1) During 2001-2002 I was applying polynomial interpolation and trigonometric interpolation , and exponential interpolation, of paths of market's prices as a way of forecasting next steps. In general the polynomial interpolation forecasting was of positive feedback and good to forecast the trend pattern (see post 32) (what was the previous move sign, was forecasted as the next move sign too) while the trigonometric interpolation was mainly of negative feedback and good to forecast the waving pattern (see post 32) (the sign of the next step move was more often opposite to the sign of the previous steps move). These ware my first attempts to synthesise the markets movement through a large number of base functions (e.g. trigonometric or cyclic functions). The trigonometric interpolation can be viewed also as the deterministic spectral analysis or Fourier analysis. There are many free online e-books of numerical analysis that have all the necessary formulae. All the above three types of functions polynomials, exponentials and trigonometric can be considered as the solutions of linear recursive systems (or even linear differential equations) by utilization of the least squares method. It is called linear model as the forecasted values are a linear function of the previous values (although a non-linear function of time). I tried also wavelets and wavelets analysis. But the scientific tools do not give directly success, as they appear in the books, without an inspired modification, enhancement of them, based on deep understanding of the social or physical phenomenon under study.
2) Non-deterministic stochastic spectral analysis is conceptually different as the Fourier transform and spectrum of a path of prices is understood as the sequence of autocorrelations of an underlying (stationary) stochastic process of many paths. Periodogram is the term that is used my econometricians. I worked with these tool from 2002 to 2003. The forecasting with such models is always with linear autoregressive linear equations. I was using mainly the book by Lambert H. Koopmans The spectral Analysis of Time series
Academic press 1995. Another very useful book is by L.D.Lutes and S. Srakani: Random Vibrations
At that time I had not discovered yet that the market is vividly influence by cycles only at particular frequencies or periods, that I have tabulated in previous posts as the rainbow frequencies.
3) During 2001-2002, I worked also with models of Probabilistic Linear Oscillations of Quantum Mechanics as stochastic process with very rich and interesting probabilistic structure compared to deterministic linear harmonic oscillations. In these models I was interpreting the probability density through the volumes of transactions of the prices. I found valuable information of numerical solutions of the probabilistic linear harmonic quantum oscilator and animated movies of it in the books: a) by B. Thaller Visual Quantum Mechanics, and b) by S. Brandt/H.D. Dahmen The picture book of Quantum Mechanics
4) After the discovery of the (12 for simplicity) rainbow frequencies, through spectral analysis, and other statistical tools, I conceived a new idea:
I realized that I was interested in creating a universal model of behavior of the markets that would capture not all of the real behavior but only that which is rooted in eternal natural cosmic and planetary rhythms, and standard social evolutionary habitual rhythms. This was giving me a psychological security when investing or trading. In this way I would not need special cases backtests of models and the anxiety that the model might not hold anymore. Non-repeatable news effects and temporary fashions though mass media was not my target to capture, as they mainly create more chaos and shifting sands of forecasting, rather than adding real meaning to the financial developments. Instead I should incorporate (through the law 7 of compensation and law 9 of correspondence) the deeper laws of growth of financial organisations and domestic economies (fundamental analysis) through the Pareto distribution of average returns and volumes transactions in all the rainbow time-frames and corresponding organisation scales.
My estimate was that this hidden regularity in the prices and volumes (created by the cyclic behavior of the physical and social world) would be more than enough to permit an abundant successful trading, with which I could apply the law 1 of elimination of household money (see the post of the 12 laws of the financial markets).
So I would leave some of the still intriguing behavior of the financial markets out! So what! The idea of the nature's cyclic behavior is simple, (almost emotional) and the model would be universal for all markets, securities (stocks), interest rates, commodities, currencies. In addition I would avoid the "over-fitted models". Such over-fitted models, are "animals" that live very efficiently in particular environments but as time changes, or if we change for them the environment they go extinct.
5) And this model I did create. I called it the Rainbow Stochastic Process of the global financial markets. Actually it applies to the growth of enterprises that may not be in stock exchange markets, and to the prices of consumer products too, so to the growth of the wealth of nations.
As I described in the post 5, while the cyclic behavior of some natural or social magnitude at a particular period, goes up, down, up, down etc, the effect in the prices was rather ups and downs in an irregular order, but all of average duration equal to the half-period or integer multiples of half-periods. I call such ups, and downs trend-vectors. The had the characteristic duration of a half-period and there could be runs of them (repetitions e.g. of ups in sequence). And at each rainbow-color or frequency of the 12 one such stochastic process was defined called rainbow walk. By superimposing all 12 rainbow walks at the different periods we create all the market. So each rainbow-walk is not directly observable, as what is observable is the superposition of all of them at at any time step. (the time steps are those of the fastest rainbow frequencies).
