Exponential Smoothing (ETS)

Suppose an aquarium operator wants to forecast next quarter’s visitor numbers. Visitors tend to rise during summer vacation and fall in winter, but total visitor numbers also grow a little every year. If you simply carry last month’s visitor count over to next month, you miss the seasonal changes, and if you use only the overall historical average, it is hard to reflect recent growth.

A classic time series forecasting method for this situation is exponential smoothing. It gradually incorporates recent observations and information accumulated in the past, and, depending on the data, models the level, trend, and seasonality. In particular, the ETS (Error, Trend, Seasonality) framework includes the form of the error as well, letting you compare many exponential smoothing models systematically.

In this article we use aquarium visitor numbers as an example to cover simple exponential smoothing, the Holt and Holt–Winters methods, ETS notation, what smoothing parameters mean, calculating forecasts, model selection, and forecast validation.

The influence of older observations decays exponentially over time, hence the name exponential smoothing. It does not use only the most recent values, though. Information from past observations also remains, accumulated in the updated level and trend states.

For example, if the aquarium’s monthly visitor count were perfectly constant, a simple model that estimates only the level might be enough. But if visitors increase over the long run, you need to add a trend, and if the summer peak season repeats every year, you need to add seasonality, so that the forecasts better reflect real operating conditions.

This aquarium is crowded every summer vacation, and visitor numbers are relatively low the rest of the year. To plan staffing and facility operating costs, the operator wants to forecast visitor numbers for January through March 2026.

Below is a simplified example of the observations for the last few months.

Because the gap between summer and late fall is large, a model that captures a pattern repeating every year is likely to do better than a simple overall average. Even so, rather than settling on a seasonal model right away, you should first look at the time series plot and the seasonal variation.

It updates the current level estimate as a weighted average of the new observation and the previous level estimate.

Here y_t is the observation at time t, ℓ_t is the updated level, and α is a smoothing parameter usually between 0 and 1. For example, if last month’s level estimate was 15,000 visitors, this month’s actual visitors were 16,000, and α = 0.30, we get the following.

So the new level estimate is 15,300 visitors. In simple exponential smoothing, the point forecasts for all future periods stay constant at this level. For strongly seasonal data like the aquarium’s, SES alone may not be enough.

It addresses the limitation of simple exponential smoothing, whose forecasts stay flat, by modeling a steady rise or fall in visitor numbers. A typical form with an additive trend is as follows.

Here b_t is the slope of the trend estimated at time t, and h is the number of periods ahead being forecast. For example, if the estimated level is 18,000 visitors and the monthly trend is +200 visitors, the point forecast for next month is 18,200 and for two months ahead it is 18,400.

But if monthly visitors rise every summer, the linear trend of Holt’s method alone has a hard time representing that movement. When seasonality is observed, consider a method that includes a seasonal component.

If monthly data have a 12-month seasonal period, the seasonal characteristics of each month are set up to cycle. Additive seasonality fits when the size of seasonal increases and decreases is roughly constant. For example, about 8,000 more people visit in summer than in other months, and this stays about the same even when the overall number of visitors changes.

Multiplicative seasonality is useful when seasonal differences grow or shrink in proportion to the level. For example, if summer visitor numbers stay at about 1.5 times the usual level, multiplicative seasonality may be natural.

Whether additive or multiplicative fits better should be judged from the actual variation in the time series and the model diagnostics. In particular, when using a multiplicative structure on data where visitor numbers can be 0, look carefully at whether it is well defined and can be estimated.

It is a state space model framework that distinguishes exponential smoothing models by specifying the form of each of the three components.

For example, ETS(A,N,N) is a model with additive errors and no trend or seasonality. ETS(A,A,A) uses additive errors, an additive trend, and additive seasonality, and ETS(M,Ad,M) uses multiplicative errors, a damped additive trend, and multiplicative seasonality.

The important point is that the traditional Holt–Winters forecasting method and the ETS notation for error-based state space models are related, but they are not exactly the same naming system. Because ETS also specifies the structure of the error distribution, it can be used for likelihood-based model comparison and for computing prediction intervals.

