Technique of mathematical modeling of the program of development of the agricultural enterprise. Mathematical methods of forecasting in enterprise management

Yurashev Vitaly Viktorovich c.f.-m. PhD, scientific director of the company "Gradient"

Shelest Igor Vladimirovich system architect at Jet Infosystems

The forecast in business is important because of the possible use of it for the stabilization effect. Reasonable forecasts encourage people to act more rationally and prevent them from "overreacting" towards pessimism or optimism. good forecast provides the firm with rational decision-making regarding the goods or services produced by the firm. The lack of a forecast forces the company's management to take unnecessary precautions.

Forecasting methods usually cost a lot of time and money. However, a businessman needs methods that do not require complex deductions in everyday work and can be presented in the form of programs. It is necessary to find forecasting methods without detailed individual analysis. In addition, it is desirable that the knowledge of the situation on the market, which people who constantly work on it, be used in such models.

Since forecasting is a difficult problem, it is clear that a firm must have several series of forecasts other than a simple descriptive forecast. This will help to take more decisive action, the result of which is an increase in profits, an increase in the efficiency of the organization and the growth of its prestige.

The input data for making a forecast using time series are usually the results of sample observations of variables - either intensity (for example, demand for a product) or state (for example, price). Decisions to be made in this moment, will affect in the future after a certain period of time, the value of which can be predicted.

Time series are data sorted in time. In accordance with this, we will henceforth denote the period of time by t, and the corresponding data value by y(t). Note that the members of the time series are either sums or numerical information received at a certain point in time. For example, the sum of weekly store sales at the end of each week for a year forms a time series.

Trend means the general direction and dynamics of the time series. In this definition, the emphasis is on the concept of “general direction”, since the main trend must be separated from short-term fluctuations, which are cyclical and seasonal fluctuations. Examples of cyclic fluctuations: prices for industrial raw materials, stock prices, sales volumes in wholesale and retail trade, etc. Seasonal fluctuations occur in time series describing sales, production, employment, etc. Important role seasonal fluctuations are played by weather conditions, fashion, style, etc. We especially note that irregular or random fluctuations in time series do not follow any pattern and there is no theory that can predict their behavior.

From the point of view of making the right decision by the management of the company, the inclusion of periodic (cyclical and seasonal) fluctuations in the overall model can improve the efficiency of the forecast and allow predicting the expected high and low values ​​of the predicted variables. It must be borne in mind, however, that "business" or economic cycles cannot be reproduced with accuracy that allows in practice to draw conclusions about future ups and downs based on an analysis of the past.

The paper presents linear, cyclic and "exponential" trends. A few words about the exponential trend. An analysis of the life cycle of goods, services, innovations and reflections on the processes taking place around showed that the model of development and death of biological systems is an effective tool for studying many phenomena in business. Moreover, as in business, the indicators of the functioning of a biological system over time are not linear at all stages of its development. The life cycles mentioned above have been simulated and their time elasticity has been found to be a linear function. The coefficients of this function make it possible to take into account not only the nonlinear mechanisms of life cycles, but also to predict their occurrence. As a result, we got a trend, which we called "exponential" because it includes a temporary exponent.

Consider a time series y(1), y(2),...(y(i),...y(T). It is required to represent the function for which this series is given by a trigonometric polynomial. The periodic components of the polynomial are unknown. The advantage of such model is that it ensures the stability of the forecast by enumeration of frequencies Coefficients are calculated using the entire data set.

In practice, such a model turns out to be difficult for the user. Therefore, a computer program was developed. Checking for compliance with the background is carried out using the least squares method (see: Taha A. Operations Research. M .: Williams, 2005). In many cases, changes in the process under study can be foreseen in advance and included in the presented forecast model. After all, experienced leaders can predict the nature of change. The program includes trend matching due to the optimal choice of frequencies in the presented series. To correct the forecast, one can vary not only trends, but also take into account the results of a subjective forecast.

We will look for a trend in the form: Y(t) = C + Asin(wt) + Bcos(wt).

Since the values ​​of this function at points 1, 2, ... T are known, we get a system of T linear equations with respect to the coefficients A, B, C, w is a parameter.

We solve this system by the least squares method (T>3) and obtain the values ​​of the coefficients A, B, C, depending on w. It is necessary to choose the values ​​of w in such a way that the trend values ​​would best approximate the values ​​of the time series. Optimization is carried out by the method of successive approximations. The initial value of w, which is the beginning of successive approximations, is found according to the formulas presented, for example, in the reference book on mathematics by the authors G. Korn, T. Korn, (M .: Nauka, 1989. Ch. 20).

We subtract from the actual (i.e., initially given as members of the time series) values ​​y(1), y(2),...y(i),....y(t) the found theoretical values ​​y(t) at times t =1, 2,...,i,...T. For the data obtained (considering them to be actual, i.e., members of the time series), we repeat the above procedure.

Forecast accuracy is 1-3%, sometimes it fluctuates up to 5-10%. It all depends on the presence of noise, which can significantly affect the forecast. If the retrospective series is large, then the program well identifies the regular components of the process. With a small retrospective time series (up to 5-8 values), exponential smoothing should be used. The exponential smoothing method is based on the moving average. But it eliminates the disadvantage of the moving average method, which is that all the data used to calculate the average has the same weight. In particular, the exponential smoothing method assigns a larger weight to the most recent observation. It, as well as the method presented in this paper, is especially effective in forecasting time series with cyclical fluctuations without strong random fluctuations (see: Taha A. Operations Research).

Let's give an example of calculating the forecasted sales volume (Tables 1, 2).

Table 1. Initial data

Table 2. Forecast Calculation Using Sinusoidal Trend

The results of the calculation are presented in the form of graphs in Figure 1 (the theoretical function is a black dash, the initial data is black, the trend is gray).

Rice. 1. Calculation of the forecasted sales volume according to the sinusoidal trend

Here is an example of using an exponential trend to calculate a sales forecast.

This example considers the change in sales during and after the advertising campaign (Tables 3, 4).

Table 3 Initial data

Table 4 Forecast calculation using exponential trend

The results of the calculation are presented in the form of graphs in Figure 2 (the theoretical function is a gray stroke, the initial data is black, the trend is gray).

Rice. 2. Calculation of the forecasted sales volume according to the exponential trend

The software product developed by us, adapted to work in specific conditions, has versatility, reliability and resistance to changing conditions. In addition, and this is essential, it is possible to increase the number of tasks to be solved. So, for example, when forecasting sales volumes, it is possible to solve the problem of the influence of each indicator (advertising, exhibitions, the Internet) on the amount of profit.

One of the advantages of the project is its cheapness. Therefore, the results obtained can be compared with those obtained by other methods. Their difference will give management a reason to conduct more in-depth research.

The program is easy to use, it is enough to enter the necessary data from the information field into the program. The only difficulty may be in obtaining personal data. Difficulties arise when creating an information field in which to work.

It all depends on the conditions in which the data must be obtained (in the field or in the laboratory). The ability of experts to build a quasi-information field simplifies the work at the preliminary stage of the study, but at the same time the “field” highlight of the project is lost.

The value of the project is also in the mobility of solving the tasks, quick response to changes environment, easy correction of changes and additions when working on a specific task.

Special courses and special seminars in the spring semester 2018/2019 academic year

03/25/2019:14:35 - 16:10 s / c masters "Analysis of graphs, networks, similarity functions", Maisurase A.I., 507 lesson will not take place March 25 (Monday), the lecturer is sick;
16:20 – 17:55 s/c bachelors “Analytical SQL”, Maysuradze A.I., 582 class will not take place March 25 (Monday), the lecturer is sick.
02/27/2019: Educational and research seminar "Data Mining: New Tasks and Methods", leaders S.I. Gurov , A.I. Maisuradze Wednesdays in aud. 704, start at 18-05. March 04 (Monday) I. S. Balashov will report at the special seminar (VVO, 3rd year) "Study of the microbiome during pregnancy using graph theory methods". It is known that microorganisms living in different loci of the body interact with each other and form communities called the microbiome, and the totality of these microorganisms is called the microbiota. For a number of diseases, the microbiota has been shown to be a risk factor for the development of certain diseases. Data on the composition of the microbiota can be presented in the form of a graph, and then explore the features of this graph in normal and pathological conditions. The paper will present the features of the subject area and their influence on the choice of methods for describing and analyzing data, and present basic models that describe the microbiome.

