from disease mapping to archaeology and presence-only modelling
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From disease mapping to archaeology and presence-only modelling
Elena Moltchanova, PhDCanterbury 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
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BYM:
Population at risk or expected counts
)Poisson( ~ iiY
Observed cases
risk
iiiii NX log)log( 0
Background level Area-specific spatial residual
Systematic part
Non-spatial residual
Back to BYM: Conditional AutoRegressive (CAR)
),(~ ,00 iii mN
Areas close together have similar values
Neighborhood Matrix W
BYM model DAG W j h i Ni Yi Xi
Nik Yik Xi
Nik Yik Xi
Applying BYM model to diabetes incidence data:
ObservedEstimated 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)
9000-6400 BP
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
6400-5100 BP
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
5100-4000 BP
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
4000-3500 BP
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
3500-2500 BP
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
2500-1500 BP
Presence only data…? We only find where we dig We only dig where we’ve found
something Similar to ecological niche modelling?
MaxEnt modelingMaximize
Subject to
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:
),multinom(p ~ N:1:1 XY N
Observed distribution of occurrences
probability
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!
𝑋=∑𝑖=1
𝑁
𝑌 𝑖
Will it work? A very simple example.
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
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