ARIMA·SARIMA·SARIMAX

An airport needs to know in advance how many bags it will have to handle tomorrow. If it expects too few, staffing falls short; if it expects far too many, equipment and staff may sit idle unnecessarily. But daily baggage volume doesn't simply stay at about yesterday's level. Weekends and weekdays follow different patterns, and demand can change a lot with holidays and flight schedules.

The standard method for forecasting the future using the movement of values observed over time is ARIMA (AutoRegressive Integrated Moving Average). Explicitly adding a repeating seasonal pattern gives SARIMA (Seasonal ARIMA), and adding external variables that affect the outcome, such as flight schedules, lets you use SARIMAX (Seasonal ARIMA with eXogenous variables).

The three methods are not completely separate, competing analyses; they are models that expand, step by step, the range of information used for forecasting. In this article we explain how each model works, setting the orders, a real example, diagnostics, forecast accuracy, prediction intervals and points to watch when using exogenous variables.

It can be applied to data recorded at equal intervals, such as daily or monthly, and is used for data where time order matters. It builds into the model the facts that today's baggage volume is related to yesterday's, and that the shock of a day with more bags than expected can linger afterward.

What matters in ARIMA is that, unlike regression analysis, it does not assume the observations are independent of each other. Instead, it directly models the temporal dependence among observations. However, if a particular repeating seasonality is strong, or you need to use external information known about the future, basic ARIMA alone may not be enough.

The three numbers determine what temporal structure the model reflects.

For example, ARIMA(1,1,1) is a model that differences the series once and, in the differenced series, uses the value and the shock from one time point earlier. The moving average (MA) here is different from a moving average that simply averages the last 7 days of baggage volume. ARIMA's MA terms mean a linear combination of past errors.

A simplified equation for understanding the form of the model is as follows.

Here B is the lag (backshift) operator, φ(B) is the autoregressive polynomial and θ(B) is the moving average polynomial. In actual forecasting, also check how the constant and trend terms are included according to the software's model definition.

It is usually written as SARIMA(p,d,q)(P,D,Q,s). The lowercase orders in front are the nonseasonal structure, and the uppercase orders after them are the seasonal structure. s is the seasonal period.

If an airport's daily baggage volume has a similar pattern every week, you can consider s = 7. If a pattern repeats every year in monthly data, you would usually consider s = 12. However, if both weekly and annual seasonality matter in daily data, a SARIMA with a single seasonal period can hardly explain both adequately. You can additionally consider calendar variables or other multiple-seasonality models.

For example, SARIMA(1,1,1)(1,0,1,7) reflects seasonal dependence at a 7-day interval on top of ordinary short-term time dependence. The appropriate model setup can differ depending on whether the structure is explained just by mean differences by day of the week or whether autocorrelation remains at lag 7.

When forecasting an airport's baggage volume, if you know the airlines' number of scheduled departures, you can use not only past volume but also the number of scheduled flights. For example, you could set up a model like this.

Here x_t is the number of scheduled departures on that date, and u_t is the regression residual following an ARIMA-type time dependence structure. This is one common regression-with-ARIMA-errors representation of SARIMAX, and the parameterization can differ by implementation.

To use an exogenous variable, you must also know, or separately forecast, its values for the future dates you want to forecast. The number of scheduled flights can be checked in advance, but the actual number of departures can change due to same-day delays and cancellations. For an operational forecast, use only information actually available at the time of forecasting.

The airport wants to plan the handling staff needed for the first week of 2026. The data record the number of bags handled each day, the number of scheduled departures on that date, the day of the week and whether it was a public holiday.

In this split, the last 56 days were set aside as a holdout for final performance evaluation. In an actual analysis, you should choose the orders by running separate time series cross-validation within the training period, and use the holdout to evaluate that choice. If you repeatedly pick the model that does best on the holdout, you risk judging final performance too optimistically.

A single-day outlier may be a recording error, but it may also be real demand caused by a concentration of flights or a special event, so look into the cause before deleting it.

Next, review the autocorrelation function (ACF) and partial autocorrelation function (PACF). If autocorrelation repeats at lags of 7, 14 and 21 days, you can suspect weekly seasonality. If you suspect nonstationarity due to a nonseasonal trend, refer to results such as the ADF unit root test and consider whether differencing is needed.

However, the differencing order d is not decided automatically by a single ADF p-value. You need to check the time series plot, the ACF and variability after differencing, the model's residuals and forecast performance together, and you should also avoid over-differencing.

