Grid spacing and quality of spatially predicted speciesabundances
A case-study for zero-inflated spatial data
Olga Lyashevska* Dick Brus** Jaap van der Meer*
*Royal Netherlands Institute for Sea ResearchDepartment of Marine Ecology
**Alterra, Wageningen University and Research Centre
July, 2 2014
Lyashevska et al, 2014 [email protected] July, 2 2014 1 / 16
Problem
Sampling is expensive, therefore it is important to statisticallyevaluate sampling designs prior to implementation ofmonitoring network;
This has been done before . . . (Bijleveld et al., 2012; Brus andde Gruijter, 2013), but. . .
spatial empirical ecological data are typically zero-inflated
and accounting for spatial dependence of such data is notstraightforward.
Lyashevska et al, 2014 [email protected] July, 2 2014 2 / 16
Problem
Sampling is expensive, therefore it is important to statisticallyevaluate sampling designs prior to implementation of monitoringnetwork;
This has been done before . . . (Bijleveld et al., 2012; Brus andde Gruijter, 2013), but. . .
spatial empirical ecological data are typically zero-inflated
and accounting for spatial dependence of such data is notstraightforward.
Lyashevska et al, 2014 [email protected] July, 2 2014 2 / 16
Problem
Sampling is expensive, therefore it is important to statisticallyevaluate sampling designs prior to implementation of monitoringnetwork;
This has been done before . . . (Bijleveld et al., 2012; Brus andde Gruijter, 2013), but. . .
spatial empirical ecological data are typically zero-inflated
and accounting for spatial dependence of such data is notstraightforward.
Lyashevska et al, 2014 [email protected] July, 2 2014 2 / 16
Problem
Sampling is expensive, therefore it is important to statisticallyevaluate sampling designs prior to implementation of monitoringnetwork;
This has been done before . . . (Bijleveld et al., 2012; Brus andde Gruijter, 2013), but. . .
spatial empirical ecological data are typically zero-inflated
and accounting for spatial dependence of such data is notstraightforward.
Lyashevska et al, 2014 [email protected] July, 2 2014 2 / 16
Aim
1. To work out a methodology for statistical evaluation ofsampling designs for zero-inflated spatially correlated countdata;
2. To test proposed methodology in a real-world case study.
Lyashevska et al, 2014 [email protected] July, 2 2014 3 / 16
Aim
1. To work out a methodology for statistical evaluation of samplingdesigns for zero-inflated spatially correlated count data;
2. To test proposed methodology in a real-world case study.
Lyashevska et al, 2014 [email protected] July, 2 2014 3 / 16
Methodology
Postulate a statistical model of the spatial distribution of thevariable;
Use prior data to calibrate such model;
Simulate a large number of pseudo-realities;
Sample each pseudo-reality repeatedly with candidate samplingdesigns;
Predict variable of interest at validation points;
Compute performance statistics;
Select the best candidate design out of evaluated candidates
Lyashevska et al, 2014 [email protected] July, 2 2014 4 / 16
Methodology
Postulate a statistical model of the spatial distribution of the variable;
Use prior data to calibrate such model;
Simulate a large number of pseudo-realities;
Sample each pseudo-reality repeatedly with candidate samplingdesigns;
Predict variable of interest at validation points;
Compute performance statistics;
Select the best candidate design out of evaluated candidates
Lyashevska et al, 2014 [email protected] July, 2 2014 4 / 16
Methodology
Postulate a statistical model of the spatial distribution of the variable;
Use prior data to calibrate such model;
Simulate a large number of pseudo-realities;
Sample each pseudo-reality repeatedly with candidate samplingdesigns;
Predict variable of interest at validation points;
Compute performance statistics;
Select the best candidate design out of evaluated candidates
Lyashevska et al, 2014 [email protected] July, 2 2014 4 / 16
