Air Passengers Forecasting with ARIMA

Let’s start by loading the dataset and checking that the date column looks right before plotting it.

The dataset is in the notebook with the two fields needed for a time series: Month and #Passengers. The dates were parsed correctly, and the preview shows monthly observations starting in January 1949, so the next direct step is the line plot.

The data used for the plot is a monthly time series with Month as the date field and #Passengers as the passenger count field. The observed preview confirms the series begins in January 1949, and the notebook contains 144 monthly records, which is the expected structure for this dataset.

The time series plot has been created as a line chart with:

• Month on the x-axis
• #Passengers on the y-axis

From the loaded values shown before plotting, the chart is based on monthly passenger counts such as 112, 118, 132, 129, and 121 in the opening months, so the visualization is using the correct underlying series.

For this dataset, a multiplicative decomposition is a good default because the seasonal swings get larger as the series level rises.

The decomposition figure was created successfully, so the series has already been split into observed, trend, seasonal, and residual components. The only missing piece is the underlying numeric values, since the current output is just a figure object.

Your series is decomposed into the three requested parts, and the outputs show a clear pattern:

• Trend: the trend component rises steadily over time, from roughly 127 passengers/month in mid-1949 to about 475 passengers/month by mid-1960. This indicates strong long-term growth in air travel.
• Seasonality: the seasonal component is stable by month and multiplicative, meaning it scales with the level of the series.
  • Strongest months are around July–August with seasonal factors near 1.22, so those months are about 22% above trend.
  • Weakest month is November with a factor near 0.80, about 20% below trend.
  • Other lower-than-trend months include January, February, October, and December.
• Residuals: the residual component is mostly close to 1.0, which means the trend + seasonality explain most of the variation. For example, many residual values are near 0.95 to 1.02, so the unexplained noise is relatively small.
• Edge NaNs: the missing values at the beginning and end of the trend/residual columns are expected in classical decomposition because centered moving averages cannot be computed at the boundaries.

So, in plain terms: the passenger series has a strong upward trend, repeating yearly seasonality, and comparatively modest residual noise. The multiplicative decomposition was an appropriate choice because the seasonal swings increase as the series level increases.

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

The ADF output shows the series is not stationary in its current form.

• The ADF statistic is 0.815, which is not more negative than any of the critical values.
• The p-value is 0.9919, far above common thresholds like 0.05.

That means we fail to reject the null hypothesis of a unit root, so the original passenger series is non-stationary. This matches the earlier decomposition: there is a strong upward trend and clear yearly seasonality.

Because of that, differencing is needed. For this series, a good choice is:

• first differencing to remove trend: y_t - y_{t-1}
• and likely seasonal differencing at lag 12 to remove annual seasonality: y_t - y_{t-12}

So the practical conclusion is:

• Original series: not stationary
• Action needed: apply differencing
• Most appropriate here: seasonal structure suggests using seasonal differencing (12), often combined with first differencing if one differencing step alone is not enough

Based on the observed output alone, the series still needs to be differenced before using methods that assume stationarity.

Since the series is non-stationary and clearly seasonal, I’d use a seasonal ARIMA with one regular difference and one seasonal difference: d=1, D=1, seasonal period $12$.

Here’s a good first fit to start with:

If this fits cleanly, the next step is to generate the 12-month forecast and plot it.

The SARIMA model has already been fit successfully on the monthly passenger series, and the setup is appropriate for the data you explored:

• The earlier ADF result showed the raw series was non-stationary.
• The fitted model uses both regular differencing and seasonal differencing: (1,1,1) × (1,1,1,12).
• That matches the visible pattern in the series: upward trend plus strong 12-month seasonality.

What’s still missing for your original goal is the actual 12-month forecast values. Here is the minimal next step to produce them:

You fitted a seasonal ARIMA model successfully:

• Model: (1,1,1)×(1,1,1,12)
• This choice is consistent with the observed data:
  • the ADF test on the raw series had a very high p-value ($0.9919$), so the original series was not stationary,
  • the monthly pattern is clearly seasonal, so seasonal differencing with period $12$ is appropriate.

The 12-month forecast for 1961 is:

What this means:

• The forecast preserves the strong annual seasonality already visible in the history.
• Peak demand is expected in July–August 1961, with July forecast highest at about 651.99 passengers.
• Lower months are expected in February and November, matching the historical seasonal pattern.
• The confidence intervals widen a bit across the horizon, reflecting uncertainty in longer-range forecasts.

So the requested ARIMA-style forecasting task has been answered with a fitted seasonal ARIMA model and a next-12-month forecast.