1. MEDIAN SMOOTHING

2.  Smoothing involves replacing the original time series with another one where most of the variation has been removed, in order to see if there is a secular trend. There are three basic smoothing techniques.  (a) Moving-average smoothing works best with an odd number of points. For a 3-point smooth, one point is lost at either end of the time series.  (b) Moving-average smoothing with an even number of points is a 2-step process. First perform a 4-point moving average, then centre by averaging pairs of the 4-point smooth. For a 4-point centred smooth, two points are lost at each end of the time series.

3. Timey-value 4-point average (smoothed value)4-point average after centring CalculationResultCalculationResult 2003 6 200410 (6 + 10 + 14 + 12) ÷ 410.5 200514 (10.5 + 11.75) ÷ 211.125 (10 + 14 + 12 + 11) ÷ 411.75 200612 (11.75 + 13) ÷ 212.375 (14 + 12 + 11 + 15) ÷ 413 200711 (13 + 13.5) ÷ 213.25 (12 + 11 + 15 + 16) ÷ 413.5 200815 200916

4.  An alternative technique using the median of each group.  Use odd-point median smoothing as it doesn’t require any calculations.  Often id can be done directly on the graph of a time series.  Generally, the effect of median smoothing is to remove some random fluctuations. It performs poorly on cyclical or seasonal fluctuations-unless the size of the range being used (3, 5, 7..points) is chosen carefully.

5.  For a 3-point medians, we look at each group of three points and choose the middle value.  Progress through the table one point at a time. As with other methods, points will be lost at the beginning and end of the table.

6.  Perform a 3-point median smoothing on the data in the table below. The table shows the cost of an airline ticket between Perth and Melbourne over an 8-month period. Construct a time-series plot from the data. Time12345678 Cost ($)340350320340300330350310

7.  Perform a 3-point median smoothing on the graph of a time series below. The 1st data points are 12, 18, 16 — so median = 16. The 2nd data points are 18, 16, 8 — so median = 16. The 3rd data points are 16, 8, 12 — so median = 12. The 4th data points are 8, 12, 16 — so median = 12. The 5th data points are 12, 16, 12 — so median = 12 The 6th data points are 16, 12, 8 — so median = 12. The 7th data points are 12, 8, 10 — so median = 10. The 8th data points are 8, 10, 14 — so median = 10.

8.  Exercise 4E pg 178 Q’s 1-6.