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Methods of Definition Almost-Periods in Empirical Data


V.I. Kuzmin, A.F. Gadzaov

During the research of empirical data it is often required to solve the problem of movements dividing that is to divide an initial empirical sequence into trend and oscillating component. For mechanical systems, where the trend is described by the equation of movement of the centre of weights, its exception is not complicated. Solving the problem of division of movements, when the equations for a basic trajectory are not known, connected with great difficulties. Regular methods for the solution of this problem actually are not present. Standard methods of an exception of a trend are based on approximation of an initial sequence by certain dependence. After definition of parameters of the used model, it is considered that the trend equations are known now. Actually, use of this method does not guarantee an exception of a trend without loss of the essential information on process. It occurs because of discrepancy of used models to properties of real processes. The method of an exception of a trend for allocation of oscillations comparing to a constant level is considered. It is shown, that transformation of the empirical data to co-ordinates , where – value of the current characteristic at the moment of time t, leads to an exception of sectionally-exponential trend areas from initial dependence in the issue of that, the fluctuations comparing to a constant level remain. Researching the oscillations remained after an exception of a trend, similar problems occur. The certain structure is imposed to investigated fluctuations, for example, Fourier series. During the investigations, the physical meaning of received parameters is often lost. Oscillations, received as a result of an exception of a trend, are processed by a Johnson method, in which, for a discrete case, at n total numbers of readouts of the function f(t), which is set by experimental values, the average absolute difference for some test period is: It allows to receive a full set of the almost-periods as minimum of Johnson function without imposing to the data of some fixed functional dependence. Capabilities of the presented algorithms are illustrated by consideration of characteristics of the electrocardiogram and the USA Dow–Jones index of business activity. The possible mechanism of effect of change of the charge of a liquid current on a round pipe, under action of an external electromagnetic field is offered

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