• 02/27/2019: Logical data analysis in recognition, (Logical data analysis in recognition) lecturer E.V. Dyukova, takes place on Mondays in room. 645, beginning at 16-20. First session February 25th. The course will present general principles, underlying discrete methods of information analysis in the problems of recognition, classification and forecasting. Approaches to the design of recognition procedures based on the use of the apparatus of logical functions and methods for constructing coverages of Boolean and integer matrices will be considered. The main models will be studied and issues related to the study of the complexity of their implementation and the quality of solving applied problems will be considered. Special course for bachelors of 2-4 courses. A textbook has been published for the special course.

• 02/27/2019: Probabilistic Thematic Modeling(Probabilistic topic modeling), lecturer, professor of the Russian Academy of Sciences, Ph.D. K.V. Vorontsov, takes place on Thursdays in room. 510, starting at 18-05. First session February 14th. Topic modeling is a modern area of ​​research at the intersection of machine learning and computational linguistics. The topic model defines which topics are contained in a large text collection, and which topics each document belongs to. Topic models allow you to search for texts by meaning, and not by keywords, and create a new type of information retrieval services to systematize knowledge. The special course deals with topic models for classification, categorization, segmentation, summation of natural language texts, as well as for recommender systems, analysis of banking transactional data and biomedical signals. From mathematics, we need probability theory, optimization methods, matrix expansions. For programming enthusiasts, there is an opportunity to participate in the BigARTM.org open source project. For those who are especially enthusiastic, there are additional seminars in the evenings at the Yandex office. The assignments for the course will be solving problems from real life who don't have the correct answer at the end of the tutorial. A special course for undergraduates, but second-year students will also understand everything :) 18+ (for students who have known the theory).

• 02/27/2019: Problems and algorithms of computational geometry(Computational Geometry: Problems and Algorithms), L.M. Mestetsky, takes place on Fridays in the room. 607, beginning at 18-05. First session February 15th. Efficient algorithms for working with geometric information are an indispensable attribute of all modern systems of machine vision, image analysis and recognition, computer graphics and geoinformatics. Geometric algorithms provide a good field for the development of algorithmic thinking, which is necessary in applied mathematics. In the first part of the special course, classical topics of computational geometry will be considered: geometric search, convex hulls, intersection and proximity of objects, Voronoi diagrams, Delaunay triangulations. The second part of the course is devoted to skeletons, generalizations of Voronoi diagrams for polygons, and problems of medial image shape analysis. Bachelors are invited.

• 02/27/2019: Machine learning and search of regularities in data, lecturer O.V. Senko, takes place on Thursdays in the auditorium. 507, starting at 18-05. First session February 14th. The course discusses the main problems that arise when using precedent-based learning methods (machine learning). Given short review existing methods of recognition and regression analysis. It tells about the methods for assessing the accuracy on the general population (generalizing ability). Various ways to increase the generalizing ability of machine learning methods are discussed. Bachelors are invited.

• 02/27/2019: Analysis of graphs, networks, similarity functions(Graphs, Network, Distance Function Analysis), A.I. Maisuradze, takes place on Mondays in room. 582, beginning at 16-20. First session February 18th. Problems and methods of systems analysis are considered, the description of which is based on pairwise or multiple interaction of objects. These objects can be of the same type or of different types. When the very presence or absence of interaction is important, formalization is carried out in the language of graph theory. Extending the graph description with quantitative characteristics leads to networks. If it is believed that each set of objects can be numerically characterized, one speaks of distances or similarities. The theoretical basis for the formalization of tasks and the construction, implementation and analysis is presented. a wide range models and methods of IAD. We study heuristic data models that describe the initial information about recognition objects based on various implementations of the concept of similarity. Problems that need to be solved when implementing these models are considered. We study special data structures and algorithms that allow you to effectively configure and use the studied models. The idea of ​​similarity is inherent in human thinking; this has given rise to a whole range of approaches for all fundamental tasks of IAD - the so-called metric methods. Methods for constructing and calculating similarity functions, matching similarity on different sets of objects, and synthesizing new ways of comparing objects based on existing ones are considered. A set of techniques designed for efficient representation and processing of metric information by computer systems is considered. The characteristics of graphs that are actively used in their analysis are considered. Algorithms on graphs are studied - both theoretically and from the point of view of efficient implementation. Various graph growth models. Construction of representative samples on graphs. Generation of graphs with given characteristics. Significant attention in the course is given to numerous formalizations of cluster analysis. It is shown what problems are solved by common methods. The typology of a wide range of clustering problems for homogeneous and heterogeneous systems (biclustering, coclustering) has been carried out. Special course for undergraduates.

• 02/27/2019: Analytical SQL(Analytical SQL), A.I. Maisuradze, takes place on Mondays in room. 507, starting at 2:35 p.m. First session February 18th. Nowadays, automation and optimization of many activities is impossible without the collection and subsequent analysis of large amounts of information. At the same time, over time, it became clear that some data models are especially convenient for people - such models have become a universal language of communication with a variety of technologies. In this sense, SQL turned out to be one of the most widely used languages, and today a variety of technologies (not only relational ones) make it possible to use it. In the know on practical examples knowledge will be given and skills will be developed that will be needed by almost any analyst when working with data sources. The emphasis is on analytical activities: the analyst uses data collection and storage systems, but is not going to administer them. Classes involve interactive execution of tasks on real databases. Special course for bachelors.

Mathematical forecasting methods can be developed on the basis of various functions, time series and analytical dependencies. For mathematical modeling and forecasting of currency markets, both price dynamics and its derivatives (indicator values, significant levels, etc.) and market data can serve as input information. macroeconomic indicators. In mathematical models for forecasting financial time series, price dynamics is used as input. However, the work with time series information models, which are descriptions of original objects using diagrams, graphs, formulas, drawings, etc., is different. One of the most important types of information modeling is mathematical, when descriptions are formulated in the language of mathematics. Accordingly, the study of such models is carried out using mathematical methods.

Mathematically, the problem of forecasting the exchange rate can be reduced to the problem of approximating multidimensional functions and, consequently, to the problem of constructing a multidimensional mapping. Depending on the type of output variables, the approximation of functions can take the form of: classification or regression. Therefore, in forecasting models exchange rates, two major subtasks can be distinguished: 1. building a mathematical model; 2nd training of expert networks that implement the solution of the problem. As a result of studying the subject area, a mathematical forecasting model should be developed, including a set of input variables; the method of forming input features and the method of training the expert system.

Analytical dependencies

Consider the features forecasting models based on analytical dependencies.

This model is based on the analysis of the exchange rate formation mechanism. The type of formula in this case will depend on the nature and type of interacting factors influencing the formation of the exchange rate. The model is based on the hypothesis of purchasing power parity. Further, in the process of considering real economic systems, new factors will be added, and the generalized model will select the main factors influencing the formation of the exchange rate.

Improving the efficiency of short-term currency transactions is one of the important tasks in the activities of banks and other investors who sell and buy various currencies in significant volumes, seeking to give movement to available free reserves in order to avoid losses from market fluctuations in exchange rates and receive additional profit. And currency operations are carried out at high speed via the Internet, since it is very important to enter the foreign exchange market with an offer before competitors. All this is an integral part of the continuous process of forming the optimal structure of foreign exchange reserves.

The effectiveness of foreign exchange transactions depends to a large extent on the reliability of forecasts of currency fluctuations. That is why short-term forecasting of rates is of great practical importance for the operational activities of banks and other investors. And the question of the possibility of using statistical methods for this purpose seems relevant and natural. Problem short-term Forecasting exchange rates using statistical models is considered based on the fact that for the successful conduct of foreign exchange transactions, it is required to obtain forecasts for one day in advance. As, for example, in the film "Pi" mathematician Max Cohen has been trying for many years to find and decipher the universal digital code, according to which the rates of all change. As you get closer to the solution, the world around Max turns into a dark nightmare: he is pursued by powerful analysts from Wall Street in order to discover the code of the universal universe. On the brink of madness, Max must make a decisive choice between order and chaos and decide whether he is able to cope with the powerful force that his brilliant mind has now awakened. But this is fantasy. In reality, it is not hard work, but the train of thought that determines investment income, and only adequate mathematical modeling can serve to evaluate the effectiveness of an idea.

