FEDERAL AGENCY FOR EDUCATION
State Educational Institution of Higher Professional Education "Ural State University. »
History department
Department of Documentation and Information Support of Management
Mathematical methods in scientific research
Course program
Standard 350800 "Documentation and documentation management"
Standard 020800 "Historical and Archival Studies"
Yekaterinburg
I approve
Vice-rector
(signature)
The program of the discipline "Mathematical Methods in Scientific Research" is compiled in accordance with the requirements university component to the mandatory minimum content and level of training:
graduate by specialty
Document management and documentation management support (350800),
Historical and archival science (020800),
on the cycle "General humanitarian and socio-economic disciplines" of the state educational standard of higher professional education.
Semester III
According to the curriculum of the specialty No. 000 - Documentation and documentation support for management:
The total labor intensity of the discipline: 100 hours,
including lectures 36 hours
According to the curriculum of the specialty No. 000 - Historical and archival science
The total labor intensity of the discipline: 50 hours,
including lectures 36 hours
Control measures:
Examinations 2 persons/hour
Compiled by:, Ph.D. ist. Sciences, Associate Professor, Department of Documentation and Information Support of Management, Ural State University
Department of Documentation and Information Support of Management
dated 01.01.01 No. 1.
Agreed:
Deputy chairman
Humanitarian Council
_________________
(signature)
(C) Ural State University
(WITH) , 2006
INTRODUCTION
The course “Mathematical Methods in Socio-Economic Research” is intended to familiarize students with the basic techniques and methods of processing quantitative information developed by statistics. Its main task is to expand the methodological scientific apparatus of researchers, to teach how to apply in practical and research activities, in addition to traditional methods, based on logical analysis, mathematical methods that help to quantitatively characterize historical phenomena and facts.
At present, the mathematical apparatus and mathematical methods are used in almost all areas of science. This is a natural process, it is often called the mathematization of science. In philosophy, mathematization is usually understood as the application of mathematics to various sciences. Mathematical methods have long and firmly entered the arsenal of research methods of scientists, they are used to summarize data, identify trends and patterns in the development of social phenomena and processes, typology and modeling.
Knowledge of statistics is necessary to correctly characterize and analyze the processes taking place in the economy and society. To do this, it is necessary to master the sampling method, summary and grouping of data, be able to calculate average and relative values, indicators of variation, correlation coefficients. An element of information culture is the ability to correctly format tables and plot graphs, which are an important tool for systematizing primary socio-economic data and visual presentation of quantitative information. To assess temporary changes, it is necessary to have an idea about the system of dynamic indicators.
The use of the methodology for conducting a selective study allows you to study large amounts of information provided by mass sources, save time and labor, while obtaining scientifically significant results.
Mathematical and statistical methods occupy auxiliary positions, supplementing and enriching the traditional methods of socio-economic analysis, their development is a necessary part of the qualifications of a modern specialist - a document specialist, a historian-archivist.
Currently, mathematical and statistical methods are actively used in marketing, sociological research, in collecting operational management information, compiling reports and analyzing document flows.
Quantitative analysis skills are required for the preparation of qualification papers, abstracts and other research projects.
The experience of using mathematical methods shows that their use should be carried out in compliance with the following principles in order to obtain reliable and representative results:
1) the general methodology and theory of scientific knowledge play a decisive role;
2) a clear and correct statement of the research problem is necessary;
3) selection of quantitatively and qualitatively representative socio-economic data;
4) the correctness of the application of mathematical methods, i.e. they must correspond to the research task and the nature of the data being processed;
5) a meaningful interpretation and analysis of the results obtained is necessary, as well as a mandatory additional verification of the information obtained as a result of mathematical processing.
Mathematical methods help to improve the technology of scientific research: increase its efficiency; they save a lot of time, especially when processing large amounts of information, they allow you to reveal hidden information stored in the source.
In addition, mathematical methods are closely related to such a direction of scientific and information activities as the creation of historical data banks and archives of machine-readable data. It is impossible to ignore the achievements of the era, and information technology is becoming one of the most important factors in the development of all spheres of society.
COURSE PROGRAM
Topic 1. INTRODUCTION. MATHEMATIZATION OF HISTORICAL SCIENCE
Purpose and objectives of the course. The objective need to improve historical methods by attracting the techniques of mathematics.
Mathematization of science, main content. Prerequisites for mathematization: natural science prerequisites; socio-technical prerequisites. The boundaries of the mathematization of science. Levels of Mathematization for the Natural, Technical, Economic and Human Sciences. The main regularities of the mathematization of science are: the impossibility of fully covering the areas of study of other sciences by means of mathematics; the correspondence of the applied mathematical methods to the content of the science being mathematized. The emergence and development of new applied mathematical disciplines.
Mathematization of historical science. The main stages and their features. Prerequisites for the mathematization of historical science. Significance of the development of statistical methods for the development of historical knowledge.
Socio-economic research using mathematical methods in pre-revolutionary and Soviet historiography of the 20s (, etc.)
Mathematical and statistical methods in the works of historians of the 60-90s. Computerization of science and dissemination of mathematical methods. Creation of databases and prospects for the development of information support for historical research. The most important results of the application of mathematical methods in socio-economic and historical-cultural research (, etc.).
Correlation of mathematical methods with other methods of historical research: historical-comparative, historical-typological, structural, systemic, historical-genetic methods. Basic methodological principles for the application of mathematical and statistical methods in historical research.
Topic 2 . STATISTICAL INDICATORS
Basic techniques and methods of statistical study of social phenomena: statistical observation, the reliability of statistical data. Basic forms of statistical observation, purpose of observation, object and unit of observation. Statistical document as a historical source.
Statistical indicators (indicators of volume, level and ratio), its main functions. Quantitative and qualitative side of a statistical indicator. Varieties of statistical indicators (volumetric and qualitative; individual and generalizing; interval and moment).
The main requirements for the calculation of statistical indicators, ensuring their reliability.
The relationship of statistical indicators. Scorecard. General indicators.
Absolute values, definition. Types of absolute statistical values, their meaning and methods of obtaining. Absolute values as a direct result of a summary of statistical observation data.
Units of measurement, their choice depending on the nature of the phenomenon under study. Natural, cost and labor units of measurement.
Relative values. The main content of the relative indicator, the form of their expression (coefficient, percentage, ppm, decimille). Dependence of the form and content of the relative indicator.
Comparison base, choice of base when calculating relative values. Basic principles for calculating relative indicators, ensuring the comparability and reliability of absolute indicators (by territory, range of objects, etc.).
