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Solution development method. Some decisions, as a rule, are typical, repetitive, and can be successfully formalized, i.e. taken according to a predetermined algorithm. In other words, a formalized decision is the result of performing a predetermined sequence of actions. For example, when drawing up a schedule for the repair of equipment, the shop manager may proceed from a standard that requires a certain ratio between the amount of equipment and maintenance personnel. If there are 50 pieces of equipment in the shop, and the maintenance standard is 10 pieces per repair worker, then the shop must have five repairmen. In the same way, when a financial manager decides to invest free funds in government securities, he chooses between different types of bonds, depending on which of them provide the greatest return on invested capital at that time. The choice is made on the basis of a simple calculation of the final yield for each option and the establishment of the most profitable one.

Formalization of decision-making increases the efficiency of management by reducing the likelihood of error and saving time: no need to re-develop a solution every time a corresponding situation arises. Therefore, the management of organizations often formalizes solutions for certain, regularly recurring situations, developing appropriate rules, instructions and regulations.

At the same time, in the process of managing organizations, there are often new, atypical situations and non-standard problems that are not amenable to a formalized solution. In such cases, intellectual abilities, talent and personal initiative of managers play an important role.

Of course, in practice, most decisions fall somewhere between these two extremes, allowing for both personal initiative and formal procedure in the process of their development. Specific methods used in the decision-making process are discussed below.

· Number of selection criteria.

If the choice of the best alternative is made according to only one criterion (which is typical for formalized decisions), then the decision made will be simple, single-criteria. And vice versa, when the chosen alternative must satisfy several criteria at the same time, the decision will be complex, multi-criteria. In management practice, the vast majority of decisions are multi-criteria, since they must simultaneously meet such criteria as: profit volume, profitability, quality level, market share, employment level, implementation period, etc.

· Decision Form.

The person making the choice from the available alternatives of the final decision can be one person and his decision will be, accordingly, the sole one. However, in modern management practice, complex situations and problems are increasingly encountered, the solution of which requires a comprehensive, comprehensive analysis, i.e. participation of a group of managers and specialists. Such group, or collective, decisions are called collegial. Increasing professionalization and deepening the specialization of management lead to widespread collegial forms of decision-making. It must also be borne in mind that certain decisions are legally classified as collegial. So, for example, certain decisions in a joint-stock company (on the payment of dividends, distribution of profits and losses, major transactions, election of governing bodies, reorganization, etc.) are referred to the exclusive competence of the general meeting of shareholders. The collegial form of decision-making, of course, reduces the efficiency of management and “blurs” responsibility for its results, but it prevents gross errors and abuses and increases the validity of the choice.

· Solution fixing method.

On this basis, management decisions can be divided into fixed, or documentary (i.e., executed in the form of a document - an order, instruction, letter, etc.), and undocumented (not having a documentary form, oral). Most decisions in the management apparatus are documented, however, small, insignificant decisions, as well as decisions made in emergency, acute, urgent situations, may not be documented.

· The nature of the information used. Depending on the degree of completeness and reliability of the information that the manager has, managerial decisions can be deterministic (taken under conditions of certainty) or probabilistic (taken under risk or uncertainty). These conditions play an extremely important role in decision making, so let's look at them in more detail.

Deterministic and probabilistic solutions.

Deterministic Solutions are taken under conditions of certainty, when the manager has almost complete and reliable information regarding the problem being solved, which allows him to know exactly the result of each of the alternative choices. There is only one such result, and the probability of its occurrence is close to one. An example of a deterministic solution would be the choice of 20% federal loan bonds with a constant coupon income as a tool for investing free cash. The financial manager in this case knows for sure that, except for extremely unlikely extraordinary circumstances, due to which the government of the Russian Federation will not be able to fulfill its obligations, the organization will receive exactly 20% per annum on the invested funds. Similarly, when deciding to launch a particular product, a manager can accurately determine the level of production costs, since rental rates, materials and labor costs can be calculated quite accurately.

Analysis of managerial decisions under conditions of certainty is the simplest case: the number of possible situations (options) and their outcomes are known. You must select one of the available options. The degree of complexity of the selection procedure in this case is determined only by the number of alternative options. Let's consider two possible situations:

a) There are two possible options;

In this case, the analyst must choose (or recommend to choose) one of two possible options. The sequence of actions here is as follows:

The criteria by which the choice will be made are determined;

· the method of “direct counting” calculates the values ​​of the criterion for the compared options;

There are various methods for solving this problem. As a rule, they are divided into two groups:

methods based on discounted estimates;

methods based on accounting estimates.

(deterministic - certain, causally determined by previous events; from lat. Determino - I determine)

Stochastic systems are systems in which changes are random.

