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.

1. Grid spacing and quality of spatially predicted species
abundances
A case-study for zero-inflated spatial data
Olga Lyashevska* Dick Brus** Jaap van der Meer*
*Royal Netherlands Institute for Sea Research
Department of Marine Ecology
**Alterra, Wageningen University and Research Centre
olga.lyashevska@nioz.nl
July, 2 2014
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 1 / 16

2. Problem
Sampling is expensive, therefore it is important to statistically
evaluate sampling designs prior to implementation of
monitoring network;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 2 / 16

3. Problem
Sampling is expensive, therefore it is important to statistically
evaluate sampling designs prior to implementation of monitoring
network;
This has been done before . . . (Bijleveld et al., 2012; Brus and
de Gruijter, 2013), but. . .
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 2 / 16

4. Problem
Sampling is expensive, therefore it is important to statistically
evaluate sampling designs prior to implementation of monitoring
network;
This has been done before . . . (Bijleveld et al., 2012; Brus and
de Gruijter, 2013), but. . .
spatial empirical ecological data are typically zero-inflated
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 2 / 16

5. Problem
Sampling is expensive, therefore it is important to statistically
evaluate sampling designs prior to implementation of monitoring
network;
This has been done before . . . (Bijleveld et al., 2012; Brus and
de Gruijter, 2013), but. . .
spatial empirical ecological data are typically zero-inflated
and accounting for spatial dependence of such data is not
straightforward.
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 2 / 16

6. Aim
1. To work out a methodology for statistical evaluation of
sampling designs for zero-inflated spatially correlated count
data;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 3 / 16

7. Aim
1. To work out a methodology for statistical evaluation of sampling
designs for zero-inflated spatially correlated count data;
2. To test proposed methodology in a real-world case study.
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 3 / 16

8. Methodology
Postulate a statistical model of the spatial distribution of the
variable;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 4 / 16

9. Methodology
Postulate a statistical model of the spatial distribution of the variable;
Use prior data to calibrate such model;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 4 / 16

10. 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;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 4 / 16

11. 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 sampling
designs;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 4 / 16

12. 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 sampling
designs;
Predict variable of interest at validation points;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 4 / 16

13. 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 sampling
designs;
Predict variable of interest at validation points;
Compute performance statistics;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 4 / 16

14. 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 sampling
designs;
Predict variable of interest at validation points;
Compute performance statistics;
Select the best candidate design out of evaluated candidates
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 4 / 16

15. Case Study
Dutch Wadden Sea;
Area: 2483 km2;
Abundance of Baltic tellin
(M. balthica);
Centrifuge tube (17.3 – 17.7
cm) to a depth of 25 cm
June–October 2010
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 5 / 16

16. Field data - Species Abundance
0
1000
2000
3000
0 25 50 75
Species abundance
Counts
90% observations are zeros
max 100 individuals
µ = 1.39 individuals
var = 24 individuals
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 6 / 16