Adaptive forecasting methods

It is difficult to draw a clear line separating adaptive forecasting methods from non-adaptive ones. Even forecasting by the method of extrapolation of ordinary regression curves contains some element of adaptation, when with each new receipt of actual data, the parameters of the regression curves are recalculated and refined. After a sufficiently long period of time, even the type of the curve can be changed. However, here the degree of adaptation is very small; moreover, over time, it decreases along with an increase total observation points and, accordingly, with a decrease in the proportion of each new point in the sample.

The sequence of the adaptation process is as follows. Let the model be in some initial state, and a prediction is made on it. When one time unit (simulation step) expires, we analyze how far the result obtained by the model is from the actual value of the series. Prediction error through feedback enters the system input and is used by the model in accordance with its logic for the transition from one state to another in order to better coordinate its behavior with the dynamics of the series. The model must respond to changes in the series with compensating changes. Then a prediction is made for the next point in time, and the whole process is repeated. Thus, the adaptation is carried out interactively with the receipt of each new actual point of the series. However, what should be the rules for the transition of the system from one state to another, what is the logic of the adaptation mechanism?

In essence, this question is solved by each researcher intuitively. The logic of the adaptation mechanism is given a priori and then tested empirically. When constructing a model, we inevitably endow it with innate properties and, at the same time, for greater flexibility, we must take care of the mechanisms of conditioned reflexes that are acquired or lost with a certain inertia. Their totality constitutes the logic of the adaptation mechanism. Due to the simplicity of each individual model and the limited initial information, often represented by a single series, one cannot expect that any one adaptive model is suitable for predicting any series, any behavioral variations. Adaptive Models flexible enough, but their versatility cannot be counted on. Therefore, when constructing and explaining specific models, it is necessary to take into account the most probable patterns of development of the real process, and correlate the dynamic properties of the series with the capabilities of the model. It is necessary to put into the model those adaptive properties that are enough for the model to track the real process with a given accuracy.

However, one cannot hope for a successful model self-adaptation, more general in relation to the one that is necessary to reflect this process, because an increase in the number of parameters makes the system excessively sensitive, leads to its buildup and deterioration of the forecasts obtained from it. Thus, when building an adaptive model, one has to choose between a general and a particular model and, weighing their advantages and disadvantages, give preference to the one from which one can expect the smallest forecast error. Therefore, it is necessary to have a certain stock of specialized models, diverse in structure and functional properties. To compare possible alternatives, a model utility criterion is needed. While such a criterion is generally controversial, in the case of short-term forecasting, the accepted criterion is usually the mean square of the prediction error. The quality of the model is also judged by the presence of autocorrelation in the errors. In more advanced systems, the process of trial and error is carried out as a result of the analysis of both sequential in time and parallel (competing) modifications of the model.

Short-term exchange rate forecasting

Information about the dynamics of exchange rates creates the impression of a chaotic movement: falling and rising rates replace each other in some random order. Even if over a long period of time there is a trend, for example, to growth, then on the chart you can easily see that this trend makes its way through complex movements. exchange rate time series. The direction of the series changes all the time under the influence of irregular and often unknown forces. The object under study is fully exposed to the elements of the world market, and there is no exact information about the future movement of the exchange rate. You need to make a prediction. At the same time, it is quite obvious that predict even the sign of the growth rate very difficult. This is usually done by experts who analyze the current market conditions and also try to identify factors that are regularly associated with the movement of the exchange rate (fundamental analysis). When building formal models, they also try to identify a range of significant factors and construct some kind of indicator on their basis, but neither expert practitioners nor formal methods have so far given good stable results. We believe this is explained, first of all, by the fact that if there really is any circle of factors that influence the exchange rate in a stable way, then their impact is reliably hidden by a superimposed random component and control actions.

As a result, these factors and their influence are difficult to isolate. Therefore, it is necessary to consider short-term rate forecasting as a matter of fact the task of predicting the consistent movement of an isolated time series, the reason for which is mainly the massive behavior of small and large financial players in the foreign exchange market, which make the bulk of financial transactions with the currency. This approach can be attributed to Of course, a single participant in the currency game is free to completely arbitrarily change his strategy. And yet it can be assumed that the behavior of the entire mass of participants through the supply and demand ratio, which affects the exchange rate, has in the current period of time some certain dominant logic, which is revealed through the law of large numbers. For example, when the exchange rate falls, they can buy it, expecting further appreciation in the future. And such a massive demand for the currency really leads to an increase in its exchange rate. Or vice versa, if, after the fall of the currency, the confidence in it falls and its further depreciation is expected, then the mass supply prevails and the rate falls even lower. Note that with such a simplified approach, the very dynamics of the time series can be read as a chronological record of the mass behavior of currency market participants. This makes it possible, when building a model, to proceed from the series itself, without involving additional information, and to use all arguments about the mass behavior of market participants only for a qualitative interpretation. If it were possible to find in the dynamics of the series at least short-term patterns that are realized with a probability of more than 50%, then this would give reason to count on success. Then it would be possible to apply statistical methods to predict rates, capturing more or less stable relationships of successive events in the time series.

In this case, the following task is posed. First, find out the applicability for short-term forecasting of exchange rates of any statistical methods, the purpose of which is to describe recurring events or situations characterized by relatively stable relationships. Secondly, if statistical methods are applicable to solve the problem, then establish their most promising class, indicate characteristics these methods, pay special attention to the simplest of them. Thirdly, to show practical results by example. It should be noted that much attention has always been paid to the issues of forecasting exchange rates. From publications on a related topic, we point out, for example, the work of K. Granger and O. Morgenstern (Granger Clive W.J., Morgenstern Oscar. Predictability of stock market prices. Massachusetts, 1970), which examines the dynamics of stock prices and provides an extensive bibliography. This monograph actually concludes that if there is any in series of this kind, then it is most likely that it exists between adjacent rate increments. However, the question arises whether we are trying to predict completely random fluctuations in exchange rates. The answer to this question is in a special study.

Modern forecasting

A new look at the role of forecasting has established itself as an indispensable element of the decision-making process. The logical consequence of the strengthening of the role of forecasting was an increase in the requirements for the validity and reliability of forecast estimates. However, the level of compliance of the apparatus of modern prognostication with these new requirements remains extremely low. Even the use of adaptive models, with the help of which, as a rule, it is possible to achieve the required level of adequacy in the description of predicted processes, only partially solves the problem of increasing reliability. The modern economy generates processes with such complex dynamics that the identification of its patterns by the apparatus of modern forecasting often turns out to be an insoluble task. The improvement of this apparatus, first of all, needs new ideas and new approaches, on the basis of which it is possible to implement the mechanisms and ways of reflecting the dynamics formed under the influence of effects, the possibility of which in the future is not found in the data of the historical period. There is a clear contradiction, the overcoming of which will contribute to the formation of a new view of forecasting as a proactive reflection in a probabilistic environment ideas about the process under study in the form of a trajectory built on the basis of objective trends and subjective expectations.

Within the framework of economic forecasting, the development of an adaptive approach occurs in three directions. The first one is mainly focused on complications adaptive predictive models. The idea behind the second direction is improvement adaptive mechanism of forecasting models. In the third direction, the approach is implemented sharing adaptive principles and other forecasting methods, in particular, simulation modeling. The works of V.V. Davnis.

Market development is determined, but the opposite is also true - fundamental factors are determined by the market, i.e. the behavior of market participants, their assessments and expectations. At the same time, the ability to give a correct assessment of the development of market situations depends on the ability to anticipate the prevailing expectations of market participants, and not on the ability to predict changes in real world. Therefore, the ideas for the development of the mathematical apparatus of forecasting do not sufficiently take into account the properties of the activity of economic systems, which reduces the level of plausibility of forecast estimates even with high interpolation accuracy. At the same time, forecasts based only on subjective information are focused on predicting qualitative characteristics, and therefore their use is possible only in special cases. This brings to the fore the problem of building forecasts based on a combination of extrapolation and subjective estimates. Studies were carried out in this area, however, the analysis of the results of these studies showed the predominance of a creative nature in them, which, in fact, indicates the initial level of development of the problem of constructing combined forecasts.