(stochastic - random, probabilistic; from Greek stochastikós - able to guess)

In a deterministic system, from its previous state and some additional information, its subsequent state can be quite definitely predicted. In a probabilistic system, based on the same information, one can only predict a set of future states and determine the probability of each of them.

7. Complex systems and their features. Control systems as objects of research.

Think the system is complex, if it consists of a large number of interconnected and interacting elements, each of which can be represented as a system. As the content of the theory of the development of complex systems, one can consider a set of methodological approaches that make it possible to build models of the processes of development of complex systems using the achievements of various sciences, as well as methods for analyzing the resulting models.

Management system of any organization is a complex system designed to collect, analyze and process information in order to obtain the maximum end result under certain restrictions. Most processes are so complex that in the current state of science it is very rarely possible to create their universal theory, which is valid at all times and in all areas of the process under consideration.

Studying the control system as an object of study, it is necessary to highlight the requirements for control systems, which can be used to judge the degree of organization of systems. These requirements include:

Determinism of the elements of the system;

The dynamism of the system;

Presence of a control parameter in the system;

The presence of a control parameter in the system;

The presence in the system of channels (at least one) feedback.

8. Modern methods of research of control systems.

The whole set of research methods can be divided into three large groups: methods based on the use of knowledge and intuition of specialists; methods of formalized representation of control systems and integrated methods.

The first group - methods based on the identification and generalization of the opinions of experienced experts, the use of their experience and non-traditional approaches to the analysis of the organization's activities include: the "brainstorming" method, the "scenarios" type method, the method of expert assessments (including SWOT analysis), "Delphi" type method, "goal tree" type, "business game" type methods, morphological methods and a number of other methods.

The second group - methods of formalized representation of control systems, based on the use of mathematical, economic and mathematical methods and models for the study of control systems.

The third group - in an effort to more adequately reflect the problem situation, in some cases it is advisable to use statistical methods, with the help of which, on the basis of a sample study, statistical patterns are obtained and extended to the behavior of the system as a whole

9. System analysis as the main method for studying complex systems and solving complex management problems.

System analysis

System analysis is used in cases where they seek to explore an object from different angles, in a complex manner. The most common area of ​​systems research is considered to be system analysis, which is understood as a methodology for solving complex problems and problems based on concepts developed within the framework of systems theory. Systems analysis is also defined as "the application of systems concepts to management functions related to planning", or even to strategic planning and the target planning stage.

The ultimate goal of system analysis is the development and implementation of the selected reference model of the management system.

FROM system analysis begins with the clarification or formulation of the goals of a particular management system (enterprise or company) and the search for an efficiency criterion, which should be expressed as a specific indicator. As a rule, most organizations are multipurpose. A set of goals follows from the characteristics of the development of an enterprise (company) and its actual state in the period under consideration, as well as the state of the environment (geopolitical, economic, social factors).

Clearly and competently formulated goals for the development of an enterprise (company) are the basis for system analysis and development of a research program.

10. Approaches and logic of research from the standpoint of system analysis. The main stages (logical steps) of system analysis.

System analysis is a scientific method for studying complex, multi-level, multi-component systems and processes, based on an integrated approach, taking into account the relationships and interactions between the elements of the system, as well as a set of methods for developing, making and justifying decisions in the design, creation and management of social, economic, human-machine and technical systems.

It is necessary to perform the following systemic studies:

1) identify general trends in the development of this enterprise and its place and role in the modern market economy;

2) establish the features of the functioning of the enterprise and its individual divisions;

3) identify the conditions that ensure the achievement of the goals;

4) determine the conditions that impede the achievement of goals;

5) collect the necessary data for analysis and development of measures to improve the current management system;

6) use the best practices of other enterprises;

7) study the necessary information to adapt the selected (synthesized) reference model to the conditions of the enterprise under consideration.

The main stages of system analysis are:

1. Goal setting;

2. Search for ways to achieve goals;

3. Selection of criteria for evaluating alternatives for achieving goals.

11. Problems and their features. Problematics and formulation of problems.

A problem is a situation in which previously set goals are not achieved.. Those. when monitoring the results achieved, it turns out that they are much worse than planned, respectively, it is required to take certain measures to correct the situation. Such a fairly natural way of control is called mismatch control. Mismatch management is effective only with a purely quantitative, well-predictable development of the process in advance.

Problem situation- it is a "gap" in activity, a "mismatch" between the goals and capabilities of the subject, i.e. conditions that give rise to the problem. A problem situation is the conditions that give rise to a problem.

Problem conditions - these are objectively arising contradictions between certain actions, in particular due to ignorance of the methods for their implementation; between the need for new knowledge and its insufficiency.

