Nonlinear Ehlers filter construction from momentum and distance weights

Coefficients from a chosen statistic

The nonlinear construction forms each coefficient from a chosen statistic of the price series, multiplies those coefficients by the corresponding prices, and outputs the ratio of that numerator to the coefficient sum.

That procedure is the nonlinear Ehlers filter: a weighted-average construction whose coefficients come from a selected statistic, then normalizes the weighted prices by the sum of those coefficients. The coefficient sum is the total of the current lookback of coefficients and is the denominator of the filter.

A five-bar change on median price

A worked example uses a five-bar absolute price change as the coefficient statistic and a default lookback of 15 samples on the midrange of high and low. Median price is that midrange, taken as the average of high and low.

Lookback length is the number of past samples that enter both the coefficient vector and the weighted average. In the worked example that length is 15.

Distance weights in the same shell

The same weighted-average shell accepts a distance coefficient formed by summing squared differences between the current price and each earlier price in the lookback.

A generic implementation treats the coefficient statistic as an input, so replacing an absolute five-bar midrange change with a distance-squared function yields the distance-coefficient variant inside the same procedure.

A spreadsheet layout

A spreadsheet construction first builds the squared-difference series, then forms the filter as the product-sum of price and those coefficients divided by the coefficient sum, copying the row downward once five squared-difference values exist.

The same ratio on other platforms

Independent platform reconstructions keep the same ratio of weighted price to coefficient sum, including a midrange price series, a 15-bar summation window, and a coefficient line that can be an absolute five-bar change or a summed squared-distance term.

Editorial: three uses of one construction

Editorial: TradersWeek reads this construction as a trend filter, an adaptive filter, and a testable momentum strategy, because the same weighted-average shell rebuilds its weights from a chosen statistic at each bar. That reading is editorial and is not a claim of the archive workflow.

A trend filter is a smoother that weights recent price, volume, or breadth observations so the output can be compared with a later out-of-sample path. The nonlinear Ehlers filter is that smoother: it weights recent prices by the current coefficients and returns the normalized ratio.

An adaptive filter is a filter whose coefficients are recomputed from a chosen statistic of the same series, so the weights change as the data change. Forming each coefficient from a chosen statistic, and treating that statistic as an input, is that recomputation.

A momentum strategy is a complete entry, exit, and abstention procedure whose signals come from a momentum-based coefficient rule and a defined holding period. Editorial: the five-bar absolute price change is a momentum-based coefficient rule. Placing that rule inside a defined holding period makes entry, exit, and abstention testable as one procedure.