17. Field data - Species Occurrence
q
qqqqqqq
q
qqqqqqq
q
qqqqq
q
q
q
q
q
q
q
q
q qqq
qqqqqq
qqqq
qqqqq
q
qqqqq
q
q
q
qqqqq
qqqqqqqqqqqq
qqqqqqq qqqq
q
q
q
q
q
q
qq
qq
q
q
q
q
q
qqqqqq
q
q
q
q
qq
q
qqqqqqqq
q
q
q
q
q
q
qq
q
q
q
q
qqq
qqq
q
qqq
q qqq
qqqqq
q
qqqqqqq
qq
q
q
q
qqq
qqqqqqq
qqqq q
q
qq
qqq
qq
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
qq
q
qq
q
q
q
q
q
q
q
qq
q
q q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
qq
q
q
qqq
qqq
q
qqqqqq
qqqqqqqqqq
qqqqq
q
qq
q
q
q
q
q
q
q
q
qq qqqqq
q
qqq
qqqqqqq qqq
qqqqqqq
qqqqqqq qqqqq
qq
q
q
q
q
q
q
q
qq
q
qqqqqqq
qqqq qqqq
qqqqq
q
q
q
q
q
qq
q
qqqqq
qq
q
qq
q
q
q
q
q
q
qqqq
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q qq
q
q
q
qq
q
qq
q
q
q
q
q
q
qq
q
qqq
q
q
q
q
qq
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
qq
q
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
qq
q
qq
qqqqq
q
q
q
qq
q
qq
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
qqqqq
q
q
q
q q
q
q
q
qq
q
qq
q
q
q
q
q
qq
q
q
q
qqq
q
q
qqqq
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
qqqqq
qq
q
q
q
qq
qqqq
q
q
q
q
qq
qqq
qqqqqqq
q
q
q
q
q
q
qq
qqq
q
q
q
qqq
qq
qq
q
q
q
qq
qqqqq
qq
qq
qq
q
q
q
q
q
q
qq
qqqqq
q
q
q
q
q
q
q
q
qq
q qqqqq
qq
q qqqqqqq
qqqqq
q
q
qq
q
q
qq
q
qqqqqq
q qqqqqqq
qqqq
qq
qq
q
qqq
qqq
qq
q
q
q
qq
qqqq
qqqqq
q
qq
qq
qqqq
qqqqqqq
qqq
q
q
q
q
q
q
qqqq
q
qq
qq
q
q
q
q
qqq
qqq qqqqqq
qq
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
qq
q
qq qq
q
qq
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
qq
q
q
q
q
qq
q
q
q
qq
q
q
qq
q
q
q
q
q
q
qq
q
q
q
qqq
q
q
q
q
q
q
qqqqqqq qqqq q
qqqq
qqqqqqqqqqq
q
q
q
q
qqqq
qqqq q
q
q
q
qq
qqq
qqqqqqqqqqq
qq
qq
qqq qqqqq qq
qq
qq
q
qqqqqqqqq
qqqqq
q
q
q
qqq
q
q
qqqqqqqqqqqqq
qq
qqqqqqqqqqqqq
qqqqqqqq
qqqqqqq qqq
qqqqqqqqqq
qq
qqq
qq
qqq
q
qqqqqqqq
q
q qq q
qq
q
qq
qqqqq
q
qqqqqq
qq
qq
qqqq
qqqq
qq
qq
qqq
qqqqqqq
qqqqq
qq
qqqqq
qqqqq
q
qqq
qqqq
qq
q
q
qqqq
q
q
q
q
qqqq
qq
qqqq
qqqqq
qqqqqqqqq
qqqqq q
qqqqq
qq qqq
qqqqqqq
q
q
q
qqqq
qqqqqqqqq
q
qqqq
qqqqq qqq
qqqqq
qqqqqqqqqq qqqqqqqqqq
qqqqqqqqq
qq
q
q
q q
q
q
qq
qqqqqqqqqqqqq
q
qq
q q
q
q
qqqqqqq
q
qqq q
q
qqq
q
q
q
q
q
q
q
q
qqqqqqqqq
qqq qqq
q
q
q
q
q
q
q
qqqqqqq
qq
qqqq
q
q
q
qqqqqq
qqqq
qqq q
q
q
q
q q
q
q
q q
qq
q
q
q q
q q
q
q
q
qqqq
q
q
qq
q
q
q
q
q
qq qq
qqq
qqq
q
q
q
q
q
q
q
q
qq
q
qq
qq
q
q
q
q
qq
q
qqq
q
q
qq
q
q
q q
q
q
q
qq
q
qq
q
qq
q
qq
q
q q
q q
qq
q
q
q
q
q
q
q
q
q qqq
q
q
q
q
q
q
qq
q q
qqq
qq q
q
q
q
q
q qq
qq
q
q
q
q q
q
qqq
q
qq
qqq
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
qq
qq
q
q
q
q
qq
q
q
q
qqq
qq
q
qq
q
qq
q
q q
qqq
qqq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
q q
q
q
q
qq
q
qq
q
q
q
q
qq
qq
qqqqqq
qqqqq
q q
q
qqqq
q
q
q
qqq
qqqqqq
qqq
qqqq
qqq
q
qq
qqq
qq
qqq
q
qqq q
q
q
q
qqq qqqqq
qqqq
q q
q
q
qqq
q
q q
q qqq
qq
q
q
qq
q
q
qq
qqqqq
qqqqqq
q
qqqq
qqq
qqq
q
qqq
qqqqq qqqq
qqqqq
q
q q
q
q
q
q q
qqqqqqqq
qq
q
q
q
q
q
q
q
q
q
q
qq
qqqqqqqq
qqqq
q
q
q
q
q
qqqq
qqqq
qq qq
qqqqqqq
qqq
qq
q
q
qqq q
q
qq
q
q
qqq
qq
q
qqqq
q
q
qq
q
q
q
qq
q
q
qqqqq
qqqqqqqq
qqqq
qqqqqqq qq
qqqqq
qq
q qqqq
qqqqqqq
qq
qqqq
qqq
q
qqq
qqq
qqqqqq
q
qq
qq
qq
qq
qq
qq
qq
q
qq
