Select and compare smoothing methods practice questions | General Mathematics (Year 12, VCE Units 3 & 4)

Selecting a smoothing method and window

Choose the method from the type of variation and choose the window from the feature being suppressed. Wider windows smooth more movement but lose more endpoint positions

\bar{x}_t=\frac{x_{t-1}+x_t+x_{t+1}}{3}

Match the method to the variation Use a median to resist one isolated extreme and a mean to average distributed short-term movement \(\displaystyle \text{isolated extreme}\Rightarrow\text{moving median}\)
Order values for a median A moving median is the middle response after the window values are ordered \(\displaystyle \operatorname{median}(8,11,40)=11\)
Match a repeated period A window used to reduce a repeating pattern should span one complete repeated period \(\displaystyle \text{period}=4\Rightarrow\text{use a 4-term window}\)
Count retained values An odd k-term smoother retains N − k + 1 values from N observations \(\displaystyle N=12,\quad k=5\Rightarrow12-5+1=8\)

Worked example

Weekly visitor counts are 42, 45, 100, 47 and 44. Use a moving median to reduce the effect of the isolated value and find the smoothed value at week 3

x_1=42,\quad x_2=45,\quad x_3=100,\quad x_4=47,\quad x_5=44

\(\displaystyle 42,45,100,47,44\to42,44,45,47,100\)Order the five responses from least to greatest before selecting the median
\(\displaystyle \operatorname{median}(42,44,45,47,100)=45\)The middle ordered response is 45
\(\displaystyle m_3=45\text{ visits}\)The median from the window is placed at its central time, week 3

Common mistakes

Using a mean for an isolated extreme

✕\(\displaystyle \operatorname{mean}(12,14,60)=\frac{86}{3}\)✓\(\displaystyle \operatorname{median}(12,14,60)=14\)

A median is less affected by one isolated extreme; a mean is pulled towards it

Thinking a wider window keeps more values

✕\(\displaystyle N=12,\quad7\text{-term window}:12-7+1=6\text{ retained}\)✓\(\displaystyle N=12,\quad5\text{-term window}:12-5+1=8\text{ retained}\)

A wider window removes more positions at the two ends, so it retains fewer smoothed values

Ignoring the repeated period

✕\(\displaystyle \text{4-period repetition}\Rightarrow3\text{-term window}\)✓\(\displaystyle \text{4-period repetition}\Rightarrow4\text{-term window}\)

A window matching the repeated period spans one complete repetition for the mean to reduce

Difficulty levels

introA short stated purpose is matched with a mean or median and a retained-value count

standardTwo window widths of the same method are compared on one series

challengeA method is chosen from an isolated extreme or a window is matched to a repeated period

examA compact choice tests the best method, width and reason for the stated purpose

Questions about this topic

What is a moving mean?+

It is the average of each consecutive window of observations, placed at the central period when the window is odd

What is a moving median?+

It is the middle response after the values in an odd-sized consecutive window have been ordered

Why can a wider window hide a short change?+

It combines more periods into each smoothed value, so a short level change is spread across the wider window and more endpoint positions are lost

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