213. Weighted Moving Average

213.1. WMA (Weighted Moving Average)

Same finite-window structure as SMA, but each observation in the window gets a custom weight — usually with more weight on more recent observations.

where ${𝑤}_{𝑖}$ is the weight on the observation $𝑖$ steps before $𝑡$ (so ${𝑤}_0$ weights the most recent value, ${𝑤}_{𝑀-1}$ the oldest in the window).

The denominator $\sum {𝑤}_{𝑖}$ normalizes the weights so the WMA is on the same scale as the data; if the weights already sum to 1, the denominator drops out.

213.1.1. Common weight schemes

• Linearly decreasing (most common): ${𝑤}_{𝑖}=𝑀-𝑖$. So ${𝑤}_0=𝑀$, ${𝑤}_1=𝑀-1$, …, ${𝑤}_{𝑀-1}=1$. Most weight on the most recent.
• Triangular: linearly increasing then decreasing — emphasizes the middle of the window.
• Custom: any non-negative weights you choose.

213.1.2. Comparison to SMA

• SMA: equal weight $1/𝑀$ on every observation in the window.
• WMA: more weight on recent observations (with linearly-decreasing weights).

WMA is more responsive to recent changes than SMA at the same window size $𝑀$, because the new observation gets a relatively large weight (e.g., $𝑀/\left(𝑀\left(𝑀+1\right)/2\right)=2/\left(𝑀+1\right)$ for linearly decreasing — compared to $1/𝑀$ for SMA).

213.1.3. Comparison to SES

WMA with linearly-decreasing weights is conceptually similar to SES, with two differences:

• WMA truncates at $𝑀$; SES uses the entire history (with geometrically-decaying weights).
• WMA’s most-recent weight depends on $𝑀$ (e.g., $2/\left(𝑀+1\right)$); SES uses the explicit smoothing parameter $𝛼$.

For $𝑀\to \infty$ with linearly-decreasing weights, WMA approaches a geometric-weighted form similar to SES — but with polynomially-decaying tails rather than exponentially-decaying.

weights = [3, 2, 1]
pl.col('X').rolling_map(
    lambda s: sum(w * x for w, x in zip(weights, s)) / sum(weights),
    window_size=len(weights),
)