Trend–Seasonal Decomposition and Rolling Statistics for Time Series

Suppose we are looking at the monthly number of trips on a city bike-share service. Usage tends to rise in summer and fall in winter, but the service itself may also become more popular year after year.

That is why, just because this July's trip count is higher than last December's, we cannot say the business grew rapidly. The two months fall in different seasons, and they come at different points in the service's operation.

Data recorded in time order like this are called a time series. To understand a time series, you need to separate the trend, which shows the long-run direction; seasonality, which repeats over a fixed cycle; and the remainder that neither of these explains.

On top of that, using rolling statistics such as the moving average and the rolling standard deviation lets you check how persistent and how unstable changes over a given period are. In this article, we use monthly bike-share data to calculate moving averages, additive and multiplicative decomposition, STL, seasonal adjustment, and rolling volatility, and look at how to interpret the results.

So the order and spacing of observations matter. If you shuffle the order of the monthly trip counts, it becomes hard to see the pattern of rising in spring and falling in winter.

A time series can show a level, a trend, seasonality, and irregular variation all at once. For example, a bike-share service's average usage might be about 100,000 trips a month, but it grows over the long run, with July–August above average and January–February below. Fluctuations caused by weather or a temporary service outage should be treated separately from these repeating patterns.

Decomposition is first and foremost a tool for understanding the data. The decomposition itself does not guarantee future trip counts or prove the causes of changes in usage.

The figures are in thousands of trips.

The values below are some of the observations, to help understand the example.

Looking at just these 12 months, you can see a pattern within the year: high in summer and low in winter. But to judge the long-run trend, you need to compare the full 2022–2025 data. Seasonality is not established from a single year's changes; you need to examine whether it repeats across several years.

For example, suppose the bike-share service's annual trips increased from 960 thousand in 2022 to 1,080 thousand in 2023, 1,280 thousand in 2024, and 1,400 thousand in 2025. These values are evidence that usage has grown over the long run.

Year-over-year growth is calculated as (this year − previous year) ÷ previous year × 100. The growth rate for 2024 is (1,280 − 1,080) ÷ 1,080 × 100 ≈ 18.5%.

However, using annual totals like this can hide monthly fluctuations. In a real analysis, it is good to plot the raw series together with a smoothly estimated trend line, and to check for level shifts or sudden structural changes.

It is used to smooth out short-term fluctuations so that the course of the series is easier to see. For example, trips in April–June 2024 were 110, 123, and 132 thousand, so the 3-month moving average for this window is (110 + 123 + 132) ÷ 3 ≈ 121.7 thousand.

The moving averages in this table are trailing moving averages that use the current month and the two months before it. So the July value is calculated from May–July only and uses no future values. A centered moving average, by contrast, uses observations before and after a given month, so carelessly using it as an input for real-time forecasting can cause future information leakage.

A 3-month moving average reflects recent changes relatively quickly, but much of the seasonal up-and-down can remain. For monthly data, a 12-month moving average, on the other hand, is useful for smoothing out the annual seasonal cycle but can be slow to show a change in the trend's direction.

Since 12 months is an even-length window, traditional classical decomposition can use a centered trend obtained through two stages of moving averages. But the 12-month moving average itself is not always an accurate seasonally adjusted value. When observations are irregularly spaced or the seasonal pattern changes, a different approach is needed.

The typical structures are additive decomposition and multiplicative decomposition.

An additive model can be suitable when the absolute size of the seasonal swings is similar regardless of the level. For example, if summer usage is roughly 20,000 trips higher every year, an additive structure is intuitive. On the other hand, if the gap between summer and winter grows in proportion as usage grows, you can consider a multiplicative structure.

Multiplicative decomposition is generally used with positive data. When a log transformation is possible, you can also take logs to handle a multiplicative relationship in additive form. But if the data include zeros, be especially careful with how logs and transformations are handled.

Its advantage over classical decomposition is that it can flexibly handle situations where the seasonal pattern changes gradually.

STL provides an additive decomposition by default. So if the seasonal swings grow in proportion to usage, you can consider an appropriate transformation, such as a log transformation.