qq
qq
qq
qqq qq
qq
qq
q
qq
q
qqqq
qq
qq
q
qq
qqqqqq
qq
q qq
qq
qqq
qq
qqqq
qq
qq
qq
q qqqq
qq
qq
qqqqqq
q
q
q
q
qq
qq
q
q
qq
q
q
q
q
qq
q
q
qq
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
qq q
q
q
q
q q
q
q
q
qq
q
q
q
q
q q
q
q
qqq
q
q
qqq
qq
q
qq
q
q
qq
q
q
qqqq
qq
qq
q
q
q
q
q qq
qq
qq
qq
qq
q q
qqqq
qqqq qqq
qqq
qq
qq
qq
q qq
qq
q
qq
q
q
q
q
q
qq
qq
q
qq
qq qq
qqqq
qqqqqq qqq
qqqq
qq
qq
qqqqqqq qq
q
qq
qq
q
qq qq
q
qq
qqqq
q
qqq
qqq
qq
qq q
qq
q q
qqq
q
qq
qq
qq
qq
qq
q
qqqqqqqqqqq
q
qqqqqqq
q
qqq
qqqqqqq
qqqq
qqqq q
qqqqq
q q qqqqq
qq qqqq
q
qqq
qqqqqqqqqqqqq qqqq
qq
qqq qq
q
qq
q
q
qqqqqqqqqqqqqqqqq
qqqqqq
qqq qqqqqq
q
q
qq
q
q
qq
qqqqqqq
qqqq
qq
qqqqqqqqqqqqq
q
q
q
q
qq
q
q
q q
q q
q
q
q
q
qq
q
q
q
qq
q
q
q
q q q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q q
q
q
q
q
qqqqqq
q q
qq
q
q qq
qq
qq
q
q
q
q
q
q
q
q
q
q
qq
qqq
q
qq q
q
q
qq
q
qqqq
qqqqqq
qqqqq
q qqq
qqqqqqqqqq
qqqqqqqqqqqqqq
qqqqqqqqqqqqqq
qqqqqqqqqqqqq qqqqqqqqqqqqqq
qq
qqqq
qqq
qqqq
qqqq
qqqqqqqqqq
qqqq
q
q
qqqqqqq
q
qqq
qqqq
q
q
q
q
qq
q
qqqqqqqqqq
q
q
q
qq
q
qq
qq
q
q
q
q
q
q
qq
qqqqq
qqqq qq
q
qqqqqqqqq
q
qqqq
q
q
q
qq
qq
qqq
qqqqq
qqqqqqq
qqqq
qqq
q
q
q
q
q
qq
qq
q
q
qq
q
qqq
qqqqq
q
q
q
q
qqqq
q
q
qq
qq
q
q
q
q
qq
q
qq
qq q q
qqqqq
qqqq
qq
qq
qqq
qq
qq
q
qq
qq
qq
qq
qq
q
q
q
qqq
q
qq
q
qqq
q
qqqqq
qqq
q q
q
qq
q
q
q
q q
qq
qq
qq
q
q
q
q q
q
q
q
q
q
q
qq
qqq
qq
q
q
q
qq
qq
qqq
q
q
q
q
q
q
qq
qqq
qq
qqq
q
q
q q
q
q
q
q
qqqq
q
q
q
q
q
qqq
qqqq
qqq
q
q
q
q
qq
q
q
q
q
q
qqq
qqqqqqqq
qqqq qq
qqqqqqq qq
qqqqqqqqq
qqq q
q
qqq
qqqqqq
qq
q
q
qqqqqq
qqqq q
q
q q
qqq
q
qq
q
q
qqq
qqqqq
qqqq qqq
q
q
qqq
q
qqq
qqqqqqqq
qq
qqqqqqq
qqq
qq qq
qq
qqqqqqqq
qqqqqqqq
q
qqqqq
q
qqqqqq
qqqqq
qq
qq
q
qq
q
qqqqqqq
qq
q
q
qq
q
q qqq
q
q q
q
q
q
qqqq qqqqqqqq
qqqqq
q
q
q
q
q q
qq
q
q
q
q q
q
q
q
q
q q
q
q
q
qqqqq
qq
q
q
q q
qqqqqqqqq
qqqqq
q
q
q
q q
q
q
q
q
q q
q
qq q
q
q
q
q
qqqqqqqqqqqqqq
qqqqqqqqqqqqqq
q
qq
q q
qq qqqq
qqqqqqqqq
qqq
qqqqqqqqqqqq
qq
q
q qqqqqq qqqqqq
qqqqqq
qqqqqqqqqqqq
qqqqq
q q
qq
qq
q
q
qqqqqqqqq
qqqq qqqq
qqqqqqqq qqq
qqqq
qqq q
qqqqqq
qqqqq
q
q
q
q
q
qq
q
q
q
q
q
qq
q
q
q
q
q
qqqqq
qqq
qqq
q
q
q
q
q q
qq
qqq
q
qq
q
q
qq
q
q
q
q
qq
q
qq
qq
q
qqqqq
q
q
q
qq
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
qq
q
q
qq
q
qqqq
qqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
qqqqqq
qqqqq
qqqq
qq
qqqqqq
qqqqq
qqq
qqqqq
qq
q
qq
qqq
q
q q
q
q q
qqqqq
qqqqqq
q
q
q q
q
q
q q
q
q
q q
q q
q
q
q
qqqqqqq
qqqqq
q
qqqqqqqq
q
qqqqqqq qq
qqqq
qq
q
q
qqqqqqq
qqq
qqqqq
qq
q q
qq
qqqqq
qqqqqqq
qqq q
qqqqqq
qqqqqq
qqq
q
q
q q
q
qqqq
qqqq
qqqqqq
qqqqq
qqq
q
q
q
q
q
q q
qqqqqqq
qqq
qqqqqq
q
qqqqqq
q
q
q
q
q
qq q
q
qqq
qqqqqqqq
qqq
qqqqqqq
qqqq
qqq
q
q
q
qq
q
q q
qqq
q
qqq
q
qq
q
qqq
q
qq
q
q
q
qq
qqqqqqqqqq qqqqqqqqq
qq
qq qqq
q
qqq
qq
qq
q
q
q
q
q
q
q
q
qq
q
q
q
qqqqq
qq
qq
qqqq
qq
qqqqq
qqq
qqq
q
q
q
qqq
q
q
q
qq
qq
q
q
q
q
q
qq
q
q
q
q q
q
q
qqqqq
q
q
q
qq
qq
q
q
q
qq
q
q
q
q
q
qq
q
q
q
q
q
q
qq
qq
qq
q
q
q
qqq
qq
q
q
qq
q
qq
3320
3340
3360
3380
4000 4050 4100
Easting (km)
Northing(km)
4100 samples
500 m grid + 10% random points
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 7 / 16

