a trading strategy based on Elliott Wave Theory - page 283

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<br / translate="no"> I started to write out formulas for interpolation of time series in general case by polynomials of degree n and do you know, Yura, what I got as a result? - Taylor's series expansion (RT) in the vicinity of some point! I was astonished of my genius :-) and having thought a little, I came to a conclusion that it should be so. After all, in fact RT is an approximation to the initial function at a point by summing polynomials of higher and lower powers with smaller and lower weights, which model behavior of the first, second, ..., n-1 derivatives. By definition, this apparatus can be used if the initial series is smooth, i.e. derivatives up to n-1 are defined and exist. BP of financial instruments does not belong to the smooth class, so we cannot apply RT decomposition or, what is the same, use extrapolation by polynomials.
By the way, the smoothness of the series is nothing but the positivity of CA! That is, the series is more likely to continue the movement started than to change direction. Yes, that's it! Looks like we need to create a section of mathematics in the study of NOT smooth functions and methods of their analysis...


Information for thought. A wavelet transform can be applied to any BP. The resulting wavelet image makes it possible to reconstruct with any accuracy the original VR. A wavelet image (with a known choice of wavelet transform function) is continuous and infinitely differentiable.

Maybe I was illiterate and did not express myself correctly somewhere. But the meaning is correct.
 
to Andre69
Finally there is a free window of time and I want to continue the post about wavelets.

What a beauty!
The first picture looks like a dive into an ever smaller scale of price change - a kind of digital microscope with variable magnification:-) I think that a very similar map (if not the same) can be obtained by subtracting at each step of the original BP series, obtained by influencing it with a gently decreasing filter bandwidth...
 
Yes, the picture is good. Indeed, the fractality of the market is presented in a textbook way. I also personally saw it as an illustration of market disequilibrium. The problem, however, is that in history we clearly see repeating structures for many different representations (at least the same channels). But in real time, by the time the structure is identified, it is usually impossible to reliably judge its future fate.
 
And here is the first result of VR processing by an algorithm that has nothing to do with the wavelet transform (see post above)! For comparison, on the right, here is Andre69's picture:



I'd say the match is satisfactory. By the way the code in MathCad contains ONLY the recurrence formula for VLF - 10 lines and that's all, while the counting time is 1 sec.
It is pleasant, that results received by absolutely different methods are similar.
 
Another picture.
A fine structure of the same BP (high frequency region).
 
This is the impression that emerged from the sum of the pictures: a regular structure exists in a certain range of frequencies. Clutter dominates both too high and too low. I wonder if this is a property of this section of BP or of the market in general.
 
This is the impression that emerged from the sum of the pictures: a regular structure exists in a certain range of frequencies. Clutter dominates both too high and too low. I wonder if this is a property of this section of BP or of the market in general.

I think the regular structure emerges on the temporary lag of large capital injection. This process is gradual and it causes a local regular market disturbance.



This is an even finer structure (minutiae). On the vertical axis is the averaging window and on the horizontal axis is the current minute bar.
 
I think a regular structure emerges on a temporary lag of large capital injection. This process is incremental and causes a local regular market disturbance.

I had a different theory - maybe it's an "adiabatic window"?
 
to Neutron

<br / translate="no">.
to Andre69
...You are first of all differentiating the price range. By doing so, mind you throw out a good chunk of the low and medium frequency harmonics of the range! For statistics, of course, this approach is sensible. But aren't we splashing the baby out with the water here...?


When differentiating, information about the low-frequency component of the signal is not lost. Indeed, after integrating the residual series, we get the original time series with all trends plus some constant. Hence, residualization of the original series by differentiation is quite correct from the mathematical point of view. Here, however, there is another trap: it generates false correlation of neighbour samples, but that is a separate story.
Otherwise, Andre69, I agree with you. And thanks for informative answers.


I agree, we do not lose information but we distort it very much. It is in this sense that I have expressed myself. Actually differentiation is the application of a high pass filter to a series. The lower harmonics are very much clipped. The constant component... To hell with it, we don't need it. But the rest... I took another look at the spectra (Fourier and wavelets) of the price series and its derivative. As the saying goes - feel the difference...
Otherwise, I agree.
 
to Neutron

To Andre69
Finally there is a free window of time and I want to continue the post about wavelets.

What a beauty!
The first picture looks like a dive into an ever smaller scale of price change - a kind of digital microscope with variable magnification:-) I think that a very similar map (if not the same) can be obtained by subtracting, at each step, from the original BP series, the series obtained by exposing it to a gently decreasing bandwidth LPF...


Glad you like it!

What you describe next is another wavelet method (different in details, but basically correct). It's called undecimated wavelet transform with a trous algorithm.

Congratulations on your discovery!
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