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Howell - Quick introduction to Puetz Universal Wave Series (UWS) fractal time)

"... The true power of fractals only emerges when time is one of the dimensions. ..."

Benoit Mandelbrot, 2004, actually, I couldn't find this quote in the Mandlebrot&Hudson book (see references below)

Table of Contents

Related background information on my other [webPage, file]s

Quick introduction to Puetz 'Universal Wave Series' temporal nested cycles

In his book "The misbehaviour of markets", Benoit Mandelbrot said that "fractals don't reach their true power until used as "multifractals", where time itself is a "fractal dimension". It just so happens that Stephen Puetz came up with by far the most stunning relationship for common temporal cycles that I have ever seen. It is like expanding the Mayan system of 20+ calendars expanded [far, far] beyond anything that I would have ever imagined. Normally, I would not have considered Puetz UWS to be a fractal system (more like a simple power series), but on closer inspection of definitions, perhaps it does qualify?

UWS is fixed, not flexible, across an unbelievable timescale that is presumed to be "infinitely extendable". If I remember correctly, the UWS has been confirmed for many long-known [natural, biological human] cycles, making it very broadly applicable.

Luckily, the UWS is dead simple : cycles are simply are factor of 3 times the previous (lower) value in the series. Furthermore - the wavelengths (lamda) of the Puetz UWS series are fixed across themes such as [physics, astronomy, geology, climate, evolution, biology, psychology, sociology, financial markets, etc], so one only has to pick some time period and that defines the rest of the entire series. Each time series has a phase angle (theta - equivalent to a time lag). The formulae for Puetz UWS are provided in the next sub-section below.

As one complication, a number of standard periodicities do not fit the UWS according to Puetz's tough criteria, but may be somewhat related. Some of these do fit a Double-UWS series (or Half-UWS), but most recent papers of Puetz focus on the Double-UWS. (See below for the JW Hurst stock-market 2* series.)

At the present moment, two cycles of 3 (almost) come to mind from history :
  1. Thucydides ~409 BC "Pelopponesian Wars" (Athens versus Sparta) - oracles predicted at the outset that the war would last 3*9 = 27 years
  2. Ibn Khaldun ~1410 AD "The Muqquadimah" - societies tend to decay within three (or four) generations of heroic times
Other periods, from Bible, (should do Mayan calendar, other [religions, philosophies] too) : This doesn't necessarily mean anything, but what if a "generation" in Ibn Khaldun's time was approximately 27 years? 3*9*3 = 3^4 years = 81 years, which doesn't seem to fit much except perhaps some climate (ocean) cycles? That would kind of fun. In any case, Puetz has collected a large number of UWS cycles across a very broad range of themes.

Quick questions that arise are :

In any case as stated by Puetz (section below "Formulae for Puetz UWS") "... The Universal Cycle model is not intended as a model for all cycles found in nature. Instead, it describes an important fractal subset of natural cycles. ...".

Another popular series of similar nature is JM Hurst's time-period fractal of 2*, which as far as I know targets only financial markets. That website link does mention fractals (at least of the market), and applies "Laguerre Hurst Cycles" because "... Laguerre moving filters have significantly less lag and the setback required is only three days/bars. ...". Very interesting. Note that the Puetz UWS-HUWS combination practically incorporates the Hurst cycles in terms of halfing, but I don't know if the Hurst cycles are "pegged" to a single basic lamba P(0,0). I suspect not.

Formulae for Puetz UWS

Stephen J. Puetz a,∗ , Glenn Borchardt b "Quasi-periodic fractal patterns in geomagnetic reversals, geological activity, and astronomical events" Chaos, Solitons and Fractals 81 (2015) 246–270
a Progressive Science Institute, Honolulu, HI 96815, USA
b Soil Tectonics, Box 5335, Berkeley, CA 94705, USA

The Universal Cycle model, consisting of an unlimited number of individual UWS cycles, has hypothetical periods of :
(1) P(k,n) = (3^k / 2^n) * P(0,0)
where :
k is a positive or negative integer corresponding to a cycle in the primary period-tripling sequence
n is one of eight period-halving harmonics where n ∈ {0, 1, 2, 3, 4, 5, 6, 7}
P(0,0) is a base cycle with a period of 2.82894367327307 solar years.

The composite stochastic Universal Cycle model consists of a superposition of cosine waves, with periods of P(k,n) from Eq. (1), corrupted by Gaussian white noise, and are of the form :
(2) yi = sum[k=k1 to K: sum[n=n1 to N: A(k,n)*cos(2*π*(ti + φ) / P(k,n)) + σ*Zi ]]
i = 1, 2, 3,..., I are the records in a time-series
K is a set of consecutive integers
N is the integer set {0, 1, 2, 3, 4, 5, 6, 7}
A(k,n) are non-negative amplitude factors
ti are negative numbers for times in the past and positive for times in the future
φ is a phase adjustment so that all UWS cycles peak synchronously at time φ

The Universal Cycle model is not intended as a model for all cycles found in nature. Instead, it describes an important fractal subset of natural cycles. Accordingly, Zi are independent random variables with standard normal distributions, with σ > 0, representing noise and other non-UWS cyclical variation in the signal. The simplest model for the noise is additive Gaussian white noise, but more sophisticated models can be used.

[Fibonacci, Fourier, Elliot, Puetz] series comparisons

"... Patterns are the fools' gold of financial markets. ..."
Mandlebrot&Hudson, 2004, page 21

16May2022 very quick notes so I won't forget later, huge number of other points, things to discover
  1. Puetz is quite close to the Fibonacci series, some applications they might be interchangable (with a mind twist)
  2. Puetz has [time, price] "mirror reflectivity", with infinite "baseline levels" (Glenn Borchardt)
    [Fibonacci, Elliot] don't, but like Puetx easily extend [up, down] to different [time, length] scales
  3. Puetz partly reduces frequency [scavenge, split, harmonic, ?one I can't remember just now?] to non-Puetz periods (cycles).
    So does Fibonacci to some extent. I doubt it's as much an effect as for Puetz.
    Inclusion of the Puetz "OR[Half, Double] Wave Series"
  4. Elliot is the only series that :
  5. Fourier - comments at a later date (eg, good starting point, wavelets better)