Exponential Smoothing Forecast (SES & Holt)

Understanding the exponential smoothing forecast — concept and principle

Exponential smoothing produces a short-term forecast from a series by averaging past values with weights that shrink the further back you go. This calculator runs two classic versions on your data: simple exponential smoothing (SES), which tracks a level only, and Holt's linear method, which also tracks a trend. They are used for demand, sales, traffic and inventory planning.

SES stays flat at 140.27; Holt climbs the trend staircase to 156.98 three steps ahead.

How it is calculated

SES: start with L = y₁, then for each value Lₜ = α × yₜ + (1 − α) × Lₜ₋₁. The forecast for every future step is the last L.
Holt: start with level L₁ = y₁ and trend T₁ = y₂ − y₁. For t = 2 … n:
Lₜ = α × yₜ + (1 − α) × (Lₜ₋₁ + Tₜ₋₁)
Tₜ = β × (Lₜ − Lₜ₋₁) + (1 − β) × Tₜ₋₁
Holt forecast h steps ahead = Lₙ + h × Tₙ.

α and β lie between 0 and 1. Forecasts are in the same unit as the data.

The principle

Unrolling the SES recursion shows that yₜ gets weight α, yₜ₋₁ gets α(1 − α), yₜ₋₂ gets α(1 − α)², and so on: a weighted average whose weights decay geometrically. A large α reacts quickly to new data; a small α smooths noise. SES assumes no trend, so on a rising series it lags behind. Holt's method fixes this by smoothing the slope as well: the level is pulled toward the new observation and toward last period's level plus trend, and the trend is pulled toward the latest change in level.

Worked example

Data: 120, 125, 130, 128, 135, 140, 145; α = 0.5, β = 0.3, h = 3.
SES levels: 120, 122.5, 126.25, 127.125, 131.0625, 135.53125, 140.265625. SES forecast = 140.265625.
Holt start: L = 120, T = 5.
Levels and trends: (125, 5), (130, 5), (131.5, 3.95), (135.225, 3.8825), (139.55375, 4.016375), (144.2850625, 4.23085625).
Holt forecast = 144.2850625 + 3 × 4.23085625 ≈ 156.9776312.

Each point is the calculator re-run with only “α (level)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

α (level)	SES forecast
0.125	128.4712639
0.25	134.0407715
0.375	137.743206
0.5	140.265625
0.625	142.0413017
0.75	143.329834
1	145

How to read the result

SES says "about 140 from now on", below the last actual value of 145, because it ignores the upward drift. Holt projects about 4.2 units per step and gives roughly 157 three steps ahead. On trending data, Holt is usually closer; on flat, noisy data, SES is often as good and simpler. Judge methods by forecast error on held-out recent periods, not by how well they fit the past.

Common mistakes

Using SES on a clearly trending or seasonal series; seasonality needs Holt-Winters.
Choosing α and β by eye instead of minimising forecast error on past data.
Extending a Holt trend far into the future; linear trends rarely persist.

Keep in mind: These are point forecasts with no prediction interval, and results depend on the simple start values used here (first value, first difference). Software that optimises starting values will give somewhat different numbers.

Frequently asked questions

What α should I use?

Typical values are 0.1 to 0.3 for noisy data. Ideally, choose α (and β) that minimise one-step-ahead squared errors on your history.

How is this related to ARIMA?

SES gives the same point forecasts as an ARIMA(0,1,1) model, and Holt's linear method corresponds to ARIMA(0,2,2).

Sources: C. C. Holt, "Forecasting seasonals and trends by exponentially weighted moving averages" (1957), reprinted in International Journal of Forecasting 20 (2004) · R. J. Hyndman and G. Athanasopoulos, Forecasting: Principles and Practice, 3rd ed., OTexts, 2021, Chapter 8

Checked 8 Nov 2026. Standard formulas and RFC/IEEE definitions, computed locally in your browser — nothing you type is sent anywhere.

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