1. Adam Butler, Stijn Bierman & Glenn Marion Biomathematics & Statistics Scotland CEH Bush, April 2008 ALARM: a statistical perspective

2. ALARM: Assessing Large-scale risks to biodiversity with tested methods Project of the 6 th framework programme of the European Union Key objectives Develop an integrated risk assessment for biodiversity in terrestrial and freshwater ecosystems at the European scale Focus on four key pressures – climate change, invasive species, chemical pollution, pollinator loss – and on interactions Contribute to the dissemination of scientific knowledge and to the development of evidence-based policy

3. “BioSS undertakes research, consultancy and training in mathematics and statistics as applied to agriculture, the environment, food and health…” Our role in ALARM - Training and consultancy Dissemination: development of a Risk Assessment Toolkit Methodological research to support scientific work - - Analysis of species-level data on distribution & traits - Quantifying uncertainty in complex mechanistic models

4. Example 1 Spread of Giant Hogweed (Heracleum Mantegazzianum) in the UK Cook, A., Marion, G., Butler, A. and Gibson, G. (2007) Bayesian inference for the spatio-temporal invasion of alien species. Bulletin of Mathematical Biology, 69(6), 2005-2025. Data: National Biodiversity Network

5. By 1910

6. By 1920

7. By 1930

8. By 1940

9. By 1950

10. By 1960

11. By 1970

12. By 1980

13. By 1990

14. By 2000

15. Data: year for which the species was first recorded in each 10x10km grid cell of the UK national grid (if it has been recorded) Aim: Estimate the probability of colonisation by a particular year (e.g. 2010) for grid cells that are not yet occupied Statistical approach: Use a spatio-temporal model to describe the spread of the species across the landscape in terms of various unknown parameters (dispersal rate, suitability etc.) Use MCMC to draw inferences about these parameters

16. Arrival rate = Sum of dispersal rates from all currently occupied cells Dispersal rate from cell i to cell j = 2 d -2 d = distance between cells =decay parameter j i Colonization rate = Arrival rate * Suitability No decolonization

17. For each grid cell: Suitability = exp(  * Temperature +  * Altitude) *  (  k * Proportion of grid cell covered by land use class k) Land uses: sea, coastal, arable, broadleaf, built, conifer, grassland, open water, semi-natural, upland Land use, climate & altitude all treated as constant over time

18. The parameters (  1,…,  10, , , ) and colonisation history are unknown We adopt a Bayesian approach, which involves treating both sets of quantities as random Plausible values are simulated using a computer-intensive algorithm known as Markov chain Monte Carlo (MCMC)

19. Colonization suitability Colonization probability: 10 year prediction Posterior mean

20. Example 2 Spatial variations in the prevalance of pollen vectors in German flora Kühn, I., Bierman, S.M., Durka, W. & Klotz, S. (2006) Relating geographical variation in pollination types to environmental and spatial factors using novel statistical methods. New Phytologist, 172(1), 127-139. Data: Florkart (spatial distribution), Biolflor (traits)

21. Context: assessing and predicting the impact of environmental change upon pollinators and pollination services Question: how are environmental variables related to the proportion of species that adopt particular pollen vectors? Approach: we cannot study how proportions change over time (no data), so instead we look at variability across space – similar to the “climate envelope modelling” approach

22. % insect % wind % selfing Altitude Wind speed

23. We are interested only in the proportion of species that adopt each pollen vector – the data are compositional, and must sum to one Standard statistical methods are unsuitable for compositional data Instead of analysing the raw data, we can analyse the log-ratios, e.g. log(selfing / wind), log(insect / wind) There are two log-ratios for each site, so this is a multivariate problem

24. The data also exhibit residual spatial autocorrelation: data at nearby sites tend to be more similar than data at locations which are far apart, and this patterns persists even in the residuals – so even after accounting for the effects of environmental variables Possible sources: Species not in equilibrium with their environment We lack data on some critical environmental variable Spatial autocorrelation leads us to underestimate uncertainty

25. Conditional autogressive (CAR) models provide a general methodology for dealing with spatial dependence In this application, we apply a multivariate CAR model to the log-ratios of the proportions

27. Example 3 Model averaging to combine simulations of future vegetation carbon stocks Butler, A., Doherty, R. M. and Marion, G. (submitted to Environmetrics). Model averaging to combine simulations of future global vegetation carbon stocks.

28. Dynamic Global Vegetation Models (DGVMs) allow us to simulate global stocks of vegetation carbon These simulations are contingent upon spatially explicit inputs – soil, climate and CO 2 concentrations Aim: prediction of annual global vegetation carbon stocks for the 21 st century, under a particular climate change scenario (SRES A2) and using a particular vegetation model (LPJ)

29. The issue: different climate models (GCMs) lead to different predictions of vegetation carbon… Scenario (SRES A2) Climate model (GCM) Vegetation model (LPJ DGVM) Vegetation carbon Temperature Rainfall Sunshine CO 2 concentration Soils

30. Control run, y based on observed climate data f i = simulation using GCM i

31. Model averaging across climate models: P(Y = y) =  i w i P(Y = y | f i,g) g is a statistical model describing the relatonship between each GCM run and the control run (i.e. between f i and y) w i is the weight associated with GCM i f

32. Model averaging across climate models and statistical models: P(Y = y) =  i  j w ij P(Y = y | f i,g j ) g is one of number of possible statistical models describing the relatonship between each GCM run and the control run w ij is the weight associated with GCM i and statistical model j f

33. 2.5% quantile Median Probabilistic predictions of trends in future vegetation carbon stocks, based on model averaging Change since 1961-1990, in gtC Year

34. Summary - key themes Species atlas data Environmental heterogeneity & environmental change Spatial & phylogenetic dependence Use of data on traits and invasive history False absences, missing data on traits Complex models Dealing with parameter uncertainty Calibration of model outputs Model averaging