Relative values of structure, dynamics, comparison, coordination and intensity. Ways to calculate them.
Relationship between absolute and relative values. The need for their complex application.
Topic 3. DATA GROUPING. TABLES.
Summary indicators and grouping in historical studies. Tasks solved by these methods in scientific research: systematization, generalization, analysis, convenience of perception. Statistical population, units of observation.
Tasks and the main content of the summary. Summary - the second stage of statistical research. Varieties of summary indicators (simple, auxiliary). The main stages of the calculation of summary indicators.
Grouping is the main method of processing quantitative data. Tasks of grouping and their significance in scientific research. Grouping types. The role of groupings in the analysis of social phenomena and processes.
The main stages of building a grouping: determining the population under study; the choice of a grouping attribute (quantitative and qualitative characteristics; alternative and non-alternative; factorial and effective); the distribution of the population into groups depending on the type of grouping (determining the number of groups and the size of the intervals), the scale for measuring signs (nominal, ordinal, interval); selection of the form of presentation of grouped data (text, table, graph).
Typological grouping, definition, main tasks, construction principles. The role of typological grouping in the study of socio-economic types.
Structural grouping, definition, main tasks, construction principles. The role of structural grouping in the study of the structure of social phenomena
Analytical (factorial) grouping, definition, main tasks, principles of construction, The role of the analytical grouping in the analysis of the relationship of social phenomena. The need for the integrated use and study of groupings for the analysis of social phenomena.
General requirements for the construction and design of tables. Development of the table layout. Table details (numbering, heading, names of columns and lines, symbols, designation of numbers). The method of filling in the information of the table.
Topic 4 . GRAPHIC METHODS FOR ANALYSIS OF SOCIO-ECONOMIC
INFORMATION
The role of graphs and graphic representation in scientific research. Tasks of graphical methods: providing clarity of perception of quantitative data; analytical tasks; characteristics of the properties of signs.
Statistical graph, definition. The main elements of the chart: chart field, graphic image, spatial references, scale references, chart explication.
Types of statistical graphs: line chart, features of its construction, graphic images; bar chart (histogram), defining the rule for constructing histograms in the case of equal and unequal intervals; pie chart, definition, construction methods.
Feature distribution polygon. Normal distribution of a feature and its graphic representation. Features of the distribution of signs characterizing social phenomena: oblique, asymmetric, moderately asymmetric distribution.
Linear relationship between features, features of a graphical representation of a linear relationship. Features of linear dependence in the characterization of social phenomena and processes.
The concept of a dynamic series trend. Identification of a trend using graphical methods.
Topic 5. AVERAGES
Average values in scientific research and statistics, their essence and definition. Basic properties of average values as a generalizing characteristic. Relationship between the method of averages and groupings. General and group averages. Conditions for the typicality of averages. The main research problems that averages solve.
Methods for calculating averages. Arithmetic mean - simple, weighted. Basic properties of the arithmetic mean. Peculiarities of calculating the average for discrete and interval distribution series. The dependence of the method of calculating the arithmetic mean, depending on the nature of the source data. Features of the interpretation of the arithmetic mean.
Median - an average indicator of the structure of the population, definition, basic properties. Determination of the median indicator for a ranked quantitative series. Calculation of the median for the indicator represented by the interval grouping.
Fashion is an average indicator of the population structure, basic properties and content. Determination of the mode for discrete and interval series. Features of the historical interpretation of fashion.
The relationship of the arithmetic mean, median and mode, the need for their integrated use, checking the typicality of the arithmetic mean.
Topic 6. INDICATORS OF VARIATION
The study of the fluctuation (variability) of the values of the attribute. The main content of measures of dispersion of the trait, and their use of research activities.
Absolute and average indicators of variation. Variational range, main content, methods of calculation. Average linear deviation. Standard deviation, main content, calculation methods for discrete and interval quantitative series. The concept of feature dispersion.
Relative indicators of variation. Oscillation coefficient, main content, methods of calculation. The coefficient of variation, the main content of the calculation methods. The meaning and specificity of the application of each indicator of variation in the study of socio-economic characteristics and phenomena.
Topic 7.
The study of changes in social phenomena over time is one of the most important tasks of socio-economic analysis.
The concept of dynamic series. Moment and interval time series. Requirements for the construction of dynamic series. Comparability in the series of dynamics.
Indicators of changes in the series of dynamics. The main content of the indicators of the series of dynamics. row level. Basic and chain indicators. Absolute increase in the level of dynamics, basic and chain absolute increases, methods of calculation.
Growth rates. Basic and chain growth rates. Features of their interpretation. Growth rate indicators, main content, methods for calculating basic and chain growth rates.
The average level of a series of dynamics, the main content. Techniques for calculating the arithmetic mean for moment series with equal and unequal intervals and for an interval series with equal intervals. Average absolute growth. Average growth rate. Average growth rate.
Comprehensive analysis of interrelated time series. Identification of a general development trend - a trend: the method of moving average, enlargement of intervals, analytical methods for processing time series. The concept of interpolation and extrapolation of time series.
Topic 8.
The need to identify and explain the relationships for the study of socio-economic phenomena. Types and forms of relationships studied by statistical methods. The concept of functional and correlation. The main content of the correlation method and the tasks solved with its help in scientific research. The main stages of correlation analysis. Peculiarities of interpretation of correlation coefficients.
Linear correlation coefficient, feature properties for which the linear correlation coefficient can be calculated. Ways to calculate the linear correlation coefficient for grouped and ungrouped data. Regression coefficient, main content, calculation methods, interpretation features. Coefficient of determination and its meaningful interpretation.
Limits of application of the main varieties of correlation coefficients depending on the content and form of presentation of the initial data. Correlation coefficient. Rank correlation coefficient. Association and contingency coefficients for alternative qualitative features. Approximate methods for determining the relationship between features: Fechner coefficient. Autocorrelation coefficient. Information coefficients.
Correlation coefficient ordering methods: correlation matrix, pleiades method.
Methods of multidimensional statistical analysis: factor analysis, component analysis, regression analysis, cluster analysis. Prospects for modeling historical processes for the study of social phenomena.
Topic 9. SAMPLE RESEARCH
Reasons and conditions for conducting a selective study. The need for historians to use methods of partial study of social objects.
The main types of partial survey: monographic, main array method, sample survey.
Definition of the sampling method, the main properties of the sampling. Sample representativeness and sampling error.
Stages of sampling research. Determination of the sample size, basic techniques and methods for finding the sample size (mathematical methods, table of large numbers). The practice of determining the sample size in statistics and sociology.
