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Spatial and spatio-temporallog-Gaussian Cox processes:
re-defining geostatistics
Peter J Diggle
Lancaster University and University of Liverpool
combininghealth information,
computation and statistics
CHICAS
Geostatistics
traditionally, a self-contained methodologyfor spatialprediction, developed at Ecole des Mines,Fontainebleau, France
nowadays, that part of spatial statisticswhich isconcerned with data obtained by spatiallydiscretesampling of a spatially continuous process
Model-based Geostatistics(Diggle, Moyeed and Tawn, 1998)
the application of general principles of statisticalmodelling and inference to geostatistical problems
which means:
− formulate a model for the data
− use likelihood-based methods of inference
− answer the scientific question
Geostatistical data and model : lead pollution in Galicia
5.0 5.5 6.0 6.5 7.0
46.5
47.0
47.5
48.0
X Coord
Y C
oord
S(x) pollution surface
xi sampling location
Yi measured pollution
Yi|S(·) ∼ N(S(xi), τ 2)
Point processes
Stochastic models for arrangements of points in space and/ortime
Scientific focus on understanding why the points are wherethey are:
− absolutely
− and/or relatively
Point process data: two examples
400000 420000 440000 460000 480000
1000
0012
0000
1400
0016
0000
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Cox process
Poisson process
λ(x) : intensity function
event-locations mutually independent, probability densityproportional to λ(x)
Requires: λ(x) ≥ 0 and∫
A λ(x)dx <∞, for any finite region A
Cox process
λ(x) is unobserved realisation of stochastic process Λ(x)
Trans-Gaussian Cox process
Cox process with:
Λ(x) = F{S(x)}
S(x) ∼ Gaussian process
Log-Gaussian Cox process: F(·) = log(·)
introduced by Møller, Syversveen and Waagepetersen (1998)
popular because of analytic tractiability (moments, etc)
but any F(·) OK if using Monte-Carlo methods of inference
LGCP as a geostatistical model
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LGCP as a geostatistical model
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Re-defining geostatistics
Old:
unobserved, real-valuedstochastic processS = {S(x) : x ∈ A}
pre-specified locations{xi ∈ A : i = 1, ..., n}measurements
Y = {Yi : i = 1, ..., n} atlocations xi
use model for [S,Y] = [S][Y|S]to predict S
New:
unobserved, real-valuedstochastic processS = {S(x) : x ∈ A}
data D
use model for [S,D] = [S][D|S]to predict S
New definition shifts focus from data to problems
LGCP model-fitting
Ingredients:
latent Gaussian process S
data D
parameters θ
Predictive inference via MCMC or INLA
[θ, S,D] = [θ][S|θ][D|S, θ]⇒ [S|D] =
∫[S|D; θ][θ|D]dθ
∫[S|D; θ][θ|D]dθ ≈ [S|D; θ] ?
Applications
intensity estimation: hickories in Lansing Woods
spatial segregation: BTB in Cornwall
disease atlases: lung cancer mortality in Spain
real-time spatial health surveillance: AEGISS
Intensity estimation: hickories in Lansing Woods
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use data to construct non-parametricestimate of λ(x)
lots of existing methods, butinferential standing unclear
LGCP approach enables predictiveinference
Λ(x) = exp{S(x)}S(·) ∼ SGP(β, σ2, ρ(u))
ρ(u) = exp(−u/φ)
Intensity estimation: Lansing Woods results
Parameter estimation
β
Den
sity
−15 −10 −5 0 5 10 15
0.0
0.5
1.0
1.5
2.0
σ
Den
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0.0 0.5 1.0 1.5 2.0 2.5
01
23
4φ
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0 10 20 30 40
0.00
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Prediction
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Spatial segregation: BTB in Cornwall
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Genotype 9Genotype 12Genotype 15Genotype 20
Λk(x) = exp{βk + S0(x) + Sk(x)} : k = 1, . . . ,m
S0(x) not identifiable:
pk(x) = {Λk(x)}/{∑m
j=1 Λj(x)} = exp[−∑
j6=k{βj + Sj(x)}]
BTB in Cornwall: estimated type-specific probabilitysurfaces
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BTB in Cornwall: areas of type-specific (0.8+) dominance
P(type k dominates)> 0.6
Dominant type defined as p_k>0.8
type 1type 2type 3type 4
P(type k dominates)> 0.7
P(type k dominates)> 0.8 P(type k dominates)> 0.9
Disease atlases: lung cancer mortality in Spain
Spatially discrete approach: population denominators andrisk-factor information aggregated to area-level averages(MRF model)
Disease atlases: lung cancer mortality in Spain
Spatially continuous approach: population denominators andrisk-factor information in principle at point level (LGCP model)
Λ(x) = d(x) exp{z(x)′β + S(x)}
0.6
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1.2
1.4
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Predicted risk surface P(relative risk > 1.1)
Building spatio-temporal dependence
Separable
ρ(u, v) = ρS(u;α)ρT(v;β)
Non-separable empirical
ρ(u, v) = ρS(u;α;β)ρT(v){1 + f(u, v; θ)}
Non-separable mechanistic
S(x, t) = limδ→0
{∫hδ(u)S(x− u, t− δ)du + Zδ(x, t)
}
In conclusion
1 Statistical modelling should be driven by the underlyingscientific problem, rather than by the format of available data
2 Most (but not all) natural phenomena are spatially continuous
3 Surprisingly many of the problems to which spatial methodscan make a useful contribution reduce to an application ofBayes’ Theorem
[S,D] = [S][D|S]⇒ [S|D]
4 Modelling an unobserved stochastic process and assigning aprior distribution to an unknown parameter are mathematicallyequivalent but scientifically different activities
5 Modelling as a route to empirical prediction and modelling asa route to understanding a physical/biological mechanismdemand different approaches
Acknowledgements
CHICAS, Lancaster University : Paula Moraga, Barry Rowlingson,Ben Taylor
MRC: Methodology Research Grant G0902153
Diggle, P.J., Moraga, P., Rowlingson, B. and Taylor, B. (2013).Spatial and spatio-temporal log-Gaussian Cox processes: extendingthe geostatistical paradigm. Statistical Science, 28, 542–563.