[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 385
You are missing trading opportunities:
- Free trading apps
- Over 8,000 signals for copying
- Economic news for exploring financial markets
Registration
Log in
You agree to website policy and terms of use
If you do not have an account, please register
More seriously, I assume that the average swing and RMS are related by a constant coefficient.
I don't think this is possible in principle. If it is true for a normal distribution, I would be very surprised. But for other distributions such a ... ?
By the way, if you assume this for the mean range, then what should be the definition for it? What does it represent?
Although, I'm lying, it's quite possible. Suffice it to say that the spread = 2*SCO. There it is, a genius solution!
If the value is not bounded (e.g. normal distribution), then the range will still have to be estimated somehow from some boundary probability. For example, take and define the spread as the difference between the percentiles 0.99 and 0.01. But percentiles can only be calculated analytically in some exceptional cases of distributions.
I think any assumptions we make will still hang in the balance until the spread is defined.
This should probably have been done in a practical way. Am I right in remembering that Peters divided series into equal intervals and for each interval he counted the spread and then averaged it over all intervals and for the resulting pair of average spread - interval he plotted a point on the log-log graph? Or did he do it for each interval, and averaged the logarithms?
Perhaps Peters was "averaging" an already constructed graph. But I haven't checked.
About the definition of the spread: well, what do you think is the spread of the normal distribution N(0,1)?
I don't understand the problem with the definition. We have a certain number of measurements, i.e. we have a time interval. The range is the difference between the maximum and minimum of a function in that range.
That is, if we consider a bar, it is High-Low, and the deviation on the same segment is Close-Open.
If we speak about one-dimensional random walk, the spread is the same High-Low, i.e. the difference between the extreme points reached during the time of the walk at the top and bottom. And the deviation is still Close-Open, i.e. the difference between the current position and the initial one.
By the way, one-dimensional random walk is one of the textbook topics of probability theory. And there was something about it here, for example in the roulette thread.
If the value is not bounded (e.g. a normal distribution), then the spread would still have to be estimated somehow based on some boundary probability. For example, take and define the spread as the difference between the percentiles 0.99 and 0.01. But percentiles are only calculated analytically in some exceptional cases of distributions.
Well no one is talking about infinite time. The RMS for SB also tends towards infinity.
Feller remembers exactly about SB.
Yurixx:
More seriously, I assume that the mean spread and RMS are related by a constant coefficient.
I think this is impossible in principle. If it is true for a normal distribution, I would be very surprised. But for other distributions it is ... ?
I don't understand the problem with the definition. We have a certain number of measurements, i.e. we have a time interval. The range is the difference between the maximum and minimum of a function in that range.
About the definition of the range: well, what do you think is the range of a normal distribution N(0,1)?
There is a theoretical notion of "spread". It is defined by its definition. If there is no definition, there is no notion - you cannot calculate, do anything and say nothing. Therefore, for any theoretical action (e.g. to obtain a formula in a general form), a definition is required in the first place.
There is a practical notion of scope. Its definition was given above by Nikolai. However, the process described by the function he mentioned is stochastic, random. Therefore our measure of the spread on another segment, even of exactly the same length, will be different. And on a third, it will be a third. And so on. So we can't deal with specific measurements, only their statistical derivatives - mo, sko, etc.
Trendiness, returnability, Wiener SB are all mathematical models that are essential for us TC builders. Identifying the currently relevant model allows us to choose the right strategy. Since the Hurst index allows us to distinguish between these market states, it turns out to be quite important. But we can do something only if we connect the experimentally determined practical range with the theoretical one from which the Hearst ratio is derived.
I haven't said anything new here. But since there's a question...
The spread of the normal distribution theoretically, according to Einstein's formula, is proportional to the square of the travel time. А practically it must be determined on the basis of the Max-Min difference data, to which an appropriate (what ?) averaging procedure has been applied.
.
If by spread we mean the maximum distance from the starting point (which, if this point is chosen correctly, is equivalent to Max-Min), then the calculation of the spread seems to rest with the summation of a random series of increments. If the distribution of the increments is known, then the distribution of the sum can in some cases be calculated. Suppose this is done and there is a distribution of the sum of N increments. Which of the moments or other statistical measures of this distribution gives the value of the spread derived practically from the experiment?
The spread is also a statistical quantity. With a finite sample, knowing only the pdf but having no experimental points, it can be estimated, but not calculated accurately.
Nikolai suggested a practical, straightforward procedure: just calculate the difference between max and min values.
What I propose (the difference of two percentiles) is not an exact value of the spread, but only an estimate of it. Frankly, I am not aware of any finer methods of estimating the spread. Feller probably has results concerning the distribution of extremes.
In fact, because it really was a stochastic quantity, for practical application expectation or mean was of course assumed. But it seemed to me, that if I give a definition of magnitude, then a separate definition for its expectation is no longer needed.
So I think that my definition of variance is not only practical but also quite exhaustive.
Scope of normal distribution. theoretically, according to Einstein's formula, is proportional to the square of the time of motion. А practically it must be determined on the basis of the Max-Min difference data, to which an appropriate (what ?) averaging procedure has been applied.
I may of course have forgotten, but I remember Einstein's formula is derived precisely for the RMS from the initial position, not for the spread. That's why to relate it to Hearst you need to determine the coefficient linking the RMS to the spread.
Besides, it seems to me that there is a certain confusion of notions, the matter concerns the range not for the normal distribution but for the random walk with normal distribution of increments, these are essentially different values. By the way, the original problem did not provide for any normal distribution, there were ticks, i.e. unit increments.
P.S. I'll add some links:
Random walk.
Brownian motion
Why should a process of non-random nature, although it has a distribution of incretions close to normal, have a sweep like Brownian motion? Don't the Honourable Men think that there is a substitution of notions - some properties inherent to the random process are attributed to the non-random process only because the other properties of these processes are identical?
So far there is no substitution.
Let me remind you of the logic of the reasoning. We find a certain indicator that is supposed to somehow characterize the degree of randomness of the market at the moment. We need to know which values of this indicator will correspond to a trend market, which ones will be flat and which ones will be unpredictable. In physics this is called calibration. We are supposed to be able to calibrate on artificially generated series with given properties.
I, for example, think that it is faster and in some sense more reliable to do exactly this, to generate needed series and to study behavior of a characteristic on them. Moreover, one should start with series sliced from the suitable parts of the real price series. But Yuri is a supporter of analytical solutions. And we (well, at least I) do our best to help him in this difficult task.
I should also note that the long-run averaged characteristics of real price series are very close to those of random ones. This actually suggests that random series can be used for calibration.
The spread is also a statistical quantity. With a finite sample, knowing only pdf, but having no experimental points, it can be estimated, but not calculated accurately.
There are, however, several useful theorems concerning the investigation of the trajectory of a Wiener process. One of them, the "law of the repeated logarithm" (proved by Hinchin, perhaps correctly written), reveals the structure of the behaviour of the trajectory of the process, namely, it defines dependence of the spread on time: The theorem defines the limit beyond which the process will not go (local extrema) during its evolution.
You can get a good approximation for the increments of quotes, even an analytical expression if you "make assumptions" :о).
Addendum: I forgot to add, not for Wiener processes such studies are done by "asymptotic analysis of random walks", including processes for which the distribution of increments with heavy tails is peculiar.