Problem. How to design an efficient spatial sampling method to estimate an
occurrence (0/1) map when. / spatial structure. / no evolution during the sampling
...
Echantillonnage spatial pour la construction de carte d’occurrence
M. Bonneau, N. Peyrard et R. Sabbadin
– p. 1
Problem How to design an efficient spatial sampling method to estimate an occurrence (0/1) map when X spatial structure X no evolution during the sampling period X imperfect observations X set of sites which can be sampled is finite X sampling has a cost
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Motivation X Control of invasive species: construction of fire ants invasion maps in Queensland, Australia X Weeds management: construction of grown plants and seed bank maps Question: Where to allocate supplementary sampling effort to estimate an invasion map of satisfying quality, taking into account sampling cost?
– p. 3
Proposed method Based on • a stochastic modeling of binary maps and sampled observations • tools from sequential decision under uncertainty • same model and same criterion for selection of sampled area and for map reconstruction Two modeling options • Image analysis: HRMF and Maximum Posterior Marginals • Geostatistics: boolean model and kriging – p. 4
Three steps method 1) From a first arbitrary sampling, estimate model parameters and compute initial knowledge = map distribution conditionally to observations 2) Based on this initial knowledge choose a sampling strategy specifying which sites to sample, in order to improve knowledge - static or adaptive sampling - trade cost/expected map quality 3) build invasion map from the improved knowledge – p. 5
Table of Contents 1- Image analysis approach 2- Geostatistics approach 3- Experimental comparison 4- Methodological comparison 5- Conclusion/perspectives
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Image analysis approach Hidden variables
Observations
Hidden Markov Random Field on G = {V, E}
– p. 7
The model • X = {Xi , i ∈ V }, Xi ∈ {0, 1}, HMRF 1Y P (X = x) = ψc (xc ) Z c∈C • Y = {Yi , i ∈ V }, Yi ∈ {0, 1} P (Y = y | X = x) =
Y
P (Yi = yi | Xi = xi )
i∈V
Example: P (Yi = 0 | Xi = 1) = θ > 0 (false positive) – p. 8
Map reconstruction • a = {ai , i ∈ V } ∈ {0, 1}: sample action ya : observation on sample set a • MPM estimator of the hidden map o n MP M PM = arg max Pa (Xi = xi | ya ) , x xM P M = xM i i xi
→ Value of knowledge V
MP M
(a, ya ) =
X i∈V
max Pa (Xi = xi | ya ) xi
– p. 9
Static sampling • Given an initial sample a0 and observations sampling action ya0 , we want to optimise next sample action a • Value of a sample a U
MP M
(a) =
X
Pa,a0 (ya |ya0 )V M P M (a, ya , a0 , ya0 )
ya
• Optimisation problem arg
max
a,|a|≤Amax
U M P M (a)
Computational complexity is NP-hard – p. 10
Heuristic static sampling (1) • Explore sites where initial knowledge is the most uncertain: marginal Pa0 (xi | ya0 ) closest to 12 arg
max
a,|a|≤Amax
X
[min(Pa0 (Xi = 1 | ya0 ), Pa0 (Xi = 0 | ya0 ))]
i,ai =1
• Marginals computation is itself NP-hard ⇒ approximation using sum-prod algorithm
– p. 11
Heuristic static sampling (2) The approximation corresponds to two simplifying assumptions X Additional observations are reliable: θ = 1 X Joint probability approximated as a product of “approximate Qnconditional marginals”: Pa (x|ya ) ∼ i=1 Pa (xi |ya )
– p. 12
Adaptive spatial sampling (1) • Principle: - Sampling locations not settled once for all before the sampling campaign - Intermediate observations are taken into account to design next sampling step - Possibility to visit a cell more than once
– p. 13
Adaptive spatial sampling (2) • a sampling strategy δ is a tree • a trajectory in τ = (a1 , y 1 , . . . , aK , y K )
δ:
Value of a trajectory
U (τ ) =
K X
V M P M (Pe,a0 ,a1 ,...,aK (X|λ, y 0 , y 1 , . . . , y K ))
k=1
Value of a strategy
V (δ) =
P
τ
U (τ )P (τ | δ) – p. 14
Heuristic adaptive spatial sampling • Exact computation is PSPACE-hard ⇒ Heuristic algorithm - online computation - approximate method for static sampling at each step
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Evaluation of HMRF method • Evaluation on simulated data (Potts model with external field) • Comparison of behavior of - random sampling (RS) - static heuristic sampling (SHS) - adaptive heuristic sampling (AHS)
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Procedure • repeat 10 times - simulate hidden map x from P (x) (50 × 50 sites) - apply regular sampling (about 10% of area): a0 - simulate ya0 from P (yi | xi ) - apply RS, SHS, AHS, 10 times
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• Comparison on reconstruction errors
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Error rates 15
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12 static adaptive random
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static adaptive random
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α = −1, β = 0.4
θ = 0.8
– p. 18
Where do we sample? Hidden map 5 10 15 20 25 30 35 40 45 50 5
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α = (1, −1), β = 0.4, θ = 0.8
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Where do we sample? Static sampling: A and O
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Where do we sample? Static sampling:marginals
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Where do we sample? Adaptive sampling: A and O (cumul)
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Where do we sample? Adaptive sampling: marginals
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General behavior • Adaptive HS ≥ Static HS ≃ Random S • Discrepancy between Adaptive HS and Static HS increases with - sampling ressources - hidden map structure
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Geostatistics approach The hidden map is a realisation of a Boolean Model Ξ. ⇒ X(s) = 1{s∈Ξ} , ∀s ∈ W compact
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Map reconstruction versus sampling choice • Map restoration: from observations, using kriging • Sample choice: given an initial sample a0 and observations ya0 , we want to optimise next sample action a Previous sample steps should influence the choice of a → conditional kriging
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Conditional kriging • Compute an estimator pλ∗ i (Ya , ya0 ) of E[Xi | Ya , ya0 ], as λ 2 ∗ λ = arg max E (pi (Ya , ya0 ) − Xi ) | ya0 . λ
with
pλi (Ya , ya0 )
= λi +
• Kriging estimator xkrig i
= 1 if
P
k∈a
λk Yk +
λ∗ pi (Ya , ya0 )
P
l∈a0
γl y l .