Methodology
Postulate a statistical model of the spatial distribution of the variable;
Use prior data to calibrate such model;
Simulate a large number of pseudo-realities;
Sample each pseudo-reality repeatedly with candidate samplingdesigns;
Predict variable of interest at validation points;
Compute performance statistics;
Select the best candidate design out of evaluated candidates
Lyashevska et al, 2014 [email protected] July, 2 2014 4 / 16
Methodology
Postulate a statistical model of the spatial distribution of the variable;
Use prior data to calibrate such model;
Simulate a large number of pseudo-realities;
Sample each pseudo-reality repeatedly with candidate samplingdesigns;
Predict variable of interest at validation points;
Compute performance statistics;
Select the best candidate design out of evaluated candidates
Lyashevska et al, 2014 [email protected] July, 2 2014 4 / 16
Methodology
Postulate a statistical model of the spatial distribution of the variable;
Use prior data to calibrate such model;
Simulate a large number of pseudo-realities;
Sample each pseudo-reality repeatedly with candidate samplingdesigns;
Predict variable of interest at validation points;
Compute performance statistics;
Select the best candidate design out of evaluated candidates
Lyashevska et al, 2014 [email protected] July, 2 2014 4 / 16
Methodology
Postulate a statistical model of the spatial distribution of the variable;
Use prior data to calibrate such model;
Simulate a large number of pseudo-realities;
Sample each pseudo-reality repeatedly with candidate samplingdesigns;
Predict variable of interest at validation points;
Compute performance statistics;
Select the best candidate design out of evaluated candidates
Lyashevska et al, 2014 [email protected] July, 2 2014 4 / 16
Case Study
Dutch Wadden Sea;
Area: 2483 km2;
Abundance of Baltic tellin(M. balthica);
Centrifuge tube (17.3 – 17.7cm) to a depth of 25 cm
June–October 2010
Lyashevska et al, 2014 [email protected] July, 2 2014 5 / 16
Field data - Species Abundance
0
1000
2000
3000
0 25 50 75Species abundance
Cou
nts
90% observations are zeros
max 100 individuals
µ = 1.39 individuals
var = 24 individuals
Lyashevska et al, 2014 [email protected] July, 2 2014 6 / 16
Field data - Species Occurrence
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3320
3340
3360
3380
4000 4050 4100Easting (km)
Nor
thin
g (
km)
4100 samples
500 m grid + 10% random points
Lyashevska et al, 2014 [email protected] July, 2 2014 7 / 16
Modelling of the spatial distribution
1. Calibrate zero-inflated Poisson mixture model (assuming independentdata);
2. Use fitted model to classify each zero either as a Bernoulli or aPoisson zero;
3. Model the Bernoulli and Poisson variables separately (accounting forspatial dependence).
Lyashevska et al, 2014 [email protected] July, 2 2014 8 / 16
Modelling of the spatial distribution
1. Zero inflated Poisson mixture model (Lambert, 1992);
P(y |x) =exp(−µ)µy
y !(1)
logit(ψ) = log(ψ
1− ψ) = xTβ (2)
P(Y = y)
{ψ + (1− ψ)exp(−µ) y=0
(1− ψ) exp(−µ)µy
y ! for y = 1, 2, 3, . . .(3)
Lyashevska et al, 2014 [email protected] July, 2 2014 9 / 16
Modelling of the spatial distribution
2. Bernoulli/Poisson zeros;
Compute the ratio of the probability of a Bernoulli zero to the totalprobability of a zero;
ψ
ψ + (1− ψ)exp(−µ)(1)
Randomly allocate each zero to a Bernoulli zero or a Poisson zero.
Lyashevska et al, 2014 [email protected] July, 2 2014 9 / 16
Modelling of the spatial distribution
3. Bernoulli and Poisson variables are modelled separately by GLGM(Diggle et al., 1998; Christensen, 2004)
GLGM is GLM for dependent data (spatial random effect);Transformed model parameters, logit(ψ) and log(µ) are modelled withGaussian Random Field.
S1 = logit(ψ) = x1β1 + ε1 (1)
S2 = log(µ) = x2β2 + ε2 (2)
The model parameters are obtained through Marcov Chain MonteCarlo (MCML);MCML is computationally prohibitive for large data sets.
