Time Series Data (VCE SSCE General Mathematics): Revision Notes

Time Series Data

What is time series data?

When we collect, observe, or record data about a variable at regular intervals over time, we call this time series data. This type of data is special because the explanatory variable is always time, which distinguishes it from other types of numerical data.

Time series data appears in many real-world contexts. For example, we might track annual road accident fatalities, daily temperatures, monthly electricity bills, or quarterly business sales. The key feature is that measurements are taken at successive time intervals.

A classic example is the annual road accident fatalities for Australia from $1982$ to $2021$. This data set shows how road deaths have changed over a $40$-year period:

When we plot this data, we can visualise the pattern more clearly:

This plot immediately reveals a clear downward trend in road fatalities over time, suggesting that road safety initiatives in Australia have been effective.

Constructing time series plots

A time series plot is essentially a special type of scatterplot where time is always placed on the horizontal axis. The key difference from a regular scatterplot is that consecutive data points are joined by line segments in time order, making it easier to see patterns and trends.

Steps for creating a time series plot

Let's work through an example. Suppose maximum temperature was recorded each day for a week in a certain town:

Important considerations for time variables

When working with time series data, we often need to convert time periods into numerical values. For example:

• Months can be numbered $1, 2, 3, \ldots$ for January, February, March, etc.
• If we have monthly data for a two-year period, we'd use values $1, 2, \ldots, 24$
• Days of the week can be numbered $1$ to $7$
• Quarters can be numbered $1, 2, 3, 4$

Using technology

For larger data sets, technology tools like spreadsheets or CAS calculators are invaluable. Consider the Australian birth rate data from $2011$ to $2020$:

This data can be plotted using a calculator to produce a time series plot showing the declining birth rate over the decade:

Looking for patterns in time series plots

When analysing time series data, we look for six main features:

• Trend: long-term increase or decrease
• Cycles: recurring patterns lasting more than one year
• Seasonality: recurring patterns within one year or less
• Structural change: sudden shifts in the pattern
• Outliers: unusual individual values
• Irregular fluctuations: random, unexplained variations

Trend

When examining a time series plot, we often notice a general upward or downward movement over time. This long-term change is called a trend. A trend represents the overall direction in which the data is moving, ignoring short-term fluctuations.

To identify trends, we can draw trend lines on the graph. These lines smooth out the fluctuations and show the overall increasing or decreasing nature of the data. An upward-sloping trend line indicates an increasing trend, while a downward-sloping trend line shows a decreasing trend.

Multiple trends in one time series

Sometimes a time series displays different trends during different time periods. This is particularly common when external events or changing circumstances affect the data.

Consider the Australian birth rate from $1931$ to $1990$:

This plot shows three distinct trends:

Cycles

Cycles are recurring movements in a time series that generally last longer than one year. Unlike the regular patterns of seasonality, cycles may vary in height and may not repeat at exactly regular intervals.

The variations in cycles can be caused by various factors, such as economic conditions, natural phenomena, or long-term patterns in human behaviour.

Example: Sunspot cycles

A classic example of cycles appears in sunspot activity. Sunspots are darker, cooler areas on the sun's surface. The following plot shows sunspot activity from $1940$ to $2020$:

The recurring pattern is clearly visible in the time series plot. The years of lowest sunspot activity occur at approximately $1954$, $1964$, $1975$, $1986$, $1996$, and $2008$. This shows a cycle that repeats roughly every $11$ years.

Seasonality

When cycles have calendar-related periods of one year or less, we call them seasonality. Seasonal patterns are periodic movements related to specific time periods such as years, months, weeks, or even days.

Seasonal movements tend to be more predictable than other time series features. They occur because of:

• Variations in weather (e.g., ice-cream sales increase in summer)
• Institutional factors (e.g., unemployment increases at the end of the school year)
• Cultural patterns (e.g., retail sales peak before Christmas)

Example: Hotel occupancy rates

Consider the quarterly hotel room occupancy rates in Australia from $2012$ to $2016$:

The regular peaks and troughs occurring at the same time each year clearly indicate seasonality. In this case:

• Demand for accommodation is lowest in the June quarter (winter)
• Demand is highest in the December quarter (summer/holiday season)

Structural change

A structural change occurs when there is a sudden shift in the established pattern of a time series. This is different from gradual trends or regular cycles because it represents an abrupt change at a specific point in time.

Structural changes often result from significant external events or changes in circumstances, such as:

• New legislation or regulations
• Major technological innovations
• Economic crises
• Changes in management or ownership
• Natural disasters

Example: Household electricity use

Consider the monthly electricity use (in kilowatt-hours, kWh) for a rental house over $12$ months:

This plot shows an abrupt change in power usage between June and July. During the first half of the year (January to June), monthly power use averaged around $300$ kWh. From July onwards, it suddenly decreased to approximately $175$ kWh per month.

Outliers

Outliers are individual data values that stand out from the general pattern of the data. They represent unusual observations that don't fit the established pattern of the time series.

Outliers can occur for various reasons:

• Measurement errors
• Recording mistakes
• Unusual one-off events
• Equipment malfunctions
• Extraordinary circumstances

Example: Daily electricity use

The following plot shows daily electricity use for a household over a fortnight (14 days):

For this household, daily electricity use follows a regular pattern, fluctuating around an average of about $10$ kWh per day. However, day $4$ is a clear outlier, with less than $2$ kWh of electricity used.

Irregular (random) fluctuations

Irregular fluctuations represent all the variations in a time series that we cannot reasonably attribute to systematic patterns like trend, cycles, seasonality, structural change, or outliers. These are the random, unpredictable movements that exist in all real-world data.

Key characteristics of irregular fluctuations:

• They are always present in any real-world time series
• They are unpredictable and cannot be forecast
• They have many sources, mostly unknown
• They can obscure underlying patterns

Summary of time series patterns

When analysing any time series, systematically look for these six features:

Remember!