1. Department of Bioscience
Spatio-temporal structural equation modeling in a hierarchical Bayesian framework: ecology of wet heathlands
Christian Damgaard
Department of Bioscience
Aarhus University

2. Erica tetralix is decreasing on Danish wet heathlands
Characteristic species of wet heathlands
What is the cause?
What management
actions may reverse
the trend?
Cover (%)

3. Wet heathland vegetation data
The cover of the three dominating species on wet heathlands, Erica tetralix, Calluna vulgaris and Molinia caerulea, as well as all other higher plant species was determined by the pin-point method
Time series data (2007 – 2014) from 39 sites with a total of 1322 plots
Important: pin-point cover data allows the aggregation of species

4. Cover data are L or U - shaped
Plant species are patchy distributed ⇒
cover data are L or U - shaped

5. Joint distribution of cover data
The Dirichlet-multinomial distribution models pin-point cover data of n species – accounts for spatial aggregation
𝑦 𝑖 : pin-point hits of species i; 𝑛 𝑦 𝑖 ≥ # of grid points
𝑝 𝑖 : relative cover of species i; 𝑛 𝑝 𝑖 =1
𝐸 𝑝 1 ,… ,𝑝 𝑛−1 = 𝑞 1 ,… ,𝑞 𝑛−1
𝛿: intra-plot correlation due to spatial aggregation
𝑌~𝑀𝑛 𝑛 𝑦 𝑖 , 𝑝 1 ,… ,𝑝 𝑛−1 ,1− 𝑝 1 −…− 𝑝 𝑛−1
Λ 𝑝 1 ,… ,𝑝 𝑛−1 ~𝐷𝑖𝑟 𝑞 1 −𝑞 1 𝛿 𝛿 ,…, 𝑞 𝑛−1 − 𝑞 𝑛−1 𝛿 𝛿 , 1−𝛿 𝛿 − 𝑞 1 −𝑞 1 𝛿 𝛿 −…− 𝑞 𝑛−1 − 𝑞 𝑛−1 𝛿 𝛿

6. Spatial and temporal model (SEM)
Green oval: data
Black box: latent variables
Black arrow: spatial proc.
Red arrow: temporal proc.
Only two years shown

7. Spatial and temporal model (SEM)
The latent variables allow the separation of measurement errors and process errors.
Only process errors are needed for predictions
Only two years shown

8. Estimation Hierarchical Bayesian modelling approach
MCMC - Metropolis-Hastings algorithm
Statistical inferences on the parameters of interest were based on the marginal posterior distribution of the parameters

9. Spatial process (2007)
Large uncertainty – especially Calluna (yellow dots) – history?
Dwarf shrubs (Erica and Calluna) have same qualitative response, and opposite to Molinia and other plants
Positive spatial effects of nitrogen deposition, pH, sandy soils, and low precipitation on dwarf shrubs
Standardized regression coefficients

10. Geographic latent factors
Regional geographic variation (50 km scale)
South Jutland behave qualitatively different
Information may be used to generate new hypotheses

11. Temporal process (2007 - 2014) Good fit!
Dwarf shrubs (Erica and Calluna) have same qualitative response as the spatial effects, and opposite to Molinia and other plants
Negative effect of grazing on Erica – insufficient data resolution
Standardized regression coefficients

12. Application: local prediction
Collect data from 20 to 40 plots
Only temporal process is relevant (good model fit)
Prediction of the effect of local management actions
Set-up of an adaptive management plan

13. Conclusions
Important to take local spatial aggregation of plant abundance into account
Plant cover data are U-shaped
Possible to fit ecological models to multivariate abundance data instead of summarizing the variation by ad hoc distance methods
e.g. Bray-Curtis distances in ordination techniques
Important to separate measurement errors from structural uncertainty when making predictions
Ecological processes are best studied using time-series data