The Finance Book Club

1. A band-pass filter is both a detrender and a smoother combined
2. Attenuation of out-of-band frequencies is superior to high-pass and low-pass filters because the ‘rejection’ is scaled to the bandwidth of the filter
3. A band-pass filter tuned to a ‘consistent dominant cycle’ (a unicorn!) has no lag
4. A narrow pass-band filter is slower to react to changes in the input data. A wider pass-band filter adapts more quickly to changes in the input data.
5. The dominant cycle period of the data can be estimated by counting the number of bars between zero crossings of the band-pass filter output.

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1. White noise has no correlation in time
2. Random walk (Brownian motion) has no correlation between increments (of what?)
3. Brownian walks can be generated
   1. With a function whose spectral density (defined later) is 1/F α where
      1. F is frequency
      2. α is a power law
      3. α = 1 signifies a long memory

      
      

   2. 1/F noise is synonymous with long-range dependence

   
   

4. The ubiquity of 1/F α noise is one of the oldest puzzles of contemporary physics and science in general.

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1. Economic events some distance in the past continue to have an influence on current prices (proved by Mandelbrot)
2. The Hurst coefficient is one way to attempt to get a handle on the slope of the power density of market data. (again a favorite of Mandelbrot)
   1. Varies from 0 to 1
   2. H = 1 - α/2; more ‘estimated’ than computed
   3. H = 2-D where D is the fractal dimension
   4. No predictive value for trading
   5. The H coefficient changes dramatically with the length of data used to estimate

   
   

3. A value of 0.5 indicates a true random walk (a Brownian time series). In a random walk there is no correlation between any element and a future element. A coefficient between 0.5 and 1 indicates “persistent behavior,” that is, a long-term dependence.
4. 0-0.5 is ‘antipersistent’ or countertend

The Hurst Coefficient in Action
Ehlers tells us that “the Hurst coefficient can be used to define trend modes (> 0.5) and cycle modes (< 0.5) in the market relative to the selected input length.” Cycle periods shorter than the hurst coefficient input parameter display as antipersistent; longer cycle periods display as persistent.

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Attached Image (click to enlarge)

1. Linear system theory says any problem can be divided into three boxes
   1. Input box
   2. Transfer box
   3. Output box

   
   

2. Ehlers almost defines ‘spectral dilation’ here but not quite. You be the judge.
3. “The transfer box can be as complex as desired and can be composed of many less complex boxes wired together. For this reason, I assumed that if I applied a filter to input data the output would contain only frequency components defined by the filter. I have certainly been aware of the fractal nature of market data and the log spiral used by Fibonaccians, but it just did not occur to me that the concept would extend to the frequency range of swing-trading filters.

I coined the term Spectral Dilation to encapsulate the significance of the effect. (what effect??) This effect has a profound influence on all technical indicators and is commonly neglected because these indicators are not viewed from the perspective of their frequency content.” (???)

Attached Image (click to enlarge)

Attached Image (click to enlarge)

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1. Spectral Dilation is inherently part of market data, arising from the fractal nature of the data. That is, longer cycle periods necessarily have correspondingly larger amplitude swings.
2. Spectral Dilation is virtually ignored in all common technical indicators. (Is it ignored or there’s simply no known way to deal with infinite variance?)
3. Ignoring Spectral Dilation has led to erroneous, and often wacky, interpretations of market activity.
4. Spectral Dilation can be corrected by placing a roofing filter before the computation of indicators. (From the testing I’ve done so far I think ‘corrected’ should be in quotes. Still, it was hastily done. Please report if you find a use for it.)
5. A roofing filter establishes a near zero mean of detrended data by allowing only cyclic components within the filter pass band to be passed on for further analysis.
6. The roofing filter, itself, can be a successful technical indicator. (It would be great if he explained how.)
7. Insertion of the roofing filter will dramatically change the use and interpretation of standard indicators, basically making them all new indicators. (New, but the same dependable uselessness for live trading?)

Next: hopefully before the new year: the autocorrelation periodogram which is the 'main course' of this book(?)

Simple as pie, Johnny. That’s what you do, you keep it simple for us stupids.

1. Peter Swerling, engineer
   1. Developed a class of statistically fluctuating target scattering models in the 1950s for pulsed radar systems
   2. He noticed return radar echoes were noisy
   3. This was caused by reflections from the different parts of the airplane as its aspect changed relative to the radar transmitter

   
   

2. The Swerling model is entirely consistent with the 1/F α spectral model that uses random inputs with long-term memory.
3. There has been a mountain of opinion regarding the randomness of the market
   1. It is reasonable to apply a Swerling-like model toward the generation of synthetic data.

