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1994issue C011-3

Constructing the double exponential average from lag cancellation

On a noise-free linear series that has reached steady state, twice the first exponential average minus the second recovers the original series at every time index. Formal seeds from a linear-regression intercept and slope make that match exact; a first-observation seed does not.

  • On 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.
  • On the worked linear series both exponential averages are straight lines parallel to the data, each delayed by the same lag of 5 sampling intervals.
  • Formal seeds from a linear-regression intercept and slope make the constructed series coincide with the linear data, while a first-observation seed does not.
  • A short sample paired with a long window makes a first-value seed less faithful, so the construction should not be judged over the first two windows unless that early segment is the object of study.
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Current trackLinear regression112 readingsExponential smoothing117 readingsMoving average449 readings

The identity on a linear series

The double exponential average is the constructed series equal to twice the first exponential average minus a second exponential average of that first average. Exponential smoothing is a recursive smoother that blends the newest observation with the previous smoothed value through a constant alpha. On a noise-free exact linear series that has already reached steady state, twice the first exponential average minus the second exponential average recovers the original series at every time index.

The worked construction

The worked linear construction uses intercept 25, slope 5, window 11, smoothing constant 0.167, and a single-average lag of 5 sampling intervals. The smoothing constant is alpha, defined as two divided by one plus the averaging window. Lag is the time shift of an exponential average on a linear series, equal to one minus alpha over alpha, or to half of one less than the window.

On that linear series both exponential averages are straight lines parallel to the data. The first is delayed by one lag and the second is delayed by another lag of the same size.

How the two lags cancel

A moving average is a lookback smoother whose window sets both responsiveness and the lag that a second smoother is meant to offset. Under the linear model the first exponential average equals the series minus slope times lag and the second equals the first minus slope times lag. That pairing rearranges to the double-exponential identity.

Editorial: read the second smoother as an estimate of the first smoother's displacement on a locally linear series. Putting that estimated displacement back is the construction itself, not a later test of the filter.

Initialization and seeds

Initialization is the starting values chosen for the two recursive averages before they reach a steady state. Formal seeds are taken from a linear-regression intercept and slope on a subset of the sample. Linear regression is a straight-line fit whose intercept and slope supply formal seed values for the two nested exponential averages. Those seeds make the constructed series coincide with the linear data, while a first-observation seed does not.

With a first-observation seed the first exponential average still lags by 5 intervals and the second by 10. The double exponential average lags by about one interval at time equal to the window and shows no significant lag after two windows.

At two windows the linear series equals 135 and the first-value-seeded construction equals 133.34, a -1.2 percent reconstruction error. At the sample end the figures are 230 and 229.903.

The first two windows

A short sample paired with a long window makes a first-value seed less faithful, so the construction should not be judged over the first two windows unless that early segment is the object of study.

EMA1 and EMA2 lag a noise-free linear series by one and two windows

On this noise-free ramp the first exponential average sits a constant five bars behind the observed series and the second sits five bars behind the first. Twice EMA1 minus EMA2 therefore rebuilds y at every index once the averages are in steady state. Every level is taken from the sidebar model y = 25 + 5t with window 11 and the formal intercept-slope seeds, not from tracing the printed figure.
On this noise-free ramp the first exponential average sits a constant five bars behind the observed series and the second sits five bars behind the first. Twice EMA1 minus EMA2 therefore rebuilds y at every index once the averages are in steady state. Every level is taken from the sidebar model y = 25 + 5t with window 11 and the formal intercept-slope seeds, not from tracing the printed figure.Synthetic linear series y = 25 + 5t · Index 0 to 40 (N = 41)

The sidebar assumes no noise and i > w. Window w = 11 so α = 2/(w+1) = 1/6 and lag L = (w−1)/2 = 5. Formal seeds are EMA1 at t = 0 ≈ 0 and EMA2 at t = 0 = −25, from intercept a = 25 and slope b = 5. Under those seeds EMA1(t) = 5t and EMA2(t) = 5t − 25 exactly, so DEMA1 = 2·EMA1 − EMA2 coincides with y.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
17 of 43 in the Linear regression track
19941-6 pp.Next on Linear regressionEvaluating money supply as a linear leading-index baselineA reconstruction test first places the leading composite and money supply on a common z-score scale, then asks how much of the published index a one-variable linear baseline already recovers.
All readings on this track · 43 readings
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  5. 1990Constructing least-squares polynomial smoothers
  6. 1991Endpoint growth rates versus linear-regression consistency
  7. 1991Out-of-sample checks for linear growth fits
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  9. 1991Quadratic trend, residual oscillator, and a secondary cycle calendar
  10. 1991Time-origin offset and residual-price divergence on a quadratic least-squares fit
  11. 1991A least-squares trendline from ordered prices
  12. 1992Constructing log-linear growth and reliability screens
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  14. 1992Next-session high, low, and close from rolling linear regression
  15. 1993Auditing an index price-earnings multiple with short-rate regression
  16. 1994Regression-seeded nested exponential price filter
  17. 1994Constructing the double exponential average from lag cancellation
  18. 1994Evaluating money supply as a linear leading-index baseline
  19. 1995Constructing least-squares trend channels
  20. 1995Linear baseline holdout checks for annual bill-rate forecasts
  21. 1995Projection bands from high and low regression slopes
  22. 1995Evaluating a least-squares end-point moving average on a known test series
  23. 1996Constructing an endpoint moving average from a least-squares line
  24. 1996Scoring equity path consistency with a k-ratio overlay
  25. 1996Evaluating month-end yield gaps for equity regimes
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  27. 1998Evaluating linear regression baselines for index valuation
  28. 1998R-squared as a two-state trend filter from a price-time fit
  29. 2000Second-order moving-average lag correction
  30. 2002Price regression line versus beta for index tracking
  31. 2003Regression slope with an r-squared trend confidence gate
  32. 2003Constructing finite-volume-element divergence with slope comparison
  33. 2004Building a daily score from regression, retracement, and volume
  34. 2004Constructing least-squares trendlines from ordered prices
  35. 2007Rectangle breakout targets beyond height
  36. 2007Confirming a price trend with regression slope and r-squared
  37. 2008A linear-regression angle assembled as one trend filter
  38. 2010A two-state swing machine from four running extremes
  39. 2016Score oil-complex tightness before divergence or regression
  40. 2017Nikkei-yen intermarket divergence as a regime case study
  41. 2017Constructing Calmar ratio and linear regression baselines
  42. 2019Pair-trade layer construction versus average-spread management
  43. 2020A convolution slope built from nested linear regression
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