1990issue C081
Lag-aligned MACD from triangular moving averages
A triangular moving average is treated as symmetric, so its lag sits near the midpoint of the lookback window. A MACD-style difference advances each average by its own lag, then a cycle-alignment-check on a known sine series tests whether the oscillator extrema fall in the same sampling periods as the cycle.
- A triangular moving average is a symmetric smoother whose weight peaks at the center of a fixed lookback window and declines toward both ends, so lag sits near the midpoint and equals half the window length minus one.
- Lookback windows of 10 and 20 periods imply lags of 4 and 9 sampling intervals. The MACD reading at time t is the 10-period triangular average at t+4 minus the 20-period triangular average at t+9.
- On a 40-observation sine series of amplitude 10, the first 10-period triangular average is 7.70 after nine observations and the first 20-period average is 8.02 after nineteen observations.
- After those offsets, the highs and lows of the difference line fall in the same sampling periods as the highs and lows of the sine series.
Symmetric windows and lag
A triangular moving average is a symmetric smoother of ordered observations whose weight peaks at the center of a fixed lookback window and declines toward both ends. Because that smoother is treated as symmetric, its lag sits near the midpoint of the window. Lag is the sampling delay between a cycle extreme and the matching extreme in the moving average, and it equals half the window length minus one.
A MACD reading here is a two-average difference used as a timing signal. Each triangular average is read at a date that offsets its own lag before subtraction.
Worked values on a sine series
On a 40-observation sine series of amplitude 10, the first 10-period triangular average is 7.70 after nine observations and the first 20-period average is 8.02 after nineteen observations. Those lookback windows imply lags of 4 and 9 sampling intervals.
Advancing each average before subtraction
The MACD reading at time t is the 10-period triangular average at t+4 minus the 20-period triangular average at t+9, so each series is advanced by its own lag before the difference is taken. When t is 20, that difference is -0.22, equal to -1.49 minus -1.27, taken from the shorter average at period 24 and the longer average at period 29.
Cycle-alignment check
A cycle-alignment-check asks whether an oscillator's highs and lows fall in the same sampling periods as those of a controlled sine series. After the lag offsets above, the highs and lows of the difference line fall in the same sampling periods as the highs and lows of the sine series.
All readings on this track · 8 readings
- 1989Constructing lag-matched triangular moving averages
- 1990Lag-aligned MACD from triangular moving averages
- 1990Constructing triangular moving average weights
- 1990Constructing a lag-aligned triangular MACD
- 1998Paired tests for moving-average lag and smoothness
- 1999Fixed-lag construction of moving-average smoothers
- 2003Weighting recipes inside one moving-average lookback
- 2010Assembling a smoothed percent-b oscillator from Heikin-Ashi and stacked averages