Methods for forming a sample population: proper random sampling, mechanical sampling, typical and nested sampling. Methodology for organizing selective censuses of the population, budget surveys of families of workers and peasants.
Methodology for proving the representativeness of the sample. Random, systematic sampling errors and observational errors. The role of traditional methods in determining the reliability of the sample results. Mathematical methods for calculating the sampling error. The dependence of the error on the volume and type of sample.
Features of the interpretation of the results of the sample and the distribution of indicators of the sample population to the general population.
Natural sample, main content, features of formation. The problem of the representativeness of a natural sample. The main stages of proving the representativeness of a natural sample: the use of traditional and formal methods. The method of the criterion of signs, the method of series - as ways of proving the property of the randomness of the sample.
The concept of a small sample. Basic principles of its use in scientific research
Topic 11. METHODS FOR FORMALIZING INFORMATION OF MASS SOURCES
The need to formalize information from mass sources to obtain hidden information. The problem of measuring information. Quantitative and qualitative features. Scales for measuring quantitative and qualitative features: nominal, ordinal, interval. The main stages of measuring source information.
Types of mass sources, features of their measurement. Methodology for building a unified questionnaire based on the materials of a structured, semi-structured historical source.
Features of measuring information of an unstructured narrative source. Content analysis, its content and prospects for use. Types of content analysis. Content analysis in sociological and historical research.
Interrelation of mathematical-statistical methods of information processing and methods of formalization of source information. Computerization of research. Databases and data banks. Database Technology in Socio-Economic Research.
Tasks for independent work
To consolidate the lecture material, students are offered tasks for independent work on the following topics of the course:
Relative indicators Average indicators Grouping method Graphical methods Indicators of dynamics
The performance of tasks is controlled by the teacher and is a prerequisite for admission to the test.
An indicative list of questions for the test
1. Mathematization of science, essence, prerequisites, levels of mathematization
2. Main stages and features of the mathematization of historical science
3. Prerequisites for the use of mathematical methods in historical research
4. Statistical indicator, essence, functions, varieties
3. Methodological principles for the use of statistical indicators in historical research
6. Absolute values
7. Relative values, content, forms of expression, basic principles of calculation.
8. Types of relative values
9. Tasks and main contents of the data summary
10. Grouping, main content and tasks in the study
11. The main stages of building a grouping
12. The concept of a grouping attribute and its gradations
13. Types of grouping
14. Rules for the construction and design of tables
15. Dynamic series, requirements for the construction of a dynamic series
16. Statistical graph, definition, structure, tasks to be solved
17. Types of statistical graphs
18. Polygon feature distribution. Normal distribution of the feature.
19. Linear relationship between features, methods for determining linearity.
20. The concept of a dynamic series trend, ways to determine it
21. Average values in scientific research, their essence and main properties. Conditions for the typicality of averages.
22. Types of average indicators of population. The relationship of averages.
23. Statistical indicators of dynamics, general characteristics, types
24. Absolute indicators of changes in time series
25. Relative indicators of changes in time series (growth rates, growth rates)
26. Average indicators of the dynamic series
27. Indicators of variation, main content and tasks to be solved, types
28. Types of non-continuous observation
29. Selective study, main content and tasks to be solved
30. Sample and general population, basic properties of the sample
31. Stages of sampling research, general characteristics
32. Determining the sample size
33. Ways of forming a sample population
34. Sampling error and methods for its determination
35. Representativeness of the sample, factors affecting representativeness
36. Natural sampling, the problem of representativeness of natural sampling
37. The main stages of the proof of the representativeness of a natural sample
38. Correlation method, essence, main tasks. Features of interpretation of correlation coefficients
39. Statistical observation as a method of collecting information, the main types of statistical observation.
40. Types of correlation coefficients, general characteristics
41. Linear correlation coefficient
42. Autocorrelation coefficient
43. Methods of formalization of historical sources: the method of a unified questionnaire
44. Methods of formalization of historical sources: the method of content analysis
III.Distribution of course hours by topics and types of work:
according to the curriculum of the specialty (No. 000 - document management and documentary support of management)
Name
sections and topics
Auditory lessons
Independent work
including
Introduction. Mathematization of science
Statistical indicators
Grouping data. tables
Average values
Variation indicators
Statistical indicators of dynamics
Methods of multivariate analysis. Correlation coefficients
Sample study
Information formalization methods
Distribution of course hours by topics and types of work
according to the curriculum of the specialty No. 000 - historical and archival science
Name
sections and topics
Auditory lessons
Independent work
including
Practical (seminars, laboratory work)
Introduction. Mathematization of science
Statistical indicators
Grouping data. tables
Graphic methods for analyzing socio-economic information
Average values
Variation indicators
Statistical indicators of dynamics
Methods of multivariate analysis. Correlation coefficients
Sample study
Information formalization methods
IV. Form of final control - offset
v. Educational and methodological support of the course
Slavko methods in historical research. Textbook. Yekaterinburg, 1995
Mazur methods in historical research. Guidelines. Yekaterinburg, 1998
additional literature
Andersen T. Statistical Analysis of Time Series. M., 1976.
Borodkin statistical analysis in historical research. M., 1986
Borodkin informatics: stages of development // New and recent history. 1996. No. 1.
Tikhonov for the humanities. M., 1997
Garskov and data banks in historical research. Göttingen, 1994
Gerchuk methods in statistics. M., 1968
Druzhinin method and its application in socio-economic research. M., 1970
Jessen R. Methods of statistical surveys. M., 1985
Jeannie K. Average values. M., 1970
Yuzbashev theory of statistics. M., 1995.
Rumyantsev theory of statistics. M., 1998
Shmoylova study of the main trend and relationship in the series of dynamics. Tomsk, 1985
Yeats F. Sampling method in censuses and surveys / per. from English. . M., 1976
Historical informatics. M., 1996.
Kovalchenko historical research. M., 1987
Computer in economic history. Barnaul, 1997
Circle of Ideas: Models and Technologies of Historical Computer Science. M., 1996
Circle of Ideas: Traditions and Trends in Historical Computer Science. M., 1997
Circle of Ideas: Macro- and Micro Approaches in Historical Computer Science. M., 1998
Circle of Ideas: Historical Computer Science on the Threshold of the 21st Century. Cheboksary, 1999
Circle of Ideas: Historical Computer Science in the Information Society. M., 2001
General theory of statistics: Textbook / ed. and. M., 1994.
