Moving median smoothing

Calculating moving medians

A moving median uses an odd number of consecutive responses. Order the responses, select the middle one and place it at the central time of the window

1. Use a complete odd window Select the stated odd number of consecutive responses around the target period \(\displaystyle M_3=\operatorname{median}(x_1,x_2,x_3,x_4,x_5)\)
2. Order the response values The median is the middle response after the values in the window are sorted \(\displaystyle \operatorname{median}(18,42,27,35,21)=\operatorname{median}(18,21,27,35,42)=27\)
3. Keep the central time The selected response is placed at the time coordinate in the centre of the window \(\displaystyle M_3=27\text{ is placed at period }3\)
4. Move through complete windows Move the window one period at a time and leave unsupported end periods without a median \(\displaystyle N=9,\ k=5\Rightarrow N-k+1=5\text{ supported values}\)

Worked example

A council depot records material collected over five consecutive weeks as 72, 55, 89, 61 and 78 tonnes. Find the 5-term moving median for Week 3

1. \(\displaystyle 55<61<72<78<89\)Order the five responses from least to greatest
2. \(\displaystyle \operatorname{median}(55,61,72,78,89)=72\)The third ordered response is the middle response
3. \(\displaystyle M_3=72\text{ tonnes}\)Keep the central time of the window, which is Week 3

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