   
   

4. We can gain insight into market activity by correlating current prices with prices in the recent history to take advantage of the memory part of the model.
5. “One of the underlying principles of technical analysis is that market data do not follow this power law of an efficient market, and we therefore can extract information from the partial correlation of the autocorrelation function.”
6. “Market data are considerably messier than purely random numbers or perfect sine waves but contain features of both. However, the characteristics that are uncovered by autocorrelation offer unique trading perspectives. Since the process is relatively complicated, describing how the computer code in Code Listing 8-2 works is the best path toward understanding.” (It’s easier to show than to tell, sometimes. However in this case, neither is easy.)
7. Market cycles are not consistent
8. From what I can understand the indicator is made up of a heat map, which is meant to show cycle strength from black (weakest) to yellow (strongest). There is also a ‘spectral estimate’ of the dominant cycle made using a centre-of-gravity algorithm. I’m going to skip all the technical details which run for several pages.
9. The dominant cycle is a value that varies with time and can be used to automatically tune other indicators such as the band-pass filter, commodity channel index (CCI), relative strength index (RSI), Stochastic, and so on. And now we get some sense of Ehlers grand plan. He has built some cool indicators, and the ‘tuning fork’ with which to optimize them (protect them from ‘spectral dilation’), at least in theory!
10. The vertical scale is the cycle length in terms of number of bars.
11. The spectrum display is equally applicable to intraday data as it is to daily or weekly data.
12. “It is obvious from the spectrum display that market cycles are evanescent—they come and go and change their periodicity over time.” (uh-oh!!)

Attached Image (click to enlarge)

Attached Image (click to enlarge)

I have a few things to say about this, but it will have to be after the new year. I will leave you with a copy of the autocorrelation periodogram to ponder. I cannot locate an autocorrelation reversals indicator, so unless someone can give us a tip, it will have to be constructed from scratch. If anyone wants to attempt that under time duress, please be my guest.

{
    Autocorrelation Reversals
    :copyright: 2013 John F. Ehlers
}
Inputs:
    HPLength(48),
    LPLength(10),
    AvgLength(3);
Vars:
    alpha1(0),
    HP(0),
    a1(0),
    b1(0),
    c1(0),
    c2(0),
    c3(0),
    Filt(0),
    M(0),
    N(0),
    X(0),
    Y(0),
    Lag(0),
    count(0),
    Sx(0),
    Sy(0),
    Sxx(0),
    Syy(0),
    Sxy(0),
    SumDeltas(0),
    Reversal(0);
Arrays:
    Corr[48, 2](0);
//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 / LPLength);
b1 = 2*a1*Cosine(1.414*180 / LPLength);
c2 = b1;
c3 = -a1*a1;
c1 = 1 - c2 - c3;
Filt = c1*(HP + HP[1]) / 2 + c2*Filt[1] + c3*Filt[2];
//Pearson correlation for each value of lag
For Lag = 3 to 48 Begin
    Corr[Lag, 2] = Corr[Lag, 1];
    //Set the averaging length as M
    M = AvgLength;
    If AvgLength = 0 Then M = Lag;
    //Initialize correlation sums
    Sx = 0;
    Sy = 0;
    Sxx = 0;
    Syy = 0;
    Sxy = 0;
    //Advance samples of both data streams and sum Pearson components
    For count = 0 to M - 1 Begin
       X = Filt[count];
       Y = Filt[Lag + count];
       Sx = Sx + X;
       Sy = Sy + Y;
       Sxx = Sxx + X*X;
       Sxy = Sxy + X*Y;
       Syy = Syy + Y*Y;
    End;
    //Compute correlation for each value of lag
    If (M*Sxx - Sx*Sx)*(M*Syy - Sy*Sy) > 0 Then Corr[Lag, 1] = (M*Sxy - Sx*Sy)/SquareRoot((M*Sxx - Sx*Sx)*(M*Syy - Sy*Sy));
    //Scale each correlation to range between 0 and 1
    Corr[Lag, 1] = .5*(Corr[Lag, 1] + 1);
End;
SumDeltas = 0;
For Lag = 3 to 48 Begin
    If (Corr[Lag, 1] > .5 and Corr[Lag, 2] < .5) Or (Corr[Lag, 1] < .5 and Corr[Lag, 2] > .5) Then SumDeltas = SumDeltas + 1;
End;
Reversal = 0;
If SumDeltas > 24 Then Reversal = 1;
Plot1(Reversal);

Wishing you the best in 2023!