Time Series

Readings taken in order of time

A time series is a set of readings of one quantity, taken at regular times and kept in the order they were taken: a temperature every hour, a town’s population every year, a shop’s sales every month.

To draw one, put time along the horizontal axis, earliest on the left, and the reading up the side. Plot one point for each reading and join the points in order with straight lines. The lines show the order of the readings, and how each one changed from the one before. Nothing was measured between two points, so a value read off a joining line is only a guess.

Here are six monthly readings: 3 in January, 5 in February, 4 in March, 6 in April, 5 in May and 7 in June. To read one month, go up from its label to its dot, then across to the scale. The May dot is level with 5.

One reading per month, plotted in time order and joined from left to right.

The trend under the wiggle

Eight readings, one each quarter of a year, are 3, 5, 4, 6, 5, 7, 6 and 8. From one quarter to the next the series goes up, then down, then up again, so no single step tells you where it is heading.

Look at the low points instead: 3, then 4, then 5, then 6. Every dip lands higher than the dip before. The high points do the same: 5, 6, 7, 8. Under the wiggle, the series rises by 1 every two quarters. That long-run direction is the trend.

A trend can also fall, when each peak tops out lower than the one before, or be flat, when the ups and downs cancel and the series ends about where it started.

The dips are 3, 4, 5 and 6, and the peaks are 5, 6, 7 and 8. Each is higher than the one before, so the trend is rising.

A moving average

In this series the wiggle repeats every two quarters: one reading low, the next high. Average each pair of neighboring readings, and every average holds one low reading and one high one, so the wiggle cancels out.

The averages are (3 + 5) ÷ 2 = 4, then (5 + 4) ÷ 2 = 4.5, then (4 + 6) ÷ 2 = 5, then 5.5, 6, 6.5 and 7. They rise by exactly 0.5 a quarter. This is a moving average: the average of a fixed number of readings in a row, moved along one reading at a time.

A moving average must cover one full cycle of the wiggle. When sales rise and fall with the four seasons, the cycle is four quarters long, and the moving average is taken over four readings in a row.

The trend line, and each reading’s distance from it

Each average sits halfway between the two quarters it covers, so the first one, 4, belongs at quarter 1.5, and the last one, 7, at quarter 7.5. They lie on a straight line that rises 0.5 a quarter. Working back from quarter 1.5 to quarter 0 takes off 1.5 × 0.5 = 0.75, so the line starts at 4 − 0.75 = 3.25. The trend line is y = 0.5n + 3.25, where n is the quarter number.

Now compare each reading with the trend. At quarter 1 the trend is 0.5 × 1 + 3.25 = 3.75 and the reading is 3, which is 0.75 below it. At quarter 2 the trend is 4.25 and the reading is 5, which is 0.75 above it. The pattern holds all the way along: every odd quarter is 0.75 below the trend and every even quarter is 0.75 above it.

So each reading is made of two parts: the trend, which carries the long-run change, and a distance from the trend that repeats in a cycle.

The trend line y = 0.5n + 3.25 through the eight readings. The gaps alternate: 0.75 below the line, then 0.75 above it.

Forecasting the next reading

To estimate quarter 9, find the trend there and put back the distance for that point in the cycle. The trend at quarter 9 is 0.5 × 9 + 3.25 = 7.75. Quarter 9 is an odd quarter, so it sits 0.75 below the trend: 7.75 − 0.75 = 7.

Check with the dips. They run 3, 4, 5, 6 at quarters 1, 3, 5 and 7, rising by 1 each time, so the next dip, at quarter 9, is 7.

A forecast like this assumes the trend and the cycle carry on as before. One step past the data is a reasonable estimate; many steps past it is extrapolation that nothing measured supports.

Seasons: compare like with like

Many series rise and fall with the time of year. Ice cream sells more in summer than in winter, every year. The readings here are 2 in January, 7 in July, 3 the next January and 8 the next July.

From July to January the sales fall by 5, and from January to July they rise by 5. Neither change says anything about how the business is doing: it is the season turning, and it happens every year.

A fair comparison takes the same season in each year. This July against last July is 8 against 7, a rise of 1. This January against last January is 3 against 2, also a rise of 1. With the season the same on both sides, a change that is left over is a real change.

Each July is far above the January before it. July against July, 7 then 8, shows the real change of 1.

The usual mistakes

Reading the neighboring dot. Go straight up from the month’s own label; the dot one step to the left or right is a different month.

Comparing a summer month with a winter month, or with the month just before it. The season changes between them, so the difference mixes the trend with the season.

Reading the trend from the first and last readings alone. In the quarterly series, the first reading, 3, is a dip and the last, 8, is a peak, so they suggest a rise of 5 over 7 quarters, about 0.71 a quarter. The trend is 0.5 a quarter; the rest is the wiggle.

Taking the trend value as the forecast. The trend is the middle of the cycle, so the distance for that point in the cycle has to be put back.

Sales with a trend and a season

In the application below, a garden center’s sales rise from year to year and also rise and fall within each year. The trend line is given. The distance of each quarter from it, below the trend in the first quarter of each year and above it in the third, is what turns the trend into an estimate for the next quarter.