Smoothing and seasonality

Five years of monthly data, each year laid over the last. Watch their average become a seasonal pattern, then disappear.

Taking out the seasons

Ice-cream sales rise every summer; heating bills rise every winter. A pattern that repeats with the calendar is a season, and it can hide what the series is really doing. Is this July’s rise the usual summer, or something new?

The picture answers by stacking. Lay each year over the last, and the noise of one year is not the noise of the next, while the season is the same every year. So the month-by-month average keeps the season and shrinks the noise: averaging 5 years cuts the noise’s standard deviation from 2 to $2/\sqrt5=0.894$. That average is the seasonal profile. Subtract it, and what is left is the seasonally adjusted series.

After the clip, drag the number of years. With one year the profile is mostly that year’s noise; with five it settles.

Moving averages

Replace each value by the average of itself and its neighbours. A centred moving average of width $2r+1$ is

The wider the window, the more noise it removes: for white noise, a width of $w$ divides the standard deviation by $\sqrt w$. But it also blurs: peaks are flattened and sharp steps are smeared over the window’s width, and the first and last $r$ values have no full window.

Exponential smoothing

A moving average forgets values all at once when they leave the window. Exponential smoothing forgets gradually:

with a smoothing constant $\alpha$ between 0 and 1. Unrolled, the weights on the present and past values are $\alpha$, $\alpha(1-\alpha)$, $\alpha(1-\alpha)^2$ and so on: a fading memory, about $1/\alpha$ samples long. It needs only the last level, so it runs as data arrive, but it looks only backwards, so it trails a rising series. For white noise it multiplies the standard deviation by $\sqrt{\alpha/(2-\alpha)}$.

Here $\alpha$ is this page’s smoothing constant; on the testing pages it was a false-alarm chance.

Kernel smoothing

A kernel smoother takes a weighted average with weights that fall off smoothly with distance, for example the Gaussian kernel $K_h(u)=e^{-u^2/(2h^2)}$:

The bandwidth $h$ sets how far the weights reach, as the window width does for a moving average. Choosing it is the overfitting question of Overfitting and model choice: too small follows the noise, too large loses the shape.

The seasonal profile, step by step

The classical recipe for monthly data has four steps.

1. Find the trend with a centred moving average one period wide, so that every month of the cycle gets equal weight. A period of 12 is even, so the centred version averages two neighbouring 12-month windows: weights $\tfrac1{24}$ at the two ends and $\tfrac1{12}$ on the 11 months between, adding to 1. For quarters it is the same with 4.
2. Remove the trend and average what is left month by month: that is the profile $g_k$.
3. Recentre the profile so its values over one period add to 0, so that removing it does not shift the level.
4. Subtract it: the adjusted series is $x_k-g_k$.

A quicker way to remove a season of period $P$ is the seasonal difference $x_k-x_{k-P}$: this January minus last January. It removes any pattern that repeats every $P$ samples and turns a straight-line trend into a constant.

These are filters

Every smoother on this page is a filter: a moving average is an FIR filter with equal taps, and exponential smoothing is a one-pole recursive filter. Their frequency responses, drawn in the second picture, explain their side effects: what they let through, what they hold back, and what they delay.

Worked example

1. Smoothing a short series. For the series $1,2,4,5,4,3,2,3$ of Time series and autocorrelation, the moving average of width 3 gives, for the six inner values: 2.333, 3.667, 4.333, 4, 3, 2.667. The peak of 5 falls to 4.333, and the series loses a value at each end.

Exponential smoothing with $\alpha=0.5$, starting from $m_1=1$, gives these exact values: 1, 1.5, 2.75, 3.875, 3.9375, 3.46875, 2.734375, 2.8671875. It trails the data: it peaks one step after the series does, below the series’ peak.

2. The same noise reduction. Exponential smoothing at $\alpha=0.5$ cuts white-noise variance to a third, $0.5/1.5=\tfrac13$, as the 3-point average does. Both pass a third of the fastest wiggle, the one that flips sign every sample. The 3-point average also blocks one frequency completely, a wiggle with period 3 samples; the exponential smoother blocks nothing completely.

3. A kernel at one point. For the five values $2,4,5,4,5$ of Linear regression, smooth at the middle value. A box of width 3 gives $13/3=4.333$. A Gaussian kernel with $h=1$ weights the offsets $0$, $\pm1$, $\pm2$ by $1$, $0.6065$ and $0.1353$, and gives $10.7996/2.4837=4.348$.

4. Quarterly data, no noise. Three years of quarterly values with trend $10+k$ and season $-4,\ 2,\ 6,\ -4$ are 7, 14, 19, 10, then 11, 18, 23, 14, then 15, 22, 27, 18. The centred average over four quarters at $k=3$, with half weights at the ends, is $(3.5+14+19+10+5.5)/4=13$, exactly the trend; at $k=4$ it is 14. Removing the trend leaves $-4,\ 2,\ 6,\ -4$ in every year, already adding to 0. The adjusted series is $11,12,13,\dots,22$, a straight line. The seasonal difference $x_k-x_{k-4}$ is 4 for every $k$ from 5 on: the season is gone, and what is left is the trend’s rise per year.

Try it yourself. The weights of exponential smoothing at $\alpha=0.5$ are 0.5, 0.25, 0.125 and so on. A width-9 moving average cuts white noise of standard deviation 3 to 1. A seasonal difference at period 4 turns a straight line rising 0.5 per sample into the constant 2.

Where you’ll meet this

Economic and energy data. Unemployment, retail sales and electricity demand are published seasonally adjusted, so that one month can be compared with the last.

Meters and gauges. A battery gauge, a speedometer’s display or an audio level meter smooths its readings, often exponentially, so the number does not flicker.

Trends under cycles. A slow drift in a sensor, hidden under a daily temperature cycle, shows up once the daily profile is subtracted.

Reference card