Can one series have several seasonal patterns?

Yes, such as weekly and annual patterns in daily data.

Does a repeated pattern prove a cause?

No; that requires additional evidence.

AI Glossary / Time series and signal processing / Advanced

Seasonal variation

Definition

Seasonal variation is recurring movement in a time series associated with a known seasonal or calendar period. It can appear as an annual pattern in monthly or quarterly observations, a weekly pattern in daily observations, or another relevant repeating schedule. Its timing is structured, although its magnitude and shape can evolve.

What it means

A series can rise over several years while also rising and falling within each year. The longer movement and the seasonal pattern describe different aspects of the data. Seasonal variation also differs from an economic cycle whose duration is not fixed to a known calendar frequency.

Seasonal does not mean that weather must be the cause. Recurring work schedules, school calendars and other regular activities can produce seasonal patterns.

How it works

Inspect observations in time order, then compare matching positions across repeated cycles. A seasonal plot aligns successive cycles; a seasonal subseries plot groups observations by their position within the cycle. These views can reveal recurring peaks and changing seasonal behavior.

Distinguish changes in the overall level from changes in the seasonal pattern. Approximately constant absolute swings suggest an additive representation, while swings proportional to the level may suggest a multiplicative one. Neither choice is established merely by seeing one high and one low observation.

Consider unusual events, calendar alignment and multiple seasonal periods. A daily series may contain both weekly and annual variation.

Three practical examples

1. Operations planning: prepare for recurring peaks in service requests while separately allowing for overall growth.

2. Performance reporting: avoid interpreting an expected seasonal decline as conclusive evidence that the business is deteriorating.

3. AI forecasting evaluation: examine errors by seasonal position to identify a system that repeatedly misses a busy quarter despite an acceptable overall average.

Think of it this way

Think of a repeating melody played at increasing volume. The positions of the louder and quieter notes can remain the same while their absolute differences grow.

Worked example

Consider an invented quarterly activity index with nonseasonal levels of 100, 120 and 140 in three successive years. Within each year, assume constant level and no irregular component. Let the quarterly seasonal multipliers be [0.8,1.0,1.4,0.8], unchanged across all three years.

Multiplying each year's level by those factors gives:

Year 1: [80,100,140,80].

Year 2: [96,120,168,96].

Year 3: [112,140,196,112].

The third quarter is the peak every year. The first and fourth quarters share the lowest value. The pattern repeats at a period of four quarterly observations, while the annual level rises.

The within-year peak-to-trough ranges are 140 - 80 = 60, 168 - 96 = 72 and 196 - 112 = 84. Those ranges increase, but the seasonal multipliers have not changed. Dividing each range by its year's level gives 60/100 = 0.6, 72/120 = 0.6 and 84/140 = 0.6.

Thus the absolute seasonal swing grows with the level while its relative size stays constant in the construction.

Each year's average equals its stipulated level because the four multipliers sum to four. Dividing observations by their annual average produces [0.8,1.0,1.4,0.8] in every year. This recovery is exact because of our assumptions.

Now compare year-2 quarter 3 with year-3 quarter 3. The index rises from 168 to 196, an increase of 28 or approximately 16.67%. The nonseasonal level rises from 120 to 140 by the same percentage. The year-over-year comparison therefore reflects the level change in this particular example; it does not reveal a new seasonal effect.

For contrast, hold an additive quarterly pattern fixed at [-20,0,40,-20] points. It reproduces year 1 exactly at level 100. At level 140, however, it predicts [120,140,180,120], with a range of 60. The original multiplicative construction instead gives [112,140,196,112], with a range of 84. The two models describe different assumptions about how seasonal variation changes with scale.

Introduce a separate one-off event that adds 70 points to year-3 quarter 2. That observation becomes 210, exceeding the quarter-3 value of 196. The observed peak has moved to quarter 2, but the stipulated seasonal multipliers remain unchanged. A single year's highest observation need not identify the peak of its recurring seasonal component.

If the event were known and appropriately accounted for, removing its stipulated contribution would restore quarter 2 to 140 in this illustration. In real data, however, an analyst must investigate the event and the model; the example does not justify deleting every observation that fails to follow the usual pattern.

Finally, suppose a separate model forecasts a nonseasonal level of 160 for the next year and the same multipliers are assumed to continue. The quarterly projections are [128,160,224,128], with an absolute range of 96. These are conditional teaching projections.

Illustrative case study

Fictional Cedar Services interprets the growing range in the example as evidence that its seasonal customers have changed behavior. Its analyst compares both raw observations and ratios to the yearly level, finding an unchanged relative pattern in the constructed data.

The team separately investigates the one-off second-quarter event before revising its recurring staffing schedule. This invented case illustrates how level growth and unusual events can affect the appearance of seasonality without proving that the seasonal pattern itself changed.

Common misconceptions and mistakes

"Seasonal variation must be weather-driven." Regular calendars and schedules can also contribute.

"Larger absolute swings prove stronger relative seasonality." The overall level may have grown.

"The largest observation always identifies the seasonal peak." An unusual event can change that ranking.

"Seasonal and cyclical variation are identical." Their timing structures differ.

Business use

Show original values alongside a clearly explained seasonal analysis. Identify the relevant period and whether variation is measured in units or proportions. Compare forecasting performance across seasons and review unusual events. Retain a record of assumptions so that changes in the pattern can be distinguished from changes in the analysis method.

Check your understanding

What is the example's seasonal period?

Answer: Four quarterly observations.

Why does the range grow?

Answer: The fixed multiplicative pattern scales with a growing level.

Why does the observed year-3 peak move after the event?

Answer: Seventy irregular points are added to quarter 2.

Related terms

Seasonal effect
Seasonal index
Seasonality
seasonal plot
trend
cyclical variation
irregular component
multiplicative decomposition

Learn next

Keep the original multipliers but set the next year's nonseasonal level to 80. Calculate the quarterly projections and their range, and compare the range as a proportion of the level with the original three years.

Sources

Rob J Hyndman and George Athanasopoulos, Forecasting: Principles and Practice, Time series patterns and seasonal plots: https://otexts.com/fpp3/graphics.html

Rob J Hyndman and George Athanasopoulos, Forecasting: Principles and Practice, Time series decomposition: https://otexts.com/fpp3/decomposition.html

U.S. Census Bureau, Seasonal Adjustment Questions and Answers, seasonal and irregular components: https://www.census.gov/data/software/x13as/seasonal-adjustment-questions-answers.html

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