Transcript From disease mapping to archaeology and presence

From disease
mapping to
archaeology
and presenceonly modelling
Elena Moltchanova, PhD
Canterbury Statistics Day
Disease Mapping. A Bit of
History:
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Besag J (1974) ‘Spatial Interaction and the
Statistical Analysis of Lattice Systems’ JRSS B
36(2) 192-236
Besag J (1975) ‘Statistical Analysis of NonLattice Data’ JRSS D 24(3) 179-195
Besag J (1986) ‘On the Statistical Analysis of
Dirty Pictures’ JRSS B 48, 259-302
Besag J, York J, and Mollie A (1991) ‘Bayesian
image restoration, with two applications in
spatial statistics’. Annals of the Institute of
Statistical Mathematics 43(1) 1-20
Fig 1. Observed
incidence of
childhood diabetes
(T1DM) in Finland in
1987-1996.
Incidence =
number of cases/
population at risk*
100 000
BYM:
Observed
cases
risk
Yi ~ Poisson(  i )
Background level
Area-specific spatial residual
Systematic part
log(  i )     0 i   X i  log N i   i
Population at
risk or
expected
counts
Non-spatial
residual
Back to BYM: Conditional
AutoRegressive (CAR)
 0 i ~ N ( 0 , i ,m i )
Areas close together
have similar values
Neighborhood Matrix
W
BYM model DAG
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W
j
i
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Nik
Yik
h
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Nik
Yik
Xi
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Xi
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Ni
Yi
Xi
Applying BYM model to
diabetes incidence data:
Observed
Estimated by BYM
model
Argeopop project
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http://www.helsinki.fi/bioscience/argeopop
aims to shed new light on the prehistory of the
Finns by integrating evidence from genetic and
archeological data within a Bayesian statistical
framework.
From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T,
Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian
Spatiotemporal Analysis of Radiocarbon Dates in Eastern
Fennoscandia” Radiocarbon (in press)
From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T,
Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian
Spatiotemporal Analysis of Radiocarbon Dates in Eastern
Fennoscandia” Radiocarbon (in press)
www.helsinki.fi/bioscience/argeopop
From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T,
Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian
Spatiotemporal Analysis of Radiocarbon Dates in Eastern
Fennoscandia” Radiocarbon (in press)
www.helsinki.fi/bioscience/argeopop
From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T,
Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian
Spatiotemporal Analysis of Radiocarbon Dates in Eastern
Fennoscandia” Radiocarbon (in press)
www.helsinki.fi/bioscience/argeopop
From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T,
Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian
Spatiotemporal Analysis of Radiocarbon Dates in Eastern
Fennoscandia” Radiocarbon (in press)
www.helsinki.fi/bioscience/argeopop
From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T,
Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian
Spatiotemporal Analysis of Radiocarbon Dates in Eastern
Fennoscandia” Radiocarbon (in press)
www.helsinki.fi/bioscience/argeopop
Presence only data…?
 We
only find where we dig
 We only dig where we’ve found
something
 Similar to ecological niche modelling?
MaxEnt modeling
Maximize
−𝑝𝑖 log(𝑝𝑖 )
Subject to
𝑥𝑖 𝑦𝑖 =
𝑥𝑖 𝑝𝑖
Where
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x[i] is a ‘feature’ i.e. value of the covariate
y[i]=1 for presence and 0 for absence
p[i] is (multinomial) probability of presence
i=1,…,N areas
BYM model recast:
Observed
distribution of
occurrences
probability
𝑁
Y1: N ~ multinom(p
1: N
,X)
𝑋=
𝑌𝑖
𝑖=1
Y[i]=1 if there is an observation in area I
… and is missing otherwise
X is therefore also missing, with lower limit known
Placing a suitable prior either on X produces an identifiable
Bayesian spatial CAR model!
Will it work? A very simple
example.
𝑌~𝐵𝐼𝑁 𝑛, 𝑝
𝑝~𝐵 𝑎, 𝑏
𝑛~𝑃𝑜𝑖𝑠𝑠𝑜𝑛(𝜃)
Further Work:
• Implement multinomial BYM model (MCMC algorithm)
with various spatial autocorrelation structures:
• None
• CAR prior only
• CAR prior + non-spatial residual
• Perform sensitivity analysis
• Compare to MaxEnt performance