Random Flow Theory and FOREX
One of the very first options that come to mind is to take the first difference of the initial process (i.e. calculate return). But I still have not got around to proving or disproving the hypothesis of stationarity, at least in a broad sense. It would be interesting to collaborate on this.
Yes this approach (approximately) to analysis was successfully done more than ten years ago
It all started with this image
It visualises the process of 'chaotic'
mixing of gases in motion.
You have to read the whole thing in English.
about the history of the Prediction Company
I would like to point out that the input stream is a tick, any transformation of it changes the characteristics of the stream. If returns[i] = Close[i] - Close[i+1], as you said in another thread, it is a linear transformation and does not affect the characteristics of the flow. But calculation of bars is a non-linear transformation that can be clearly seen when trying to test it; we take the history of 1-minute bars and generate a law inside a 1-minute bar for proper reproduction.
Unfortunately, we have to put up with this, because of connection breakdowns and the possibility of reconstructing only minute bars. In other words, it is better to use minute bars and other timeframes will increase nonlinearities, but we have already got plenty of nonlinearities with minute bars :)
First of all it's plotting ACF of a process. My attempts to compare ACF of ticks and minutes show that there are differences in a quiet market, but at the beginning of the strong movements there are. I think that Williams with his squat bar has paid attention to it and wanted to use it in his trading system.
For those who want to move in this direction of market research be sure to note that the concept of correlation and covariance are different in the foreign literature and in our literature, especially it is evident in the construction of the ACF process.
Well... Now instead of resting for the next month, I'll be doing equations and functions... :(((
Prival, let's forget about ticks: this is clearly a futile case. The tick distribution laws are very different for different instruments - and the ticks themselves come in extremely unevenly in time. Here are my tiny attempts: 'Ticks: amplitude and delay distributions'. It's really better to start with at least one minute.
The most important question I have for you is this. You have a history of, say, one year's closing prices, a transposed vector (r1, r2, ..., rN), where N is something like 6000. During the year, the oyra (EURUSD) has run 20-25 figures, i.e. 2000-2500 points. Hence, expectation on this interval (on the strongest trend) is about 0.3-0.4 points. At the same time, dispersion on the watch is dozens of times larger, somewhere near 10-15 points, i.e. not less than 25 times. So what we are counting here, covariance or correlation, is not too important, as the distribution itself is not too sharp, and its m.o. is many times smaller than s.c.o.
Do you have somewhere a unified procedure for checking the stationarity of this process in a broad sense? Oddly enough, there is very little information about it on the internet.
2 geometrr: I read the article, very fascinating. But there, however, it's more about chaos than a random process.
2 Red.Line: well, say something positive, if the subject is of interest...
While riding the underground, it occurred to me that we are talking about the same thing. Returns/delta_t is velocity, i.e. the increase in price over a certain time interval - velocity. If there is speed, then there is also the acceleration of the first, second and other derivatives. I'll try to get matrix F for the simplest variant and make a MathCad walk.
Statistical research will of course represent the process but what to do with it is not clear to me as well, because both the GIR and variance depend on the selected analysis interval and I don't know how to use them later when building a trading system. We must investigate the dynamics of the process, not the statics. And it's not right to use the hourly chart for analysis, IHMO, it's difficult to analyze it without 0.5, or to drive yourself into a non-linear transformation a priori, it may completely ruin your kidneys :-).
Concerning stationarity, yes definitions are available, but there is no criterion, and the main thing is a number, allowing to take a decision. (S>5 -> stationary, S<5 -> non-stationary). At least I have not come across.
I have used these concepts in practice, but it was long ago, and not for Forex. The idea is as follows ACF allows us to determine the time during which the process is correlated and it enables us to predict further movements with a certain accuracy. Here is an example on the picture, let us suppose it is 0.707 level and then non-stationarity.
Here, the correlation or covariance may be important for drawing ACF. I do not remember ACF being built once again.
I would like to build the indicator in MQL and run it to see how it behaves.
I wish Rosh could help, he just sends me links and nods :-)
I understand that any idea must be well presented (drawn, written). Don't scold the pianist, he plays as well as he can.
But the question keeps bugging me, maybe this topic is of interest only to me and the mathematician. At least write up, everything will be on top of the topic.