> 1/2, 0 otherwise
Possible because X and Y are of same nature – p. 27
Static sampling • Value of a sample a: X 2 kri λ∗ U (a) = − E (pi (Ya , ya0 ) − Xi ) | ya0 i∈V
• Same form than in the HMRF case: X kri Pa0 ,a (ya | ya0 )V kri (a0 , a, ya0 , ya ) U (a) = ya
with V
kri
0
(a , a, ya0 , ya ) = −
X i∈V
.
λ∗ 2 E (pi (ya , ya0 ) − Xi ) | ya0 , ya – p. 28
Heuristic sampling • Static samplig - same assumptions lead to the same heuristic than for the HMRF approach - rank sites according to conditional marginals - marginals computation using simple kriging • Adaptive sampling - idem HMRF approach - online + static heuristic – p. 29
Evaluation of geostatical method • Evaluation on simulated data (Boolean model with disk of constant diameter as grains) • Comparison of behavior of - random sampling (RS) - static heuristic sampling (SHS) - adaptive heuristic sampling (AHS)
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Procedure • repeat 10 times: - simulate hidden map x on W = [0; 500] × [0; 500] - V = 50 × 50 regular grid - apply regular sampling (about 10% of area): a0 - simulate ya0 from P (yi | xi ) - apply RS, SHS, AHS, 10 times
– p. 31
Error rates 7
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15 static adaptive random
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Adaptive HS >> Static HS ≃ Random S except when structure is high – p. 32
Where do we sample? Static HS
Adaptive HS
– p. 33
Comparison on Lavender map Global error 55 HMRF−static−true HMRF−static−init HMRF−adapt−true HMRF−adapt−init HMRF−adapt−reest krig−static−true krig−static−init krig−adapt−true krig−adapt−init krig−adapt−reest
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5 80 2
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Provided by A. Calonnec, INRA Bordeaux
– p. 34
Comparison on Hevea map Global error 20 HMRF−static−true HMRF−static−init HMRF−adapt−true HMRF−adapt−init HMRF−adapt−reest krig−static−true krig−static−init krig−adapt−true krig−adapt−init krig−adapt−reest
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Provided by D. Nandris and F. Pellegrin, IRD – p. 35
Theoretical comparison • Differences - Boolean models continuous space covariance - HMRF discrete space or structured space conditional independences can handle categorial variables
– p. 36
Theoretical comparison • Similarities - Sampling same assumptions lead to the same heuristic - Restoration kriging amount to maximising conditional marginals in 0/1 case They differ by the choice of the approximation method - HMRF: marginals computed using sum-product - Boolean models: marginals computed using kriging
– p. 37
Conclusion • Two methods for spatial sampling in any binary spatial process
• Geostatistical approach seems to perform better. Why? - Model? - Criterion? - Approximation method?
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Around this work • Exact algorithms for small (MAP) problems: combining variable elimination and search in trees • Mathieu Bonneau: PhD thesis on adaptive spatial sampling for adventices mapping in crop fields. With Sabrina Gaba (INRA Dijon) - development on a more general framework: knowledge gathering in graphical models - developement of approximate resolution algorithms using reinforcement learning - application to mapping of (multiple?) species abundances from spatial sampling of grown plants
– p. 39
Around this work and many other interesting questions • How to model cost? • Other choices for the sample value? • And if the process to map evolves during the sampling period? • How to combine sampling actions and management actions?
– p. 40
