Bayesian time series modelling of environmental health indicators

The Environmental Performance Indicators project is a long-standing research project that collates and analyses data on the quality of the environment, ecology and human health, developing a global ranking every two years. The rankings can be found at http://epi.yale.edu/ together with the data used to generate the rankings.

Look for the time series data on access to clean water (for the 2014 rankings it was called Water + Sanitation Raw Data File; the name should be similar for the 2016 ranking). For these data, a possible model is:

y_ct ∼ N (µ_ct, σ²)

µ_ct ∼ N (µ_c,t-1, λ²)

where y_ct is the water quality index for country c for year t. The model is structured to have an autoregressive prior, with the ‘true' level of water quality (µ_ct) changing from year to year, with its ability to change determined by the smoothing parameter λ. The smoothing parameter could be country specific (λ_c) or not.

Obviously, the best fitting model from the perspective of the likelihood would have λ high, to allow µct = yct and an infinite likelihood. This would, however, have poor out of sample performance. If λ were zero, or close to zero, on the other hand, the fitted model would be a constant or essentially a constant level, and again would have poor predictive performance. In between these extremes is a model that changes with trends in the data, but not too rapidly.

As λ is essentially a tuning parameter governing the complexity of the model, it could be se- lected via a form of cross validation, and fixed at an optimal level to estimate the µct (and other) parameters.

Please fit the above model, or a variant that you find more suitable, within a Bayesian framework (i.e. setting suitable priors for the parameters other than λ), and use it to develop a rank- ing of countries on access to clean water, accounting for observation error and the uncertainty in the true values, at the most recent time point available.

Write up methods and results sections in a format that is suitable for publication, including suitable figures or tables. You do not need to write up the introduction or discussion, but should include any references for the methods and results sections. If you write more than 10 pages, you have probably written too much.