Efficient adaptive covariate modelling for extremes

Transcription

1 Efficient adaptive covariate modelling for extremes Slides at jonathan Matthew Jones, David Randell, Emma Ross, Elena Zanini, Philip Jonathan Copyright of Shell December / 23

2 Structural damage Ike, Gulf of Mexico, 28 (Joe Richard) North Sea, Winter (The Inertia) Copyright of Shell December / 23

3 Motivation Rational and consistent design and assessment of marine structures Reduce bias and uncertainty in estimation of structural integrity Quantify uncertainty as well as possible Non-stationary marginal, conditional, spatial and temporal extremes Multiple locations, multiple variables, time-series Multidimensional covariates Improved understanding and communication of risk Incorporation within established engineering design practices Knock-on effects of improved inference The ocean environment is an amazing thing to study... especially if you like to combine beautiful physics, measurement and statistical modelling! Copyright of Shell December / 23

4 Fundamentals Environmental extremes vary smoothly with multidimensional covariates Model parameters are non-stationary Environmental extremes exhibit spatial and temporal dependence Characterise these appropriately Uncertainty quantification for whole inference Data acquisition (simulator or measurement) Data pre-processing (storm peak identification) Hyper-parameters (extreme value threshold) Model form (marginal measurement scale effect, spatial extremal dependence) Statistical and computational efficiency Slick algorithms Parallel computation Bayesian inference Copyright of Shell December / 23

5 A typical sample Typical data for South China Sea location. Sea state (grey) and storm peak (black) H S on season and direction Copyright of Shell December / 23

6 Outline Directional-seasonal covariate models for H sp S Introductory example using P-splines Adaptive splines Partition models South China Sea example as connecting theme Focus on the generalised Pareto (GP) inference Copyright of Shell December / 23

7 Simple gamma-gp model Copyright of Shell December / 23

8 Simple gamma-gp model Sample of peaks over threshold y, with covariates θ θ is 1D in motivating example : directional θ is nd later : e.g. 4D spatio-directional-seasonal Below threshold ψ y follows truncated gamma with shape α, scale 1/β Hessian for gamma better behaved than Weibull Above ψ y follows generalised Pareto with shape ξ, scale σ ξ, σ, α, β, ψ all functions of θ ψ for pre-specified threshold probability τ Generalise later to estimation of τ Frigessi et al. [22], Behrens et al. [24], MacDonald et al. [211] Randell et al. [216] Copyright of Shell December / 23

9 Simple gamma-gp model Density is f (y ξ, σ, α, β, ψ, τ) = { τ f TG (y α, β, ψ) for y ψ (1 τ) f GP (y ξ, σ, ψ) for y > ψ Likelihood is L(ξ, σ, α, β, ψ, τ {y i } n i=1 ) = f TG (y i α, β, ψ) f GP (y i ξ, σ, ψ) i:y i ψ Estimate all parameters as functions of θ i:y i >ψ τ n B (1 τ) (1 n B) where n B = i:y i ψ Copyright of Shell December / 23 1.

10 Standard P-spline model Physical considerations suggest α, β, ρ, ξ, σ, ψ and τ vary smoothly with covariates θ Values of η {α, β, ρ, ξ, σ, ψ, τ} on some index set of covariates take the form η = Bβ η For nd covariates, B takes the form of tensor product B θn... B θκ... B θ2 B θ1 Spline roughness with respect to each covariate dimension κ given by quadratic form λ ηκ β ηκ P ηκβ ηκ Pηκ is a function of stochastic roughness penalties δ ηκ Brezger and Lang [26] Copyright of Shell December / 23

11 P-splines Kronecker product Periodic P-splines Copyright of Shell December / 23

12 Priors and conditional structure Priors Conditional structure density of β ηκ exp ( 12 ) λ ηκβ ηκ Pηκβ ηκ λ ηκ gamma ( and τ beta, when τ estimated ) f (τ y, Ω \ τ) f (y τ, Ω \ τ) f (τ) f (β η y, Ω \ β η ) f (y β η, Ω \ β η ) f (β η δ η, λ η ) f (λ η y, Ω \ λ η ) f (β η δ η, λ η ) f (λ η ) η Ω = {α, β, ρ, ξ, σ, ψ, τ} Copyright of Shell December / 23

13 Inference Elements of β η highly interdependent, correlated proposals essential for good mixing Stochastic analogues of IRLS and back-fitting algorithms for maximum likelihood optimisation used previously Estimation of different penalty coefficients for each covariate dimension Gibbs sampling when full conditionals available Otherwise Metropolis-Hastings (MH) within Gibbs, using suitable proposal mechanisms, mmala where possible Roberts and Stramer [22], Girolami and Calderhead [211], Xifara et al. [214] Copyright of Shell December / 23

14 Season median posterior Season median posterior p-splines: GP parameter estimates 36 Prediction Prediction Direction Direction Copyright of Shell December / 23

15 Inference with adaptive splines Advantages Arbitrary location of knots, and number of knots Estimate number, location, coefficient of knots Reversible-jump MCMC: Birth-death Split-combine (local birth-death) Detailed balance Biller [2], Zhou and Shen [21], DiMatteo et al. [21], Wallstrom et al. [28] Copyright of Shell December / 23

