Smoothing a Time Series Using Moving Means (VCE SSCE General Mathematics): Revision Notes

Smoothing a Time Series Using Moving Means

Introduction to smoothing

When analysing time series data, we often encounter various sources of variation including trends, cycles, seasonality, structural changes, outliers, and irregular fluctuations. These variations, particularly irregular fluctuations and seasonality, can make it difficult to identify underlying trends in the data. Smoothing is a statistical technique used to overcome this problem by reducing the impact of these short-term fluctuations.

Moving mean smoothing is a method that involves replacing individual data points in a time series with the average (mean) of that data point and some of its neighbouring points. This process helps to "smooth out" random variations and makes trends easier to see. The simplest approach is to smooth over a small odd number of data points, such as three or five, though any number of points can be used depending on the needs of your analysis.

The three-moving mean

The three-moving mean is the simplest form of moving mean smoothing. To apply this method, you replace each data value with the average of three consecutive values: the data point itself, plus the value immediately before it and the value immediately after it.

Formula:

$\text{smoothed } y_2 = \frac{y_1 + y_2 + y_3}{3}$

where $y_1$, $y_2$, and $y_3$ are three sequential data values in the time series.

Worked example: Three-moving mean

Let's look at temperature data recorded at a weather station at 9:00 a.m. each day for a week:

The five-moving mean

The five-moving mean works on the same principle as the three-moving mean, but uses five consecutive values instead. You replace each data value with the average of five values: the data point itself, plus the two values immediately before it and the two values immediately after it.

Formula:

$\text{smoothed } y_3 = \frac{y_1 + y_2 + y_3 + y_4 + y_5}{5}$

where $y_1$, $y_2$, $y_3$, $y_4$, $y_5$ are five sequential data values in the time series.

Worked example: Five-moving mean

Using the same temperature data from the previous example:

Smoothing a complete time series

When smoothing an entire time series, you apply the moving mean calculation to every possible data point. The table below demonstrates how to calculate both three-moving mean and five-moving mean values for monthly birth data at a country hospital:

Notice how the calculations work:

• For the three-moving mean, we lose the first and last data points (January and December)
• For the five-moving mean, we lose the first two and last two data points (January, February, November, and December)
• Each smoothed value is calculated by taking the mean of the specified number of consecutive values

Comparing smoothing methods

The graph below shows the original (raw) data alongside both the three-moving mean and five-moving mean smoothed data:

From this visualization, we can observe two important points:

Moving mean smoothing with centring

When we smooth using an even number of data points (such as 2, 4, or 6), we encounter a problem: the centre of the set of points doesn't align with any actual time point in our original series. Instead, the calculated mean falls between two time points. To solve this problem, we use a technique called centring.

Centring involves taking a two-moving mean of the already smoothed values. This extra step ensures that our smoothed values line up properly with the original data points.

Two-moving mean with centring

Let's understand the centring process using a simple example with three days of temperature data:

When we calculate consecutive two-moving means for this data, we get:

• Mean of Monday and Tuesday: $(18.1 + 24.8) \div 2 = 21.45$
• Mean of Tuesday and Wednesday: $(24.8 + 26.4) \div 2 = 25.60$

The problem is that these means fall between the days rather than on specific days. The diagram below illustrates this:

To centre these values on Tuesday, we take the average of the two means:

$\text{Centred mean} = \frac{21.45 + 25.60}{2} = 23.525$

Now we have a smoothed value that properly aligns with Tuesday.

Worked example: Two-moving mean with centring

Using the complete week of temperature data:

Four-moving mean with centring

The process for smoothing with larger even numbers follows the same principles as two-moving mean smoothing, but uses larger groups of values.

Six-moving mean with centring

For six-moving mean smoothing with centring, follow the same process but use seven data values.

Smoothing a complete time series with centring

When applying four-moving mean smoothing with centring to an entire time series, it's helpful to organize your work in a table format. This keeps track of all the intermediate calculations. Here's an example using monthly birth data from a country hospital:

The table shows:

• Column 1: The month
• Column 2: Number of births (raw data)
• Column 3: Four-moving means (averaging groups of four consecutive months)
• Column 4: Centred four-moving means (averaging consecutive pairs from column 3)

Exam tips

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