Workshop on the theory of statistics: Proc. allowance M., 2000
Eliseev statistics. M., 1990
Slavko-statistical methods in historical and research M., 1981
Slavko methods in the study of the history of the Soviet working class. M., 1991
Statistical Dictionary / ed. . M., 1989
Theory of Statistics: Textbook / ed. , M., 2000
Ursul Society. Introduction to social informatics. M., 1990
Schwartz G. Sampling method / per. with him. . M., 1978
Mathematical Methods of Operations Research
regression analysis model programmatic
Introduction
Description of the subject area and statement of the research problem
Practical part
Conclusion
Bibliography
Introduction
In economics, the basis of almost any activity is forecasting. Already on the basis of the forecast, a plan of action and measures is drawn up. Thus, we can say that the forecast of macroeconomic variables is a fundamental component of the plans of all economic entities. Forecasting can be carried out both on the basis of qualitative (expert) and quantitative methods. The latter by themselves can do nothing without a qualitative analysis, just as expert assessments must be supported by sound calculations.
Now forecasts, even at the macroeconomic level, are of a scenario nature and are developed according to the following principle: what happens if… , - and are often a preliminary stage and justification for major national economic programs. Macroeconomic forecasts are usually made with a lead time of one year. The modern practice of the functioning of the economy requires short-term forecasts (half a year, a month, a decade, a week). Designed for the tasks of providing advanced information to individual participants in the economy.
With changes in the objects and tasks of forecasting, the list of forecasting methods has changed. Adaptive methods of short-term forecasting have received rapid development.
Modern economic forecasting requires developers to have versatile specialization, knowledge from various fields of science and practice. The tasks of a forecaster include knowledge of the scientific (usually mathematical) apparatus of forecasting, the theoretical foundations of the forecasting process, information flows, software, interpretation of forecasting results.
The main function of the forecast is to substantiate the possible state of the object in the future or to determine alternative paths.
The importance of gasoline as the main type of fuel today is difficult to overestimate. And it is just as difficult to overestimate the impact of its price on the economy of any country. The nature of the development of the country's economy as a whole depends on the dynamics of fuel prices. An increase in gasoline prices causes an increase in prices for industrial goods, leads to an increase in inflationary costs in the economy and a decrease in the profitability of energy-intensive industries. The cost of petroleum products is one of the components of the prices of goods in the consumer market, and transportation costs affect the price structure of all consumer goods and services without exception.
Of particular importance is the issue of the cost of gasoline in the developing Ukrainian economy, where any change in prices causes an immediate reaction in all its sectors. However, the influence of this factor is not limited to the sphere of the economy; many political and social processes can also be attributed to the consequences of its fluctuations.
Thus, the study and forecasting of the dynamics of this indicator is of particular importance.
The purpose of this work is to forecast fuel prices for the near future.
1. Description of the subject area and statement of the research problem
The Ukrainian gasoline market can hardly be called constant or predictable. And there are many reasons for this, starting with the fact that the raw material for the production of fuel is oil, the prices and volume of production of which are determined not only by supply and demand in the domestic and foreign markets, but also by state policy, as well as special agreements between manufacturing companies. In conditions of strong dependence of the Ukrainian economy, it is dependent on the export of steel and chemicals, and the prices for these products are constantly changing. And speaking of gasoline prices, one cannot fail to note their upward trend. Despite the restraining policy pursued by the state, their growth is habitual for the majority of consumers. Prices for petroleum products in Ukraine today change daily. They mainly depend on the cost of oil on the world market ($ / barrel) and the level of the tax burden.
The study of gasoline prices is very relevant at the present time, since the prices of other goods and services depend on these prices.
In this paper, we will consider the dependence of gasoline prices on time and such factors as:
ü oil prices, US dollar per barrel
ü official exchange rate of the dollar (NBU), hryvnia per US dollar
ü consumer price index
The price of gasoline, which is a product of oil refining, is directly related to the price of the specified natural resource and the volume of its production. The dollar exchange rate has a significant impact on the entire Ukrainian economy, in particular on the formation of prices in its domestic markets. The direct connection of this parameter with gasoline prices directly depends on the US dollar exchange rate. The CPI reflects the general change in prices within the country, and since it is economically proven that a change in the prices of some goods in the vast majority of cases (in conditions of free competition) leads to an increase in the prices of other goods, it is reasonable to assume that a change in the prices of goods across the country affects the studied indicator at work.
Description of the mathematical apparatus used in the calculations
Regression analysis
Regression analysis is a method of modeling measured data and studying their properties. The data consists of pairs of values of the dependent variable (the response variable) and the independent variable (the explanatory variable). Regression model<#"19" src="doc_zip1.jpg" />. Regression analysis is the search for a function that describes this relationship. Regression can be represented as a sum of non-random and random components. where is the regression dependence function, and is an additive random variable with zero mat expectation. The assumption about the nature of the distribution of this quantity is called the data generation hypothesis<#"8" src="doc_zip6.jpg" />has a Gaussian distribution<#"20" src="doc_zip7.jpg" />.
The problem of finding a regression model of several free variables is posed as follows. A sample is given<#"24" src="doc_zip8.jpg" />values of free variables and the set of corresponding values of the dependent variable. These sets are denoted as the set of initial data.
A regression model is given - a parametric family of functions depending on parameters and free variables. It is required to find the most probable parameters:
The probability function depends on the data generation hypothesis and is given by Bayesian inference<#"justify">Least square method
The method of least squares is a method of finding the optimal parameters of linear regression, such that the sum of squared errors (regression residuals) is minimal. The method consists in minimizing the Euclidean distance between two vectors - the vector of recovered values of the dependent variable and the vector of the actual values of the dependent variable.
The task of the least squares method is to choose a vector to minimize the error. This error is the distance from vector to vector. The vector lies in the column space of the matrix, since there is a linear combination of the columns of this matrix with coefficients. Finding a solution using the least squares method is equivalent to the problem of finding a point that lies closest to and is located in the column space of the matrix.
Thus, the vector must be a projection onto the column space, and the residual vector must be orthogonal to this space. Orthogonality is that each vector in the column space is a linear combination of columns with some coefficients, that is, it is a vector. For everything in space, these vectors must be perpendicular to the residual:
Since this equality must be true for an arbitrary vector, then
The least squares solution of an inconsistent system consisting of equations with unknowns is the equation
which is called the normal equation. If the columns of a matrix are linearly independent, then the matrix is invertible and the only solution
The projection of a vector onto the column space of a matrix has the form
The matrix is called the projection matrix of the vector onto the column space of the matrix. This matrix has two main properties: it is idempotent, and it is symmetric, . The converse is also true: a matrix with these two properties is a projection matrix onto its column space.
