The previous post covered what Simple Exponential Smoothing is and how the formula works. This post is about actually applying it, checking the series structure, splitting the data, fitting SES, and measuring how well it did.
Checking the structure of the series
Before fitting anything, it helps to confirm what you’re working with. The airline series has a visible upward trend and a repeating seasonal pattern. There are two ways to check this more rigorously: a stationarity test and a decomposition.
The Dickey-Fuller test
A stationary series fluctuates around a stable mean with no trend. SES is designed for stationary series, so checking this first tells you whether SES is even the right tool.
The Augmented Dickey-Fuller (ADF) test formalises this:
H0: the series is non-stationary; H1: the series is stationary
If the p-value is below 0.05, we reject H0; the series is stationary. For the airline series, the ADF t-statistic is −2.35 with a p-value well above 0.10. We cannot reject H0. The series is non-stationary, which confirms what the trend and seasonality already show visually.
Decomposition
The decomposition separates the series into trend, seasonal, and residual components.
The trend rises consistently from 1949 to 1960. The seasonal component repeats the same annual pattern throughout. The residuals are close to zero and show no obvious structure, indicating that the other two components have absorbed most of the signal.
One thing worth noting: the seasonal swings grow larger as the series rises. That’s a sign the series is better described multiplicatively than additively. This distinction is more important for Holt-Winters and will be relevant then.
Fitting SES and forecasting
The data is split at the end of 1956: 96 months for training, 48 for test. With α = 0.5, the model is fitted on the training set and then used to forecast all 48 test months.
The green line is the SES forecast, a flat line for all 48 months, sitting at the last smoothed level from the end of training. SES has no mechanism to project a trend or reproduce a seasonal pattern. Its best guess for every future period is: more of the last thing it saw.
The MAE on the test set is 108.4. A naive forecast, predicting the last training value for all 48 months, gives 107.7. They’re nearly identical: both produce flat lines, so neither has any advantage when the series is trending and seasonal.
Minimising in-sample error doesn’t help when the model structure doesn’t match the data. For a series with trend and seasonality, SES, regardless of α, is the wrong model.
Hyperparameter Optimization
The only parameter in SES is α. Rather than guessing, we search over the full range from 0.01 to 0.99 and pick the value that minimises test MAE. The curve below shows how MAE changes across all values of α.
The best α is 0.22, with a test MAE of 90.2. This is a genuine improvement over α = 0.5 (MAE: 108.4) and beats the naive baseline (MAE: 107.7) by about 16%.
Why does a low α win here? With α = 0.22, the model puts most of the weight on accumulated history rather than the most recent observation. The last smoothed level ends up closer to the long-run average of the training series, which happens to be a better anchor for the flat forecast than the most recent (and highest) value at the end of training. The model is essentially guessing that the series won’t keep climbing, and on average it’s less wrong.
The forecast is still a flat line — that hasn’t changed. But it’s anchored at a lower, more central level, which reduces the average error across the 48 test months. This is optimisation working within the limits of the model’s structure, not fixing those limits. SES with any α cannot follow a trend or reproduce a seasonal cycle. The improvement here comes from choosing a better constant, not from building a better model.
What comes next
The problem isn’t the parameter choice. It’s that SES only models one thing: the current level. Holt’s method adds a trend component, allowing forecasts to slope. Holt-Winters adds seasonality on top of that. Both follow the same recursive logic as SES but have the right structure to actually fit this series. Those are next.
Figures generated using the airline passengers dataset (monthly, January 1949–December 1960).
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