SPECTRAL DILATION And What To Do About It

John Ehlers [email protected] StockSpotter.com 1 SPECTRAL DILATION And What To Do About It The information provided is for informational pur...
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John Ehlers [email protected]

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SPECTRAL DILATION And What To Do About It

The information provided is for informational purposes only. Trading is risky and is not for everyone. For a full disclosure statement, please go to: http://www.stockspotter.com/In/Legal.aspx

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OUTLINE

John Ehlers [email protected]

• Theoretical Basis of Market Data Structure – Market Data is Fractal – Aliasing Noise Swamps Short Period Data

• Indicator Dynamics – Super Smoother Eliminates Aliasing Noise – Roofing Filter Mitigates Spectral Dilation – Improving Conventional Indicators

• Using Indicators to Predict Market Turning Points

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Drunkards Walk

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• Described in “MESA and Trading Market Cycles” – Drunk steps right or left with each step forward • Random Variable is position • Results in the famous Diffusion Equation • Describes the shape of a plume of smoke (or a trend)

– Drunk steps in the same or opposite direction as the last step with each step forward • Random Variable is momentum • Results in the famous Wave Equation • Describes a meandering river (or a cycle)

• The 2nd Order Partial Differential Equations are nearly identical • Results are that cycles and trends can coexist in a complex mixture

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Swerling Model

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• Peter Swerling statistically described radar echoes – Pulses were noisy over time – due to complex airplane shapes and changes in aspect from the fixed radar site. – Model described as pure noise with memory

• I have synthesized market data as noise with an EMA – Not bad for a simple model

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Hurst Exponent

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• The Hurst Exponent describes the randomness of a data series

“Hurst Exponent and Financial Market Predictability” By Bo Qian and Khaled Rasheed University of Georgia

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Logarithmic Spiral

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Fibonaccians use the Golden Spiral to show the relationship between cycle period and cycle amplitude

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Modeled Market Spectrum

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Signal Amplitude scales as 1/Fα Aliasing noise swamps signal power

• – –



Cycle periods less than 10 bars must be ignored Cycle components whose periods are less than 10 bars must be removed by filtering

Filtering components longer than 10 bars just reduces signal power –

Data contains systemic noise

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Aliasing Noise

power

Spectral Dilation

Noise Intercepts Data

frequency

SMOOTHING FILTERS

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John Ehlers [email protected]

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The real reason to use filters is to remove aliasing noise Only the SuperSmoother Rejects Aliasing Noise 10 Bar EMA

10 Bar SuperSmoother Noise Removed

Only 13 dB Rejection

(10 Bar Period)

(10 Bar Period)

Longer Moving Averages Do Give More Rejection, But – Increase Indicator Lag – Attenuate the Real Signal – Not Just Aliasing Noise

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John Ehlers SuperSmoother Filter Code [email protected]

SuperSmoother Filter © 2013 John F. Ehlers a1 = expvalue(-1.414*3.14159 / 10); b1 = 2*a1*Cosine(1.414*180 / 10); c2 = b1; c3 = -a1*a1; c1 = 1 - c2 - c3; Filt = c1*(Close + Close[1]) / 2 + c2*Filt[1] + c3*Filt[2]; Code Conversion Notes: 1) Filter is tuned to a 10 Bar Cycle (attenuates shorter cycle periods) 2) Arguments of Trig functions are in degrees 3) [N] means value of the variable “N” bars ago

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• •

Spectrum power models as 1/Fα Wave amplitude doubles for each doubling of cycle period – –



Modeled Market Spectrum

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Increases 6 dB per octave This is Spectral Dilation

Indicators must compensate for spectral dilation to get an accurate frequency response

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Aliasing Noise

power

Spectral Dilation

Noise Intercepts Data

frequency

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HighPass Filter

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• HighPass Filters are oscillators (detrenders) because they attenuate low frequency components

A classical oscillator or ordinary HighPass filter does not produce a zero mean Because low frequency spectral dilation components are “leaking” through The one pole HighPass Filter response

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Roofing Filter

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• Comprised of a two pole HighPass Filter and a SuperSmoother

• The Roofing Filter guarantees only the desired frequency components will be passed for analysis • Establishes a true zero mean for swing signals