Let we have statistical data about the parameter y depending on x. We present these data in the form
xx1 X2 …..Xi…..Xny *y 1*y 2*......y i* …..y n *
The least squares method allows for a given type of dependence y= ?(x) choose its numerical parameters so that the curve y= ?(x) displayed the experimental data in the best way according to the given criterion. Consider the justification from the point of view of probability theory for the mathematical definition of the parameters included in ? (x).
Suppose that the true dependence of y on x is exactly expressed by the formula y= ?(x). The experimental points presented in Table 2 deviate from this dependence due to measurement errors. The measurement errors obey the normal law according to Lyapunov's theorem. Consider some value of the argument x i . The result of the experiment is a random variable y i , distributed according to the normal law with mathematical expectation ?(x i ) and with standard deviation ?i characterizing the measurement error. Let the measurement accuracy at all points x=(x 1, X 2, …, X n ) is the same, i.e. ?1=?2=…=?n =?. Then the normal distribution law Yi looks like:
As a result of a series of measurements, the following event occurred: random variables (y 1*,y 2*, …, yn *).
Description of the selected software product
Mathcad - computer algebra system from the class of computer-aided design systems<#"justify">4. Practical part
The task of the study is to forecast gasoline prices. The initial information is a 36-week time series - from May 2012 to December 2012.
Statistics data (36 weeks) are presented in the Y matrix. Next, we will create the H matrix, which will be needed to find the vector A.
Let's present the initial data and the values calculated using the model:
To assess the quality of the model, we use the coefficient of determination.
First, let's find the average value of Xs:
The part of the variance, which is due to regression, in the total variance of the indicator Y characterizes the coefficient of determination R2.
Determination coefficient, takes values from -1 to +1. The closer its value of the coefficient modulo to 1, the closer the relationship of the effective feature Y with the studied factors X.
The value of the coefficient of determination serves as an important criterion for assessing the quality of linear and nonlinear models. The greater the share of the explained variation, the less the role of other factors, which means that the regression model approximates the initial data well and such a regression model can be used to predict the values of the effective indicator. We obtained the coefficient of determination R2 = 0.78, therefore, the regression equation explains 78% of the variance of the effective feature, and 22% of its variance (i.e., residual variance) falls to the share of other factors.
Therefore, we conclude that the model is adequate.
Based on the data obtained, it is possible to make a forecast of fuel prices for the 37th week of 2013. The formula for the calculation is as follows:
The calculated forecast using this model: the price of gasoline is UAH 10.434.
Conclusion
In this paper, we have shown the possibility of conducting a regression analysis to predict gasoline prices for future periods. The purpose of the course work was to consolidate knowledge in the course "Mathematical Methods of Operations Research" and gain skills in developing software that allows you to automate operations research in a given subject area.
The forecast for the future price of gasoline, of course, is not unambiguous, which is due to the peculiarities of the initial data and the developed models. However, based on the information received, it is reasonable to assume that, of course, gasoline prices will not fall in the near future, but most likely will remain at the same level or will grow slightly. Of course, factors related to consumer expectations, customs policy and many other factors are not taken into account here, but I would like to note that they are largely mutually repayable . And it would be quite reasonable to note that a sharp jump in gasoline prices at the moment is really extremely doubtful, which, first of all, is connected with the policy pursued by the government.
Bibliography
1.Buyul A., Zöfel P. SPSS: the art of information processing. Analysis of statistical data and restoration of hidden patterns. - St. Petersburg: OOO "DiaSoftUP", 2001. - 608 p.
2. Internet resources http://www.ukrstat.gov.ua/
3. Internet resources http://index.minfin.com.ua/
Internet resources http://fx-commodities.ru/category/oil/
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Always and in all spheres of his activity, a person made decisions. An important area of decision-making is related to production. The larger the volume of production, the more difficult it is to make a decision and, therefore, it is easier to make a mistake. A natural question arises: is it possible to use a computer to avoid such errors?
The answer to this question is given by a science called cybernetics. Cybernetics (derived from the Greek "kybernetike" - the art of management) is the science of the general laws of receiving, storing, transmitting and processing information.
The most important branch of cybernetics is economic cybernetics - the science that deals with the application of ideas and methods of cybernetics to economic systems.
Economic cybernetics uses a set of methods for studying management processes in the economy, including economic and mathematical methods.
At present, the use of computers in production management has reached a large scale. However, in most cases, with the help of computers, so-called routine tasks are solved, that is, tasks related to the processing of various data, which, before the use of computers, were solved in the same way, but manually. Another class of problems that can be solved with the help of computers are decision-making problems. To use a computer for decision-making, it is necessary to make a mathematical model. Is it necessary to use computers when making decisions? Human capabilities are quite diverse. If you put them in order, Man is so arranged that what he possesses is not enough for him. And the endless process of increasing its capabilities begins. To raise more, one of the first inventions appears - a lever, to make it easier to move the load - the wheel. For the time being, only the energy of the person himself is used in these tools. Over time, the use of external energy sources begins: gunpowder, steam, electricity, atomic energy. It is impossible to estimate how much the energy used from external sources exceeds the physical capabilities of a person today.
As for the mental faculties of a person, then, as they say, everyone is dissatisfied with his condition, but satisfied with his mind. Is it possible to make a person smarter than he is? To answer this question, it should be clarified that all human intellectual activity can be divided into formalizable and non-formalizable.
Formalizable is an activity that is performed according to certain rules. For example, the performance of calculations, searches in directories, and graphic work can undoubtedly be entrusted to a computer. And like everything that a computer can do, it does it better, that is, faster and better than a person.
Non-formalizable is such an activity that occurs with the application of some rules unknown to us. Thinking, reasoning, intuition, common sense - we still do not know what it is, and naturally, all this cannot be entrusted to a computer, if only because we simply do not know what to entrust, what task to put before a computer.
Decision making is a kind of mental activity.
It is generally accepted that decision-making is a non-formalized activity. However, this is not always the case. On the one hand, we do not know how we make a decision. And explaining some words with the help of others like "we make a decision with the help of common sense" does not give anything. On the other hand, a significant number of decision-making tasks can be formalized. One of the types of decision-making problems that can be formalized are optimal decision-making problems, or optimization problems. The optimization problem is solved with the help of mathematical models and the use of computer technology.
