212. Simple Moving Average

212.1. SMA (Simple Moving Average)

For a time series ${𝑦}_{𝑡}$, the SMA of window size $𝑘$ at time $𝑡$ is:

The average of the last $𝑘$ values. Equal weights $1/𝑘$ on each. The forecast is ${\widehat{𝑥}}_{𝑡,𝑡+1}={𝑠}_{𝑡}$.

212.1.1. Choosing the window size $𝑘$

The window size trades two effects:

• Small $𝑘$ (1–3): forecast tracks recent observations closely. Responsive to changes; noisy.
• Large $𝑘$ (10+): forecast smooths out short-term noise. Stable; lags behind real changes.

Rules of thumb:

• For non-trending, smooth series: $𝑘$ around 5–10 typically works.
• For high-frequency noise: bigger $𝑘$ to filter the noise.
• For responsive forecasting (regime detection): smaller $𝑘$.

212.1.2. Lag

SMA lags the true level. If the underlying series is rising linearly, the SMA tracks roughly $𝑘/2$ time steps behind — because the average of the last $𝑘$ values has a center of mass at $𝑡-\left(𝑘-1\right)/2$. Larger $𝑘$ → larger lag.

This is why SMA is poor for trending series — it’s structurally late. Use ETS or ARIMA when there’s trend.

212.1.3. Visualization

The SMA filters the high-frequency noise and produces a smoother curve, but with a horizontal lag relative to the data.

• Model parameters

  • $𝑘\in {\mathbb{Z}}_{\ge 1}$: window size (number of past observations to average)

• Observation

  • ${𝑦}_{𝑖}$: observed value at time $𝑖$, for $𝑖\in \left\{𝑡-𝑘+1,\dots ,𝑡\right\}$

• Output

  • ${𝑠}_{𝑡}={\widehat{𝑥}}_{𝑡,𝑡+1}$: one-step-ahead forecast for time $𝑡+1$, given information up to $𝑡$

pl.col('X').rolling_mean(window_size)