spatial model Bayesian nonstationary kriging model Y i N(g(x i ), η 2 ), x i R 2 g( ) GP(µ, σ 2 R NS (, ; ν, Σ( )) Let R NS be the nonstationary Matérn correlation Kernels (Σ x ) constructed based on stationary GP priors define multiple kernel matrices, Σ x, x X smoothly-varying (element-wise) in domain positive definite Fit via MCMC, including parameters determining Σ( ) 38

40 Smoothly-varying kernel matrices Spectral decomposition for each Σ x = Γ T x Λ x Γ x in R 2, parameterize each kernel using unnormalized eigenvector coordinates (a x, b x ) and the second eigenvalue (log λ 2,x ) define stationary GP priors for Φ( ) {(a( ), b( ), log(λ 2 ( ))} efficiently parameterize each GP using basis function approximation (Zhao & Wand, 2004) a x, b x λ 2x x λ 1x 39

41 Colorado precipitation characterization

42 Estimating Colorado precipitation Posterior mean Posterior std dev a c b d

43 Why doesn t Bayes overfit? Fourier basis involves k 2 (=4096, e.g.) coefficients Nonstationary covariance involves very highly-parameterized covariance structure No direct penalty on complicated spatial functions marg. lik. (prior predictive) P(Y M2) data space P(Y M1) P (y M i ) y Y 2 x y Y 1 x y x Model 1 ρ = 2.5 Model 2 ρ =

44 What does a Bayesian approach give us? ability to create rich hierarchical models that reflect our understanding of the system in environmental health applications the ability to incorporate time, latent variables, misaligned data a natural penalty on overfitting a recipe (perhaps slow) for estimation proper characterization of the uncertainty challenge lies less on the modelling side than with computations, model comparison and evaluation, and reproducibility 43

45 Future methodological work collaborative work on spatio-temporal modelling computational approaches for applying existing methodological ideas to real health data GP computations and parameterization: flexibility + efficiency + hierarchical modelling computational tricks for the nonstationary covariance; e.g., knotbased approaches for faster computation use of a wavelet basis with irregular spatial data in a similar framework as the spectral basis combining deterministic and stochastic models, e.g. for air pollution useful, practical methods for designing spatial monitoring networks and determining power 44