Literature

1. Sobolev V.V. Currency dealing in financial markets / Yuzh.-Ros. state tech. un-t (NPI). - Novocherkassk, 2009. - 442 p.
2. Lukashin Yu. P. Adaptive methods of short-term forecasting of time series: Proc. allowance. - M.: Finance and statistics, 2003. - 416 p.
3. Davnis V.V., Tinyakova V.I. Adaptive Models: Analysis and Forecast in Economic Systems. - Voronezh: Voronezh Publishing House. state un-ta, 2006.– 380 p.
4. Mishkin F. Economic theory of money, banking and financial markets: Tutorial for universities / Per. from English. D.V. Vinogradov, ed. M.E. Doroshenko. – M.: Aspect Press, 1999. – 820 p.
5. Lukashin Yu.P. On the Possibility of Short-Term Forecasting of Currency Rates Using Simple Statistical Models // Bulletin of Moscow State University. -1990. — Ser. 6. Economy. -No. 1.-S. 75-84.
6. Sobolev V.V. Financiers / South-Ros. state tech. un-t (NPI).–Novocherkassk, 2009.–315 p.
7. Soros J. Alchemy of Finance: Translated from English. – M.: “Infra-M”, 1996. – 416 p.

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Question number 25. Mathematical forecasting methods .

Forecasting methods- scientific prediction based on the analysis of the actual data of the past and present of the object under study. The set of special rules, techniques and methods is forecasting technique. The forecast in the control system is a pre-planned development of multivariant models for the development of the control object. The main forecasting methods include: economic and mathematical, analog, expert, etc. ^ Economic and mathematical methods of forecasting:

linear programming, allowing to formulate the optimization problem in the form of linear constraints (inequalities or equalities) and a linear objective function;

dynamic programming, designed for solving multistage optimization problems;

integer Programming, allowing to solve optimization problems, including problems of optimal resource allocation, with discrete (integer) values ​​of variables, etc.;

probabilistic and statistical models – are implemented in the methods of queuing theory;

game theory – modeling of such situations, decision-making in which should take into account the discrepancy between the interests of various departments;

simulation models – make it possible to experimentally check the implementation of solutions, change the initial prerequisites, and clarify the requirements for them.

pattern (PATTERN - Planning Assistance Through Technical Evaluation Relevance) - the technique was developed in 1963, it is used in planning research and development under conditions of uncertainty (i.e., in complex, inconsistent systems). The main elements of the pattern structure: selection of the forecast object; identification of the internal patterns of the object; script preparation; formulation of the task and general goal of the forecast; hierarchy analysis; formulation of goals; adoption of internal and external structure; questioning; mathematical processing of questionnaire data; quantitative assessment of the structure; verification; development of a resource allocation algorithm; allocation of resources; evaluation of distribution results. The technique makes it possible to obtain a pre-forecast orientation, to form the internal structure of the object (“tree of goals”), the external structure (system of local criteria), to develop options for resource provision of the elements of the object.

Method of exploratory forecasting.

One of the main methods used in exploratory forecasting is the extrapolation of time series - statistical data about the object of interest to us. Extrapolation methods are based on the assumption that the law of growth that took place in the past will continue in the future, taking into account corrections due to the possible saturation effect and the stages of the object's life cycle. Among the curves that accurately reflect the change in the predicted parameters in a number of common situations is the exponential, that is, a function of the form: y=a*ebt, where t is the time, a and b are the parameters of the exponential curve. Among the most famous exponential curves used in forecasting are the Pearl curve, derived from extensive research in the field of growth of organisms and populations, and having the form: Y = L / (1 + a * (e-bt), where L is the upper limit of y variable.

No less common is the Gompertz curve, derived from the results of research in the field of income distribution and mortality (for insurance companies), where k is also an exponential parameter.

The Pearl and Gompertz curves were used to predict parameters such as efficiency gains steam engines, the growth of the efficiency of radio stations, the growth of the tonnage of merchant fleet ships, etc. Both the Pearl curve and the Gompertz curve can be classified as so-called S-shaped curves. Such curves are characterized by an exponential or close to exponential growth at the initial stage, and then, when approaching the saturation point, they take on a flatter shape.

Many of the mentioned processes can be described using the corresponding differential equations, the solution of which is the Pearl and Gompertz curves. As an example, we can cite a differential equation that describes the increment in the amount of information (knowledge) I depending on the number of researchers N, the average productivity coefficient of one researcher q per unit time t, and C- a constant coefficient characterizing the dynamics of changes in the amount of information.

Extrapolation uses regression and phenomenological models. Regression models are built on the basis of the established patterns of development of events using special methods for selecting the type of extrapolating function and determining the values ​​of its parameters. In particular, the least squares method can be used to determine the parameters of the extrapolating function.

Assuming the use of one or another extrapolation model, one or another distribution law, it is possible to determine confidence intervals that characterize the reliability of predictive estimates. Phenomenological models are built on the basis of the conditions of maximum approximation to the trend of the process, taking into account its features and limitations, and accepted hypotheses about its future development.

With a multi-factor forecast in phenomenological models, it is possible to assign large weighting factors to factors that in the past had a greater influence on the development of events in the past.

If, when forecasting, a retrospective period is considered, consisting of several periods of time, then, depending on the nature of the forecasted indicators, less distant from the moment of forecasting on the time scale, etc. It should also be taken into account that often, when forecasting, experts' assessments of the near future may be overly optimistic, and assessments of the more distant future may be overly pessimistic.

If several different technologies can participate in the predicted process, each of which is represented by a corresponding curve, then the envelope of partial curves corresponding to individual technologies can be used as the resulting expert curve.

scripting method.

In the development of managerial decisions, the scenario method is widely used, which also makes it possible to assess the most probable course of events and the possible consequences of the decisions made. Scenarios for the development of the analyzed situation developed by specialists allow, with one level of certainty or another, to determine possible development trends, relationships between acting factors, to form a picture of possible states that the situation may come to under the influence of certain influences. Professionally developed scenarios allow you to more fully and clearly determine the prospects for the development of the situation, both in the presence of various control actions, and in their absence.

On the other hand, scenarios of the expected development of the situation make it possible to realize in a timely manner the dangers fraught with unsuccessful managerial actions or unfavorable developments.

Currently, various implementations of the scenario method are known, such as: obtaining a consensus opinion, a repeating procedure of independent scenarios, using interaction matrices, etc. The method of obtaining a consensus opinion is, in fact, one of the implementations of the Delphi method, focused on obtaining the collective opinion of various groups of experts relatively major events in a particular area in a given period of the future. The disadvantages of this method include insufficient attention paid to the interdependence and interaction of various factors influencing the development of events, the dynamics of the development of the situation.

The method of iterative combination of independent scenarios consists in the compilation of independent scenarios for each of the aspects that have a significant impact on the development of the situation, and the repeated iterative process of coordinating scenarios for the development of various aspects of the situation.

The advantage of this method is a more in-depth analysis of the interaction of various aspects of the development of the situation.

Its disadvantages include the insufficient development and methodological support of scenario coordination procedures.

The method of mutual influence matrices, developed by Gordon and Helmer, involves the determination, based on expert assessments, of the potential mutual influence of events in the population under consideration.

Estimates linking all possible combinations of events by their strength, distribution in time, etc., make it possible to refine the initial estimates of the probabilities of events and their combinations. The disadvantages of the method include the complexity of obtaining a large number estimates and their correct processing.

The paper proposes a methodology for compiling scenarios, which involves a preliminary definition of the space, parameters that characterize the system. The state of the system at time t is the point S(t) in this parameter space. Determination of possible trends in the development of the situation makes it possible to determine the probable direction of the evolution of the position of the system in the space of the identified parameters S(t) at various points in time in the future S(t+l), S(t+2), etc.

If there are no control actions, then it is assumed that the system will evolve in the most probable direction.

Control actions are equivalent to the action of forces capable of changing the direction of the trajectory S(t). Naturally, the control actions should be considered taking into account the limitations imposed by both external and internal factors.

The proposed technology for developing scenarios involves considering the position of the system at discrete times t, t+1, t+2, ... .

It is assumed that the point corresponding to the system S(t) in the parameter space is located in a cone that expands with distance from the initial time t. At some time t+T, the system is expected to be located in the section of the cone corresponding to time t+T.

