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

- Quick introduction to Puetz 'Universal Wave Series' temporal nested cycles
- Formulae for Puetz UWS
- [Fibonacci, Fourier, Elliot, Puetz] series comparisons

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 :

- Thucydides ~409 BC "Pelopponesian Wars" (Athens versus Sparta) - oracles predicted at the outset that the war would last 3*9 = 27 years

- Ibn Khaldun ~1410 AD "The Muqquadimah" - societies tend to decay within three (or four) generations of heroic times

- Geological [eon, era, period, epoch]s (OK this isn't human history) - have there been calls to split the Paleozoic era into two eras of 3 periods each, at the time of the Great Silurian extinction event? That would make perfect series of three ages per era since the dawn of the Cambrian! (yeah, so what, says you)
- Not a factor of 3, brother Steve found : Abraham (2nd millenium BC) -> 14 generations -> King David (1000 BC? correct this) -> 14 generations -> Babylon conquest of Jews (600 BC? correct this).

Quick questions that arise are :

*"Universal Function Approximation"*(UFA) - to what degree does the Puetz UWS approach this? In other words, is it just a curve-fitting formula like [polynomial, Fourier, neural network] systems?*Fourier Series*- is the Puetz UWS just a greatly simplified Fourier Series, especially when the half-waves are included (Puetz [H]UWS)?- Clearly "yes" in that the Puetz [H]UWS periods are a subset of Fourier series all-inclusive integer periods, and Puetz [H]UWS does have a more restrictive set of rules of application.
- Clearly "no", as Fourier Series has NO concept of a DOMINANT and PREDICTIVE fractal series of periodicities that applies across a hugely diverse set of themes : [physics, astronomy, geology, 5 major biological extinction events, climate, weather, history, war, pandemic, brain waves, behaviour, etc, etc, etc]. However, Fourier Series could be used to [discover, apply] such a [dominant, predictive, broad-theme] fractal series.
- Actually, this seems a bit like a [political argument, open Platonic dialectic, any dialectic materialism] where the whole point is to get intellectuals to chase their own tails in trenches that they have built for themselves ("... Monsters of their own making ...").

- What "causes" the Puetz [H]UWS to work so well? Broad explanations include :
- Unseen drivers - why not. All modern fundamental theoretical physics ("existentialist" in words of Bill Lucas) is based on inventing concepts that are critical to save [dumb-ass, failed] theory. Anybody can do the same thing, even day-care children. Ignore the scientists when they tell you that you have no right to do the same as they. Actually, just ignore the scientists [any, all]ways?.
- Dynamics of complex systems - for example [oscillators, electrial engineering signal processing, Fractional Order Calculus (FOC) of memrister arrays, etc, etc]. My own simplistic analysis looks at frequency [harmonic, sympathetic resonance, split, scavange, etc].
- [Electric, Plasma] Universe concepts - are by far the most scalable experiments in [time, space] (even if gravity isn't an electro-magnetic force, it drops off too fast). After a modest amount of looking (laz), I haven't yet come across any frequency-series [data, model]s to support this explanation. Mythologists have always been far ahead of the scientists, but the fun today is about how they got together, what that has already accomplished, where it may wander the future.
- Other? - there are probably thousand of other explanations. Can you think of some?

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.

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 ]]

where

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.

Mandlebrot&Hudson, 2004, page 21

16May2022 very quick notes so I won't forget later, huge number of other points, things to discover

- Puetz is quite close to the Fibonacci series, some applications they might be interchangable (with a mind twist)
- 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 - 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" - Elliot is the only series that :
- specifically formalises human market behaviour,
- forces retracements and your thinking of them, more than [Fibonacci, Puetz]
- beats sociology at its own game (core parts of it) as per Robert Prechter's "Socionomics"
- lacks a clock

- Fourier - comments at a later date (eg, good starting point, wavelets better)