[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 386
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Random Wandering
Oooh! Thanks a lot. Finally something intelligible on the subject.
1. The mean square of the distance a process travels in N steps is what the spread is. Let's not forget that the concept of the spread was introduced by Hearst, who looked at the Nile spill. Einstein, who considered the Brownian motion of the particle, spoke of the path it traveled from its starting position. These are all physical quantities. And I was looking for a definition - that is, their mathematical meaning. Now the question is clearer. The span in the Nile, the path of the Broin particle, the maximum gain in a game of eagle are all the same concept as defined in this link.
2 There also, literally in two lines, a formula (a special case of Hurst formula for pure SB) is derived from which it follows that for pure SB with the same unit increments at each step the coefficient in Hurst formula = 1. That is what I asserted and tried to illustrate on my fingers to Nikolai in his thread. From the physical point of view it is clear: in fact this coefficient is needed where values have dimensionality.
3. Now the meaning of S in the formula R/S = c*(T^h) is also clearer. As it is written everywhere, S is the RMS. I, because of my stupidity, could not understand the RMS of what series. Now I understand - the series of increments, but not the SB itself. And the point is just to normalize the gradients by RMS, i.e. sort of reduce the gradients to +/- 1.
4. And in the end, I understood why the calculation of the index, which I described in my branch, did not give 0.5 value on sgenerated model series. I calculated it for intervals of astronomical time M1, M10, H1. And averaged over all data. But each of identical astronomical intervals had its own number of ticks (i.e. steps of the process). To average the number of ticks in order to fit it to Hurst's formula is quite contrary to the definition. But now it turns out not only that. I was averaging the spread as well. I should have averaged the square of the spread and then extracted the root from it. So there were two mistakes.
Well, that cleared it up. I'll have to recalculate it properly. :-)
And the question about the theoretical derivation of the formula for a given SB distribution can now be more substantive.
Oops. If you knew, you wouldn't have given the link :). In Feynman's picture everything is drawn, well imagine that these are ticks inside a bar.
Let's take the uppermost, point trajectory. For it Open = 0, Low = -2, High = 3, Close = 2. D for it equals 2, i.e. D = Close-Open. About what I wrote you in your topic at once (that Einstein did his formula for Close-Open). And for Close-Open the coefficient in your approach will really equal 1. But you take High-Low, and it is equal to 5 in this case. It means that it is not equal to D and therefore the coefficient is not equal to 1. I can see that you want to substitute the spread for the final deviation at all costs. But then be kind enough to come up with a term for High-Low, so that I can tell you in your own words that for this quantity the coefficient will not equal one and the slope of the ray from the origin will not be Hurst.
Oops. If you knew, you wouldn't have given the link :). Feynman has a picture of everything, imagine that these are ticks inside a bar.
Take the uppermost, point trajectory. For it Open = 0, Low = -2, High = 3, Close = 2. D for it equals 2, i.e. D = Close-Open. About what I immediately wrote in your topic (that Einstein did his formula for Close-Open). And for Close-Open the coefficient in your approach will really be equal to 1. But you take High-Low, and it is equal to 5 in this case. That is, it is not early D and for this reason, the coefficient will not equal 1. I can see that you want to substitute the spread for the final deviation at all costs. But then be kind enough to come up with a term for High-Low so that I can tell you in your own words that for that value the coefficient will not equal one and the slope of the beam from the origin will not be Hurst.
You've migrated here from tick volumes, haven't you...
That's where you would have practiced.
:)
Albert himself suggests that the plausibility of the HL in a bar should be estimated from the number of ticks in it!
;)
Nikolai, I also wrote you that I think the Close-Open thing is wrong. And the fact that you have equated this difference with D is even more wrong. To understand the definition of "the path taken by the process" as this simple difference contradicts the formulation of the problem in general. Then there is no reason to square it. You average, you get zero and you're happy.
Think about it in terms of diffusion. There, the phenomenon itself produces the averaging - huge number of particles (molecules) spread by Brownian motion. The path that the process has taken is the diffusion boundary. Which particle has reached it at this moment, and which particle has reached it and already returned to the starting point does not play the role.
In general, I will recalculate all over again, then it will be possible to speak more reasonably.
Alexei, I know the distribution of the series. I want to know the spread of the extremes. That's what you said. How ?