16 Inference with adaptive splines : e.g. birth-death Copyright of Shell December / 23

17 Inference with adaptive bases: birth-death Acceptance probability α(m m) = min { 1, f (m ) f (m) f (y m ) f (y m) q(m m ) q(m m) m m } Dimension-jumping proposals: β 1 (p-vector) β 2 ((p + 1)-vector) η = B 1 β 1 = B 2 β2 β 2 = G ˆβ 2 = [ (B 2B 2 ) 1 B 2B ] 1 β1 = Gβ 1 u N(, ). 1 [ ] β1 u Copyright of Shell December / 23

18 Season median posterior Season median posterior Adaptive splines: GP parameter estimates 36 Prediction Prediction Direction Direction Copyright of Shell December / 23

19 Partition model Pros & cons Naturally local, nd Piecewise constant Estimate Number of cells Centroid locations Cell coefficients Reversible-jump MCMC Birth-death Detailed balance Green [1995], Heikkinen and Arjas [1998], Denison et al. [22], Costain [28], Bodin and Sambridge [29] Copyright of Shell December / 23

20 Season median posterior Season median posterior Partition model: GP parameter estimates 36 Prediction Prediction Direction Direction Copyright of Shell December / 23

21 Season median posterior Season median posterior Season median posterior Season median posterior Season median posterior Season median posterior Qualitative comparison of different estimates P-splines: n ξ = 6 6, n ν = 6 6 Adaptive splines: n mo ξ = 3 3, nmo ν = 4 4 Partition: nmo ξ = 1, nmo ν = 7 36 Prediction Prediction Prediction Prediction Direction Prediction Direction Prediction Direction Direction Direction Direction Copyright of Shell December / 23

22 Summary Covariate effects important in environmental extremes Need to tackle big problems need efficient models Need to provide solutions as end-user software stable inference P-splines: straightforward, global roughness per dimension Adaptive splines: optimally-placed knots All splines: nd basis is tensor product of marginal bases Partition: piecewise constant, naturally nd Partition mixture model Combinations useful Conditional, spatial and temporal extremes Copyright of Shell December / 23

23 References C N Behrens, H F Lopes, and D Gamerman. Bayesian analysis of extreme events with threshold estimation. Stat. Modelling, 4: , 24. C. Biller. Adaptive bayesian regression splines in semiparametric generalized linear models. J. Comput. Graph. Statist., 9:122 14, 2. Thomas Bodin and Malcolm Sambridge. Seismic tomography with the reversible jump algorithm. Geophysical Journal International, 178: , 29. A. Brezger and S. Lang. Generalized structured additive regression based on Bayesian P-splines. Comput. Statist. Data Anal., 5: , 26. D. A. Costain. Bayesian partitioning for modeling and mapping spatial case?control data. Biometrics, 65: , 28. D. G. T. Denison, N. M. Adams, C. C. Holmes, and D. J. Hand. Bayesian partition modelling. Comput. Stat. Data Anal., 38: , 22. I. DiMatteo, C. R. Genovese, and R. E. Kass. Bayesian curve-fitting with free-knot splines. Biometrika, 88: , 21. A. Frigessi, O. Haug, and H. Rue. A dynamic mixture model for unsupervised tail estimation without threshold selection. Extremes, 5: , 22. M. Girolami and B. Calderhead. Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. Roy. Statist. Soc. B, 73: , 211. P.J. Green. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82: , Juha Heikkinen and Elja Arjas. Non-parametric bayesian estimation of a spatial poisson intensity. Scand. J. Stat., 25:435 45, A. MacDonald, C. J. Scarrott, D. Lee, B. Darlow, M. Reale, and G. Russell. A flexible extreme value mixture model. Comput. Statist. Data Anal., 55: , 211. D. Randell, K. Turnbull, K. Ewans, and P. Jonathan. Bayesian inference for non-stationary marginal extremes. Environmetrics, 27:439 45, 216. G. O. Roberts and O. Stramer. Langevin diffusions and Metropolis-Hastings algorithms. Methodology and Computing in Applied Probability, 4: , 22. G. Wallstrom, J. Liebner, and R. E. Kass. An implementation of Bayesian adaptive regression splines (BARS) in C with S and R wrappers. Journal of Statistical Software, 26, 28. T. Xifara, C. Sherlock, S. Livingstone, S. Byrne, and M Girolami. Langevin diffusions and the Metropolis-adjusted Langevin algorithm. Stat. Probabil. Lett., 91(22):14 19, 214. S. Zhou and X. Shen. Spatially adaptive regression splines and accurate knot selection schemes. J. Am. Statist. Soc., 96: , 21. Copyright of Shell December / 23

24 Supporting material Copyright of Shell December / 3

25 Season Threshold Partition model: ψ 36 Threshold Direction Copyright of Shell December / 3

26 Partition model: ξ and ν traces basis trace plots basis trace plots basis trace plots.4 basis trace plots k (# centroids) 8 k (# centroids) k (# centroids) k (# centroids) Loc (dim 2) iteration Copyright of Shell Loc (dim 2) Loc (dim 2).6 3 Loc (dim 1) 1.8 Loc (dim 2) 2 Loc (dim 1) Loc (dim 1) Loc (dim 1) iteration December 218 3/3