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.

我们考虑一个新的框架，其中一个连续但有界的随机变量具有随时间变化的未观察到的边界，并在有界随机变量的分布的参数上考虑边界。我们引入了扩展对数似然估计，并设计算法通过在线最大似然估计来跟踪边界。由于所得到的优化问题不是凸的，我们利用最近关于凸优化的规范化梯度下降（NGD）的理论结果，最终导出在线规范化梯度下降算法。我们基于仿真研究和实际风电功率预测问题说明和讨论了我们方法的工作原理。