18. Modelling of the spatial distribution
1. Calibrate zero-inflated Poisson mixture model (assuming independent
data);
2. Use fitted model to classify each zero either as a Bernoulli or a
Poisson zero;
3. Model the Bernoulli and Poisson variables separately (accounting for
spatial dependence).
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 8 / 16

19. 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
) = xT
β (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 olga.lyashevska@nioz.nl July, 2 2014 9 / 16

20. Modelling of the spatial distribution
2. Bernoulli/Poisson zeros;
Compute the ratio of the probability of a Bernoulli zero to the total
probability 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 olga.lyashevska@nioz.nl July, 2 2014 9 / 16

21. 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 with
Gaussian Random Field.
S1 = logit(ψ) = x1β1 + 1 (1)
S2 = log(µ) = x2β2 + 2 (2)
The model parameters are obtained through Marcov Chain Monte
Carlo (MCML);
MCML is computationally prohibitive for large data sets.
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 9 / 16

22. Simulation of the pseudo-realities
Simulate signals S (linear combination of covariates and
Gaussian noise) with GLGM models for Bernoulli and Poisson
variables at sampling locations (original grid);
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 10 / 16

23. Simulation of the pseudo-realities
Simulate signals S (linear combination of covariates and Gaussian
noise) with GLGM models for Bernoulli and Poisson variables at
sampling locations (original grid);
Use sequential Gaussian simulation to simulate signals at very
fine grid (100 m x 100 m) supplemented with validation points;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 10 / 16

24. Simulation of the pseudo-realities
Simulate signals S (linear combination of covariates and Gaussian
noise) with GLGM models for Bernoulli and Poisson variables at
sampling locations (original grid);
Use sequential Gaussian simulation to simulate signals at very fine
grid (100 m x 100 m) supplemented with validation points;
Combine pairwise the simulated fields of Bernoulli indicators
and Poisson counts to pseudo-realities of zero-inflated Poisson
counts;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 10 / 16

25. Simulated data vs Original
Figure : Simulated data, species occurrence
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 11 / 16