The main settings include the seasonal period, the smoothness of the seasonal component, the smoothness of the trend component, and a robust decomposition option that reduces the influence of outliers. When analyzing an annually repeating pattern in monthly data, it is natural to set the period to 12.

However, 48 months of data contain only four annual cycles, which limits how precisely you can judge the stability of the seasonal pattern and the long-run trend. STL results should be treated as estimates that describe the data rather than as established facts.

Since an additive decomposition is used here, 140 = 108 + 26 + 6. A seasonal component of +26 thousand for July means the model estimated a pattern in which that month's trips run about 26 thousand above the trend level for seasonal reasons.

The remainder of +6 thousand is the difference that trend and seasonality could not explain. However, it cannot be immediately interpreted as the effect of a specific event, because many factors, such as weather conditions, events, service outages, and data errors, can affect the remainder.

In an additive decomposition, you subtract the estimated seasonal component from the actual observation.

In the July 2024 example above, actual trips were 140 thousand and the seasonal component was +26 thousand, so seasonally adjusted trips are 140 − 26 = 114 thousand.

This figure shows "the level with July's seasonal boost removed." It does not mean only 114 thousand trips were actually taken. Also, seasonally adjusted values can still include temporary shocks and estimation error, so they should not be confused with the trend component.

The rolling standard deviation shows how much the observations have fluctuated over a recent fixed period. For example, if trips over the last 3 months were 110, 123, and 132 thousand, the sample standard deviation is about 11.1 thousand. Unlike the mean, this figure summarizes the size of the fluctuations in that window.

However, in strongly seasonal data such as bike usage, a high rolling standard deviation of the raw observations does not necessarily mean the service has become more unstable. Volatility may look large simply because of seasonal rises and falls.

If the goal is to spot abnormal fluctuations, it may be more appropriate to look at the rolling standard deviation of the remainder after removing trend and seasonality. Even then, you should also examine the window length and the influence of outliers.

A decomposition plot that stacks the raw series, trend, seasonality, and remainder from top to bottom lets you compare the size of each component and how it changes over time.

If a clear seasonal pattern or autocorrelation persists in the remainder, the chosen decomposition method may not have adequately explained the structure of the data. However, a descriptive decomposition can be useful even if the remainder is not perfect white noise. How important remainder diagnostics are depends on the purpose of the analysis.

A 12-month cycle in monthly data differs from a roughly 52-week cycle in weekly data, and daily data can have multiple patterns, such as day-of-week and annual seasonality. Basic STL, which handles only one seasonal period at a time, cannot explain every kind of complex seasonality.

Check how the moving average is aligned in time.

A centered moving average can use past and future observations together, so in real-time forecasting or policy evaluation, take care that future information does not get mixed in.

Do not interpret trend and seasonality as causes.

High summer usage does not let you conclude that temperature alone is the cause, and a rising trend does not let you conclude that marketing was effective. Causal relationships require a separate study design.

With short data, decomposition results can be unstable.

In 48 months of data, the annual seasonal cycle repeats only four times. To judge in detail whether seasonality has changed, a longer observation period helps.

Distinguish forecasting from descriptive decomposition.

Decomposition is useful for understanding a time series, but by itself it does not prove the performance of a forecasting model. If forecasting is the goal, you need separate time series validation and an evaluation of forecast error.

If there were log transformations, outlier treatment, or adjustments for missing values, you should also describe those steps.

In real studies, it is good to present the full time series, the decomposition plots, and the remainder diagnostics together.

This is because observations mix trend, seasonality, and temporary fluctuation together.

Moving averages smooth out short-term wobbles to show the overall course, and decomposition methods such as STL help you interpret observations by splitting them into trend, seasonality, and remainder. Seasonal adjustment makes values from different months easier to compare, and the rolling standard deviation summarizes the level of fluctuation over a given period.

However, decomposition describes the patterns that appear in the data; it does not prove the causes of change or accurately predict the future. It is important to choose the period, cycle, and decomposition method that fit the purpose of the analysis and to check the results together with the raw data.