Can seasonal periods differ from twelve?

Yes; choose the period appropriate to the observation interval and cycle.

Does changing the order of ordinary and seasonal differencing change these values?

No, when applied to the same complete data and aligned dates.

AI Glossary / Time series and signal processing / Advanced

Seasonal difference

Definition

A seasonal difference subtracts an earlier observation from the same position in a seasonal cycle from the current observation. With seasonal period m, the first seasonal difference is y_t - y_(t-m). For quarterly observations with an annual cycle, m is four; for monthly observations with an annual cycle, it is twelve.

What it means

The transformed value describes a change across one seasonal cycle, in the original measurement units. It is not the current level and is not automatically a percentage change. Comparing corresponding seasons can help remove a recurring pattern when preparing a time-series model.

Seasonal differencing is distinct from seasonal adjustment. Differencing compares observations separated by a lag; adjustment estimates and removes a seasonal component to retain an adjusted level. A seasonal difference may still contain patterns that require modeling, and it does not automatically make every series stationary.

How it works

First identify the observation frequency and the seasonal period of interest. Align each observation with the value m positions earlier, then subtract. In backshift notation, the operation is (1 - B^m)y_t, where B shifts the series back one observation.

Ordinary first differencing uses lag one. Combining it with seasonal differencing gives y_t - y_(t-1) - y_(t-m) + y_(t-m-1). On complete, regularly spaced data, the two operations commute: their order does not change the values at the dates where both are defined.

Deciding whether to use either transformation is a separate modeling question. Review the transformed series and relevant diagnostics, and avoid unnecessary repeated differencing. Preserve dates and missing values rather than allowing a missing period to silently change the meaning of a lag.

Three practical examples

1. Quarterly activity analysis: compare each quarter with the same quarter in the preceding year.

2. Monthly forecasting: create annual differences as a possible input to a seasonal ARIMA specification.

3. Model interpretation: explain whether an AI-assisted forecast predicts original demand levels or changes from the corresponding earlier season.

Think of it this way

Think of comparing this year's spring activity with last year's spring activity, rather than with the winter immediately before it. The comparison aligns the season, but the difference can still reflect growth, unusual events and measurement changes.

Worked example

Consider an invented activity index observed for three complete years, with four quarters per year:

Year 1: [100,130,150,120].

Year 2: [112,145,166,137].

Year 3: [128,162,185,156].

To calculate first seasonal differences at lag four, start with the first quarter of year 2 because no earlier year is supplied for year 1.

The year-2 differences are 112 - 100 = 12, 145 - 130 = 15, 166 - 150 = 16 and 137 - 120 = 17. The year-3 differences are 128 - 112 = 16, 162 - 145 = 17, 185 - 166 = 19 and 156 - 137 = 19.

The transformed sequence is [12,15,16,17,16,17,19,19]. It has eight values, each aligned with its later observation. The first four undefined differences should not be entered as zeros: zero would claim no change, whereas the necessary comparison observation is unavailable.

For the first quarter of year 3, the seasonal difference is 16 index points. Its year-over-year percentage change is instead 16/112 × 100, approximately 14.29%. The same numerator produces a different quantity after division by the earlier level. A seasonal difference of 16 does not mean growth of 16%.

Now apply an ordinary first difference to the eight seasonal differences. The result is [3,1,1,-1,1,2,0], beginning at the second quarter of year 2. For that first value, 15 - 12 = 3: the year-over-year increase grew by three index points between consecutive quarters.

Check the same calculation directly from the original series. For quarter 6, the combined difference is y_6 - y_5 - y_2 + y_1 = 145 - 112 - 130 + 100 = 3. Alternatively, the quarter-to-quarter changes are 145 - 112 = 33 in year 2 and 130 - 100 = 30 in year 1; their difference is again three.

This is one ordinary difference plus one seasonal difference. It is not two seasonal differences. Applying the seasonal operation twice would give y_t - 2y_(t-4) + y_(t-8). At the first quarter of year 3, that value is 128 - 2 × 112 + 100 = 4. Across year 3, the twice-seasonally-differenced values are [4,2,3,2], not the seven-value sequence above.

Suppose a forecasting model predicts seasonal differences of 18, 20, 21 and 20 for year 4. Add the corresponding year-3 observations to recover level forecasts: [128 + 18,162 + 20,185 + 21,156 + 20] = [146,182,206,176]. These difference forecasts are invented assumptions, not estimates obtained by fitting the twelve observations.

Recovering levels requires the earlier seasonal observations as anchors. A sequence of differences alone does not identify the original level: adding a constant to every original observation leaves all seasonal differences unchanged. Keep the original history available alongside the transformed values.

Illustrative case study

Fictional Alder Analytics receives eight seasonal differences and labels them quarterly activity levels. The values look much smaller than the original index, causing confusion in a planning report.

Its analyst restores the dates, identifies the four missing initial comparisons and reconstructs the example's forecast levels using the earlier seasonal observations. This invented case illustrates interpretation and data alignment, not evidence that a particular differencing choice improves accuracy.

Common misconceptions and mistakes

"A seasonal difference is a percentage." It remains in the original units unless further transformed.

"Missing initial differences are zeros." They are undefined from the supplied history.

"A second seasonal difference equals an ordinary plus seasonal difference." Their lag formulas differ.

"Differencing guarantees stationarity." The transformed data still require assessment.

Business use

Document the seasonal period, number of differencing operations and resulting units. Ensure that dashboards and forecast exports identify differences as differences. Check date alignment before modeling, and retain the observations needed to convert predictions back to levels. Evaluate the complete forecasting process on appropriately held-out periods.

Check your understanding

What is the first year-3 seasonal difference?

Answer: Sixteen index points.

How many first seasonal differences are available?

Answer: Eight.

What is the first combined ordinary-and-seasonal difference?

Answer: Three.

Related terms

First difference
Stationarity
Seasonal adjustment
backshift operator
seasonal ARIMA
seasonal period
year-over-year change
inverse transformation

Learn next

Add 50 to every original index value. Recalculate the first seasonal difference and the first year-3 percentage change. Explain why the difference is unchanged while the percentage changes.

Sources

Rob J Hyndman and George Athanasopoulos, Forecasting: Principles and Practice, Stationarity and differencing: https://otexts.com/fpp3/arima.html

Thomas B Fomby, Seasonal Differencing in the Box-Jenkins Approach, Southern Methodist University teaching notes: https://s2.smu.edu/tfomby/eco5375/data/Notes/SEASONAL%20DIFFERENCING.pdf

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