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The effect of grid spacing on spatial prediction of species abundances was estimated. Data on counts of intertidal macrofauna (M. balthica) were collected in the Dutch Wadden sea over a grid of 500 × 500 m. The first step in the procedure was modelling of the zero-inflated data without taking spatial dependency into account. The problem of excess zeros was addressed through a mixture model (Lambert, 1992) which allowed to distinguish the point mass at zero through a Bernoulli process and the count component through a Poisson process. In the second step spatial correlation in both processes was then accounted for through generalised linear geostatistical model (GLSM) (Diggle et al., 1998; Christensen, 2004). Using simulations from the conditional distribution by MCMC a Monte Carlo approximation to the likelihood function was made. In the third step the two calibrated GLSMs were used to generate 100 pseudo-realities. This was done by conditional simulation from the original grid to the nodes of a fine prediction grid (100 × 100 m) supplemented with 1000 randomly selected validation points. The simulated pseudo-realities of the Bernoulli variable and the Poisson variable were combined into 100 pseudo-realities of a zero-inflated Poisson variable. In the fourth step each simulated pseudo-reality was repeatedly sampled by grid sampling with a varying spacing. Each sample was used to predict the study variable at the validation points by inverse distance weighted interpolation, and to estimate the Mean Squared Error (MSE). By averaging the MSEs over the pseudo-realities an estimate of the model-expectation of the MSE was obtained. The results showed that the decrease in resolution of the sampling grid (upscaling) had a clear effect on the precision of the predictions. This has direct implications for decisions with respect to sampling density for ecological monitoring programmes.
Citation preview
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
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