These values determine how quickly information from new observations is reflected in the state estimates. In general, a large α reacts more sensitively to recent observations, while a small one keeps the previously estimated level relatively stable. The smoothing parameters for trend and seasonality control the update speed in a similar way, but the exact formulas and allowed ranges depend on the ETS model you use.

In real analyses, rather than fixing the smoothing parameters arbitrarily, they are often estimated from the data based on the likelihood or forecast errors of the chosen model. The initial level and initial seasonal effects also affect the forecasts, so estimate them as well or use an appropriate initialization method.

Then you can compare additive vs multiplicative seasonality, whether to include a trend, and whether a damped trend is needed. Below is a comparison of information criteria obtained by fitting several candidates to the same training data.

In this example, ETS(A,Ad,A) had the lowest AICc. AICc is an information criterion that includes a correction for sample size, and it can be used to compare the relative fit and complexity of likelihood-based candidate models fitted to the same data.

However, the lowest AICc does not mean the best future forecasting performance. If the actual purpose is forecasting, check that the errors are also stable in a validation that preserves time order. If several models have similar errors, you can also consider interpretability and the stability of the forecasts.

The point forecasts in the table below extend forward the level, trend, and seasonal states estimated from the previous 48 months.

Expected visitors are estimated at 15,800 in January, 16,500 in February, and 19,100 in March. You can interpret this as the result of reflecting both a seasonal pattern in which demand rises toward spring and a gentle trend.

A point forecast is the most representative expected value, but it does not mean the actual observation will be exactly that number. The 95% prediction interval is a range computed by reflecting the model’s errors, state estimates, and future uncertainty, and it should be interpreted in terms of its long-run coverage rate under the model’s assumptions. You should not claim that, for an individual forecast, the actual number of visitors will definitely fall inside the interval with 95% probability.

So prediction intervals for long-range forecasts are generally wider than for short-range ones. In the results above, the prediction interval is 6,800 visitors wide for January but wider, 9,000, for March. When making operating plans, it is better to examine both low-demand and high-demand scenarios rather than using only the point forecast.

For example, staffing can be based on the point forecast while securing extra staff to handle the peak season or unexpected surges in demand. However, the uncertainty included in ETS prediction intervals depends on the model and estimation method used, so they cannot be assumed to capture structural changes such as a new facility opening or a change in admission pricing.

If you split the data randomly, you can get data leakage, using future information to predict the past. In this example, you could train the model on the 36 months of 2022–2024 and use the 12 months of 2025 as the validation period. Alternatively, you can run a rolling-origin evaluation, gradually extending the training period and forecasting the next 1 or 3 months at each point.

For example, if ETS(A,Ad,A) has a validation MAE of 1,650 visitors and the seasonal naive forecast (predicting the value from the same month last year) has an MAE of 2,250, ETS showed a smaller mean absolute error in that validation period. But this alone cannot guarantee that it will be better in every future period.

If the residuals drift to one side over time or repeat at a certain interval, the model may not have captured the trend or seasonality adequately. Check the residual time series plot and the autocorrelation function (ACF), and if needed use the Ljung–Box test to assess whether the autocorrelations at several lags are jointly 0.

However, the test is affected by sample size, degrees-of-freedom corrections, and the model fitting process, so don’t judge forecast quality by a single p-value.

Because ETS extends past patterns, it has trouble accurately forecasting external events such as a sudden closure, a new competing attraction, or an admission price increase without separate inputs. If such variables matter for the forecast, consider alternatives such as regression-based time series models.

ETS takes the view of updating level, trend, and seasonality as states, while ARIMA explains time series dependence mainly through lagged relationships among past observations and errors, and differencing.

Neither is better for all data. ETS can be strong for data with easily interpreted seasonal patterns, and ARIMA can be useful for modeling the autocorrelation structure of the residuals in detail. In practice, the most practical approach is to compare both models over the same validation period.

A real study should also report the results of validation that preserves time order and residual diagnostics.

Simple exponential smoothing handles a relatively constant level, Holt’s method adds a trend, and Holt–Winters additionally accounts for seasonality. ETS adds the form of the error to these structures, letting you compare candidate models systematically.

But a more complex model does not always produce better forecasts. Check whether seasonality really exists, whether an additive or multiplicative structure is appropriate, and whether a damped trend is needed, and evaluate forecasting performance with validation that preserves time order.