Stochastic resonance - page 20
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Did I get it right, the spread is taken over the entire window N ? If so, it is difficult to count on any constancy here, imho. Rather, it may appear for differences of muvinings, for example with the highest muvinings (with maximal M).
I am of course talking about muwings, but they are not price muwings. In the very first post on this topic I wrote "the set of values of elements X is bounded from above, i.e. all X's belong to the interval [0,Xmax]". In principle, price increments also fit this definition.
N is all available history on the chart. It will not be needed in our work. But at the moment I am using it for statistics - averages, cramps, etc. The thought is that the nature of the statistics changes little and slowly, or not at all. And so the parameters of the series calculated in this way can be applied in the future.
The range over the entire window N, i.e. over the entire history, is [0,Xmax]. The range over the window M is just what I want to define theoretically, i.e. based on statistics of the main series and values of N and M only, instead of experimentally, i.e. running through all possible windows of M.
The point is simple. When moving to another t/f (with the same window M) the range of values of the series Y should not change. Then a change in the local Y values can tell us something. If, on the other hand, the area of values changes, then it is not clear to what to attribute the change in local Y values, to a change in scale or to a really significant event.
PS
By the way, I was wrong about Gauss. The normal distribution exists on the whole axis, and here we are talking about the right half-axis. But the kind of distribution doesn't really matter. I was interested in the idea or procedure of calculation, and it can be applied to any distribution.
The range over the entire window N, i.e. over the entire history, is [0,Xmax]. But the spread over the window M is exactly what I want to determine theoretically, i.e. based only on the statistics of the main series and the values of N and M, rather than experimentally, i.e. by running through all possible windows M.
OK. Suppose there is a series X already described with a known distribution function. How to construct a distribution function for series Y, which is a moving average with period M of series X ?
Yurixx, you will theoretically have a hard time building it, I can tell you that. The returns distribution itself has no explicit analytical expression, that's the issue. Besides, in this case we have to deal with a random process, not the distribution itself. And random processes have their own intricacies - the autocorrelation function, for example. Give up on this theoretical stuff...
There is no point in constructing the muving distribution function based on the distribution function of population X - simply because the consecutive price samples are not independent tests. The sum of two independent tests from the same population is one thing (the convolution of distributions theorem works here), but the sum of two adjacent tests that are not independent is another.
Yurixx, you will theoretically have a hard time building it, I can tell you that. The returns distribution itself has no explicit analytical expression, that's the issue. Besides, in this case we have to deal with a random process, not the distribution itself. And random processes have their own intricacies - the autocorrelation function, for example. Give up the theory...
...
Yeah, I've been hinting at that to Yuri for a long time, but he won't listen. He'd have long since obtained a dependence empirically and quite accurately. :о)
So, I've been thinking more :). The only way out is to consider that this is not an abstract problem, but a quite specific one. Let's say the mouwing on increments will be a sliding spread. The aim is to make it dimensionless. Experimentally the corresponding unit can be obtained simply by approximating the dependence of the spread at constant M on the timeframe. If it is the same for different M at least in some range (M1, M2) - this can be used in this range.
I also think it is a mistake to try to get something analytically, but if you still need it, the first method is to take a series of M values of one random variable as a series of unique values of M independent random variables, and then as Mathemat wrote.
P.S. In other words, look for such a scaling transformation so that the pictures in grasn's post turn into something resembling a horizontal line. look for ... Maybe in science of fractals?
P.P.S. By the way, it's hardly possible to use this dimensionless spread so easily. In the separate window of the second screenshot on my page something like that is drawn (I won't say what it really is :). There are no definite recipes for my version.
OK. Suppose there is a series X already described with a known distribution function. How do you construct a distribution function for series Y, which is a moving average with period M of series X?
Yurixx, you will theoretically get tired of building it, I'm telling you. The distribution itself has no explicit analytical expression, that's the question. Besides, in this case you have to deal with a random process, not the distribution itself. And random processes have their own intricacies - the autocorrelation function, for example. Give up on this theoretical stuff...
There is no point in constructing the muving distribution function based on the distribution function of population X - simply because the consecutive price samples are not independent tests. The sum of two independent tests from the same population is one thing (the convolution theorem of distributions works here), but the sum of two neighbouring tests which are not independent is another.
