Identify qualitative features in a time series practice questions | General Mathematics (Year 12, VCE Units 3 & 4)

Identifying time-series features

Read the complete plot before naming a feature. Trend describes long-term movement, while seasonality, events and irregular fluctuations require checks of repeated positions and later values

\text{whole plot}\Rightarrow\text{trend, repetition, events and irregular variation}

Read the long-term direction Use all periods to decide whether the series generally increases, decreases or stays near one level \(\displaystyle 100,112,121,116\Rightarrow\text{overall direction: increasing}\)
Check repeated calendar positions Seasonality is supported when a similar movement repeats at the same positions in successive periods \(\displaystyle Q_1:410,455;\quad Q_2:470,520\Rightarrow\text{compare matching quarters}\)
Separate an outlier from a level shift An isolated value returns to the nearby pattern, while a structural change persists afterwards \(\displaystyle 120,122,180,121,123\Rightarrow\text{possible outlier at period 3}\)
Recognise irregular fluctuations Short-term departures that do not repeat and do not persist are irregular fluctuations \(\displaystyle 100,102,99,101,100\Rightarrow\text{irregular fluctuations}\)

Worked example

A community pool records quarterly attendance of 410, 470, 445, 500, 455, 520, 468 and 530. Describe the trend, repeated pattern and event status

y=(410,470,445,500,455,520,468,530)

\(\displaystyle \bar{y}_1=\frac{410+470+445+500}{4}=456.25\)Find the mean for the first cycle
\(\displaystyle \bar{y}_2=\frac{455+520+468+530}{4}=493.25\)The second cycle has a higher mean than the first, supporting an increasing long-term level
\(\displaystyle Q_1:410,455;\quad Q_2:470,520;\quad Q_3:445,468;\quad Q_4:500,530\)The same high and low ordering repeats across the two cycles, supporting seasonality
\(\displaystyle \text{repeated quarterly highs and lows; no isolated departure or persistent shift}\Rightarrow\text{no outlier or structural change}\)The highs and lows repeat by quarter; no isolated departure or persistent abrupt shift appears
\(\displaystyle \text{trend: increasing},\quad\text{seasonality: present},\quad\text{event: none}\)These classifications use the complete series rather than one adjacent pair

Common mistakes

Using only the last movement

✕\(\displaystyle 116-121=-5\Rightarrow\text{decreasing trend}\)✓\(\displaystyle 100,112,121,116\Rightarrow\text{overall trend: increasing}\)

One final movement is only a small part of the plot; the overall trend uses the full interval

Calling an isolated spike a structural change

✕\(\displaystyle 120,122,180,121,123\Rightarrow\text{structural change at period 3}\)✓\(\displaystyle 120,122,180,121,123\Rightarrow\text{possible outlier at period 3}\)

A structural change persists at a new level, whereas an isolated spike is followed by a return to the nearby pattern

Calling a level shift an outlier

✕\(\displaystyle 100,102,140,142,141\Rightarrow\text{outlier at period 3}\)✓\(\displaystyle 100,102,140,142,141\Rightarrow\text{persistent level shift from period 3}\)

A persistent change affects later observations, so it is structural rather than an isolated outlier

Difficulty levels

introOne dominant trend or one isolated outlier is identified from a short plot

standardTrend is considered with irregular variation or a repeated seasonal pattern

challengeSeveral simultaneous features require an outlier and structural-change distinction

examA complete description is selected from near-neighbour feature combinations

Questions about this topic

What is a trend in a time series?+

It is the long-term direction of the response, such as generally increasing, generally decreasing or approximately level

How is seasonality identified?+

Compare the same calendar positions in more than one repetition. A similar movement at matching positions supports seasonality

How is an outlier different from a structural change?+

An outlier is an isolated departure followed by a return to the nearby pattern. A structural change persists at a new level or with a new trend

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 5: Time series

In this chapter

Construct and read a time series plot Identify qualitative features in a time series Calculate moving means Moving median smoothing Select and compare smoothing methods Interpret and complete seasonal indices Deseasonalise and reseasonalise values Calculate seasonal indices from complete cycles Fit a time trend and forecast