  In this paper, we construct a hierarchical model for spatial compositional
data, which is used to reconstruct past land-cover compositions (in terms of
coniferous forest, broadleaved forest, and unforested/open land) for five time
periods during the past $6\,000$ years over Europe. The model consists of a
Gaussian Markov Random Field (GMRF) with Dirichlet observations. A block
updated Markov chain Monte Carlo (MCMC), including an adaptive Metropolis
adjusted Langevin step, is used to estimate model parameters. The sparse
precision matrix in the GMRF provides computational advantages leading to a
fast MCMC algorithm. Reconstructions are obtained by combining pollen-based
estimates of vegetation cover at a limited number of locations with scenarios
of past deforestation and output from a dynamic vegetation model. To evaluate
uncertainties in the predictions a novel way of constructing joint confidence
regions for the entire composition at each prediction location is proposed. The
hierarchical model's ability to reconstruct past land cover is evaluated
through cross validation for all time periods, and by comparing reconstructions
for the recent past to a present day European forest map. The evaluation
results are promising and the model is able to capture known structures in past
land-cover compositions.
