janine illian r-inla
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Integrated nested Laplace approximation andfriends – fast and flexible modelling with R-INLA
Janine Illian
March 2014
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA
INLA in a nutshell
• many data sets these days are complex, resulting in complexmodels, e.g. complex spatial models
• usually Markov chain Monte Carlo (MCMC) methods havebeen used to fit these models
• (realistically) complex models result in very long running times• often impossible (or unrealistic) to fit
• INLA is an alternative to MCMC• much, much faster• R-INLA makes coding very easy• allows non-experts to fit complex models
• suitable for a specific (but very large !) class of models
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA
INLA in a nutshell
Three main ingredients in INLA
• Gaussian Markov random fields
• Latent Gaussian models
• Laplace approximations
which together (with a few other things) give a very nice tool forBayesian inference
• quick
• accurate
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA
how INLA is developed
• INLA – both theory and implementation – are under constantdevelopment within Havard Rue’s group at NTNU and beyond
• new features and models are added all the time
• queries from users and interests within the group have pushedvarious extensions
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA
R-INLA
• the R library R-INLA is updated regularly
• information on and developments may be found on thewebpage (http ://www.r-inla.org/)
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA
the SPDE approach – more flexible models
• simple models use a simple gridding approach to approximatethe continuous spatial field
• this is easy to implement
• however : this can be• computationally inefficient and• not flexible enough (complicated boundaries or domains)
⇒ use continuously specified finite dimensional Gaussian randomfields
⇒ spatial field as solution to a stochastic partial differentialequation (“SPDE approach”)
Lindgren et al., 2011, Simpson et al. 2012
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA
continuous specification
• use a continuous specification of the random field model :a finite-dimensional basis function expansion
• allows computation using the exact positions of the points
• basis functions have compact support, i.e. the field can beevaluated in O(1) operations !
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA
this allows for flexible modelling...
• non-stationary models (anisotropy)
• models on a sphere (oscillating)
• non-separable models
Intro B, W, M, & R SPDE/GMRF Example End Finite Projection Markov Lattices ... and beyond
Beyond classical Matern models
The approach can in a straightforward way be extendedto oscillating, anisotropic, non-stationary, non-separablespatio-temporal, and multivariate fields on manifolds.
(κ2 + ∇ · m −∇ · M∇)α/2(τx(u)) = W(u), u ∈ Rd
Finn Lindgren - [email protected] Matern/SPDE/GMRF
All these models – and many more – can be fitted within R-INLA
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA
this allows for flexible modelling...
• non-stationary models (anisotropy)
• models on a sphere (oscillating)
• non-separable models
Intro B, W, M, & R SPDE/GMRF Example End Finite Projection Markov Lattices ... and beyond
Beyond classical Matern models
The approach can in a straightforward way be extendedto oscillating, anisotropic, non-stationary, non-separablespatio-temporal, and multivariate fields on manifolds.
(κ2u + ∇ · mu −∇ · Mu∇)α/2(τux(u)) = W(u), u ∈ Ω
Finn Lindgren - [email protected] Matern/SPDE/GMRF
All these models – and many more – can be fitted within R-INLA
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA
this allows for flexible modelling...
• non-stationary models (anisotropy)
• models on a sphere (oscillating)
• non-separable models
Intro B, W, M, & R SPDE/GMRF Example End Finite Projection Markov Lattices ... and beyond
Beyond classical Matern models
The approach can in a straightforward way be extendedto oscillating, anisotropic, non-stationary, non-separablespatio-temporal, and multivariate fields on manifolds.
∂∂t + κ2
u,t + ∇ · mu,t −∇ · Mu,t∇(τu,tx(u, t)) = E(u, t), (u, t) ∈ Ω× R
Finn Lindgren - [email protected] Matern/SPDE/GMRF
All these models – and many more – can be fitted within R-INLA
Janine Illian
Integrated nested Laplace approximation and friends – fast and flexible modelling with R-INLA