Time series plots and components in WACE Mathematics Applications Unit 4
“Construct a time series plot and identify trend, seasonal, cyclic and irregular components.”
A time series plot graphs a variable against time, with time on the horizontal axis and the points joined in chronological order. Once plotted, identify the components: trend (long-term direction), seasonal (a regular pattern over a fixed period), cyclic (longer swings of no fixed length), and irregular (random variation). Naming the components from a plot or description is the first step of all time-series analysis.
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What this dot point is asking
You must construct a time series plot and identify its trend, seasonal, cyclic and irregular components.
Constructing a time series plot
A time series plot is a special scatterplot where the explanatory variable is time.
Plot the measured variable on the vertical axis against time on the horizontal axis, in chronological order, and join consecutive points with line segments. Time is the explanatory variable, so the plot is read left to right as the variable's history.
Joining the points (unlike an ordinary scatterplot) emphasises the movement over time, making patterns easier to see.
The four components
A time series is a blend of up to four components, and recognising them is the examinable skill.
- Trend (T): the long-term upward or downward movement across the whole series.
- Seasonal (S): a pattern that repeats over a fixed, known period (a year, quarter, week).
- Cyclic (C): longer swings of no fixed length, such as the economic boom-and-bust cycle.
- Irregular (I): random, unpredictable variation with no pattern.
A series can show several at once: monthly retail sales typically rise (trend), peak each December (seasonal), and have occasional random jumps (irregular).
The key difference is the period. Seasonal variation repeats over a fixed, known length (every months, every quarters). Cyclic variation swings over a longer, variable length tied to broader cycles, so you cannot predict exactly when the next peak falls.
Identifying components from a plot
A time series plot shows quarterly electricity demand over five years. It rises steadily overall, peaks every winter quarter, dips every spring quarter, and has one unusual spike during a heatwave.
Identify the trend
The steady overall rise across the five years is the trend component: demand is growing over time.
Identify the seasonal component
The winter peak and spring dip repeating every year (a fixed -quarter period) is the seasonal component.
Identify the irregular component
The one-off heatwave spike, which does not repeat and could not be predicted, is irregular variation.
Comment on cyclic
No long, variable-length swing is described, so there is no clear cyclic component here. (A cyclic component would show as multi-year ups and downs of no fixed length.)
Additive and multiplicative descriptions
A time series can combine its components in two ways. In an additive model the components add up, so the seasonal swing is roughly the same size each cycle regardless of the level. In a multiplicative model the components multiply, so the seasonal swing grows in proportion as the trend rises (a December lift is larger in dollars when sales are higher). On a plot, additive seasonality shows constant-height waves about the trend, while multiplicative seasonality shows waves that widen as the series climbs. Seasonal indices (which are ratios summing to the number of seasons) describe the multiplicative case, the one SCSA uses for deseasonalising.
Why identify components first
Naming the components decides the analysis that follows. A clear seasonal component means you will compute seasonal indices and deseasonalise. A clear trend means a trend line is worth fitting (after deseasonalising). Strong irregular variation suggests smoothing first. The plot is the diagnostic that tells you which tools to reach for.
Reading a plot well also guards against false conclusions: a single seasonal peak is not evidence of growth, and one irregular spike is not a change in trend. By separating the steady trend from the repeating seasonal swing and the random noise, you describe what the data is genuinely doing over time, which is the foundation for every later step of forecasting.
- Calling every repeating pattern seasonal
- Seasonal has a fixed, known period; a longer, variable-length swing is cyclic.
- Putting the variable on the horizontal axis
- Time is the explanatory variable and goes on the horizontal axis.
- Not joining the points
- A time series plot joins consecutive points to show movement; leaving them as a scatter hides the pattern.
- Confusing trend with cycle
- Trend is the single long-term direction; cyclic variation is repeated rises and falls over the long term.
Exam-style questions
Questions in the style of SCSA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.
WACE 2022-style5 marksA time series plot of monthly ice-cream sales over three years rises overall and shows a clear peak every summer and a dip every winter, with a few random spikes. (a) Identify the components present and where each shows in the description. (b) Distinguish a seasonal component from a cyclic one.Show worked answer →
Match each described feature to a named component.
(a) Trend: the overall rise over the three years. Seasonal: the summer peak and winter dip repeating each year (fixed -month period). Irregular: the random spikes. (3 marks)
(b) A seasonal component repeats over a fixed, known period (a year, a quarter); a cyclic component is a longer swing of no fixed length, such as an economic boom-and-bust cycle. (2 marks)
Markers reward correctly naming trend, seasonal and irregular from the description and the fixed-period versus variable-length distinction.
WACE 2024-style4 marksDescribe how to construct a time series plot, and explain why time always goes on the horizontal axis.Show worked answer →
A time series plot is a scatterplot in time order, usually joined.
To construct it, plot the variable on the vertical axis against time on the horizontal axis, plotting points in chronological order and joining consecutive points with line segments to show the movement. (2 marks)
Time goes on the horizontal axis because it is the explanatory variable: the measured quantity is treated as responding to the passage of time, so reading left to right shows how the variable changes over time. (2 marks)
Markers reward the axes and joined-points construction and time as the explanatory variable.