Initial statement (formulation) of the problem. The initial statement of the problem should serve as a kind of task for preparing a solution or performing a preliminary stage of development, the results of which will be considered by the decision maker and determine the further course of action.

The statement (formulation) of the problem is called the initial, or preliminary stage, because in the course of analysis and synthesis and on their basis, many initial provisions can be revised.

Formulation of goals and conditions for solving the problem. It is important to formulate the goals of solving the problem, first of all, for the correct identification of ways to achieve them and for comparing options for solving the achievement of goals.

12. Typology of problems. Problem difficulty levels

Problem

Qualitative problems- problems that are described by qualitative characteristics, properties (associated with a detailed enumeration of future or poorly defined resources and their properties or characteristics).

Quantitative problems- problems that are expressed in numbers or in such symbols, which, after all, can be expressed in numerical estimates. Features of quantitative problems: accuracy, reliability of the solution, rigor and controllability.

- Operational issues- these are problems, the solution of which is aimed at preventing, eliminating or compensating for disturbances that disrupt the current operation of the system. These are structured problems. The solution of these problems is associated with their quantitative assessment, the presence of well-established alternative sets of actions in a given situation;

problems of improvement and development of systems- these are problems, the solution of which is aimed at improving the efficiency of functioning by changing the characteristics of the control object or the control system of the object, as well as introducing new ideas. These are weakly structured problems, the solution of which is the object of study of system analysis and synthesis;

innovative problems- these are problems, the solution of which is associated with the development of new ideas and the introduction of innovations. These are very semi-structured (or unstructured) problems. The solution of these problems is connected with the generation of new ideas and the application of heuristic methods based on experience and intuition.

According to the nature of the manifestation problems are categorized as recurring, similar, new and unique.

According to the degree of connection distinguish between complex and autonomous problems.

13. Creative approach to problem solving.

Problem(from Greek - task) in a broad sense - a complex theoretical or practical issue that requires study, resolution. Essentially, a problem is a situation of discrepancy between the desired and the existing.

Creating truly innovative products and services depends a lot on how creative you are. For most project managers, this means the deliberate use of creative problem-solving in the project management process.

Methods: funny ideas; Follow the scheme "Encouragement-Pros-Risks-Decisions"; Do not be afraid of disagreements and opposing points of view.

14. The main stages of problem setting. Isolation of the problem from the external environment. Structuring the problem.

Stage 1 "diagnosis" - general knowledge of the problem as well as related matters, the study of which may be useful; drawing up a general work plan, indicating the deadline, performers and main sources that can presumably be used.

Stage 2-establishing its "symptoms". The concept of "symptom" is used here almost in a medical sense and means some indirect sign or characteristic that indicates the presence of a problem.

Stage 3- collection of factors confirming the "symptoms", those. identifying the causes of the problem.

Stage 4- interpretation of the factors i.e. analysis of all necessary internal and external information related to "symptoms".

Stage 5- problem statement includes:

¨ drawing up the initial formulation of the problem;

¨ comprehension of this formulation in relation to various parts of the problem;

¨ understanding the factors that relate to the problem;

¨ general clarification of the original formulation of the problem

Structuring the problem implies splitting it. Splitting (decomposition - see below) - the search for additional questions (sub-questions), without which it is impossible to get an answer to the central - problematic - question.

15. The process of finding and developing a solution. The specifics of the solution implementation process.

1) Problem Diagnosis. Due to its complexity, diagnosing a problem is a process consisting of a number of steps:

· awareness and identification of symptoms of difficulties or existing unused opportunities (for example, low profits, high costs, conflicts, etc.);

identifying the problem in a general way, i.e. the causes of the problem;

· collection and analysis of internal and external information, involvement of consultants.

2) Formulation of constraints and decision criteria. realism and efficiency. In order for the solution to be realistic, it is necessary first of all to formulate the existing limitations.

3) Definition of alternatives.

4) Evaluation of alternatives. In some cases, some of them may be quantitative, and some - qualitative.

5) Choice of an alternative.

6) Implementation and control over the implementation of decisions. An important condition is recognition by the team. To do this, it is necessary to convince and involve people in decision-making. Practice shows that if the team to some extent participated in the preparation of the option, considers it "their own", the resistance to the course of its implementation is significantly reduced. Then the next phase of the stage under consideration begins - control over the progress of implementation, i.e. establishing feedback to examine the consistency of actual results with expectations.

