Expanding ETS(1/2) at the final step to show the geometric weight on every past observation — the most recent gets half, then a quarter, an eighth, and so on.

highlighted = computed this step

Weight contributions at final step

The smoothed value is a weighted sum of all past observations. Weights decay geometrically — the most recent observation carries the highest weight.

weight table\text{weight table}
Weight Decomposition at Final Steptxweightcontributiont481/24t341/41t281/81t141/80.5sum16.5

Geometric weight decay (bar plot)

The weight on each past observation halves with each additional step back. The initial observation absorbs the remaining tail weight.

weight decay\text{weight decay}
Geometric weight decay — alpha one-halfobservationweight01/41/211/2t41/4t31/8t21/8t1