Modern computers meet the highest requirements. They are capable of performing millions of operations per second, they can have all the necessary information in their memory, the display-keyboard combination provides a dialogue between a person and a computer. However, one should not confuse successes in the creation of computers with advances in the field of their application. In fact, all that a computer can do is, according to a program given by a person, ensure the transformation of the initial data into a result. It must be clearly understood that the computer does not and cannot make decisions. The decision can only be made by a person-manager, endowed with certain rights for this. But for a competent manager, a computer is a great assistant, able to develop and offer a set of various solutions. And from this set, a person will choose the option that, from his point of view, will be more suitable. Of course, not all decision-making problems can be solved with the help of a computer. Nevertheless, even if the solution of a problem on a computer does not end with complete success, it still turns out to be useful, as it contributes to a deeper understanding of this problem and its more rigorous formulation.
In order for a person to make a decision without a computer, often nothing is needed. I thought and decided. A person, good or bad, solves all the problems that arise before him. True, there are no guarantees of correctness in this case. The computer does not make any decisions, but only helps to find solutions. This process consists of the following steps:
1) Selecting a task.
Solving a problem, especially a rather complex one, is a rather difficult task that requires a lot of time. And if the task is chosen unsuccessfully, then this can lead to loss of time and disappointment in the use of computers for decision-making. What are the basic requirements that the task must satisfy?
A. There must be at least one solution to it, because if there are no solutions, then there is nothing to choose from.
B. We must clearly know in what sense the desired solution should be the best, because if we do not know what we want, the computer will not be able to help us choose the best solution.
The choice of the task is completed by its substantive formulation. It is necessary to clearly formulate the problem in ordinary language, highlight the purpose of the study, indicate the limitations, raise the main questions that we want to get answers as a result of solving the problem.
Here we should highlight the most significant features of the economic object, the most important dependencies that we want to take into account when building a model. Some hypotheses for the development of the object of study are formed, the identified dependencies and relationships are studied. When a task is selected and its meaningful statement is made, one has to deal with specialists in the subject area (engineers, technologists, designers, etc.). These specialists, as a rule, know their subject very well, but do not always have an idea of what is required to solve a problem on a computer. Therefore, the meaningful formulation of the problem often turns out to be oversaturated with information that is completely unnecessary for working on a computer.
2) Compilation of the model
An economic-mathematical model is understood as a mathematical description of the studied economic object or process, in which economic patterns are expressed in an abstract form using mathematical relationships.
The basic principles for compiling a model boil down to the following two concepts:
1. When formulating the problem, it is necessary to cover the simulated phenomenon quite widely. Otherwise, the model will not give a global optimum and will not reflect the essence of the matter. The danger is that the optimization of one part can be at the expense of others and to the detriment of the overall organization.
2. The model should be as simple as possible. The model must be such that it can be evaluated, tested and understood, and the results obtained from the model must be clear to both its creator and the decision maker. In practice, these concepts often conflict, primarily because there is a human element involved in data collection and entry, error checking, and interpretation of results, which limits the size of the model that can be satisfactorily analyzed. The size of the model is used as a limiting factor, and if we want to increase the breadth of coverage, then we have to decrease the detail and vice versa.
Let's introduce the concept of model hierarchy, where the breadth increases and the detail decreases as we move to higher levels of the hierarchy. At higher levels, in turn, restrictions and goals are formed for lower levels.
When building a model, the planning horizon generally increases with the growth of the hierarchy. If the long-range planning model of an entire corporation can contain little of the day-to-day details, then the production planning model of an individual subdivision consists mainly of such details.
When formulating a task, the following three aspects should be taken into account:
1) Factors under study: The objectives of the study are rather loosely defined and depend heavily on what is included in the model. In this regard, it is easier for engineers, since the factors they study are usually standard, and the objective function is expressed in terms of maximum income, minimum costs, or, possibly, minimum consumption of some resource. At the same time, sociologists, for example, usually set themselves the goal of "public utility" or something like that, and find themselves in the difficult position of having to attribute a certain "utility" to various actions, expressing it in mathematical form.
2) Physical boundaries: The spatial aspects of the study require detailed consideration. If production is concentrated in more than one point, then it is necessary to take into account the corresponding distribution processes in the model. These processes may include warehousing, transportation, and equipment scheduling tasks.
3) Temporal boundaries: The temporal aspects of the study lead to a serious dilemma. Usually the planning horizon is well known, but a choice must be made: either simulate the system dynamically in order to obtain time schedules, or simulate static operation at a certain point in time. If a dynamic (multi-stage) process is modeled, then the dimensions of the model increase in accordance with the number of considered time periods (stages). Such models are usually conceptually simple, so that the main difficulty lies rather in the ability to solve a problem on a computer in an acceptable time than in the ability to interpret a large amount of output data. c It is often enough to build a model of the system at some given point in time, for example, in a fixed year, month, day, and then repeat the calculations at certain intervals. In general, the availability of resources in a dynamic model is often approximated and determined by factors outside the scope of the model. Therefore, it is necessary to carefully analyze whether it is really necessary to know the time dependence of the change in the characteristics of the model, or whether the same result can be obtained by repeating the static calculations for a number of different fixed moments.
In the history of mathematics, two main periods can be conventionally distinguished: elementary and modern mathematics. The milestone, from which it is customary to count the era of new (sometimes they say - higher) mathematics, was the 17th century - the century of the emergence of mathematical analysis. By the end of the XVII century. I. Newton, G. Leibniz and their predecessors created the apparatus of a new differential calculus and integral calculus, which forms the basis of mathematical analysis and even, perhaps, the mathematical basis of all modern natural science.
Mathematical analysis is a vast area of mathematics with a characteristic object of study (a variable), a peculiar research method (analysis by means of infinitesimals or by passing to the limit), a certain system of basic concepts (function, limit, derivative, differential, integral, series) and constantly improving and developing apparatus, which is based on differential and integral calculus.
Let's try to give an idea of what kind of mathematical revolution took place in the 17th century, what characterizes the transition from elementary mathematics associated with the birth of mathematical analysis to the one that is now the subject of research in mathematical analysis, and what explains its fundamental role in the entire modern system of theoretical and applied knowledge. .
Imagine that in front of you is a beautifully executed color photograph of a stormy ocean wave running ashore: a powerful stooped back, a steep but slightly sunken chest, already tilted forward and ready to fall head with a gray mane torn by the wind. You have stopped the moment, you have managed to catch the wave, and now you can carefully study it in all its details without haste. A wave can be measured, and using the tools of elementary mathematics, you will draw many important conclusions about this wave, and therefore all its oceanic sisters. But by stopping the wave, you have deprived it of movement and life. Its origin, development, run, the force with which it falls on the shore - all this turned out to be out of your field of vision, because you do not yet have either a language or a mathematical apparatus suitable for describing and studying not static, but developing, dynamic processes, variables and their interrelations.