Well, the criterion of stationarity in the broad sense is sort of known - the constancy of m.o. and the dependence of ACF only on the difference of the arguments, not on each of them. Or is that wrong?
The variance is the same ACF, only with zero shift. But I cannot understand on what subintervals (roughly speaking, windows) inside the series to count these "partial ACs". There are some criteria, right? That is: what is "dependence of AC only on the difference of arguments"? This means that for a given difference in the arguments we must build several (many) different "partial ACs", and then investigate (statistically) the stationarity of the series of partial ACs obtained. It's a vicious circle...
I can't use your zip, until I downloaded Matcad. I'll have to download it and see what kind of monster it is...
One more thing: I have no plans to apply these statistical studies directly to the TS. The plans to apply it are in its testing.
I haven't read it yet but it's already interesting :)
You're right, you say it yourself but you don't understand it. I think so :-) "the constancy of the m.o. and the dependence of the ACF only on the difference of the arguments" is the key phrase. The question is how to use it. Just now read your thread about tics, you've come to the conclusion that clearly the process is not stationary. The type of ACF will tell us what is happening to the market, for example the delta function is clearly non-stationary news output gap, etc. The market has become stationary (flat or trend anyway), i.e. the ACF's main lobe has expanded and the market has become predictable.
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The idea of applying the apparatus of random flow theory to describe various processes occurring in nature appeared long ago. The most fundamental work in this field may be considered the work by Bolshakov I.A. Statistical Problems of Signal Flow Extraction from Noise. -M: Soviet Radio, 1969.
Bottom line (I will show in brackets what I mean by this term)
There is a stream of objects (world events) which is not directly observable, there is a statistically related stream of measurements (the current rate of say, EUR/USD). Measurements are made at discrete points in time and skipping of measurements is possible (a world event has occurred, but the exchange rate has not changed).
There is a certain correspondence between the observed object parameters and the parameters of the observed measurements: the area W of the parameter values corresponds to the area S of the parameter y values.
At the output of a measuring device (MT-terminal) along with measurements generated by signals from objects () there appear measurements generated by fluctuational noise and various kinds of interference, i.e. false measurements.
Ways of describing random walks:
Multivariate probability density function, which is a compact description of a random flow
Here is an arbitrary function of some class.
Description by means of moment functions
of which a special role in flow theory is played by the first order moment function, called the flow intensity (FE):
As a model of movement (the EUR/USD exchange rate trajectory)
Different hypotheses may be considered; let us suppose independent realizations of a homogeneous Markov process with a transition probability density such that the motion can be described by a linear difference equation of the form
(1)
where F is a known transition matrix,
wk is a noise with zero expectation E(wk)=0 and covariance matrix E(wk,wj)=Qkdk,j,
dk,j is a Kronecker symbol.
What does the use of this theory suggest at first glance:
1. Determine where the exchange rate is moving, where the useful component of the movement is and where the noise is.
2. To get away from qualitative definitions of flat, trend (flat on the clock, trend on the min). It seems to me that it is often understood in terms of created trading system (a lot of losses - flat), and if there is a profit - a trend (because a trend is a friend). And if we take another TS on the same time interval, then maybe the trend is the enemy :).
3. Go to the quantitative description of the flow - it has an intensity (perhaps it is Volume) and the parameters of speed, acceleration, etc. I am somehow convinced that there is NO flat or trend. There is only multidimensional and multidimensional movement, which changes its characteristics in time.
4. A flow may be stationary (parameters are constant in some interval of time) and non-stationary (gaps, spikes, exit or expectation of important news).
5. Theory makes it possible to study and analyse correlated flows.
6. And most importantly under certain conditions to predict the direction of movement.
As an example, I will give the trajectories modelled by formula (1) these trajectories have absolutely identical motion parameters (i.e. are stationary), and their external difference is generated by noise (wk).
If you have read to this place, have you encountered such an approach to market analysis? And if you can give me a link to read it, I need to think. Not everything is that simple. Many procedures and functions obtained by Bolshakov cannot be computed as required computational resources = infinity. The description of approaches to flow analysis is also too fundamental.
I can't see the formulas very well, I'll attach it in Word.