26. Simulated data vs Original
q
qqqqqqq
q
qqqqqqq
q
qqqqq
q
q
q
q
q
q
q
q
q qqq
qqqqqq
qqqq
qqqqq
q
qqqqq
q
q
q
qqqqq
qqqqqqqqqqqq
qqqqqqq qqqq
q
q
q
q
q
q
qq
qq
q
q
q
q
q
qqqqqq
q
q
q
q
qq
q
qqqqqqqq
q
q
q
q
q
q
qq
q
q
q
q
qqq
qqq
q
qqq
q qqq
qqqqq
q
qqqqqqq
qq
q
q
q
qqq
qqqqqqq
qqqq q
q
qq
qqq
qq
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
qq
q
qq
q
q
q
q
q
q
q
qq
q
q q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
qq
q
q
qqq
qqq
q
qqqqqq
qqqqqqqqqq
qqqqq
q
qq
q
q
q
q
q
q
q
q
qq qqqqq
q
qqq
qqqqqqq qqq
qqqqqqq
qqqqqqq qqqqq
qq
q
q
q
q
q
q
q
qq
q
qqqqqqq
qqqq qqqq
qqqqq
q
q
q
q
q
qq
q
qqqqq
qq
q
qq
q
q
q
q
q
q
qqqq
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
q
q
q qq
q
q
q
qq
q
qq
q
q
q
q
q
q
qq
q
qqq
q
q
q
q
qq
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
qq
q
q
q
qq
q
q
q
q
qq
q
q
q
q
q
q
q
qq
q
qq
qqqqq
q
q
q
qq
q
qq
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q q
q
q
qqqqq
q
q
q
q q
q
q
q
qq
q
qq
q
q
q
q
q
qq
q
q
q
qqq
q
q
qqqq
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q q
q
q
q
q
q
q
q
q
qqqqq
qq
q
q
q
qq
qqqq
q
q
q
q
qq
qqq
qqqqqqq
q
q
q
q
q
q
qq
qqq
q
q
q
qqq
qq
qq
q
q
q
qq
qqqqq
qq
qq
qq
q
q
q
q
q
q
qq
qqqqq
q
q
q
q
q
q
q
q
qq
q qqqqq
qq
q qqqqqqq
qqqqq
q
q
qq
q
q
qq
q
qqqqqq
q qqqqqqq
qqqq
qq
qq
q
qqq
qqq
qq
q
q
q
qq
qqqq
qqqqq
q
qq
qq
qqqq
qqqqqqq
qqq
q
q
q
q
q
q
qqqq
q
qq
qq
q
q
q
q
qqq
qqq qqqqqq
qq
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
qq
q
qq qq
q
qq
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
qq
q
q
q
q
qq
q
q
q
qq
q
q
qq
q
q
q
q
q
q
qq
q
q
q
qqq
q
q
q
q
q
q
qqqqqqq qqqq q
qqqq
qqqqqqqqqqq
q
q
q
q
qqqq
qqqq q
q
q
q
qq
qqq
qqqqqqqqqqq
qq
qq
qqq qqqqq qq
qq
qq
q
qqqqqqqqq
qqqqq
q
q
q
qqq
q
q
qqqqqqqqqqqqq
qq
qqqqqqqqqqqqq
qqqqqqqq
qqqqqqq qqq
qqqqqqqqqq
qq
qqq
qq
qqq
q
qqqqqqqq
q
q qq q
qq
q
qq
qqqqq
q
qqqqqq
qq
qq
qqqq
qqqq
qq
qq
qqq
qqqqqqqqqqqq
qq
qqqqq
qqqqq
q
qqq
qqqq
qq
q
q
qqqq
q
q
q
q
qqqq
qq
qqqq
qqqqq
qqqqqqqqq
qqqqq q
qqqqq
qq qqq
qqqqqqq
q
q
q
qqqq
qqqqqqqqq
q
qqqq
qqqqq qqq
qqqqq
qqqqqqqqqq qqqqqqqqqq
qqqqqqqqq
qq
q
q
q q
q
q
qq
