Exponential Smoothing - Forecasting Mechanics, Honestly

Exponential smoothing is a recurrence, not a fitted guarantee. This lesson pins the recurrence weights and follows one step all the way through exact fraction arithmetic.

highlighted = computed this step

Smoothing recurrence

Exponential smoothing uses alpha 1/2 on the current observed value and 1/2 on the previous state. Motivation: the recurrence keeps a running state instead of opening a fresh window each time.

s_t=\alpha x_t+(1-\alpha)s_{\text{prev}}

One recurrence step

For period 4, the current value is 6 and the previous state is 3. Why: those are the two exact inputs to this recurrence step, so the formula can be audited locally.

s_{4}=1/2\cdot6 + 1/2\cdot3

Component sum

The two components are 3 and 3/2, so the recurrence outputs 9/2. Why: no rounding is needed to add these fractions.

3 + 3/2=9/2

All smoothed values

The smoothed outputs are 2, 3, 3, 9/2, 19/4, and 51/8. Interpretation: the plot labels are exact recurrence outputs, not smoothed evidence of future accuracy.

\text{ES}_{1/2}=2,3,3,9/2,19/4,51/8

Diagram note

Honesty note: These values are exact formula outputs on the pinned integer series; they do not validate what happens later. Pixel positions are rounded for layout; every number shown is exact.

\text{exponential smoothing is exact recurrence arithmetic}