[SOLVED] CS代考程序代写 Bayesian data structure A QUICK LOOK INTO INLA

A QUICK LOOK INTO INLA
STA465: Theory and Methods for Complex Spatial Data
Instructor: Dr. Vianey Leos Barajas

BACK TO THE AIR POLLUTION

GLOBAL PM2.5 DATA

FITTING MULTILEVEL MODELS IN INLA

WHAT IS INLA?
INLA stands for the Integrated Nested Laplace Approximation
It is a clever way to compute posterior distributions for multilevel models and their spatial generalizations
For the purposes of this course, it is magic.
This magic takes the form of an R-package that can be downloaded from http://r-inla.org/
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HOW TO CALL INLA
INLA works just like the formula in lm but with some slight differences
The f() function describe random effects (ie things that have more structure.
➤ The first two terms are μj . The second two are βjxij
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MORE INFORMATION ABOUT INLA
Sections 4.6-4.9 on Blangiardo and Cameletti.
Geospatial Health Data: Modeling and Visualization with R- INLA and Shiny
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WHAT COMES OUT?

WHAT WERE THE PRIORS?
Good question!
You’ve got to dig into the documentation to find them.
μ ∼ N(0,1000) β ∼ N(0,100)
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Are these sensible???
τμ2 ∼ Exp(100) τ2 ∼ Exp(100)
β
σ2 ∼ Exp(100)

WE CAN SIMULATE FROM THE PRIORS
Simulate
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μ β
∼ N(0,1000) ∼ N(0,100)
τμ2 τ2
∼ Exp(100) ∼ Exp(100) ∼ N ( μ , τ μ2 ) ∼ N(β, τ2)
β μ j
βj σ2
yij
β
∼ Exp(100)
∼N(μj+βjxij,σ2)

WAIT?! WHAT?
The prior model is two orders of magnitude off the real data
Two orders of magnitude on the log scale! ➤ Logdensityofneutronstaronly60 μgm−3
What does this mean practically?
The data will have to overcome the prior…
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WE CAN DO BETTER
With more sensible priors
μ ∼ N(0,1) β ∼ N(1,1)
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τμ2 ∼ N+(0,1) τ2 ∼ N+(0,1)
β
σ2 ∼ N+(0,1)

AND MAKE IT EASIER TO DEFEND YOUR MODELLING CHOICES
Non-informative
Weakly informative

A DIFFERENT VISUALIZATION
Prior predictive distribution with vague prior
Prior predictive distribution with weakly informative prior
Pallastunturi fells
Pallastunturi fells Concrete
Concrete
Neutron star
−1500 −1000
−500 0 500 1000
log(PM2.5)
−20 −10
0 10 20
log(PM2.5)
30

SPATIAL STRUCTURES AND INLA

NEXT WEEK…
Next week we will get into spatial models. We will use INLA for the remainder of the term.
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So far, we have focused on two general topics: Simulation from Bayesian models
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Maps
For the rest of the term, we’ll put those two together!
We’ll simulate from models.
We’ll plot our model results with maps.
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GENERAL SPATIAL DATA STRUCTURES
General spatial data structure: Z(s) : s ∈ D ⊂ Rd Areal Data
Geostatistical Data
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Point patterns