qqqqqqqqqqqqq
q
qq
q q
q
q
qqqqqqq
q
qqq q
q
qqq
q
q
q
q
q
q
q
q
qqqqqqqqq
qqq qqq
q
q
q
q
q
q
q
qqqqqqq
qq
qqqq
q
q
q
qqqqqq
qqqq
qqq q
q
q
q
q q
q
q
q q
qq
q
q
q q
q q
q
q
q
qqqq
q
q
qq
q
q
q
q
q
qq qq
qqq
qqq
q
q
q
q
q
q
q
q
qq
q
qq
qq
q
q
q
q
qq
q
qqq
q
q
qq
q
q
q q
q
q
q
qq
q
qq
q
qq
q
qq
q
q q
q q
qq
q
q
q
q
q
q
q
q
q qqq
q
q
q
q
q
q
qq
q q
qqq
qq q
q
q
q
q
q qq
qq
q
q
q
q q
q
qqq
q
qq
qqq
q
q
q
q
q q
q
q
q
q
q
q
q
q
q
qq
qq
q
q
q
q
qq
q
q
q
qqq
qq
q
qq
q
qq
q
q q
qqq
qqq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
q q
q
q
q
qq
q
qq
q
q
q
q
qq
qq
qqqqqq
qqqqq
q q
q
qqqq
q
q
q
qqq
qqqqqq
qqq
qqqq
qqq
q
qq
qqq
qq
qqq
q
qqq q
q
q
q
qqq qqqqq
qqqq
q q
q
q
qqq
q
q q
q qqq
qq
q
q
qq
q
q
qq
qqqqq
qqqqqq
q
qqqq
qqq
qqq
q
qqq
qqqqq qqqq
qqqqq
q
q q
q
q
q
q q
qqqqqqqq
qq
q
q
q
q
q
q
q
q
q
q
qq
qqqqqqqq
qqqq
q
q
q
q
q
qqqq
qqqq
qq qq
qqqqqqq
qqq
qq
q
q
qqq q
q
qq
q
q
qqq
qq
q
qqqq
q
q
qq
q
q
q
qq
q
q
qqqqq
qqqqqqqq
qqqq
qqqqqqq qq
qqqqq
qq
q qqqq
qqqqqqq
qq
qqqq
qqq
q
qqq
qqq
qqqqqq
q
qq
qq
qq
qq
qq
qq
qq
q
qq
qq
qq
qq
qqq qq
qq
qq
q
qq
q
qqqq
qq
qq
q
qq
qqqqqq
qq
q qq
qq
qqq
qq
qqqq
qq
qq
qq
q qqqq
qq
qq
qqqqqq
q
q
q
q
qq
qq
q
q
qq
q
q
q
q
qq
q
q
qq
q
q
q
qq
q
q
qqq
q
q
q
q
q
q
q
q
qq q
q
q
q
q q
q
q
q
qq
q
q
q
q
q q
q
q
qqq
q
q
qqq
qq
q
qq
q
q
qq
q
q
qqqq
qq
qq
q
q
q
q
q qq
qq
qq
qq
qq
q q
qqqq
qqqq qqq
qqq
qq
qq
qq
q qq
qq
q
qq
q
q
q
q
q
qq
qq
q
qq
qq qq
qqqq
qqqqqq qqq
qqqq
qq
qq
qqqqqqq qq
q
qq
qq
q
qq qq
q
qq
qqqq
q
qqq
qqq
qq
qq q
qq
q q
qqq
q
qq
qq
qq
qq
qq
q
qqqqqqqqqqq
q
qqqqqqq
q
qqq
qqqqqqq
qqqq
qqqq q
qqqqq
q q qqqqq
qq qqqq
q
qqq
qqqqqqqqqqqqq qqqq
qq
qqq qq
q
qq
q
q
qqqqqqqqqqqqqqqqq
qqqqqq
qqq qqqqqq
q
q
qq
q
q
qq
qqqqqqq
qqqq
qq
qqqqqqqqqqqqq
q
q
q
q
qq
q
q
q q
q q
q
q
q
q
qq
q
q
q
qq
q
q
q
q q q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
qq
q
q
qq
q
q
q
q
q q
q
q
q
q
qqqqqq
q q
qq
q
q qq
qq
qq
q
q
q
q
q
q
q
q
q
q
qq
qqq
q
qq q
q
q
qq
q
qqqq
qqqqqq
qqqqq
q qqq
qqqqqqqqqq
qqqqqqqqqqqqqq
qqqqqqqqqqqqqq
qqqqqqqqqqqqq qqqqqqqqqqqqqq
qq
qqqq
qqq
qqqq
qqqq
qqqqqqqqqq
qqqq
q
q
qqqqqqq
q
qqq
qqqq
q
q
q
q
qq
q
qqqqqqqqqq