I programmed this in VBA in MS-excel, utilising the in-built random number generator command. It took me some weeks to code it and a couple of years to test, elaborate, and study. This was from 2003 to 2005. At each rainbow frequency the user is defining 1) The trend-vector slope (in this way spikes of various orders can be included with their probability frequency) 2) The probability that the next trend-vector will be in the same direction or opposite (so trending or ranging markets are created) 3) Probability of intermittency, where no trend-vectors appear at all for some time 4) Correlation of higher volumes at the end and start of the trend-vectors. All the support-resistance levels are defined in this way as parallel horizontal levels and characteristic regularity at each rainbow time-frame. The slopes relative to the duration of the trend vectors, or in other words the amplitudes of the underlying yclic phenomena among all the different rainbow frequencies were regulated by the law of correspondence by the rules of n^(1/2).
The ability to define explicitly the above known phenomenology in the prices, was also a strong reason, to prefer as basic building blocks at each rainbow frequency the rainbow-walk stochastic process instead of a purely cyclical process. Although the law 2 of transfer (see the 12 laws of the markets) somehow creates correlations of the behavior between the rainbow walks of different frequencies, in the coded price generator, I programmed independence and simple superposition.
From the point of view of classical stochastic spectral analysis, I could obtain a similar result by assuming a (stationary) stochastic process with discrete spectrum at the rainbow frequencies, and narrow band continuous around the rainbow frequencies. It is proved in books of random vibrations that the effect of support-resistance boundaries of channels, appears for narrow band (stationary) processes and the distribution of the prices around the support-resistance is the Rayleigh distribution (there are also the rice formulae, and the cartwrite coefficient that measure the degree of emergence of a support-resitance)
6) Once I had coded the rainbow price generator, my whole psychology of trading and attitude towards the markets changed. I was not feeling alienated anymore from the markets, as I had captured what I want from them, right in my computer and I could reproduce it indefinitely. So 50 years data could be relatively easily generated. Also I had to win in my own game, not the game defined my other people. Furthermore I could solve it and find trading systems and know also how optimal or sub-optimal they are, how complete for all market phases or how partial and thin slices of only of some of the phases of the markets are. In other words I could satisfy my scientific intellect and subconscious, and be able to know why it is possible to be a systematic winner, of how much , with what expected MaxDrawDown etc.Furthermore I could analyse any system or robot that I could find for free or buy in the web, and assess how good or bad it was, assuming that the markets behave as in the rainbow process. Most of the trading systems were obvious solutions if we assumed that the markets behaved like the rainbow process. Rarely some of them contained very smart elements that led me to perfect or add a new feature to the rainbow process.
I had to be sure of course that the rainbow price generator was realistic. I could configure it so that it becomes a pure random walk were there is no winning trading system ever, and I could configure it so that purely cyclic behavior occurs at each rainbow frequency (this would be a quite easy market to trade) or I could configure it in a realistic way, that it is as difficult to trade it as is the real markets. So I took winning automated systems that were know to be winning for 20 years or more, and I applied them to the artificial prices of the rainbow price generator. The trading results were almost identical. Conversely when I devised a winning trading system over the artificial prices, then I was testing it over real historical prices, and the results were almost identical.
7) It was 2004 and I was even thinking to put this price generator online for free for other traders to try systems too. Tradestation was quite expensive and not free. But then the Metatrader4 appeared in 2004, and more and more forex brokers allowed free demo accounts, so this became not so critical anymore.
8) So this is the 1st part of the story of what I did with the discovery of the characteristic frequencies of the markets. It is the part that describes the creation of the universal model for the prices. The 2nd part is supposed to do with finding a relatively simple trading solution within such a universal model. The simplest of all is based on the fact that the volatility of the prices and volumes of transcations do have direct (stochastic) cycles. So the easiest method is the volatility-short/volatility-long techniques. Although this terminology comes from the trading of options it is possible to create somehow artificial options with ordinary positions (even at forex). Examples of volatility-long techniques are the break-out methods, and examples of volatility-short techniques are the B. Williams angulation-counter-trend method or other counter-trend methods that focus on retracements of spikes.
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