A seasonal difference with a 7-day period, on the other hand, takes the difference between today and the same weekday a week earlier. For example, if Monday's volume is 8,400 bags and the previous Monday's was 8,100 bags, the seasonal difference is 300 bags. The two kinds of differencing have different purposes. Ordinary differencing deals with nonseasonal long-term trends, while seasonal differencing is used to reduce repeating seasonal shifts in level.

Seasonality being present does not necessarily mean D = 1 is needed. Seasonal mean differences can be expressed with day-of-week variables, or explained by seasonal AR and MA terms. d and D should be set according to the actual data.

The following orders are candidates for illustration, not optimal orders estimated from real data.

For model selection you can use AIC and AICc on the training period, residual diagnostics, validation errors from a rolling-origin approach and so on. However, directly comparing the information criteria of models that use different differencing or likelihood definitions may not be appropriate. Also, when evaluating forecast performance, you must apply the same evaluation period and forecast origins.

SARIMAX with exogenous variables is not always the best model. If the exogenous variable carries little information or its future values are highly uncertain, a simpler SARIMA can be more stable.

Each day's forecast used only the past baggage volume and scheduled flight information available at that forecast time.

Mean absolute error (MAE) is the average of the absolute differences between actual volume and the forecasts. SARIMAX's MAE of 570 bags means that, under this evaluation method, a one-day forecast differed from actual volume by about 570 bags on average.

Root mean squared error (RMSE) is more sensitive to large errors. SARIMAX's RMSE of 760 bags was smaller than the other models', so it can be interpreted as having reduced relatively large forecast misses as well. However, the table above shows 1-day-ahead forecast results, so you cannot conclude that the same ranking holds for 7-day-ahead forecasts.

The expected volume on January 3 is 9,700 bags, with a 95% prediction interval of 8,300–11,100 bags. This interval expresses the uncertainty of a new daily observation, on the condition that the model assumptions and the specified number of scheduled departures are appropriate. It is not a guarantee that each day's actual volume will fall inside the interval; 95% is a criterion about the model's repeated forecasting properties.

Operators can set baseline staffing from the mean forecast and separately plan flexible staff or equipment to respond when demand close to the upper bound appears. Note, though, that this interval may not fully include the possibility that the scheduled flights themselves change. If large-scale cancellations or extra flights are expected, it is a good idea to calculate exogenous-variable scenarios separately.

So after fitting, check the residual series and its ACF, the Ljung–Box test, changes in residual variance and so on.

For example, suppose the p-value of the Ljung–Box test on the SARIMAX residuals was 0.28. This means there is not enough evidence to reject the null hypothesis that there is no residual autocorrelation up to the chosen lag. But it is not evidence that the residuals are completely independent or that the model is necessarily correct.

If large errors occur repeatedly on particular dates such as holidays, you can consider holiday dummies, event variables or separate calendar effects. Even then, feed only variables that can be known in advance for future dates into the forecast, and avoid leaking past information when checking performance.

With time series data, splitting training and evaluation data by shuffling rows at random can let future information leak into past forecasts. So it is best to use time-based splits and rolling-origin validation.

Check whether the exogenous variables will be available in the future.

The flight variable in SARIMAX must be the flight schedule available at the time of forecasting. Putting actual flight records confirmed later into past forecasts overstates performance.

Don't specify the seasonal period mechanically.

Daily data doesn't always have a 7-day period as its key cycle. Check whether weekly, annual and holiday effects exist at the same time and consider other models if needed.

More differencing does not mean better forecasts.

Over-differencing can remove useful level information and make forecasts unstable. Check the residuals and forecast errors together.

Don't interpret model coefficients directly as causal effects.

Even if SARIMAX shows a positive relationship between the number of scheduled flights and baggage volume, this is only a conditional relationship useful for forecasting; an exogenous causal effect has not been proven.

Both approaches can be used for many kinds of time series, and in certain cases they are closely related mathematically.

You can't say any one method is always accurate. Evaluate them over the same forecast horizon and data conditions, and also consider the interpretability needed for operations and the maintenance burden.

If you forecast several days at once, report performance separately for each forecast horizon as well.

An actual report should also present the rationale for choosing the final orders, the coefficient estimates, the residual tests, how the prediction intervals were calculated and the range of data available at the time of forecasting.

SARIMAX adds exogenous variables on top of that, letting the forecast use changes in operating plans that are hard to capture from past baggage volume alone.

When choosing among the three models, rather than comparing only the orders, it is important to review the seasonal structure of the data, the future availability of exogenous variables, residual diagnostics and errors by forecast horizon together. Also, for practical decision-making it is more useful to explain the final forecast together with a prediction interval showing its uncertainty, rather than presenting it as a single number.