I don't know what this has to do with returns, but it makes absolutely no difference whether the actual distribution of what I'm dealing with has or doesn't have an analytical form. You can construct a distribution function (if you have the data) for any process - random, Markovian, chaotic, or payroll process. :-) My premise is that the nature of the market doesn't change every day, which means that the distribution of the series I'm dealing with must be RELATIVELY stable. I've checked it on different t/fs - the assumption is confirmed, starting from M5 the distribution shapes reproduce each other quite well. In principle it should not be difficult to approximate this shape by an analytical function with 2-3 parameters.
In order to obtain a more or less smooth estimation of the market condition, this X series should be smoothed, for example, by a muving. And here appears the problem. The construction of a muving distribution function would solve the problem, because then I would know how to calculate the limits of the range of values. Naturally not exact, but statistical. "consecutive price counts" have nothing to do with series X, I've written about this before. Unfortunately I was wrong when I wrote a few pages earlier that it was a series of prices. I didn't take into account the significant difference in areas of value and nature of change. Once again I apologise.
Thanks to this discussion I understood that firstly, the sum of values in a muving can rightfully be considered the sum of any values in the series, rather than the sum of consecutive values. Reason: the evaluation of the limits of the area of change is the evaluation of the BEFITS, not the current values. In addition, the minimum (maximum) of a moving average is obtained when the X value passes its minimum (maximum) - almost all elements of the moving average are near the range boundary - quite a realistic situation. This is also true for the price.
Second, due to the above, the integral equation, the solution of which may yield values Ymax and Ymin, is S(p(x)dx) = M/N. Here S(...) is a definite integral, p(x) is a function of the probability density function of series X. To determine Ymin, the integral is taken from 0 to some X1. As the result we obtain an analytic equation (if the integral is taken in the analytic form) with respect to X1. Then, by calculating the average value of X over this interval [0,X1] we obtain Ymin.
Similarly, to determine Ymax we take the integral from X2 to infinity. By determining X2 we can then determine Ymax.
And the physical meaning of this is more than transparent. Ymin is the muving value at M lowest values of X, Ymax is the muving value at M highest values of X. It is clear that these two values are not exact. In the sense that for the existing data they are unlikely to be achieved in the calculation of the real series of muvings. However, Ymax and Ymin were originally needed as statistical marginal estimates. I hope no one will claim that they will never be achieved in the future. :-)
And the marginal estimates for the cases M=1 and M=N are the same as what I wrote earlier.
The estimates for Ymax and Ymin could be refined. But that's exactly what the muvinge distribution function is for.
So, I'm ready to listen to criticism.
Mathemat, the thing is that I am a theorist. That's my specialty. Everyone has their own shortcomings. So it's a lost cause to urge me to give up any theoretical venture. It's like urging an alcoholic to quit drinking. :-) But thank you for taking part (in my destiny). :-))
By the way, could you tell me more about convolution of distributions?
Yurixx, don't listen to anyone (discussants please, no offence).
Do what you think is right. It's a good thing if you manage to keep up your efforts. There is nothing worse than giving up. A man is born by himself, dies by himself and lives by himself; and all his own experiences are his alone. It doesn't matter so much what it turns out to be. I mean, important, of course, but the value in the movement as such is much higher. Good luck.
The convolution: see e.g. http://www. nsu.ru/mmf/tvims/chernova/tv/lec/node39.html#2933. Well, you can find a lot about convolution of distribution functions. The important thing is that here the distribution of the sum of two independent variables is calculated.
Thank you. My gut feeling was that something similar should be (I mean the solution to the problem, not the formula itself), but out of ignorance I didn't know what. :-)
2 SK
Thank you Sergei. "Movement is everything, the goal is nothing" is the slogan of the anarchists. And you and I stick to the middle way. So I accept your wishes in that sense. By the way, sometimes you have to quit. Or even very necessary. You are not going to argue that if a person has foolishly or ignorantly stumbled into some drawn-out dogma and finally, oh miracle, realizes his mistake, he shouldn't give it up anyway ?
And if there's nothing worse than quitting, is that I'm going to spend the rest of my life on forex ? :-)))