Simple Exponential Smoothing is a classical technique used for smoothing time series data by assigning exponentially decreasing weights to past observations through a recursive equation; it is sometimes presented as a rule of thumb procedure. We introduce a novel theoretical perspective where the recursive equation that defines simple exponential smoothing occurs naturally as a stochastic gradient ascent scheme to optimize a sequence of Gaussian log-likelihood functions. Under this lens of analysis, our main theorem shows that -in a general setting- simple exponential smoothing converges to a neighborhood of the trend of a trend-stationary stochastic process. This offers a novel theoretical assurance that the exponential smoothing procedure yields reliable estimators of the underlying trend shedding light on long-standing observations in the literature regarding the robustness of simple exponential smoothing.

翻译：简单指数平滑是一种经典的时间序列平滑技术，通过递归方程对历史观测值赋予指数衰减权重。它常被作为一种经验性方法使用。本文提出了一种新的理论视角：定义简单指数平滑的递归方程，可自然视为优化一组高斯对数似然函数的随机梯度上升方案。在此分析框架下，我们的主要定理表明——在一般设定中——简单指数平滑会收敛于趋势平稳随机过程趋势的邻域内。这为指数平滑方法能够生成可靠的基础趋势估计提供了新型理论保障，并揭示了文献中长期观察到的关于简单指数平滑稳健性的现象。