On tracking varying bounds when forecasting bounded time series

TL;DR

This paper introduces a novel online maximum likelihood estimation framework for tracking unobserved, time-varying bounds of bounded random variables in univariate time series, applicable to real-world forecasting tasks.

Contribution

It develops an extended log-likelihood approach and an Online Normalized Gradient Descent algorithm to estimate changing bounds in bounded time series data.

Findings

Effective bound tracking demonstrated on simulation data.

Successful application to wind power forecasting.

Algorithm converges despite non-convex optimization challenges.

Abstract

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.