The probability of reaching the extremes is sort of theorized as an SB problem with an absorption screen. I.e. in t a is a screen, once reached the particle stops moving and we need to find the probability that in time t it will reach it /go?link=https://dic.academic.ru/dic.nsf/enc_mathematics/414/%25D0%2591%25D0%2595%25D0%25A0%25D0%259D%25D0%25A3%25D0%259B%25D0%259B%25D0%2598
P.S. And Einstein's formula is really about the average deviation of a particle through time T, i.e. analogous to |Close-Open|
Think of it in terms of diffusion. There, the phenomenon itself produces averaging - a huge number of particles (molecules) spread by Brownian motion. The path that the process has taken is the diffusion boundary. Which particle has reached it at the moment, and which particle has reached it and already returned to the starting point does not play a role.
Again the spline...
If it's about boundaries. And the Nile delta is also well approximated.
:)
the probability of reaching the extremes seems to be theorized as an SB problem with an absorbing screen. I.e. in t a is a screen, on reaching which the particle stops moving and one should find the probability that in time t it will reach it /go?link=https://dic.academic.ru/dic.nsf/enc_mathematics/414/%25D0%2591%25D0%2595%25D0%25A0%25D0%259D%25D0%25A3%25D0%259B%25D0%259B%25D0%2598
P.S. And Einstein's formula is really about the average deflection of a particle through time T, i.e. analogous to |Close-Open|
A Galton board with bitumen traps on the sides?
from which row?
I see...
;)
No substitution so far.
Let me remind you of the logic of reasoning. A certain indicator is found that is supposed to characterize somehow the degree of randomness of the market at the moment. We must find out which values of this indicator will correspond to the trend market, which ones will be flat and which ones will be unpredictable.
I see.
Candid:
In physics, this is called calibration. We are supposed to be able to calibrate on artificially generated series with given properties.I, for example, believe that it is faster and in some sense more reliable to generate necessary series and study the behavior of the characteristic on them. And you should start with series sliced from suitable parts of real price series.
I once suggested generating rows with the necessary characteristics, and on them to study the survivability of TC, including the behaviour of NN. Some forum members were against this approach, but there were no specific arguments against it. There were those who agreed with me.
But as time goes on, more and more I am convinced that the approach was wrong.
Candid:
Well, we (well, at least I) try our best to help him in this difficult task.I try too, to the best of my ability. Maybe not with formulas, but with ideas and considerations.
Let me try to draw an analogy.
Writers. Blok, Pushkin, Tolstoy, Lem, Shackley. Each in its own unique and the reader can easily identify not only the genre of work, but can also identify the author (this is a kind of index, a parameter unique to each author). However, statistically, any sufficiently large text contains a constant number of each of the letters of the alphabet. It is a statistical characteristic of the language in which the work is written. If one randomly generates letters, but with preset statistical characteristics, one can get a text with the right amount of information. But this text would not have any meaning, and even more impossible (because it's not there) to identify the author of the "work".
And Yurixx's efforts are precisely to find this indicator which allows unambiguously identifying the author of the work from the statistical indicators of the text.
I had such thoughts: to collect statistics for several years for three months - October, November and December in order to generate series with the same statistical characteristics, taking into account dynamics of characteristics change by years, and to optimize Expert Advisor on this generated series, which would then expose Expert Advisor on championship....
But right now, I rather like the idea of collecting stories for these three months, normalising by one volatility, gluing these pieces into one series and going forward.
The disadvantages of both approaches are obvious. But the second one is still more promising.
Nikolai, I also wrote you that I think the Close-Open thing is wrong. And the fact that you have equated this difference with D is even more wrong. To understand the definition of "the path taken by the process" as this simple difference contradicts the formulation of the problem in general. Then there is no reason to square it. You average, you get zero and you're happy.
OK, that's my last remark on this point. If you don't agree, do what you want, I'll keep quiet :)
I didn't equate it, Feynman did. I tied everything to his picture. Feynman wrote "We should expect that there will be no mean progression at all, since we are equally likely to go forward as well as backward. However, it is felt that with increasing N we are more and more likely to wander somewhere further and further away from the starting point. So the question arises: what is the average absolute distance, i.e. what is the average value of |D|? However, it is more convenient to deal not with |D| but with D2; " Do you have a different text at my link?
And also think why Hearst guessed to normalize to RMS, but did not guess to count the degree by the ray from the origin. He did, by regression. He was a fool, is that what you're saying?