"Mathematical analysis is no less comprehensive than nature itself: it determines all tangible relationships, measures times, spaces, forces, temperatures." J. Fourier
Movement, variables and their relationships are all around us. Various types of motion and their regularities constitute the main object of study of specific sciences: physics, geology, biology, sociology, etc. Therefore, the exact language and appropriate mathematical methods for describing and studying variables turned out to be necessary in all areas of knowledge approximately to the same extent as numbers and arithmetic are necessary in describing quantitative relationships. So, mathematical analysis is the basis of the language and mathematical methods for describing variables and their relationships. Today, without mathematical analysis, it is impossible not only to calculate space trajectories, the operation of nuclear reactors, the running of an ocean wave and the patterns of cyclone development, but also to economically manage production, resource distribution, organization of technological processes, predict the course of chemical reactions or changes in the number of various species interconnected in nature. animals and plants, because all these are dynamic processes.
Elementary mathematics was basically the mathematics of constants, it studied mainly the relations between the elements of geometric figures, the arithmetic properties of numbers, and algebraic equations. To some extent, her attitude to reality can be compared with an attentive, even thorough and complete study of each fixed frame of a film that captures the changing, developing living world in its movement, which, however, is not visible on a separate frame and which can be observed only by looking tape as a whole. But just as cinema is unthinkable without photography, so modern mathematics is impossible without that part of it, which we conditionally call elementary, without the ideas and achievements of many outstanding scientists, sometimes separated by tens of centuries.
Mathematics is one, and its “higher” part is connected with the “elementary” one in much the same way as the next floor of a house under construction is connected with the previous one, and the width of the horizons that mathematics opens up to us in the world around us depends on which floor of this building we managed to reach. rise. Born in the 17th century mathematical analysis opened up possibilities for scientific description, quantitative and qualitative study of variables and motion in the broadest sense of the word.
What are the prerequisites for the emergence of mathematical analysis?
By the end of the XVII century. the following situation has arisen. First, within the framework of mathematics itself, over the years, certain important classes of problems of the same type have accumulated (for example, problems of measuring the areas and volumes of non-standard figures, problems of drawing tangents to curves) and methods have appeared for solving them in various special cases. Secondly, it turned out that these problems are closely related to the problems of describing an arbitrary (not necessarily uniform) mechanical motion, and in particular with the calculation of its instantaneous characteristics (velocity, acceleration at any time), as well as with finding the distance traveled for movement at a given variable speed. The solution of these problems was necessary for the development of physics, astronomy, and technology.
Finally, thirdly, by the middle of the XVII century. the works of R. Descartes and P. Fermat laid the foundations of the analytical method of coordinates (the so-called analytical geometry), which made it possible to formulate geometric and physical problems of heterogeneous origin in the general (analytical) language of numbers and numerical dependences, or, as we now say, numerical functions.
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NIKOLAI NIKOLAEVICH LUZIN
N. N. Luzin - Soviet mathematician, founder of the Soviet school of function theory, academician (1929). Luzin was born in Tomsk, studied at the Tomsk gymnasium. The formalism of the gymnasium course in mathematics alienated the talented young man, and only a capable tutor could reveal to him the beauty and grandeur of mathematical science. In 1901, Luzin entered the mathematical department of the Faculty of Physics and Mathematics of Moscow University. From the first years of study, questions related to infinity fell into the circle of his interests. At the end of the XIX century. the German scientist G. Kantor created the general theory of infinite sets, which has received numerous applications in the study of discontinuous functions. Luzin began to study this theory, but his studies were interrupted in 1905. The student, who took part in revolutionary activities, had to leave for France for a while. There he listened to lectures by the most prominent French mathematicians of that time. Upon his return to Russia, Luzin graduated from the university and was left to prepare for a professorship. Soon he again went to Paris, and then to Göttingen, where he became close to many scientists and wrote his first scientific papers. The main problem that interested the scientist was the question of whether there can be sets containing more elements than the set of natural numbers, but less than the set of points of the segment (the continuum problem). For any infinite set that could be obtained from segments using the operations of union and intersection of countable collections of sets, this hypothesis was true, and in order to solve the problem, it was necessary to find out what other ways of constructing sets were. At the same time, Luzin studied the question of whether it is possible to represent any periodic function, even if it has infinitely many discontinuity points, as the sum of a trigonometric series, i.e. sums of an infinite set of harmonic vibrations. Luzin obtained a number of significant results on these issues and in 1915 he defended his dissertation "The Integral and the Trigonometric Series", for which he was immediately awarded the degree of Doctor of Pure Mathematics, bypassing the intermediate master's degree that existed at that time. In 1917 Luzin became a professor at Moscow University. A talented teacher, he attracted the most capable students and young mathematicians. Luzin's school reached its heyday in the first post-revolutionary years. Luzin's students formed a creative team, which was jokingly called "Luzitania". Many of them received first-class scientific results during their student days. For example, P. S. Aleksandrov and M. Ya. Suslin (1894-1919) discovered a new method for constructing sets, which initiated the development of a new direction - descriptive set theory. Research in this area, conducted by Luzin and his students, showed that the usual methods of set theory are not enough to solve many of the problems that arose in it. Luzin's scientific predictions were fully confirmed in the 1960s. 20th century Many students of N. N. Luzin later became academicians and corresponding members of the Academy of Sciences of the USSR. Among them P. S. Aleksandrov. A. N. Kolmogorov. M. A. Lavrentiev, L. A. Lyusternik, D. E. Menshov, P. S. Novikov. L. G. Shnirelman and others. Modern Soviet and foreign mathematicians in their works develop the ideas of N. N. Luzin. |
The combination of these circumstances led to the fact that at the end of the XVII century. two scientists - I. Newton and G. Leibniz - independently managed to create a mathematical apparatus for solving these problems, summing up and generalizing individual results of their predecessors, including the ancient scientist Archimedes and contemporaries of Newton and Leibniz - B. Cavalieri, B. Pascal , D. Gregory, I. Barrow. This apparatus formed the basis of mathematical analysis - a new branch of mathematics that studies various developing processes, i.e. interrelations of variables, which in mathematics are called functional dependencies or, in other words, functions. By the way, the term “function” itself was required and naturally arose precisely in the 17th century, and by now it has acquired not only a general mathematical, but also a general scientific meaning.
Initial information about the basic concepts and the mathematical apparatus of analysis is given in the articles "Differential Calculus" and "Integral Calculus".
