Overview

209. Overview

209.1. Moving Averages

A family of forecasting methods that produce a forecast as a weighted average of past observations. Each method is a different choice of weights along a single axis: how much should past observations count?

209.1.1. The weighting-scheme axis

Two extremes and two interpolators. All five methods can be written in the form

{\hat{𝑥}}_{𝑡, 𝑡 + 1} = \sum_{𝑖 = 0}^{𝑁 - 1} 𝑤_{𝑖} 𝑥_{𝑡 - 𝑖}

— they only differ in how they choose the weights $𝑤_{𝑖}$ (and the window $𝑁$ ).

Method	Window	Weights	Formula	See
Naïve	1	(1, 0, 0, …)	${\hat{𝑥}}_{𝑡, 𝑡 + 1} = 𝑥_{𝑡}$	[naive.typ](naive.typ)
SMA	$𝑀$	$(1 / 𝑀, \dots, 1 / 𝑀, 0, \dots)$	${\hat{𝑥}}_{𝑡, 𝑡 + 1} = \frac{1}{𝑀} \sum_{𝑖 = 0}^{𝑀 - 1} 𝑥_{𝑡 - 𝑖}$	[sma.typ](sma.typ)
WMA	$𝑀$	$(𝑤_{0}, 𝑤_{1}, \dots, 𝑤_{𝑀 - 1}, 0, \dots)$	${\hat{𝑥}}_{𝑡, 𝑡 + 1} = \frac{\sum 𝑤_{𝑖} 𝑥_{𝑡 - 𝑖}}{\sum 𝑤_{𝑖}}$	[wma.typ](wma.typ)
SES	all	$𝛼 {(1 - 𝛼)}^{𝑖}$ (geometric decay)	${\hat{𝑥}}_{𝑡, 𝑡 + 1} = 𝛼 \sum_{𝑖 = 0}^{𝑡} {(1 - 𝛼)}^{𝑖} 𝑥_{𝑡 - 𝑖}$	ETS(A,N,N)
Cumulative	$𝑡$	$(1 / 𝑡, 1 / 𝑡, \dots, 1 / 𝑡)$	${\hat{𝑥}}_{𝑡, 𝑡 + 1} = \frac{1}{𝑡} \sum_{𝑖 = 1}^{𝑡} 𝑥_{𝑖}$	[cumulative.typ](cumulative.typ)

209.1.2. Reading the axis

Two endpoints and two interpolators:

Naïve is the case $𝑀 = 1$ : only the most recent observation matters. Maximum responsiveness, no smoothing.
Cumulative is the case $𝑀 = 𝑡$ : every observation since the start counts equally. Maximum smoothing, no responsiveness.

Between them, two interpolators:

SMA and WMA: finite window of size $𝑀$ , with explicit weights (equal for SMA, custom for WMA). Choosing $𝑀$ trades responsiveness vs smoothing.
SES: infinite window with geometrically-decaying weights. Choosing $𝛼$ trades responsiveness ( $𝛼 \to 1$ , looks like Naïve) vs smoothing ( $𝛼 \to 0$ , looks like Cumulative).

The full ETS family (all 30 cells, see ets/) generalizes SES by adding trend and seasonal components on top of the level.

209.1.3. When to use each

Method	Use when
Naïve	Random-walk demand (no structure to exploit). Useful baseline — every other method should beat it.
SMA	Smooth, slowly-changing demand without trend. Easy to explain to non-technical stakeholders.
WMA	Same as SMA, but you want to give more weight to recent observations without going to full geometric decay.
SES	Slowly-changing level + want infinite-history weighting. Most operational forecasting starts here.
Cumulative	Estimating a long-run mean (for stable processes) or as a slow-baseline in regime-detection.

209.1.4. Comparing them on the same data

For a stationary series, SMA, WMA, SES, and Cumulative all converge to the same long-run mean — they only differ in how fast they react to changes. For a trending series, all of them lag behind (they’re averages — they smooth out the trend); use ETS or ARIMA if you have trend.

The next four files cover Naïve, SMA, WMA, and Cumulative; SES is in [ets/a-n-n.typ](../ets/a-n-n.typ) as ETS(A, N, N).