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1994issue C011-2

Regression-seeded nested exponential price filter

A two-layer exponential filter can open both averages from one linear fit on an initial block of weekly closes. After that opening, the nested second average updates from the first average rather than from price, and the dedicated filter series is twice the first average minus the second.

  • A linear fit on an initial block of closes can supply the intercept and slope that open both exponential averages.
  • The nested second average updates from the current first average, not from the raw close.
  • At each step the filter equals twice the first average minus the second average, and that series is written as a dedicated output.
  • If the regression-seeds are omitted, both averages may instead be opened with the first close.
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Current trackLinear regression112 readingsExponential smoothing117 readings

Two nested averages and one filter series

DEMA1 is a constructed price filter equal to twice a single exponential average minus a second exponential average of that first average. The two-layer construction first forms an exponential-average of the price series, then forms a nested-second-average of that first average with the same smoothing-weight.

The worked construction uses weekly closing prices as the ordered input series. The filter series is written as a dedicated output, separate from the two nested averages used to form it.

Regression-seeds for both layers

A linear fit on an initial block of closes supplies the intercept and slope used to open both averages. A regression-seed is an opening level for an exponential average computed from that intercept and slope rather than from the first observation.

The opening value of the first average is the fitted intercept minus ((1-0.074)/0.074) times the fitted slope. The opening value of the second average uses the same intercept and slope but subtracts twice that ((1-0.074)/0.074) multiple of the slope.

Updates after the opening step

After initialization, the first average updates as 0.074 times the new close plus (1-0.074) times its previous value. After initialization, the second average updates as 0.074 times the current first average plus (1-0.074) times its previous value.

An exponential-average is a recursive level that blends a new input with the previous level using a fixed weight. The nested-second-average takes the first exponential average as its input, not the raw price. In this construction the smoothing-weight is 0.074, the fixed fraction applied to the newest input at each update.

The dedicated filter series

The filter value at each step is twice the first average minus the second average. That combination is DEMA1, written as its own series rather than left inside the two nested averages.

Both averages may instead be opened with the first close when the linear-fit seeds are omitted.

Weekly NASDAQ close against regression-seeded and close-seeded DEMA1

A trader should see the two DEMA1 openings split after the 1987 crash—the regression-seeded filter undershoots the close-seeded one—then lock together near mid-1988, while both lag the weekly NASDAQ close on the 1990–91 V-bottom and track it more tightly in the 1992–93 advance. The coordinates were read from the plotted curves; the sidebar gives formulas, not a price table.
A trader should see the two DEMA1 openings split after the 1987 crash—the regression-seeded filter undershoots the close-seeded one—then lock together near mid-1988, while both lag the weekly NASDAQ close on the 1990–91 V-bottom and track it more tightly in the 1992–93 advance. The coordinates were read from the plotted curves; the sidebar gives formulas, not a price table.NASDAQ · Weekly · 1987-08-14T00:00:00.000Z to 1993-08-27T00:00:00.000Z

The source fixes the EMA constant at 0.074 and opens both averages from LINEST on the first 26 weekly closes (D2:D27). The close-seeded series instead copies the first close into both averages. Raster reading is only good to about five index points.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
16 of 43 in the Linear regression track
19941-3 pp.Next on Linear regressionConstructing the double exponential average from lag cancellationOn a noise-free exact linear series that has already reached steady state, twice the first exponential average minus the second recovers the original series at every time index.
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