q
q
q
qq
q
qq
qq
q
q
q
q
q
q
qq
qqqqq
qqqq qq
q
qqqqqqqqq
q
qqqq
q
q
q
qq
qq
qqq
qqqqq
qqqqqqq
qqqq
qqq
q
q
q
q
q
qq
qq
q
q
qq
q
qqq
qqqqq
q
q
q
q
qqqq
q
q
qq
qq
q
q
q
q
qq
q
qq
qq q q
qqqqq
qqqq
qq
qq
qqq
qq
qq
q
qq
qq
qq
qq
qq
q
q
q
qqq
q
qq
q
qqq
q
qqqqq
qqq
q q
q
qq
q
q
q
q q
qq
qq
qq
q
q
q
q q
q
q
q
q
q
q
qq
qqq
qq
q
q
q
qq
qq
qqq
q
q
q
q
q
q
qq
qqq
qq
qqq
q
q
q q
q
q
q
q
qqqq
q
q
q
q
q
qqq
qqqq
qqq
q
q
q
q
qq
q
q
q
q
q
qqq
qqqqqqqq
qqqq qq
qqqqqqq qq
qqqqqqqqq
qqq q
q
qqq
qqqqqq
qq
q
q
qqqqqq
qqqq q
q
q q
qqq
q
qq
q
q
qqq
qqqqq
qqqq qqq
q
q
qqq
q
qqq
qqqqqqqq
qq
qqqqqqq
qqq
qq qq
qq
qqqqqqqq
qqqqqqqq
q
qqqqq
q
qqqqqq
qqqqq
qq
qq
q
qq
q
qqqqqqq
qq
q
q
qq
q
q qqq
q
q q
q
q
q
qqqq qqqqqqqq
qqqqq
q
q
q
q
q q
qq
q
q
q
q q
q
q
q
q
q q
q
q
q
qqqqq
qq
q
q
q q
qqqqqqqqq
qqqqq
q
q
q
q q
q
q
q
q
q q
q
qq q
q
q
q
q
qqqqqqqqqqqqqq
qqqqqqqqqqqqqq
q
qq
q q
qq qqqq
qqqqqqqqq
qqq
qqqqqqqqqqqq
qq
q
q qqqqqq qqqqqq
qqqqqq
qqqqqqqqqqqq
qqqqq
q q
qq
qq
q
q
qqqqqqqqq
qqqq qqqq
qqqqqqqq qqq
qqqq
qqq q
qqqqqq
qqqqq
q
q
q
q
q
qq
q
q
q
q
q
qq
q
q
q
q
q
qqqqq
qqq
qqq
q
q
q
q
q q
qq
qqq
q
qq
q
q
qq
q
q
q
q
qq
q
qq
qq
q
qqqqq
q
q
q
qq
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
qq
q
q
qq
q
qqqq
qqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
qqqqqq
qqqqq
qqqq
qq
qqqqqq
qqqqq
qqq
qqqqq
qq
q
qq
qqq
q
q q
q
q q
qqqqq
qqqqqq
q
q
q q
q
q
q q
q
q
q q
q q
q
q
q
qqqqqqq
qqqqq
q
qqqqqqqq
q
qqqqqqq qq
qqqq
qq
q
q
qqqqqqq
qqq
qqqqq
qq
q q
qq
qqqqq
qqqqqqq
qqq q
qqqqqq
qqqqqq
qqq
q
q
q q
q
qqqq
qqqq
qqqqqq
qqqqq
qqq
q
q
q
q
q
q q
qqqqqqq
qqq
qqqqqq
q
qqqqqq
q
q
q
q
q
qq q
q
qqq
qqqqqqqq
qqq
qqqqqqq
qqqq
qqq
q
q
q
qq
q
q q
qqq
q
qqq
q
qq
q
qqq
q
qq
q
q
q
qq
qqqqqqqqqq qqqqqqqqq
qq
qq qqq
q
qqq
qq
qq
q
q
q
q
q
q
q
q
qq
q
q
q
qqqqq
qq
qq
qqqq
qq
qqqqq
qqq
qqq
q
q
q
qqq
q
q
q
qq
qq
q
q
q
q
q
qq
q
q
q
q q
q
q
qqqqq
q
q
q
qq
qq
q
q
q
qq
q
q
q
q
q
qq
q
q
q
q
q
q
qq
qq
qq
q
q
q
qqq
qq
q
q
qq
q
qq
3320
3340
3360
3380
4000 4050 4100
Easting (km)
Northing(km)
Figure : Original data, species occurrence
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 11 / 16

