Lead-lag smoothing for weekly trend-channel construction

Noise damping versus filter-lag

Filters used to build a trend channel are judged by two opposing properties: how much they damp short-term noise and how many bars they lag the input series. Filter-lag is that bar count when the smoother is plotted on the latest close. A data-filter is the rule that turns an ordered price series into the smoother used as a center line or as input to a width measure.

Matching simple and exponential lag

A simple moving average of length N lags the input by (N-1)/2 bars, so a seven-bar average is three bars late when plotted on the current close. Exponential smoothing is a recursive average that updates by a fraction alpha of the gap between the latest observation and the prior average. An exponential moving average lags by (1/alpha)-1, and setting alpha equal to 2/(N+1) matches the lag of an N-bar simple average.

A lead-lag filter for the center line

Applying a first-order correction with gamma equal to 2(alpha-1) after an exponential smoother yields a lead-lag filter whose net lag can be cancelled while the same alpha still sets the smoothing. The lead-lag filter is that correction: net delay can be driven toward zero while the original alpha still sets smoothness. Lengthening the lead/lag window increases smoothness but also increases overshoot and slows the response when the input slope changes, as a seven-point filter does less of both than an 11-point filter on a ramp-and-hold series.

Modified-sigma after the shift

Volatility measured against an uncentered seven-week average spikes after breakouts, while shifting that average forward by three weeks narrows the reading but postpones the calculation until later closes exist. A weekly modified-sigma width can be built by averaging five squared deviations from the lead/lag series and then averaging those variances over 13 observations. Modified-sigma is that two-stage width, averaged again so the envelope does not jump with every bar.

Rails, slope, and the trend filter

Upper and lower channel lines are the modified lead/lag series plus or minus j times modified sigma, with the scale factor j set at or above 2.0. Those rails are the price channel: a scaled modified-sigma width added to and subtracted from the low-lag center line. Suggested weekly construction ranges are a lead/lag length of 6 to 8 and a width multiplier of 2.3 to 2.6, with a separate longer exponential average used as the trend filter compared with the lower channel rail. The trend filter is that longer exponential average, used as the baseline that the lower channel rail is compared with to mark a timing hypothesis.

A three-week exponential prefilter that is shifted for zero net lag needs the next close, so the finished channel ends one bar before the latest print and can be extended with the channel slope.