Air Passengers Forecasting with ARIMA

Let’s load it first and check the structure, then we’ll plot the time series in the next step.

The dataset is in the expected time-series shape: 144 monthly records with a date-like Month column and one numeric series, #Passengers. That means it’s ready to plot as a single passenger-count time series.

The outputs already show the dataset structure and the plotted figure was created from the correct two columns: Month and #Passengers, with 144 monthly records. The first few observed values begin in 1949 and confirm this is a monthly passenger time series suitable for plotting over time.

The time series plot uses Month converted to datetime on the x-axis and #Passengers on the y-axis, which matches the requested visualization of air passengers over time.

Since this is monthly data and the seasonal swings grow as the level rises, a multiplicative decomposition is the right choice here.

The decomposition plot is in place and it confirms the series was split into observed, trend, seasonal, and residual components over 144 monthly points. Because the output is visual-only, the missing piece is a compact table of the component values themselves. One detail to expect: the trend and residual columns will have NaN at the start and end because the 12-month decomposition needs surrounding months to estimate the centered trend.

The series has been successfully decomposed into trend, seasonality, and residuals using a multiplicative model with a 12-month period, which fits this dataset because the seasonal variation grows as the overall level rises.

From the observed component values:

• Trend captures the long-term upward movement in passenger volume.
  Early in the series, the estimated trend is around 126.8 in July 1949 and rises to about 134.9 by March 1950, showing steady growth.

• Seasonality captures the repeating within-year monthly pattern.
  The seasonal factor is:

  • above 1 in stronger months, such as July (1.2266) and August (1.2199), meaning these months are typically higher than the trend.
  • below 1 in weaker months, such as November (0.8012) and February (0.8836), meaning these months are typically lower than the trend.

• Residuals are what remains after removing trend and seasonality.
  These values are mostly close to 1, for example 0.9517, 1.0022, 1.0063, which suggests the decomposition is capturing most of the systematic structure and leaving relatively small irregular fluctuations.

• The NaN values at the beginning are expected for decomposition because the centered moving average used to estimate trend cannot be computed at the edges.

So, in plain terms:

• the trend shows passenger counts increasing over time,
• the seasonality shows a strong recurring annual cycle with summer peaks and late-autumn lows,
• the residuals are comparatively small random deviations after accounting for those two patterns.

The decomposition already suggests the series is probably not stationary because it has both a rising trend and strong seasonality. The next step is to confirm that with an ADF test on the original monthly series.

If the p-value is above 0.05, the next step will be first-order differencing.

The ADF test shows the original series is not stationary.

From the observed output:

• ADF statistic = 0.815
• p-value = 0.9919

Because the p-value is far above 0.05, we fail to reject the null hypothesis of a unit root. That means the monthly passenger series still has non-stationary structure, which matches what we already saw in the decomposition: a clear upward trend and strong seasonality.

So differencing is needed. Given this series has both trend and yearly seasonality, the practical interpretation is:

• First differencing addresses the trend:
  y_t - y_{t-1}
• Since this is monthly data with annual seasonality, it will likely also need seasonal differencing with lag 12:
  y_t - y_{t-12}

Based on the current output alone, the correct conclusion is:

• the original series is non-stationary
• differencing should be applied before modeling
• for this dataset, seasonal differencing is especially important because of the strong 12-month cycle visible in the decomposition

Because the series is non-stationary and clearly seasonal, a seasonal ARIMA (SARIMA) is the better choice here than a plain non-seasonal ARIMA.

A good first fit for this dataset is:

• regular differencing: d = 1
• seasonal differencing: D = 1 with period 12

This will fit the model and produce the next 12 monthly forecasts with confidence intervals. After that, the next useful step is to plot the historical series together with the forecast.

The fitted model is a seasonal ARIMA, specifically SARIMA(1,1,1) × (1,1,1,12), which is appropriate here because the series was non-stationary and showed strong yearly seasonality.

The 12-month forecast continues the same seasonal pattern seen in the historical data:

• lower values at the start of the year,
• rising through spring,
• a summer peak,
• then declining into autumn and winter.

The forecasted passenger counts for 1961 are:

• Jan: 447.2
• Feb: 422.7
• Mar: 457.7
• Apr: 492.3
• May: 505.6
• Jun: 565.7
• Jul: 652.0
• Aug: 639.6
• Sep: 543.1
• Oct: 493.7
• Nov: 426.9
• Dec: 470.1

The model expects the peak month to be July 1961 at about 652 passengers, with August close behind. The lowest forecast is February 1961 at about 423 passengers.

The confidence intervals widen a bit for later months, which means uncertainty grows as the forecast horizon extends. Even so, the intervals still support the same overall seasonal shape, so the forecast is consistent and plausible for this dataset.