Development of a Method for Forecasting Random Events during Instability Periods

The object of research is random events in the formation of new economic and financial models; in particular; with cardinal changes in economic and social strategies. The scope and variety of methods used in the prediction of random processes is large. Promising mathematical apparatus for solving the problem are statistical methods of analysis. Today; there are many methods for predicting random processes; but most existing models are not suitable for predicting non-stationary processes. One of the most problematic places in forecasting time series is that there is no single methodology by which to analyze the characteristics of a non-stationary random process. Therefore; it is necessary to develop special methods of analysis that can be applied to individual cases of unsteady processes. The optimal solution to the problem may be the approximation of the time series by finely rational functions or the so-called Padé approximation. Such an approach should take advantage of polynomial approximation. In polynomial approximation; polynomial can’t have horizontal asymptotes; which makes it impossible to make long-term forecasts. A rational approximation is guaranteed to tend to horizontal asymptotes; with all the poles of the finely rational function lying on the left side of the p-plane; that is; the Laplace transform plane. A method for predicting non-stationary time series with high accuracy of estimation and flexibility of settings is proposed. To ensure the stability of the method and the stability of the obtained results; it is proposed that the poles of the approximating function be introduced into the stability zone – the unit circle of the z-plane in compliance with the rules of conformal transformation. Namely; by transforming linear dimensions and preserving the angles between the orthogonal coordinates on infinitely small neighborhoods of the coordinate plane (the so-called conservatism of angles). It is shown that; subject to the conformity of the proposed transformation; the dynamic characteristics of the estimation and forecasting system are stored. This method can be especially successfully applied in the presence of non-stationarity of various natures.


Introduction
In periods of instability that are characteristic of transi tion economies, changes in economic strategy, and especially during periods of global critical changes, incidents can oc cur that will affect fundamental operating principles. For prediction, and if possible, to prevent such events, powerful statistical methods of modeling and forecasting are needed.
A feature of the changes that are considered is pre cisely their suddenness, which occurs due to the lack of a priori information. A model such as a flash of white noise could serve as an adequate model, but as a result of an ultrawide band of such a process, it does not provide practical conclusions for making operational and correct management decisions. The desire to build a model that most closely matches the real situation, the immediate and adequate reaction of decision makers, and the need to improve the quality of forecasts lead to modifications of existing models and the emergence of new classes of models. But at the same time, it is necessary to carefully analyze the weaknesses of such models that are not obvi ous at first glance.
This work is devoted to the search for answers to these pressing questions.

The object of research and its technological audit
In the formation of economic strategies from the point of view of mathematical modeling, the problem arises of forecasting time series of various origins, since the influence of many factors. In this case, it is necessary to analyze many factors that determine the behavior of an object. When forecasting economic indicators, the requirements for forecasting accuracy are rather rigidly put forward; this leads to a constant search for new simple and adequate, as a rule, formal mathematical models. Thus, the object of the study is random events in the formation of new economic and financial models, in particular, with cardinal changes in economic and social strategy.

ISSN 2664-9969
One of the most problematic places in forecasting time series is that a single methodology by which it would be possible to analyze the characteristics of an unsteady random process does not exist in principle. Therefore, it is necessary to develop special methods of analysis that can be applied to individual cases of unsteady processes.
Since a unified approach to the analysis of the charac teristics of a nonstationary random process does not exist in principle, it is necessary to look for ways to solve the problem. Alternative approaches can be used for this. The first is to develop a classification of nonstationary random processes with an exhaustive mathematical description of all the characteristics of each class. Based on this mathe matical model, it is theoretically possible to synthesize a general method and an estimation algorithm. The second is to develop special analysis methods that can be applied only to individual classes of nonstationary processes.
According to the author of this work, the general pro cessing method and algorithm, which would cover all pos sible classes of nonstationary random processes, is too complex to obtain results of acceptable quality. As always, the optimum between the two alternative approaches -to serve all random processes with zero quality or to serve an empty set of random processes with ideal qualitylies somewhere in between.

The aim and objectives of research
The aim of this research is development of a powerful and accurate method for predicting nonstationary time series with the ability to evaluate the parameters of ran dom processes.
To achieve this aim, it is necessary to complete the following objectives: 1. To analyze existing approaches to forecasting sudden events.
2. To justify and choose an approach to forecasting random processes.
3. To evaluate the local characteristics of the process nonstationarity by the current implementation and the synthesis of the corresponding approximation methods and algorithms.