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Plan

Introduction

1. Essence and classification of methods of economic and mathematical forecasting

1.1 Basic methods of economic and mathematical forecasting

1.2 Main ideas of the technology of scenario expert forecasts

2. Application of information technologies in economic and mathematical forecasting

Conclusion

List of used literature

Introduction

The economic system in our country, which took shape by the end of the 1980s, was characterized by a relatively high material and capital intensity of production, low rates of scientific and technological progress, and a significant imbalance in the economy. The emerging problems associated with low labor productivity, technical and technological backwardness, environmental degradation, low level industrial output and structural imbalances, had to solve economic reforms.

During several years of economic reforms, only a number of tactical tasks were solved, in particular, to achieve an improvement in the ratio between the money demand of the population and the supply of consumer goods. But this was achieved not by increasing the output of the latter, but by reducing the real incomes of the bulk of the population.

The current socio-economic situation in the Russian Federation is characterized by an acute structural crisis that has led to a sharp drop in living standards. This crisis is expressed, among other things, in a decrease in the output of industrial and consumer goods, and in a number of cases, in the termination of the production and economic activities of industrial enterprises. As a result of the current situation - a reduction in spending on social needs. Another important aspect of the crisis situation is the loss of not only international, but also domestic regional markets for products for domestic producers.

The fall in domestic production, of course, predetermines the need for a wide import of industrial, and especially consumer goods, in particular, such an important item as food. In turn, the expansion of imports requires the stimulation of exports in order to purchase foreign currency. But since domestic products currently do not have access to international markets (for various reasons - poor quality, lack of competitiveness, etc.), raw materials are exported - oil, gas, ores, wood, which has an extremely negative effect on the general state of the economy countries.

The problems that have arisen cannot be resolved even if the pace of the inflationary process slows down. Moreover, investments in small shares in many branches of industrial production are absolutely ineffective in the absence of clear, real planning and forecasting of economic processes.

The effectiveness of economic research and forecasts at the present time largely depends on how fully and accurately they reflect the characteristic features of economic processes. At the same time, the indicators that characterize the increase in complexity, speed, uncertainty and the possible number of alternatives for the implementation of economic processes have the most significant impact on the reliability and reliability of research.

It is necessary to prepare and make managerial decisions at the present stage in conditions of a high degree of dynamic change in economic processes, their sharply increased complexity, indeterminacy and nonlinearity. At the same time, when developing predictive options for the development of economic processes, it is necessary to take into account the complexity, consistency, multifactorial and multivariate nature of their further development.

The purpose of the work is to study the essence, classification and tools of economic and mathematical methods of forecasting.

1) to study the essence and classification of methods of economic and mathematical forecasting

2) consider the use of information technology in economic and mathematical forecasting

1. Essence and classification of methodseconomic and mathematicalforecasting

1.1 Basic methods of economic and mathematical forecasting

Let us briefly consider various methods of forecasting (prediction, extrapolation) used in the socio-economic field. There are a large number of publications on forecasting issues. As part of econometrics, there is a scientific and educational discipline "Mathematical Methods of Forecasting". Its purpose is to develop, study and apply modern mathematical methods of econometric (in particular, statistical, expert, combined) forecasting of socio-economic phenomena and processes, and the methods must be worked out to a level that allows them to be used in the practical activities of an economist, engineer and manager.

The main objectives of this discipline include the development, study and application of modern mathematical and statistical forecasting methods (including nonparametric least squares methods with estimation of forecast accuracy, adaptive methods, autoregressive methods, etc.), development of the theory and practice of expert forecasting methods, including including methods for analyzing expert assessments based on statistics of non-numerical data, forecasting methods under risk conditions and combined forecasting methods using jointly economic-mathematical and econometric (both statistical and expert) models. The theoretical basis of forecasting methods are mathematical disciplines (primarily probability theory and mathematical statistics, discrete mathematics, operations research), as well as economic theory, economic statistics, management, sociology, political science and other socio-economic sciences.

As is generally accepted since the days of the founder of scientific management, Henri Fayol, forecasting and planning are the basis of a manager's work. The essence of econometric forecasting is the description and analysis of future development, in contrast to planning, in which the future movement is set in a directive way. For example, the forecaster's conclusion may be that in an hour we can walk no more than 5 km from point A, and the planner's indication that in an hour we need to be at point B. It is clear that if the distance between A and B no more than 5 km, then the plan is real (feasible), and if more than 10 km, it cannot be implemented under the given conditions. It is necessary either to abandon the unrealistic plan, or to switch to other conditions for its implementation, for example, to move not on foot, but by car. The considered example demonstrates the possibilities and limitations of forecasting methods. Namely, these methods can be successfully applied under the condition of some stability in the development of the situation and fail with abrupt changes.

One of the applications of forecasting methods is to identify the need for changes by "reduction to the absurd". For example, if the population of the Earth doubles every 50 years, then it is not difficult to calculate how many years later there will be 10,000 people per square meter of the Earth's surface. From such a forecast it follows that the patterns of population growth must change.

Accounting for undesirable trends identified in the course of forecasting makes it possible to take the necessary measures to prevent them, and thereby hinder the implementation of the forecast.

There are also self-fulfilling predictions. For example, if an evening television program predicts the imminent bankruptcy of a certain bank, then in the morning many depositors of this bank will wish to receive their money, a crowd will gather at the entrance to the bank, and banking operations will have to be stopped. Journalists describe this situation with the words: "The bank burst." Usually, it is enough for this that at one "perfect" (for the bank) moment, depositors wish to withdraw a significant share (say, 30%) of funds from deposit accounts.

Forecasting is a particular type of modeling as the basis of knowledge and control.

The role of forecasting in the management of a country, industry, region, enterprise is obvious. It is necessary to take into account STEP factors (social, technological, economic, political), factors of the competitive environment and scientific and technological progress, as well as forecasting the costs and incomes of enterprises and society as a whole (in accordance with the life cycle of products - in time and in 11 stages of the international standard ISO 9004). The problems of implementation and practical use of mathematical methods of econometric forecasting are primarily related to the lack of a sufficiently extensive experience of such studies in our country, since for decades planning was given priority over forecasting.

Statistical forecasting methods. The simplest methods for recovering the dependencies used for predicting are based on a given time series, i.e. a function defined at a finite number of points on the time axis. In this case, the time series is often considered within the framework of a probabilistic model, other factors (independent variables) are introduced, in addition to time, for example, the amount of money supply (aggregate M2). The time series can be multidimensional, i.e. the number of responses (dependent variables) may be more than one. The main tasks to be solved are interpolation and extrapolation. The method of least squares in the simplest case (a linear function of one factor) was developed by K. Gauss more than two centuries ago, in 1794-1795. Preliminary transformations of variables may be useful.

Experience in forecasting the inflation index and the cost of the consumer basket has been accumulated at the Institute of High Statistical Technologies and Econometrics. At the same time, it turned out to be useful to transform (logarithm) a variable - the current inflation index. Characteristically, under stable conditions, the accuracy of forecasting turned out to be quite satisfactory - 10-15%. However, the significant increase in the price level predicted for the autumn of 1996 did not materialize. The fact is that the country's leadership has switched to a strategy of curbing the growth of consumer prices through massive non-payment of wages and pensions. Conditions have changed - and the statistical forecast has turned out to be unusable. The influence of the decisions of the Moscow leadership was also manifested in the fact that in November 1995 (before the parliamentary elections) prices in Moscow fell by an average of 9.5%, although usually November is characterized by a faster rise in prices than in other months of the year, except for December and January.

The most commonly used method is the least squares method with several factors. Least moduli and other extrapolation methods are less commonly used, although their statistical properties are often better. Tradition and the general low level of knowledge about econometric methods of forecasting play an important role.

Evaluation of forecast accuracy is a necessary part of the qualified forecasting procedure. In this case, probabilistic-statistical dependence recovery models are usually used, for example, they build the best forecast using the maximum likelihood method. Parametric (usually based on the model of normal errors) and non-parametric estimates of forecast accuracy and confidence limits for it (based on the Central Limit Theorem of probability theory) have been developed. Thus, we have proposed and studied methods of confidence estimation of the point of overlap (meeting) of two time series and their application to assess the dynamics of the technical level of our own products and competitors' products presented on the world market.

Heuristic techniques are also used that are not based on any theory: the method of moving averages, the method of exponential smoothing.