27. Grid spacing and Performance
Sample each pseudo-reality of zero-inflated Poisson data
repeatedly by grid-sampling with a given spacing;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 12 / 16

28. Grid spacing and Performance
Sample each pseudo-reality of zero-inflated Poisson data repeatedly
by grid-sampling with a given spacing;
Repeat it for all considered grid-spacings;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 12 / 16

29. Grid spacing and Performance
Sample each pseudo-reality of zero-inflated Poisson data repeatedly
by grid-sampling with a given spacing;
Repeat it for all considered grid-spacings;
Predict values with IDW interpolation at validation points;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 12 / 16

30. Grid spacing and Performance
Sample each pseudo-reality of zero-inflated Poisson data repeatedly
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) − ˆY (a0)
2
(3)
MMSE =
1
(R ∗ S)
R
i=1
S
j=1
MSEji (4)
N is a number of validation points, R - simulations and
S - samples.
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 12 / 16

31. MMSE and Variance of MMSE
68
72
76
80
1000 2000 3000
Spacing (m)
MMSE
●
●
●
●
●
●
0
2000
4000
6000
1000 2000 3000
Spacing (m)
varianceMMSE
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 13 / 16

32. Conclusions
Sampling design for zero-inflated spatial count data is
evaluated;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 14 / 16

33. Conclusions
Sampling design for zero-inflated spatial count data is evaluated;
A strong monotonous increase of the MMSE is observed;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 14 / 16

34. 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;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 14 / 16

35. 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 estimating
MMSE;
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 14 / 16

36. 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 estimating
MMSE;
Spatial modelling of zero-inflated spatial data is laborious and
computer-intensive.
Is there an easier way: INLA?
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 14 / 16

37. Thanks!
Acknowledgements:
This work was done in the framework of the WaLTER (Wadden Sea Long-Term
Ecosystem Research) project (WP5)
www.walterproject.nl
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 15 / 16

38. References I
Bijleveld, A. I., van Gils, J. A., van der Meer, J., Dekinga, A., Kraan, C., van der
Veer, H. W., and Piersma, T. (2012). Designing a benthic monitoring
programme with multiple conflicting objectives. Methods in Ecology and
Evolution, 3(3):526–536.
Brus, D. and de Gruijter, J. (2013). Effects of spatial pattern persistence on the
performance of sampling designs for regional trend monitoring analyzed by
simulation of spacetime fields. Computers & Geosciences, 61(0):175 – 183.
Christensen, O. F. (2004). Monte carlo maximum likelihood in model-based
geostatistics. 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 to
defects in manufacturing. Technometrics, 34(1):pp. 1–14.
Lyashevska et al, 2014 olga.lyashevska@nioz.nl July, 2 2014 16 / 16