Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Novel Theoretical Framework for Exponential Smoothing

Enrico Bernardi, Alberto Lanconelli, Christopher S. A. Lauria

Abstract

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.

A Novel Theoretical Framework for Exponential Smoothing

Abstract

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.
Paper Structure (7 sections, 1 theorem, 50 equations, 3 figures)

This paper contains 7 sections, 1 theorem, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Exponential Smoothing as Stochastic Gradient Ascent
  3. Assumptions and statement of the main result
  4. Simulations
  5. The moving average case
  6. The autoregressive case
  7. Proof of the main result

Key Result

Theorem 3.3

Let Assumptions Stationarity and ltvp hold. Then, running equation with $\alpha\in ]0,1[$ we obtain

Figures (3)

  • Figure 1: The evolution of $\{m_t^{\star}\}_{t\in\mathbb{N}}$ in subfigure \ref{['fig:1a']} is given by the equation $m_t^{\star} = m_{t-1}^* + 0.1$, starting from $m_0^* = 2$, while $a = 2$. The evolution of $\{m_t^{\star}\}_{t\in\mathbb{N}}$ in subfigure \ref{['fig:1b']} is given by the equation $m_t^{\star} = \sin( \pi t / 1000)$, starting from $t = 0$, while $a = 2$. The SES in both subfigures is iterated $1000$ times with $\hat{m}_0= 8$ and $\alpha = 0.1$.
  • Figure 2: The evolution of $\{m_t^{\star}\}_{t\in\mathbb{N}}$ in subfigure \ref{['fig2:1a']} is given by the equation $m_t^{\star} = m_{t-1}^* + 0.1$, starting from $m_0^* = 2$, while $a = -0.4$. The evolution of $\{m_t^{\star}\}_{t\in\mathbb{N}}$ in subfigure \ref{['fig2:1b']} is given by the equation $m_t^{\star} = \sin( \pi t / 1000)$, starting from $t = 0$, while $a = -0.4$. The SES in both subfigures is iterated $1000$ times with $\hat{m}_0= 8$ and $\alpha = 0.1$.
  • Figure 3: The evolution of $\{m_t^{\star}\}_{t\in\mathbb{N}}$ in subfigure \ref{['fig:3a']} is given by the equation $m_t^{\star} = 0.1 + 0.01 t$, starting from $t=0$, while $\theta = 0.2$. The evolution of $\{m_t^{\star}\}_{t\in\mathbb{N}}$ in subfigure \ref{['fig:3b']} is given by the equation $m_t^{\star} = \sin( \pi t / 1000)$, starting from $t = 0$, while $\theta = 0.2$. The SES is iterated $1000$ times with $\hat{m}_0= 8$ and $\alpha = 0.1$.

Theorems & Definitions (5)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.3
  • Remark 3.4