On tracking varying bounds when forecasting bounded time series

Many statistical applications involve response variables which are both continuous and bounded. This is especially the case when one has to deal with rates, percentages or proportions, for example when interested in the spread of an epidemic (Guolo and Varin, 2014), the unemployment rates in a given country (Wallis, 1987) or the proportion of time spent by animals in a certain activity (Cotgreave and Clayton, 1994). Indeed, proportional data are widely encountered within ecology-related statistical problems, see Warton and Hui (2011) among others. Similarly, when forecasting wind power generation, the response variable is also such a continuous bounded variable. Wind power generation is a stochastic process with continuous state space which is bounded from below by zero when there is no wind, and from above by the nominal capacity of the turbine (or wind farm) for high-enough wind speeds. More generally, renewable energy generation from both wind and solar energy are bounded stochastic processes, with the same lower bound (i.e., zero energy production) and different characteristics of their upper bound (since solar energy generation has a time-varying maximum depending on the time of day and time of year), see for example Pinson (2012) and Bacher et al. (2009). These continuous bounded random variables call for probability distributions with a bounded support such as the beta distribution, truncated distributions or distributions of transformed normal variables as discussed for example in Johnson (1949). Very often the response variable is first assumed to lie in the unit interval (0, 1) and is then rescaled to any interval (a, b) through the transformation X = (b a) X + a, where X (0, 1) and X (a, b).