In conclusion, I would like to dwell on only one principle of mathematical abstraction that is common to all mathematics and characteristic of analysis, and in this connection to explain in what form mathematical analysis studies variables and what is the secret of such universality of its methods for studying all kinds of specific developing processes and their interrelations. .
Let's look at some explanatory examples and analogies.
We sometimes no longer realize that, for example, a mathematical ratio, written not for apples, chairs or elephants, but in an abstract form abstracted from specific objects, is an outstanding scientific achievement. This is a mathematical law that experience has shown to be applicable to various concrete objects. So, studying in mathematics the general properties of abstract, abstract numbers, we thereby study the quantitative relationships of the real world.
For example, it is known from a school mathematics course that, therefore, in a specific situation, you could say: “If two six-ton dump trucks are not allocated to me for transporting 12 tons of soil, then you can request three four-ton dump trucks and the work will be done, and if they give only one four-ton dump truck, then she will have to make three flights. Thus, the abstract numbers and numerical regularities that are now familiar to us are connected with their concrete manifestations and applications.
Approximately in the same way, the laws of change of specific variable quantities and developing processes of nature are connected with the abstract, abstract form-function in which they appear and are studied in mathematical analysis.
For example, an abstract ratio may be a reflection of the dependence of the box office at the cinema on the number of tickets sold, if 20 is 20 kopecks - the price of one ticket. But if we are cycling on a highway at 20 km per hour, then the same ratio can be interpreted as the relationship of the time (hours) of our bike ride and the distance covered during this time (kilometers), you can always argue that, for example, a change by several times leads to a proportional (i.e., by the same number of times) change in the value of , and if , then the opposite conclusion is also true. So, in particular, to double the box office revenue of a cinema, you have to attract twice as many viewers, and to ride a bike at the same speed twice as far, you have to ride twice as long.
Mathematics studies both the simplest dependence and other, much more complex dependences in an abstract, general, abstract form abstracted from private interpretation. The properties of a function identified in such a study or methods for studying these properties will be in the nature of general mathematical techniques, conclusions, laws and conclusions applicable to each specific phenomenon in which the function studied in an abstract form occurs, regardless of which field of knowledge this phenomenon belongs to. .
So, mathematical analysis as a branch of mathematics took shape at the end of the 17th century. The subject of study in mathematical analysis (as it appears from modern positions) are functions, or, in other words, dependencies between variables.
With the advent of mathematical analysis, it became possible for mathematics to study and reflect the developing processes of the real world; variables and motion entered mathematics.
Mathematical methods are most widely used in conducting systematic research. At the same time, the solution of practical problems by mathematical methods is sequentially carried out according to the following algorithm:
mathematical formulation of the problem (development of a mathematical model);
choice of research method for the obtained mathematical model;
analysis of the obtained mathematical result.
Mathematical formulation of the problem usually represented as numbers, geometric images, functions, systems of equations, etc. Description of an object (phenomenon) can be represented using continuous or discrete, deterministic or stochastic and other mathematical forms.
Mathematical model is a system of mathematical relationships (formulas, functions, equations, systems of equations) that describe certain aspects of the studied object, phenomenon, process or the object (process) as a whole.
The first stage of mathematical modeling is the formulation of the problem, the definition of the object and objectives of the study, the setting of criteria (features) for studying objects and managing them. An incorrect or incomplete statement of the problem can negate the results of all subsequent stages.
The model is the result of a compromise between two opposing goals:
the model should be detailed, take into account all the really existing connections and the factors and parameters involved in its work;
at the same time, the model must be simple enough so that acceptable solutions or results can be obtained in an acceptable time frame under certain resource constraints.
Modeling can be called approximate scientific research. And the degree of its accuracy depends on the researcher, his experience, goals, resources.
The assumptions made in the development of the model are a consequence of the goals of modeling and the capabilities (resources) of the researcher. They are determined by the requirements of the accuracy of the results, and like the model itself, are the result of a compromise. After all, it is the assumptions that distinguish one model of the same process from another.
Usually, when developing a model, insignificant factors are discarded (not taken into account). Constants in physical equations are assumed to be constant. Sometimes some quantities that change in the process are averaged (for example, the air temperature can be considered unchanged over a certain period of time).
Model development process
This is a process of consistent (and possibly repeated) schematization or idealization of the phenomenon under study.
The adequacy of a model is its correspondence to the real physical process (or object) that it represents.
To develop a model of a physical process, it is necessary to determine:
Sometimes an approach is used when a model of small completeness, which is probabilistic in nature, is applied. Then, with the help of a computer, it is analyzed and refined.
Model validation begins and passes in the very process of its construction, when one or another relationship between its parameters is selected or established, the accepted assumptions are evaluated. However, after the formation of the model as a whole, it is necessary to analyze it from some general positions.
The mathematical basis of the model (i.e., the mathematical description of physical relationships) must be consistent precisely from the point of view of mathematics: functional dependencies must have the same trends as real processes; equations must have an area of existence not less than the range in which the study is carried out; they should not have special points or gaps if they are not in the real process, etc. The equations should not distort the logic of the real process.
The model should adequately, i.e., as accurately as possible, reflect reality. Adequacy is needed not in general, but in the considered range.
Discrepancies between the results of the analysis of the model and the actual behavior of the object are inevitable, since the model is a reflection, and not the object itself.
On fig. 3. a generalized representation is presented, which is used in the construction of mathematical models.
Rice. 3. Apparatus for building mathematical models
When using static methods, the apparatus of algebra and differential equations with time-independent arguments are most often used.
Dynamic methods use differential equations in the same way; integral equations; partial differential equations; theory of automatic control; algebra.
Probabilistic methods use: probability theory; information theory; algebra; theory of random processes; theory of Markov processes; automata theory; differential equations.
An important place in modeling is occupied by the question of the similarity between the model and the real object. Quantitative correspondences between the individual aspects of the processes occurring in a real object and its model are characterized by scales.
In general, the similarity of processes in objects and models is characterized by similarity criteria. The similarity criterion is a dimensionless set of parameters that characterizes a given process. When conducting research, depending on the field of research, various criteria are used. For example, in hydraulics, such a criterion is the Reynolds number (characterizes the fluidity of a liquid), in heat engineering - the Nussselt number (characterizes the conditions of heat transfer), in mechanics - Newton's criterion, etc.
It is believed that if such criteria for the model and the object under study are equal, then the model is correct.
Another method of theoretical research adjoins the theory of similarity - dimensional analysis method, which is based on two assumptions:
physical laws are expressed only by products of degrees of physical quantities, which can be positive, negative, integer and fractional; the dimensions of both parts of the equality expressing the physical dimension must be the same.
