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2017issue C098-12

Causal reverse exponential average for cycle and trend

A one-sided exponential average can run on a live bar, but its group-delay can distort mixed-frequency series. A true backward pass would cancel that phase and cannot be evaluated at the right edge. The archive rebuilds a delayed reverse cascade so one coefficient can steer the same residual toward a trend-filter or a dominant-cycle reading.

  • Exponential-smoothing blends the newest observation with the prior average using complementary weights that sum to one, so a constant input settles at the same level.
  • Group-delay varies by frequency, so the one-sided average imposes a nonlinear phase map that can distort mixed-frequency market series.
  • A backward pass would cancel that phase and double the smoothing, but it is noncausal and cannot be evaluated at the live right edge.
  • A delayed causal-reverse-path cascade makes the reverse stage evaluable, and one coefficient steers the residual toward trend-filter or dominant-cycle structure.
Entries in this reading3 entries
Exponential smoothing117 readingsCurrent trackDominant cycle detection119 readingsTrend filter137 readings

The one-sided average on a live bar

An exponential average updates as a weighted mix of the newest observation and the previous average. The weights are complementary and sum to one, so a constant input settles at the same level.

That recursion is exponential-smoothing. It blends the newest observation with the prior output and remains one-sided.

Why phase can distort the waveform

Group-delay varies across frequencies. The one-sided average therefore imposes a nonlinear phase map that can distort the waveform of mixed-frequency market series.

The backward pass is not a live-bar step

Running the same average backward after the full series is available cancels that nonlinear phase and doubles the smoothing. The procedure is noncausal. It cannot be evaluated at the live right edge.

A delayed reverse cascade

Expanding the delay-operator ratio of the exponential recurrence produces an infinite decaying coefficient series. Once later terms become negligible, that series can be cut to a finite-impulse form.

Time-reversing that finite-impulse path and inserting enough delay makes the reverse stage causal. That truncated, delayed reconstruction is the causal-reverse-path. The archive gives a cascade of eight successive modules as sufficient for ordinary market-data error.

One coefficient steers the residual

The published residual subtracts a scaled forward-and-reverse exponential response from a standard exponential average. The subtraction isolates the frequency-phase distortion component.

A single exponential coefficient steers that residual. A value of 0.05 shifts it toward trend structure, the longer-horizon emphasis of a trend-filter. A value of 0.3 shifts it toward cycle structure, the shorter-horizon emphasis of a dominant-cycle reading.

What the finished construction does

The finished construction is causal. It applies extra high-frequency smoothing to limit aliased sampled components. It also applies a low-frequency difference that declines at 6 dB per octave to reduce spectral-dilation.

Reverse EMA residual on daily SPY, July 2016–June 2017

With alpha fixed at 0.1 the residual marks SPY turning points with little delay: the mid-2016 dip, the November 2016 rally, and the February 2017 crest each print as a clear swing around the plotted zero line. Points were read from the published TradeStation pane, so amplitudes are approximate.
With alpha fixed at 0.1 the residual marks SPY turning points with little delay: the mid-2016 dip, the November 2016 rally, and the February 2017 crest each print as a clear swing around the plotted zero line. Points were read from the published TradeStation pane, so amplitudes are approximate.SPY · daily · 2016-07-01T00:00:00.000Z to 2017-06-30T00:00:00.000Z

The published figure uses a single alpha of 0.1. The same residual leans toward trend at 0.05 and toward the dominant cycle at 0.3. A magazine raster cannot carry more than about one decimal.

Educational research material, not investment advice. Historical source context does not establish present-day performance.
30 of 31 in the Dominant cycle detection track
20206-9 pp.Next on Dominant cycle detectionConstructing a cycle-plus-trend oscillator from a one-wavelength chordThe construction treats the series as a cycle sitting on a trend and draws that trend as the straight line from the current close to the close one assumed cycle period earlier.
All readings on this track · 31 readings
  1. 1982Cycle phase windows for chart signal filters
  2. 1987Constructing a cycle-scaled trend oscillator
  3. 1987Constructing a dominant-cycle grid from marked lows
  4. 1988Cycle lead from staggered exponential averages
  5. 1988Auditing the forty-month stock-price cycle
  6. 1989When long-wave dominant cycles cannot be disproved
  7. 1991Half-cycle average plot shift versus cycle attenuation
  8. 1991Half-cycle average contact as an amplitude-ratio test
  9. 1993Building a restoring-pull indicator from cycle frequency and volume
  10. 1995Regime filters for a dominant long wave
  11. 1995A cycle-tuned lead filter from bounded oscillators
  12. 1998Testable cycle rules instead of fear and greed
  13. 1999Nested Euro cycle timing as one checkable procedure
  14. 2002Constructing an instantaneous trendline from a dominant cycle
  15. 2002Half-cycle center of gravity oscillator from moving-average balance
  16. 2004Testing a locked forty-week cycle with a hold-or-sit-out rule
  17. 2005Nested timing bands for dominant-cycle confirmation
  18. 2005Dominant-cycle baselines versus policy-news narratives
  19. 2006Pairing a dominant-cycle horizon with trend and oscillators
  20. 2006A dominant-cycle split into a trend filter and residual Relative Strength Index
  21. 2007Construct a momentum difference from the dominant cycle
  22. 2007Naive dominant-cycle rules fail without crowd tests
  23. 2012Constructing a dominant-cycle forecast as a timing window
  24. 2012Open-parameter construction of dominant-cycle baselines
  25. 2013Using a second-term election to check a predeclared dominant-cycle forecast
  26. 2014Constructing a dominant-cycle forecast baseline
  27. 2014Quotient transform as an early-onset trend filter
  28. 2014Construct a trough-to-trough cycle map with the Detrended Price Oscillator
  29. 2015Dominant-cycle alignment before an earnings catalyst
  30. 2017Causal reverse exponential average for cycle and trend
  31. 2020Constructing a cycle-plus-trend oscillator from a one-wavelength chord
By fieldIndicators and filters682 readingsBy year2017100 readings
All 119 readings tagged Dominant cycle detection
Also on Dominant cycle detection5 readings
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