Adaptive forecasting methods allow you to quickly correct forecasts when new points appear. We are talking about adaptive methods for estimating model parameters and adaptive methods for nonparametric estimation. Note that with the development of computing power of computers, the problem of reducing the amount of computation loses its significance.

Multivariate regression, including the use of nonparametric distribution density estimates, is currently the main econometric forecasting tool. We emphasize that it is not necessary to use the unrealistic assumption about the normality of measurement errors and deviations from the regression line (surface). However, to abandon the normality assumption, it is necessary to rely on a different mathematical apparatus based on the multidimensional central limit theorem of probability theory and the econometric technology of linearization. It allows you to perform point and interval estimation of parameters, check the significance of their difference from 0 in a non-parametric formulation, and build confidence limits for the forecast.

The problem of checking the adequacy of the model, as well as the problem of selecting factors, is very important. The fact is that the a priori list of factors influencing the response is usually very extensive, it is desirable to reduce it, and a large area of ​​modern econometric research is devoted to methods for selecting an "informative set of features." However, this problem has not yet been finally resolved. Unusual effects appear. Thus, it has been established that commonly used estimates of the degree of a polynomial have a geometric distribution. Non-parametric methods for estimating the probability density and their application for restoring the regression dependence of an arbitrary form are promising. The most general statements in this area are obtained using non-numerical data statistics approaches.

Modern statistical forecasting methods also include autoregressive models, the Box-Jenkins model, systems of econometric equations based on both parametric and non-parametric approaches.

To establish the possibility of applying asymptotic results for finite (so-called "small") sample sizes, computer statistical technologies are useful. They also allow you to build various simulation models. Note the usefulness of data propagation methods (bootstrap methods). Computer-intensive forecasting systems combine various forecasting methods within a single forecaster workstation.

Prediction based on data that is non-numeric, in particular, prediction of qualitative features is based on the results of statistics of non-numeric data. Very promising for forecasting are regression analysis based on interval data, including, in particular, the determination and calculation of the note and rational sample size, as well as regression analysis of fuzzy data. The general formulation of regression analysis within the framework of statistics of non-numerical data and its particular cases - analysis of variance and discriminant analysis (pattern recognition with a teacher), giving a unified approach to formally different methods, is useful in the software implementation of modern statistical forecasting methods.

Expert forecasting methods. The need and general understanding of the application of expert forecasting methods in decision-making at various levels of management - at the level of the country, industry, region, enterprise. We note the great practical importance of expertise in comparing and selecting investment and innovative projects, in project management, and environmental reviews. The roles of decision makers (DM) and specialists (experts) in decision-making procedures, decision-making criteria and the place of expert assessments in decision-making procedures are discussed above. As examples of specific expert procedures widely used in forecasting, we point out the Delphi method and the scenario method. On their basis, specific procedures for preparing and making decisions are formed using methods of expert assessments, for example, procedures for distributing research funding (based on scores or paired comparisons), feasibility studies, desk marketing research (as opposed to "field" selective research ), evaluation, comparison and selection of investment projects.

In relation to the tasks of forecasting, let us recall some aspects of planning and organizing an expert study. A working group and an expert commission should be formed. Very important stages are the formation of the goals of the expert study (collection of information for the decision maker and / or preparation of a draft decision for the decision maker, etc.) and the formation of the composition of the expert commission (methods of lists (registries), "snowball", self-assessment, mutual assessment) with a preliminary solution to the problem a priori preferences of experts. Various options for organizing an expert study, differing in the number of rounds (one, several, not fixed), the order of involving experts (simultaneously, sequentially), the method of taking into account opinions (with weights, without weights), the organization of communication between experts (without communication, in absentia, face-to-face restrictions ("brainstorming") or without restrictions) allow you to take into account the specifics of a particular expert study. Computer support for the activities of experts and the Working Group, the economic issues of conducting an expert study are important for the successful conduct of an expert study.

Expert assessments can be obtained in various mathematical forms. The most commonly used are quantitative or qualitative (ordinal, nominal) signs, binary relations (rankings, partitions, tolerances), intervals, fuzzy sets, results of paired comparisons, texts, etc. Basic concepts of (representative) measurement theory: basic types of scales, acceptable transformations , adequate conclusions, etc. - are important in relation to expert evaluation. It is necessary to use average values ​​corresponding to the main scales of measurement. With regard to various types of ratings, the representative theory of measurements makes it possible to determine the degree of their adequacy in the prognostic situation, to suggest the most useful ones for forecasting purposes.

For example, an analysis of the ratings of politicians in terms of their influence, published by one of the well-known national newspapers, showed that due to the inadequacy of the mathematical apparatus used, only the first 10 places may have some relation to reality (they do not change when switching to another method of data analysis). , i.e. do not depend on the subjectivism of the members of the Working Group), the rest are "information noise", attempts to rely on them in predictive analysis can only lead to errors. As for the initial section of the rating of this newspaper, it can also be questioned, but for deeper reasons, for example, related to the composition of the expert commission.

The main procedures for processing predictive expert assessments are consistency check, cluster analysis and finding a group opinion.

Checking the consistency of expert opinions, expressed by rankings, is carried out using the Kendall and Spearman rank correlation coefficients, the Kendall and Babington Smith rank concordance coefficient. Parametric models of paired comparisons - Thurstone, Bradley-Terry-Lews - and non-parametric models of the theory of Lucians (about Lucians) are used.

In the absence of consistency, the splitting of expert opinions into groups of similar ones is carried out by the nearest neighbor method or other methods of cluster analysis (automatic classification building, pattern recognition without a teacher). The classification of Lucians is carried out on the basis of a probabilistic-statistical model.

Various methods of constructing the final opinion of the commission of experts are used. The method of average ranks stands out for its simplicity. Computer modeling made it possible to establish a number of properties of the Kemeny median, which is often recommended for use as the final (generalized, average) opinion of a commission of experts. The interpretation of the law of large numbers for non-numerical data in terms of expert survey theory is as follows: the final opinion is stable, i.e. changes little with a change in the composition of the expert commission, and with an increase in the number of experts, it approaches the "true". At the same time, in accordance with the accepted approach, it is assumed that the answers of experts can be considered as measurement results with errors, all of them are independent identically distributed random elements, the probability of accepting a certain value decreases with distance from a certain center - "truth", and the total number of experts is sufficient great.

Problems of application of forecasting methods under risk conditions. There are numerous examples of situations associated with social, technological, economic, political, environmental and other risks. It is in such situations that forecasting is usually necessary. There are various types of criteria used in the theory of decision making under conditions of uncertainty (risk). Due to the inconsistency of the decisions obtained according to various criteria, the need to apply expert assessments is obvious.

In specific forecasting tasks, it is necessary to classify risks, set the task of evaluating a specific risk, carry out risk structuring, in particular, build cause trees (other terminology, failure trees) and consequences trees (event trees). The central task is to build group and generalized indicators, for example, indicators of competitiveness and quality. Risks must be taken into account when predicting the economic consequences of decisions made, the behavior of consumers and the competitive environment, external economic conditions and macroeconomic development of Russia, the ecological state of the environment, the safety of technologies, and the environmental hazard of industrial and other facilities. The scenario method is indispensable in relation to the analysis of the technical, economic and social consequences of accidents.

There is some specificity in the application of forecasting methods in situations associated with risk. The role of the loss function and methods of its estimation is great, including in economic terms. In specific areas, probabilistic safety analysis (for nuclear power) and other special methods are used.

Modern computer technologies of forecasting. Interactive forecasting methods using econometric databases, simulation databases (including those based on the Monte Carlo method, i.e. the method of statistical tests) and economic and mathematical dynamic models combining expert, statistical and modeling blocks are promising. Let's pay attention to the similarities and differences between the methods of expert assessments and expert systems. We can say that the expert system models the behavior of an expert by formalizing his knowledge using a special technology. But the intuition of a "living expert" cannot be put into a computer, and when the expert's opinions are formalized (in fact, during his interrogation), along with the refinement of some of his ideas, others are coarsened. In other words, when using expert assessments, they directly turn to the experience and intuition of highly qualified specialists, and when using expert systems, they deal with computer algorithms for calculations and conclusions, the creation of which once upon a time involved experts as a source of data and standard conclusions.

