The Moving Average as a Filter

Press Germany, 7 days. These reports sag at the start of each week and pile up later: Mondays average 60.1% of the week around them, Thursdays 130.7%. In the 7-day average the weekday swing is 0.8 points. Every window of seven consecutive days holds exactly one Monday, one Tuesday and so on to Sunday, so any pattern that repeats weekly adds the same amount to every window, whatever shape it has. On the lower chart, the gain sin(7πf)/(7 sin πf) is exactly zero at 1/7, 2/7 and 3/7 cycles a day, the weekly cycle and the only two harmonics a series of daily values can carry. That 0.8 is not zero because the real pattern does not repeat exactly; press The weekly pattern alone and it is 0.0. Holidays break the repeat outright. Christmas Day reads 0 and the next day 39,988, the most of any day in the series, so on 22 December the centred average drops to 21,025, its window ending on the zero, and a day later, with the 39,988 inside, it is back at 22,266. Now change M by one day. Eight days hold one weekday twice and keep an eighth of the swing, 8.8 points of 70.7 in the synthetic week; six days leave one weekday out and keep a sixth, 11.8. Fourteen and twenty-one delete it again. A trailing average reports the same numbers on a later day, so it deletes the weekly cycle just as well.

Press A 5-day wave: a cosine with a period of 5 days, through the same 7-day average. Smoothing is supposed to shrink a fast wiggle, and the dashed ghost is that expectation, the wave smaller but upright. The average comes out at −0.2311 of the wave: smaller, and inverted, with its peaks on the days the input has troughs. The reason fits in a window. Any five consecutive days hold one whole cycle, which sums to zero, so a 7-day window is left with its two end days, 3 days either side of the middle. Three days is six tenths of a cycle, and the two days that far either side of any point add up to 2 cos(6π/5) = −1.618 times the wave's value there. Divided by 7, that is −0.2311. On the lower chart the gain is negative for every period between 3.5 and 7 days, most negative at −0.2330 near 4.86 days, positive again between 2.33 and 3.5. Press Day on, day off: the fastest wiggle a daily series can hold gets through at −0.1429, a seventh of its size. Steven W. Smith's DSP textbook calls the moving average "an exceptionally good smoothing filter" and "an exceptionally bad low-pass filter" in one sentence, and these lobes are why.

Press Trailing, as dashboards do. A dashboard cannot average days that have not happened, so it puts each 7-day average on the last day of its window. That is the same seven numbers and the same mean, reported 3 days after the window's middle; the dashed line is the centred average, and the solid line is that curve slid right. The highest 7-day average in these reports, 25,851 a day, belongs to the week of 18 to 24 December. Centred, it sits on 21 December. Trailing, it sits on the 24th. Neither placement is free. The centred average is on time and stops 3 days before the data does, because its last windows are not complete yet. The trailing average reaches today and describes a week whose middle was 3 days ago; on a straight-line trend it is behind by exactly 3 days of change. The gain is untouched, so the weekly cycle still vanishes: only the timing moves, by the same 3 days at every frequency. For an M-day window the delay is (M − 1)/2 days, so a 14-day trailing average is 6.5 days behind.