Lyashevska et al, 2014 [email protected] July, 2 2014 9 / 16
Simulation of the pseudo-realities
Simulate signals S (linear combination of covariates andGaussian noise) with GLGM models for Bernoulli and Poissonvariables at sampling locations (original grid);
Use sequential Gaussian simulation to simulate signals at very finegrid (100 m x 100 m) supplemented with validation points;
Combine pairwise the simulated fields of Bernoulli indicators andPoisson counts to pseudo-realities of zero-inflated Poisson counts;
Lyashevska et al, 2014 [email protected] July, 2 2014 10 / 16
Simulation of the pseudo-realities
Simulate signals S (linear combination of covariates and Gaussiannoise) with GLGM models for Bernoulli and Poisson variables atsampling locations (original grid);
Use sequential Gaussian simulation to simulate signals at veryfine grid (100 m x 100 m) supplemented with validation points;
Combine pairwise the simulated fields of Bernoulli indicators andPoisson counts to pseudo-realities of zero-inflated Poisson counts;
Lyashevska et al, 2014 [email protected] July, 2 2014 10 / 16
Simulation of the pseudo-realities
Simulate signals S (linear combination of covariates and Gaussiannoise) with GLGM models for Bernoulli and Poisson variables atsampling locations (original grid);
Use sequential Gaussian simulation to simulate signals at very finegrid (100 m x 100 m) supplemented with validation points;
Combine pairwise the simulated fields of Bernoulli indicatorsand Poisson counts to pseudo-realities of zero-inflated Poissoncounts;
Lyashevska et al, 2014 [email protected] July, 2 2014 10 / 16
Simulated data vs Original
Figure : Simulated data, species occurrence
Lyashevska et al, 2014 [email protected] July, 2 2014 11 / 16
Simulated data vs Original
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3320
3340
3360
3380
4000 4050 4100Easting (km)
Nor
thin
g (
km)
Figure : Original data, species occurrence
Lyashevska et al, 2014 [email protected] July, 2 2014 11 / 16
Grid spacing and Performance
Sample each pseudo-reality of zero-inflated Poisson datarepeatedly by grid-sampling with a given spacing;
Repeat it for all considered grid-spacings;
Predict values with IDW interpolation at validation points;
Calculate the performance statistics: the Mean Squared Error
MSE =1
N
N∑i=1
{Y (a0)− Y (a0)
}2(3)
MMSE =1
(R ∗ S)
R∑i=1
S∑j=1
MSEji (4)
N is a number of validation points, R - simulations andS - samples.
Lyashevska et al, 2014 [email protected] July, 2 2014 12 / 16
Grid spacing and Performance
Sample each pseudo-reality of zero-inflated Poisson data repeatedlyby grid-sampling with a given spacing;
Repeat it for all considered grid-spacings;
Predict values with IDW interpolation at validation points;
Calculate the performance statistics: the Mean Squared Error
MSE =1
N
N∑i=1
{Y (a0)− Y (a0)
}2(3)
MMSE =1
(R ∗ S)
R∑i=1
S∑j=1
MSEji (4)
N is a number of validation points, R - simulations andS - samples.
Lyashevska et al, 2014 [email protected] July, 2 2014 12 / 16
Grid spacing and Performance
Sample each pseudo-reality of zero-inflated Poisson data repeatedlyby grid-sampling with a given spacing;
Repeat it for all considered grid-spacings;
Predict values with IDW interpolation at validation points;
Calculate the performance statistics: the Mean Squared Error
MSE =1
N
N∑i=1
{Y (a0)− Y (a0)
}2(3)
MMSE =1
(R ∗ S)
R∑i=1
S∑j=1
MSEji (4)
N is a number of validation points, R - simulations andS - samples.
Lyashevska et al, 2014 [email protected] July, 2 2014 12 / 16
Grid spacing and Performance
Sample each pseudo-reality of zero-inflated Poisson data repeatedlyby grid-sampling with a given spacing;
Repeat it for all considered grid-spacings;
Predict values with IDW interpolation at validation points;
Calculate the performance statistics: the Mean Squared Error
MSE =1
N
N∑i=1
{Y (a0)− Y (a0)
}2(3)
MMSE =1
(R ∗ S)
R∑i=1
S∑j=1
MSEji (4)
N is a number of validation points, R - simulations andS - samples.
