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:
Besag J (1974) ‘Spatial Interaction and the
Statistical Analysis of Lattice Systems’ JRSS B
36(2) 192-236
Besag J (1975) ‘Statistical Analysis of Non-
Lattice 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
Y i
~ Poisson(
i
) log(
i
)
Area-specific spatial residual
Background level
0 i
X i
Systematic part
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
N ik
j
Y ik
N i
X i
i
Y i
N ik
W
h
Y ik
X i
X i
Applying BYM model to diabetes incidence data:
Observed
Estimated by BYM model
Argeopop project
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
−𝑝 𝑖 log(𝑝 𝑖
)
Subject to 𝑥 𝑖 𝑦 𝑖
= 𝑥 𝑖 𝑝 𝑖
Where
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:
probability
Observed distribution of occurrences Y
1 : N
~ multinom(p
1 : N
, X )
𝑁
𝑋 = 𝑌 𝑖 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