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Roofing Filter Code

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Roofing Filter © 2013 John F. Ehlers //Two Pole Highpass filter passes cyclic components whose periods are shorter than 48 bars alpha1 = (Cosine(.707*360 / 48) + Sine (.707*360 / 48) - 1) / Cosine(.707*360 / 48); HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close - 2*Close[1] + Close[2]) + 2*(1 - alpha1)*HP[1] - (1 - alpha1)*(1 - alpha1)*HP[2]; //Smooth with a Super Smoother Filter a1 = expvalue(-1.414*3.14159 / 10); b1 = 2*a1*Cosine(1.414*180 / 10); c2 = b1; c3 = -a1*a1; c1 = 1 - c2 - c3; Filt = c1*(HP + HP[1]) / 2 + c2*Filt[1] + c3*Filt[2];

Code Modification Notes: 1) HP Filter is tuned to a 48 Bar Cycle (attenuates longer cycle periods) 2) SuperSmoother is tuned to a 10 Bar Cycle (attenuates shorter cycle periods) 3) Arguments of Trig functions are in degrees 4) [N] means value of the variable “N” bars ago

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John Ehlers Impact of Spectral Dilation [email protected] 14 On Traditional Indicators

• Spectral Dilation has distorted the interpretation of virtually all indicators • Roofing Filter removes Spectral Dilation distortion

Conventional Stochastic

Stochastic preceded by a Roofing Filter

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My Stochastic Code

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My Stochastic © 2013 John F. Ehlers //Highpass filter cyclic components whose periods are shorter than 48 bars alpha1 = (Cosine(.707*360 / 48) + Sine (.707*360 / 48) - 1) / Cosine(.707*360 / 48); HP = (1 - alpha1 / 2)*(1 - alpha1 / 2)*(Close - 2*Close[1] + Close[2]) + 2*(1 - alpha1)*HP[1] - (1 - alpha1)*(1 - alpha1)*HP[2]; //Smooth with a Super Smoother Filter from equation 3-3 a1 = expvalue(-1.414*3.14159 / 10); b1 = 2*a1*Cosine(1.414*180 / 10); c2 = b1; c3 = -a1*a1; c1 = 1 - c2 - c3; Filt = c1*(HP + HP[1]) / 2 + c2*Filt[1] + c3*Filt[2]; HighestC = Filt; LowestC = Filt; For count = 0 to Length - 1 Begin If Filt[count] > HighestC then HighestC = Filt[count]; If Filt[count] < LowestC then LowestC = Filt[count]; End; Stoc = (Filt - LowestC) / (HighestC - LowestC); MyStochastic = c1*(Stoc + Stoc[1]) / 2 + c2*MyStochastic[1] + c3*MyStochastic[2];

Code Modification Notes: 1) HP Filter is tuned to a 48 Bar Cycle (attenuates longer cycle periods) 2) SuperSmoother is tuned to a 10 Bar Cycle (attenuates shorter cycle periods) 3) Arguments of Trig functions are in degrees 4) [N] means value of the variable “N” bars ago

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John Ehlers Conventional Wisdom [email protected] 16 for Confirmation

• Buy when indicator crosses over 20% level • Sell Short when indicator crosses under 80% level

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Waiting For Confirmation

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• This equity curve of using (only) Conventional Wisdom rule using My Stochastic indicator on the last 10 years of S&P Futures data • Monthly cycle is 10 bars up – 10 bars down • Lag = 2 bars Roofing + 2 bars Stochastic + 1 bar after signal + 3 bars confirmation = 8 bars • 8 bars lag into a 10 bar move = consistent loss

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Prediction Anticipates the Cyclic Turning Point

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• Buy when indicator crosses under 20% level • Sell Short when indicator crosses over 80% level

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Anticipate the Turning Point

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• This equity curve of using (only) the Prediction rule using My Stochastic indicator on the last 10 years of S&P Futures data • Monthly cycle is 10 bars up – 10 bars down • Lag = 2 bars Roofing + 2 bars Stochastic + 1 bar after signal - 3 bars confirmation = 2 bars • 2 bars lag into a 10 bar move = consistent winner

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Swing Setup Analyzer

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What You Have Learned

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• Spectral Dilation dominates market structure • Very short cycles are swamped by Aliasing Noise • Only a SuperSmoother can effectively remove aliasing noise • A Roofing filter additionally mitigates the effects of Spectral Dilation • Swing trading signals must anticipate the price turning points – A well constructed oscillator indicates the probability of reversion to the mean – In this sense an indicator can be predictive

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Contact Info

www.StockSpotter.com

John Ehlers [email protected] (805) 927-3065

John Ehlers [email protected]

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