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1984issue C031-4

Matching average age across price filters

Specify a trend filter by the average age you will allow in the average, then build that same delay as an equal-weight moving average or as exponential smoothing. The two constructions stay comparable instead of competing as branded indicators.

  • A single daily price sample, such as a close, discards information and can distort the underlying signal through recording mistakes, spillover from other instruments, rumors, political events, and amplification of small intraday moves by exchange intermediaries.
  • Filtering modifies a daily price stream to strip noise and recover directional content. The hard design choice is how much noise can be removed without destroying tradable movement.
  • The average age of a simple linear weighted moving average is (N-1)/2. Equating that vintage with single exponential smoothing gives a smoothing factor of 2/(N+1).
  • Equal-weight linear averages sit halfway back through the window and trail current price. Exponential weights reduce the influence of older prints, update recursively, and are said to plot with less lag.
Entries in this reading3 entries
Current trackExponential smoothing183 readingsTrend filter184 readingsMoving average597 readings

Daily prices need a filter

A single daily price sample, such as a close, discards information and can distort the underlying signal. The distortions named in the archive workflow are recording mistakes, spillover from other instruments, rumors, political events, and amplification of small intraday moves by exchange intermediaries.

Filtering is the modification of a daily price stream to strip noise and recover directional content. The hard design choice is how much noise can be removed without destroying tradable movement. A trend filter, in this workflow, is any averaging construction used to strip sampling noise while leaving directional movement intact enough to study.

Average age puts both constructions on one delay scale

The average age of data in a simple linear weighted moving average equals (N-1)/2, where N is the number of past days averaged. The average age of single exponential smoothing equals (1-a1)/a1. Setting those two vintages equal defines an equivalence between the weighting schemes.

Under that equivalence the single smoothing factor can be written as 2/(N+1) in terms of the familiar linear lookback N. The smoothing factor is the control on how many past observations, such as closing prices, still affect the forecast or average.

Equal-weight windows trail current price

Equal-weight linear averages are logically centered halfway back through the window and therefore trail current price. Exponential weights reduce the influence of older prints, update recursively, and are said to plot with less lag.

A well-tuned simple linear moving average is presented as often matching a much more elaborate filter. The complicated construction is sometimes only slightly better at separating noise.

Higher-order smoothing and a weekly visual filter

Higher-order recursive exponential smoothing is introduced in the same workflow. Triple, or order 3, weighting uses the prior three prices. The average-age relation is extended so a p-th order factor can be expressed from the same N used for a linear moving average.

Inspecting weekly ranges instead of daily ranges, sometimes marking only the Friday close, is treated as a simple visual weekly filter on otherwise hectic daily trading.

Smoothing factors that match an N-day average's age

Fix the lag budget first: the same average age as an N-day equal-weight window requires a smaller exponential α as N grows, and a still smaller α at each higher smoothing order. Single, double and triple factors are the exact values from Hutson's published average-age table, not a redraw of the page.
Fix the lag budget first: the same average age as an N-day equal-weight window requires a smaller exponential α as N grows, and a still smaller α at each higher smoothing order. Single, double and triple factors are the exact values from Hutson's published average-age table, not a redraw of the page.

Hutson equates average age K=(N−1)/2 of the equal-weight window with the lag of each exponential order and solves αp=1−(1−2/(N+1))^(1/p). Column heads use N+1; the formula printed above the table uses N−1.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
1 of 41 in the Exponential smoothing track
19841-3 pp.Next on Exponential smoothingResidual-impact construction of exponential smoothersSimple moving averages and exponentially smoothed moving averages are identified as the two calculation methods most often found among widely used trading systems.
All readings on this track · 41 readings
  1. 1984Matching average age across price filters
  2. 1984Residual-impact construction of exponential smoothers
  3. 1988Constructing simple and exponential moving averages as specified filters
  4. 1989Evaluating smoothed free reserves and funds-discount spreads
  5. 1990Integer construction of moving averages and exponential smoothing
  6. 1990Constructing incremental moving average filters
  7. 1991Constructing a double-smoothed stochastic oscillator
  8. 1991Constructing an exponential average from a smoothing weight
  9. 1991Constructing a double-smoothed momentum oscillator family
  10. 1991Competing certainty rules from exponential-average conditions
  11. 1991Hybrid construction of a mechanical timing system from smoothed signals
  12. 1991Constructing an exponential smoother from a moving-average length
  13. 1991Equivalent relative-strength-index constructions from a ratio and from smoothed momentum
  14. 1991Constructing the true strength index from double-smoothed momentum
  15. 1991True strength index from signed, double-smoothed momentum
  16. 1992Constructing exponential averages from simple lookbacks
  17. 1992Constructing exponential averages by the smoothing constant, startup length and lag
  18. 1992Nested range smoothing and windowed high-low damping
  19. 1992A weekly equity stance from the discount-bill spread
  20. 1993Double-smoothed stochastics momentum index construction
  21. 1993Constructing the true strength index with double exponential smoothing
  22. 1994Polarized fractal efficiency with a shared exponential smoother
  23. 1994Five-interval exponential smoothing of ten-interval signed path-efficiency
  24. 1994Hand-ranked price-volume series with stacked smoothing and stochastic
  25. 1995Range expansion filter on multi-horizon averages
  26. 1995The exponential smoothing constant from averaging lag
  27. 1997Construct a nested-average rainbow band and oscillator
  28. 1998Filter stacks that combine a moving average with exponential smoothing
  29. 2002Smoothed relative-strength construction from wilder-averaging and a price prefilter
  30. 2003Reverse-engineered RSI price targets
  31. 2004Piecewise trend quality from moving-average reversals
  32. 2004Nested smoothing recipes versus unaudited oscillator experiments
  33. 2008Weight construction beyond the N-day label
  34. 2010A consensus filter for objective RSI and smoothing construction
  35. 2012Sentiment-zone oscillator from signed closes
  36. 2012Classic RSI one-over-N smoothing versus the standard exponential average
  37. 2013Why exponential smoothing uses two over n plus one
  38. 2015Constructing ratio indicators with moving-average and exponential-smoothing weights
  39. 2015Slow relative-strength index from an exponentially smoothed price
  40. 2017Rebuilding the relative strength index as open-scale buy-sell pressure
  41. 2017Comparing Wilder and exponential smoothing inside a relative-strength index
By fieldIndicators and filters874 readingsBy year198420 readings
All 183 readings tagged Exponential smoothing
Also on Exponential smoothing5 readings
1995Range expansion filter on multi-horizon averages1997Construct a nested-average rainbow band and oscillator2004Piecewise trend quality from moving-average reversals2012Sentiment-zone oscillator from signed closes1989Alpha-beta lag parameters versus moving-average windows