Let us pay attention to the possibility of using production functions in forecasting that statistically describe the relationship between output and factors of production, to various ways of taking into account scientific and technological progress, in particular, based on trend analysis and with the help of expert identification of growth points. Examples of economic forecasts of all kinds are available in the literature. To date, computer systems and software tools for combined forecasting methods have been developed.

economic mathematical forecast informational

1. 2 Main ideas of the technology of scenario expert forecasts

As already noted, socio-economic forecasting, like any forecasting in general, can be successful only under some stability of conditions. However, the decisions of authorities, individuals, and other events change the conditions, and events develop in a different way than previously expected. Objectively, there are points of choice (furcations), after which the development considered by forecasters can go along one of several possible paths (these paths are usually called scenarios). The choice can be made at different levels - by a specific person (switch to another job or stay), a manager (produce one or another brand of products), competitors (cooperation or struggle), power structures (choice of a particular taxation system), the population of the country (choice president), "the international community" (to impose or not to impose sanctions against Russia).

Consider an example. It is quite obvious that after the first round of the presidential elections in 1996 about further development socio-economic events could be spoken only in terms of scenarios: if B.N. Yeltsin, then this and that will happen if G.A. wins. Zyuganov, then events will go this way and that way.

For example, the work was aimed at forecasting the dynamics of gross domestic product (GDP) for 9 years (1999-2007). When it was held, it was clear that various political events would take place during this time, in particular, at least two cycles of parliamentary and presidential elections (provided that the current political structure was maintained), the results of which could not be unambiguously predicted. Therefore, the forecast of GDP dynamics could only be made separately for each scenario from a certain range, covering the possible paths of the socio-economic dynamics of Russia.

The scenario method is needed not only in the socio-economic field. For example, when developing methodological, software and information support for risk analysis of chemical engineering projects, it is necessary to compile a detailed catalog of accident scenarios associated with leaks of toxic chemicals. Each of these scenarios describes an accident of its own type, with its own individual origin, development, technical, economic and social consequences, and warning capabilities.

Thus, the scenario method is a method of decomposition (dividing into parts) of the forecasting task, which provides for the selection of a set of individual options for the development of events (scenarios), which together cover all possible development options. At the same time, each individual scenario should allow for sufficiently accurate forecasting, and the total number of scenarios should be visible.

The possibility of such a decomposition is not obvious. When applying the scenario method, it is necessary to carry out two stages of the study:

Building a comprehensive but manageable set of scenarios;

Forecasting within each specific scenario in order to obtain answers to questions of interest to the researcher.

Each of these stages is only partially formalized. A significant part of the reasoning is carried out at a qualitative level, as is customary in the socio-economic and human sciences. One of the reasons is that the desire for excessive formalization and mathematization leads to the artificial introduction of certainty where it does not exist in essence, or to the use of a cumbersome mathematical apparatus. Thus, reasoning at the verbal level is considered evidence-based in most decision-making situations, while an attempt to clarify the meaning of the words used, using, for example, fuzzy set theory, leads to very cumbersome mathematical models and calculations.

In order to build an exhaustive, but visible set of scenarios, it is necessary to first analyze the dynamics of the socio-economic development of the considered economic agent and its environment. The roots of the future are in the present and the past, and often in the very distant past. In addition to macroeconomic and microeconomic characteristics, known only with errors that cannot be considered random or small, it is necessary to take into account the state and dynamics of domestic mass consciousness, political, including foreign policy realities, since in the usually considered time interval (up to 10 years), the economy often follows politics, and not vice versa.

So, for example, by the beginning of 1985, the USSR economy was in a fairly stable state with an annual growth of 3-5% on average. If the leadership of the country were in the hands of other people, then development would continue in the same conditions, and by the end of the millennium the GDP of the USSR would have increased by 50% and would have amounted to approximately 150% of the 1985 level. 15 years fell by about 2 times, i.e. amounted to about 50% compared with 1985, or 3 times less than could be expected from purely economic reasons, if the conditions of 1985 were stable.

The set of scenarios should be visible. We have to exclude various unlikely events - the arrival of aliens, the fall of an asteroid, mass epidemics of previously unknown diseases, etc.

In itself, the creation of a set of scenarios is the subject of an expert study carried out in accordance with the methodology described above. In addition, experts can assess the probabilities of the implementation of a particular scenario. It is clear that these estimates are not reliable.

A simplified approach to scenario forecasting is often used. Namely, they formulate three scenarios - optimistic, probable and pessimistic. At the same time, for each of the scenarios, the values ​​of the parameters describing the production and economic situation (in English - case) are chosen rather arbitrarily. The purpose of this approach is to calculate the scatter intervals for characteristics and "corridors" for time series of interest to the researcher (and research customer). For example, they predict a financial flow (in English - cash flow) and a net present value (in English - net present value or NPV) of an investment project.

It is clear that such a simplified approach cannot give the maximum or minimum value of the characteristic, it only gives an idea of ​​the order of the quantitative measure of the spread. However, its development leads to a Bayesian formulation in decision theory. For example, if a scenario is described by an element of a finite-dimensional Euclidean space, then any probability distribution on the set of initial parameters is transformed into a distribution of characteristics of interest to the researcher. Calculations can be carried out using modern information technology of the method of statistical tests. It is necessary, in accordance with the given distribution on the set of parameters, to select a specific vector of parameters using a pseudo-random number generator and calculate the final characteristics for it. The result is an empirical distribution on the set of final characteristics, which can be different ways analyze, find an estimate of the mathematical expectation, scatter, etc. It remains only unclear how to set the distribution on a set of parameters. Naturally, experts can be used for this.

Forecasting within each specific scenario in order to obtain answers to questions of interest to the researcher is also carried out in accordance with the forecasting methodology described above. Under stable conditions, statistical methods for forecasting time series can be applied. However, this is usually preceded by an analysis with the help of experts, and often forecasting at the verbal level is sufficient (to obtain conclusions of interest to the researcher and decision maker) and does not require quantitative clarification.

The question of using forecasting results is not related to econometrics, but to a related science - decision theory. As is known, when making decisions based on the analysis of the situation, including the results of predictive studies, one can proceed from various criteria. So, you can focus on the fact that the situation will develop in the worst, or best, or average (in any sense) way. You can try to outline activities that provide the minimum acceptable useful results in any scenario, etc.

So, the concept of a modern methodology for expert assessment by the scenario method is considered. It was used, for example, to predict the socio-economic development of Russia.

2. Application of information technologies in economics and mathematicsOpredicting

Before the advent of modern IT, there were no wide opportunities to use effective economic and mathematical models directly in the process of economic activity. In addition, the use of existing forecasting models for analytical purposes did not put forward such high requirements for their information support.

Fundamentals of forecasting technologies

When building a predictive system from scratch, it is necessary to resolve a number of organizational and methodological issues. The first ones include:

Training of users in methods of analysis and interpretation of forecast results;

Determining the directions of movement of predictive information within the enterprise, at the level of its divisions and individual employees, as well as the structure of communications with business partners and authorities;

Determining the timing and frequency of forecasting procedures;

Development of principles for linking the forecast with long-term planning and the procedure for selecting options for the results obtained when drawing up an enterprise development plan.

The methodological problems of building a forecasting subsystem are:

Development of the internal structure and mechanism of its functioning;

Organization of information support;

Development of software.

The first problem is the most difficult, since to solve it it is necessary to build a set of forecasting models, the scope of which is a system of interrelated indicators. The problem of systematization and evaluation of forecasting methods is one of the central ones here, since in order to select a specific method, it is necessary to conduct their comparative analysis. A variant of the classification of forecasting methods, taking into account the peculiarities of the knowledge system that underlies each group, can be summarized as follows: methods of expert assessments; methods of logical modeling; mathematical methods.

Each group is suitable for solving a certain range of tasks. Therefore, practice puts forward the following requirements for the methods used: they must be focused on a specific forecasting object, must be based on a quantitative measure of adequacy, and be differentiated in terms of the accuracy of estimates and the forecasting horizon.

The main tasks that arise in the process of creating a predictive system are divided into:

Building a system of predictable processes and indicators;

Development of an apparatus for economic and mathematical analysis of predicted processes and indicators;

Concretization of the method of expert assessments, selection of indicators for examination and obtaining expert assessments of some predicted processes and indicators;

Forecasting indicators and processes with indication of confidence intervals and accuracy;

Development of methods for interpreting and analyzing the results obtained.

