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If you take the point of view of Poincare, who initiated the study of such maps, one can imagine the market as a complex dynamic process of interaction of various factors, many of which are inaccessible to observation. His projection is visible market index. So is it possible, watching only for one dimensional projection of the attractor, to assess its real dimension (amount of information necessary for the forecast)?

Until recently, the question seemed impossible. But in 1981, the mathematician Teykenc proved the theory that some of the general properties of the whole attractor persist in its projections. Such a common feature is the embedding dimension of the attractor, or, as he called it Teykens, the dimension of the limiting capacity. Evaluation of embedded dimension answers the question about the true dimension of the attractor. How many detainees coordinates observed projections need to take in order to adequately project the real attractor space defined by these coordinates? By Remark Ruelle, they need to take at least twice that of the embedding dimension.

There remains the question about the choice of <correct> delay in the formation of space in detention coordinates. By Theorems Teykensa and Manet, for an infinite number of noise without delay, you can choose almost at random. However, in practice (observed sample is finite and there is noise in the a priori) choice significantly affects the quality of the projections and, as a consequence, the amount of information that can be extracted from it. To describe the d-dimensional space, which reduced phase portrait, is proposed to take d as independent coordinates of the observed series. This allows <consider> projection of the attractor at the optimum <angle>.

What's next?

Whether to close the answers all the questions in the predictability of the market? No, not close. There are still a few questions, the answers to that yet. For example, it is unclear which class of functions to look for the dependence of the future from the past. In the case of one-dimensional chaos, we were able to see that the function is approximated by a parabola. Such an opportunity will no longer be at the attractor dimension greater than three. One more question. According to our research, the market is not characterized by fixed laws and relationships. In contrast to the mathematical laws, market patterns are short term, and over time the market structure is changing. Market-related indicators at a time, not related to the other, and vice versa.

In terms of phase portraits, this means that all known parameters vary with time, the phase space, the time delay, the internal parameters of the equations and even the class of functional dependencies. Market shows clearly breathing pattern.

Pictured <interaction> between two prices - Open and Close - over time. The horizontal axis shows the time delay relative to the second of a series. The vertical axis - on the contrary, the delay time of the second row on the first. Physical time is directed from top left to bottom right. This diagonal produces no shift series with each other and reflect the instantaneous interaction of these parameters (with no lag and lead).

Green represents the strength and direction of mutual dynamics of: dark red color indicates the correlation series, dark blue - anticorrelation. If we now follow any diagonal from top left to bottom right, you can see how changing interaction parameters over time. At one time, the ranks of anti-correlated, then the time comes when they do not influence each other, then comes the period of antiphase dynamics. Such is the nature of the market. And hope to describe it as one model built once and for all, it would be naive. This places demands on adaptive models and methods used for forecasting and market analysis.

Alas, there are other unresolved issues. Nevertheless, the first step to solving the problem of the predictability of the market (including stock) are made. So, to answer the question posed in the title of this article, can affirmatively, qualifying only that not all assets are predicted equally well. Moreover, periods of <predictability> can be followed by periods <unpredictability>. Since there are mathematical <Tools> responding to these questions, experts can use them in a completely practical level - to predict the behavior of the market in general and <throwing> concrete actions with a degree of accuracy sufficient for profit.

## Unpredictable market? Per aspera ad astra

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