Lyashevska et al, 2014 [email protected] July, 2 2014 12 / 16
MMSE and Variance of MMSE
68
72
76
80
1000 2000 3000Spacing (m)
MM
SE
●
●
●●
●
●
0
2000
4000
6000
1000 2000 3000Spacing (m)
varia
nce
MM
SE
Lyashevska et al, 2014 [email protected] July, 2 2014 13 / 16
Conclusions
Sampling design for zero-inflated spatial count data isevaluated;
A strong monotonous increase of the MMSE is observed;
MSEji varies strongly between simulations and samples, especially forlarge grid spacings;
So numerous simulations and samples are needed for estimatingMMSE;
Spatial modelling of zero-inflated spatial data is laborious andcomputer-intensive.Is there an easier way: INLA?
Lyashevska et al, 2014 [email protected] July, 2 2014 14 / 16
Conclusions
Sampling design for zero-inflated spatial count data is evaluated;
A strong monotonous increase of the MMSE is observed;
MSEji varies strongly between simulations and samples, especially forlarge grid spacings;
So numerous simulations and samples are needed for estimatingMMSE;
Spatial modelling of zero-inflated spatial data is laborious andcomputer-intensive.Is there an easier way: INLA?
Lyashevska et al, 2014 [email protected] July, 2 2014 14 / 16
Conclusions
Sampling design for zero-inflated spatial count data is evaluated;
A strong monotonous increase of the MMSE is observed;
MSEji varies strongly between simulations and samples,especially for large grid spacings;
So numerous simulations and samples are needed for estimatingMMSE;
Spatial modelling of zero-inflated spatial data is laborious andcomputer-intensive.Is there an easier way: INLA?
Lyashevska et al, 2014 [email protected] July, 2 2014 14 / 16
Conclusions
Sampling design for zero-inflated spatial count data is evaluated;
A strong monotonous increase of the MMSE is observed;
MSEji varies strongly between simulations and samples, especially forlarge grid spacings;
So numerous simulations and samples are needed for estimatingMMSE;
Spatial modelling of zero-inflated spatial data is laborious andcomputer-intensive.Is there an easier way: INLA?
Lyashevska et al, 2014 [email protected] July, 2 2014 14 / 16
Conclusions
Sampling design for zero-inflated spatial count data is evaluated;
A strong monotonous increase of the MMSE is observed;
MSEji varies strongly between simulations and samples, especially forlarge grid spacings;
So numerous simulations and samples are needed for estimatingMMSE;
Spatial modelling of zero-inflated spatial data is laborious andcomputer-intensive.Is there an easier way: INLA?
Lyashevska et al, 2014 [email protected] July, 2 2014 14 / 16
Thanks!
Acknowledgements:This work was done in the framework of the WaLTER (Wadden Sea Long-TermEcosystem Research) project (WP5)
www.walterproject.nl
Lyashevska et al, 2014 [email protected] July, 2 2014 15 / 16
References I
Bijleveld, A. I., van Gils, J. A., van der Meer, J., Dekinga, A., Kraan, C., van derVeer, H. W., and Piersma, T. (2012). Designing a benthic monitoringprogramme with multiple conflicting objectives. Methods in Ecology andEvolution, 3(3):526–536.
Brus, D. and de Gruijter, J. (2013). Effects of spatial pattern persistence on theperformance of sampling designs for regional trend monitoring analyzed bysimulation of spacetime fields. Computers & Geosciences, 61(0):175 – 183.
Christensen, O. F. (2004). Monte carlo maximum likelihood in model-basedgeostatistics. Journal of Computational and Graphical Statistics, 13(3):pp.702–718.
Diggle, P. J., Tawn, J. A., and Moyeed, R. A. (1998). Model-based geostatistics.Journal of the Royal Statistical Society. Series C (Applied Statistics), 47(3):pp.299–350.
Lambert, D. (1992). Zero-inflated poisson regression, with an application todefects in manufacturing. Technometrics, 34(1):pp. 1–14.
Lyashevska et al, 2014 [email protected] July, 2 2014 16 / 16
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