The work on the information and mathematical support of the forecasting system deserves special attention. The process of creating software can be represented as the following steps:

Development of a methodology for structural identification of the object of forecasting;

Development of methods for parametric identification of the forecasting object;

Development of methods for predicting trends;

Development of methods for predicting the harmonic components of processes;

Development of methods for assessing the characteristics of random components of processes;

Creation of complex models for predicting indicators that form an interconnected system.

The creation of a forecasting system requires an integrated approach to solving the problem of its information support, which is usually understood as a set of initial data used to obtain forecasts, as well as methods, methods and tools that ensure the collection, accumulation, storage, search and transmission of data during the functioning of the forecasting system. and its interaction with other enterprise management systems.

Information support of the system usually includes:

Information fund (database);

Sources of formation of the information fund, flows and methods of data receipt;

Methods of accumulation, storage, updating and retrieval of data that form the information fund;

Methods, principles and rules of data circulation in the system;

Methods for ensuring the reliability of data at all stages of their collection and processing;

Methods of information analysis and synthesis;

Methods for an unambiguous formalized description of economic data.

Thus, the following main components are required to implement the forecasting process:

Sources of internal information, which is based on management and accounting systems;

Sources of external information;

Specialized software that implements forecasting algorithms and analysis of results.

Given the importance of solving the problem of forecasting for market participants, it is advisable to check the quality of the proposed methods and algorithms, as well as technologies in general, using specially selected (test) initial data. A similar verification method has been used for a long time in assessing the adequacy of mathematical tools designed for nonlinear optimization, for example, using the Rosenbrock and Powell functions.

Confirmation (or verification) of the quality and performance of the forecasting technology is usually carried out by comparing a priori known model data with their predicted values ​​and evaluating the statistical characteristics of forecast accuracy. Let's consider this trick in a situation where the process models are an additive set of the trend Tt, seasonal (harmonic) and random components.

At present, a wide variety of software tools have become widespread, providing, to one degree or another, the collection and analytical processing of information. Some of them, such as MS Excel, are equipped with built-in statistical functions and programming tools. Others, especially inexpensive accounting and management accounting programs, do not have such capabilities or analytical capabilities are not implemented in them sufficiently, and sometimes incorrectly. However, this is, unfortunately, inherent in some more powerful and multifunctional enterprise management systems. This situation is apparently explained by a shallow analysis on the part of the developers of the properties of the forecasting algorithms they have chosen and their uncritical application. For example, judging by the available sources, zero-order exponential smoothing is often used as the basis of predictive algorithms. However, this approach is valid only in the absence of a trend in the process under study. In fact, economic processes are non-stationary, and forecasting involves the use of more complex models than models with a constant trend.

It is interesting to trace the path of development of domestic automated banking systems from the perspective of the topic under consideration. The first banking systems were based on rigid technology, constantly requiring changes or additional software. This prompted developers of financial software, following the principles of openness, scalability and flexibility, to use industrial DBMS. However, by themselves, these DBMS turned out to be unsuitable for solving high-level analytical problems, which include the problem of forecasting. For this, it was necessary to use additional technologies for data storage and operational analytical processing, which ensured the operation of decision support systems for financial and credit institutions and for making forecasts. The same approach is used in complex enterprise management systems.

Another direction of modern applied use of forecasting methods based on IT is the solution of a wide range of marketing tasks. An illustration is the SAS Churn Management Solution for Telecommunications software. It is intended for telecommunications operators and allows, according to its developers, to build predictive models and use them to assess the likelihood of an outflow of certain categories of customers. The basis of this software is the Scalable Performance Data Server distributed database server, tools for building and administering data warehouses and data marts, Enterprise Miner data mining tools, SAS / MDDB Server decision support system, as well as auxiliary tools.

To ensure the competitiveness of newfangled CRM-systems in the list of their advanced features, as well as for automated banking systems, reporting functions are included that use OLAP technologies and allow, to a certain extent, to predict the results of marketing, sales and customer service.

There are quite a lot of specialized software products that provide statistical processing of numerical data, including individual elements of forecasting. These products include SPSS, Statistica, etc. These tools have both advantages and disadvantages, which significantly limit the scope of their practical application. It should be noted here that the assessment of the suitability of specialized mathematical and statistical software tools for solving forecasting problems by ordinary users who do not have special training requires a separate serious study and discussion.

However, solving forecasting problems for consumers from small and medium-sized businesses with the help of powerful and expensive information systems and technologies is practically impossible, primarily for financial reasons. Therefore, a very promising direction is the development of the analytical capabilities of existing and widespread low-cost accounting and management accounting systems. Additional reports developed based on specific business processes and containing the necessary analytical information for a particular user have high attitude"efficiency - cost".

Some software developers create entire lines of analytical tools. For example, Parus Corporation offers Parus-Analytics and Triumph-Analytics solutions for a wide range of users from small and medium-sized businesses. More complex tasks of analytical processing of forecast information are integrated into the Parus system in the form of a so-called situational center. According to Dmitry Sudarev, manager for the development of circulation solutions, it was decided to develop and implement software products that allow moving from simple accounting of facts in the enterprise's activities to information analysis. At the same time, a transition was planned from automating the work of accountants and middle managers to processing information for top management. Taking into account the possible circle of consumers, Parus-Analytics and Triumph-Analytics do not impose special requirements on the software and hardware environment, however, the Triumph-Analytics solution is implemented on the basis of MS SQL Server, which provides it with greater opportunities for predicting the processes under study , in particular, the harmonic component of forecasts is taken into account.

The value of the forecast increases many times when it is directly used in the management of the enterprise. Therefore, an important direction is the integration of predictive systems with systems such as Kasatka, MS Project Expert, etc. strategic planning. Such a purpose predetermines the need to identify long-term trends and take them into account in planning. The forecasting horizon is determined based on the relevant goals of the organization.

Conclusion

Thus, to date, quite a lot of research has been carried out and impressive practical solutions have been obtained for the problem of forecasting in science, technology, economics, demography, and other fields. Attention to this problem is due, among other things, to the scale modern economy, the needs of production, the dynamics of the development of society, the need to improve planning at all levels of management, as well as the accumulated experience. Forecasting is one of the decisive elements of the effective management of individual business entities and economic communities due to the fact that the quality of decisions made is largely determined by the quality of forecasting their consequences. Therefore, decisions made today should be based on reliable estimates of the possible development of the studied phenomena and events in the future.

The improvement of forecasting by many experts is seen in the development of appropriate information technologies. The need for their application is due to a number of reasons, including: the growth of information volumes; the complexity of the algorithms for calculating and interpreting the results; high requirements for the quality of forecasts; the need to use forecasting results to solve planning and control problems.

From time to time there is information about the positive results achieved by a particular company. A number of publications note that a successful assessment of trends in the market situation, demand for goods or services, as well as other economic processes and characteristics allows you to get a significant increase in profits, improve other economic indicators. At first glance, the mechanism of success is simple and clear: assuming what will happen in the future, effective measures can be taken in a timely manner, using positive trends and compensating for negative processes and phenomena.

Accuracy, reliability and efficiency, however, as well as other components of the quality of forecasting, are provided by a number of factors, among which it is necessary to highlight: software based on economic and mathematical models adequate to reality; n completeness of coverage and reliability of sources of initial information on which operation of forecasting algorithms; Efficiency of processing internal and external information; the ability to critically analyze forecast estimates; the timeliness of making the necessary changes in the methodological and information support of forecasting.

Special software is based on carefully selected models, methods and techniques. Their implementation is extremely important for obtaining high-quality forecasts when solving problems of current and strategic planning. An analysis of the current situation shows that the difficulties in introducing IT, which provide forecasting of economic processes, are not only technical or methodological, but also organizational and psychological in nature. Consumers of the results sometimes do not understand the principles of the models used, their formalization and objectively existing limitations. This, as a rule, gives rise to distrust of the results obtained. Another group of implementation problems is related to the fact that predictive models are often closed, autonomous, and therefore their generalization for the purpose of development and mutual adaptation is difficult. Therefore, a compromise solution may turn out to be a phased approach with the allocation of the main analytical tasks.

However, there are practically no ready-made replicated or corporate solutions that provide forecasting for small and medium-sized economic entities at the system level with high quality and affordable prices. Currently, automated enterprise management systems are limited mainly to elementary tasks of accounting and control.

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