Research of existing solutions of the problem
Today, many models for predicting sudden events have been developed. Mathematical models of sudden changes can be built, for example, on the basis of the theory of emissions of random processes [1] or methods based on the theory of Markov processes [2]. However, these theories, with a sufficiently high degree of abstraction, rarely give practical results that could be applied to achieve real goals.
Recently, neural network forecasting models have gained popularity. The main disadvantage of the class of neural network models is the inaccessibility of intermediate cal culations performed by the system, and, as a result, the complexity of interpreting the simulation results. In addition, the weakness of this class of models is the difficulty in choosing an algorithm for training a neural network [3,4].
The role of statistical methods of analysis is not losing significance and is growing steadily. The scope and variety of methods used in the tasks of forecasting time series is great. Moreover, most of the proposed methods [5] relate to the analysis of stationary time series. But the question remains, if the series is unsteady, then these methods do not give a reliable result. Most often, by the nature of origin, the studied time series are the result of observ ing the behavior of complex systems whose deterministic description is impossible [6].
Literary sources describe various approaches to the analysis of time series. This is primarily due to the diver sity of the origin of time series. Among the main works devoted to solving this problem, one can single out works aimed at constructing adequate mathematical models of time series, in particular, financial time series [7,8]. These models are constructed using the results of systems analysis theory and conflict theory [9,10].
To predict nonstationary processes in most of the ana lyzed works, regression models are used taking into account the trend of various types [11,12]. However, this does not take into account the significant limitations of trend based forecasting results. In particular, in cases where the process involves a slow trend or jumps in mathematical expectation. A polynomial model is usually taken as a trend. Designing a trend for a long term into the future is im practical, because sooner or later the variable must stabilize, and the polynomial can't have a horizontal asymptote. In addition, the selection of the trend and the seasonal com ponent should be carried out using an iterative process, it provides at least two estimates of each component. As a result, the amount of computation will, as a rule, be significant even for highspeed computers [13].
A polynomial of even a high degree does not give a good prognosis. In any case, it can be used in this capacity only for a fairly near future. For values located at a consider able distance, this polynomial grows, and its derivative also grows. Accordingly, the forecast error also increases [14,15].
Thus, the results of the analysis allow to conclude that approximation of a number of finely rational functions or the socalled Pad approximation can give good results [16,17]. Such an approach should take advantage of polynomial approximation. In polynomial approximation, polynomial can't have horizontal asymptotes, which makes it impos sible to make longterm forecasts. A rational approximation is guaranteed to tend to horizontal asymptotes, with all the poles of the finely rational function lying on the left side of the pplane, that is, the Laplace transform plane.

Methods of research
The Pad approximation [18] represents a function in the form of the ratio of two polynomials. Using the Pad approximation with the help of a rational (more precisely, finely rational) function, it is possible to get rid of the restrictions associated with the expansion in a Taylor series.
While let's consider the approximated function as the function of a real variable. Following Baker [19], let's specify the coefficients of the polynomials. Obviously, they are determined by the coefficients of the expansion of the function in a Taylor series. Thus, if an expansion in a power series of the form is given: then the Pad approximation is a rational function of the form: the expansion of which in the Taylor series coincides with the expansion (1). Function (2) has L +1 coefficients in the numerator and M +1 coefficients in the denominator. The entire set of coefficients is determined accurate to a common fac tor. To simplify, it is possible to put one of the constant terms (a 0 or b 0 ) equal to unity, since this does not affect the dynamic properties of the process, which is subject to approximation. Let's put for definiteness b 0 1 = . Then let's have L +1 free terms in the numerator and M in the denominator of fraction (2), that is L M + +1 free terms. Then the coefficients of the expansion of the function L M [ ] in a Taylor series at powers 1 2 , , , , x x x L M  + must coincide with the corresponding coefficients of the se ries (1). The obvious relation follows from this: Let's multiply function (3) by the denominator of the fraction and find that: Comparing the coefficients in the formula (4) at the same degrees x - . min ,  As can be seen from expression (7) with an error, tends to zero unlimited, but here two problems arise: the construction of the Pad approximation and its convergence to the function is approximated. To construct the Pad approximation by expressions (6) and (7), it is necessary to have only the coefficients of the series (8).
Let's consider the problem of stability and convergence of the obtained solutions in the next section.

Research results
As it is known, any step series has its own region of conver gence -the circle of convergence of radius R. The series co incides at z R < , when z R > -diverges. If R → ∞, the series is a function analytic everywhere in the complex plane [18]. The value of the function at an arbitrary point z can be approximately obtained by direct summation of the series, and the error of approximation with an unlimited increase in the number of terms of the series monotonously tends to zero:

