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 - finn.lindgren@math.ntnu.no 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 - finn.lindgren@math.ntnu.no 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 - finn.lindgren@math.ntnu.no 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