On tracking varying bounds when forecasting bounded time series
Amandine Pierrot Pierre Pinson
We consider a new framework where a continuous, though bounded, random
variable has unobserved bounds that vary over time. In the context of
univariate time series, we look at the bounds as parameters of the distribution
of the bounded random variable. We introduce an extended log-likelihood
estimation and design algorithms to track the bound through online maximum
likelihood estimation. Since the resulting optimization problem is not convex,
we make use of recent theoretical results on Normalized Gradient Descent (NGD)
for quasiconvex optimization, to eventually derive an Online Normalized
Gradient Descent algorithm. We illustrate and discuss the workings of our
approach based on both simulation studies and a real-world wind power
forecasting problem.