ISSN 2664-9969
If the sequence of Pad approximations of the formal power series (8) converges to a function f z ( ) in the circle of convergence   , , z ∈ in practical applications it is pos sible to safely assume that the Laurent series of type (8) corresponds which, in essence, plays the role of analytic elongation of polynomial (2) onto the plane of the complex variable z, will be stable. A positive answer to this question will take place when the convergence condition for the Pad approximation in the unit circle of the zplane is satisfied [17]. This condition is satisfied relatively simply.
In accordance with the main theorem of higher algebra [17], a polynomial of arbitrary degree (say, N degree) with real coefficients has exactly N roots that are either real or create complex conjugate pairs. Then the denominator of polynomial (2) and the purely formal denominator of polyno mial (9) have exactly M roots. These roots are the poles of the function of a complex variable (9). If these poles do not go beyond the limits of a single zplane, the object (digital filter, diffe rence equation, etc.) is stable. In other words, it returns to a stable state after completion of arousal.
Thus, when applying the Pad approximation, it is necessary to control the absolute values of the poles of the approximating polynomial. This is a kind of fee that it is possible to pay for the high accuracy of the Pad approximations and their convergence to horizontal asymptotes even with a low order of the approximating polyno mial [17,19]. In this regard, let's note that the approximating polynomial will retain its properties, the desire for horizontal asymptotes (in particu lar, the abscissa axis) when a simple condition is met. In expressions of type (9), the rule L M < must always be fulfilled; there is a degree of the numerator of function (9) should always be to a lesser extent the denominator.
At the end of this section, let's consider another subtle point in the construction of the Pad ap proximation. If in the approximating polynomial (9) one or several poles go beyond the limits of a single zplane, then the Laurent series is such that it diverges everywhere except for a point z = 0, and its application, in fact, becomes useless.
To force a polynomial of the form (9) to re turn to a stable state, it is necessary to return the «unreliable» poles inside the unit circle. To do this, it is necessary to make the pole modules less than unity, but so that the angular position of these poles does not change, that is, observe the rules of conservatism of angles [21]. The easiest way to do this is by representing the pole as a complex number in exponential form.  Fig. 1 shows the location diagram of the initial pole p m , which is modulo more than 1, which leads to instability of the function f z ( ), and the modified pole p mnew , which module is p p mnew m = 1 by definition less than 1. When a mirror reflection of the pole inside the unit circle, observing the conservatism of the angles, the dynamics of changes in the state of the object is not violated [22]. In Fig. 2, 3 as an example, the amplitudefrequency and pulse characteristics of a secondorder digital filter are shown; stable -impulse response tends to zero (Fig. 2) and unstable -impulse response increases unlimited (Fig. 3). Let's pay attention to the identity of the amplitude frequency characteristics in contrast to the difference in the impulse characteristics. Therefore, it can be argued that the proposed transformation is conformal.

SWOT analysis of research results
Strengths. To obtain the most accurate forecast results in the proposed method, an approach based on the Pad approximation is used. Using this method of identifying the parameters of random processes, the maximum forecast accuracy will be ensured.
Weaknesses. The negative side when applying the Pad approximation is that it is necessary to control the abso lute values of the poles of the approximating polynomial. However, this inconvenience is compensated by the high accuracy of the Pad approximations and their convergence to horizontal asymptotes, even with a low order of the approximating polynomial.
Opportunities. In the future, it is planned to develop a method for simplifying the analysis by decomposing a highorder approximating function into an additive com position of elementary units of the first or second order with real coefficients. To calculate the weight coefficients of elementary links, it is advisable to use the method of residue theory -a powerful and effective method of the theory of functions of a complex variable.
Threats. It is practically impossible for statistical mo dels to be «comprehensive» in the sense of including all relevant variables that affect the process para meters. The imperfect nature of statistical models is reinforced by real evidence that the output is often incomplete, inconsistently encoded, and generally «raw». The main factors influencing the characteristics of the studied processes are so different and unpredictable that the results of each new experiment are almost unique. Natu rally, statistical analysis needs to be carried out only on one sampleimplementation of a random process.

Conclusions
1. An analysis of existing approaches to fore casting sudden events has been carried out, which shows that today there are many methods for predicting random processes, including methods based on the theory of Markov chains, theories of random emissions, neural network models, etc.
2. It is shown that most of the existing mo dels are not suitable for predicting nonstationary processes. This is primarily due to the diversity of the origin of time series. It is shown that the optimal solution to the problem may be the ap proximation of the time series by finely rational functions or the socalled Pad approximation.
3. For quick and accurate prediction of global events during periods of instability, a new power ful method of statistical processing and forecast ing time series is proposed in the work, taking into account the specifics of nonstationarity of the process, assessing its local characteristics. The method for forecasting nonstationary time series is developed with high accuracy of estimation and flexibility of settings. This method can be especially successfully applied in the presence of nonstationarity of various natures. To ensure the stability of the method and the stability of the results obtained, it is proposed that the poles of the approximating function be introduced into the stability zone -the unit circle of the zplane in compliance with the rules of conformal transformation. Namely, by transform ing linear dimensions and preserving the angles between the orthogonal coordinates on infinitely small neighbor hoods of the coordinate plane (the socalled conservatism of angles). It is shown that, subject to the conformity of the proposed transformation, the dynamic characteristics of the estimation and forecasting system are stored.