16. Goals and means to achieve them. The system of values ​​as a method of choosing goals. Classification of goals.

Means to achieve goals:

1. Skills, 2. Abilities, 3. Skill

Target classification:

· by area covered(general, private goal);

· by value(main, intermediate, secondary);

· by the number of variables(single and multi-alternative);

· subject matter(calculated for a general or partial result);

· by sources of formation goals can be set from outside and formed within the organization;

· in order of importance goals are divided into: strategic and tactical;

· by time goals differ in: short-term (up to one year), medium-term (from 1 year to 5 years), long-term (over 5 years);

· according to the form of expression allocate goals that are characterized by quantitative indicators, and described qualitatively;

· according to time among the goals are strategic, current and operational;

· by hierarchy level the mission, main, general and specific (local) goals are determined;

· according to the features of interaction goals can be indifferent to each other (indifferent), competing, complementary (complimentary), mutually exclusive (antagonistic), coinciding (identical).

System of values- this is a group of programs specific to each person that determines the scheme and style of his thinking at a subconscious level. This part of the model of the world allows us to develop our personal, subjective attitude to the events happening to us, that is, determines our reaction to them. The value system helps us to distinguish with certainty what is good and what is bad, what is right and what is wrong, what is normal and what is not normal, what is important and what is not important, what is acceptable and what is not acceptable.

17. Target approach in organizational management. The "tree of goals" method and the specifics of its application.

With a targeted approach to the strategy, the problems of excessive detail, congestion and common places are more easily solved. Everything that does not relate to or does not significantly affect the main decision issues is not analyzed and prescribed in the strategy. These issues are addressed within the framework of the business planning system and other current plans and programs. Similarly, the risks of inconsistency between the plans of various departments are reduced: by discarding everything superfluous and insignificant, it is easier to focus on solving the main tasks

An effective goal setting method STRUCTURING METHOD, better known as goal tree. It allows you to identify the count-tion and qualities of the relationship and relationship between goals at different levels.

The "tree" consists of several levels of goals:

1. General goal (main goals); 2. Goals of the 2nd level; 3. Goals 3rd. Achievement of the main goal, only when the goals of the 2nd and 3rd sublevels are achieved.

The procedure for building a goal tree includes several successive steps.

· Definition of top of a tree - the general purpose of the organization. At a certain time stage, there cannot be several common goals. Depending on this goal, the final result of the activity and the effectiveness of this result are determined.

· Formation of subsequent levels in areas of activity or decomposition of goals. Each subsequent level is formed in such a way as to ensure the achievement of the goals of a higher level.

· Each "branch" of the tree describes not a way to achieve the goal, but a specific end result, expressed by some indicator.

Subgoals of one level of decomposition are independent (parallel) among themselves. Achieving the goals of a higher level is possible only if the lower ones are achieved.

18. The process of forming a set of goals. Features of the procedure for selecting targets.

Goals are subdivided according to the areas of activity of the manager, content, management hierarchy and time (short-term, medium-term and long-term). A goal that cannot be reached, but that one can strive to approach, is called an ideal.

Goal setting is the result of the alternatives considered. The fundamental rule of modern management is that the achievement of goals is possible only within the limits imposed by the environment. the management process involves making decisions, choosing alternative strategies and evaluating results in accordance with pre-set goals.

The allocation of levels of the hierarchy of goals can be carried out both on the basis of the functional principle of management, and on the basis of the commodity-market principle. Functional differentiation is associated with grouping according to the content of activities: production, personnel, marketing, finance.

For an organization built on the basis of functional division, the goal tree is built according to the principle: the goal of the enterprise - functional goals (by departments) - operational goals. For an organization according to the commodity-market principle: the purpose of the enterprise - the goals of businesses - operational goals. In practice, these two approaches are often combined, and the structure of the goal tree will look like: enterprise goal - business goals - functional goals of departments - operational goals.

19. Structuring and presentation of goals. Goal analysis. Goal measurability. Measurement scales.

The goal is the desired result.

Method goal structuring provides development of a system of organization goals (including their quantitative and qualitative formulations) and subsequent analysis of organizational structures in terms of their compliance with the system of goals. When using it, the following steps are most often performed:

Development of a system ("tree") of goals, which is a structural basis for linking all types of organizational activities, based on the final results (regardless of the distribution of these activities among organizational units and program-target subsystems in the organization);

Expert analysis of the proposed options for the organizational structure in terms of organizational security for achieving each of the goals, observing the principle of homogeneity of goals set for each unit, determining the relationship of leadership, subordination and cooperation of units based on the relationship of their goals, etc .;

Drawing up rights and responsibility maps for achieving goals both for individual departments and for complex cross-functional activities, where the scope of responsibility (products, resources, workforce, production and management processes, information) is regulated; concrete results for the achievement of which responsibility is established; the rights that the unit is given to achieve results (approval and submission for approval, approval, confirmation, control)

Measurability of goals. When we say that a goal must be measurable, what we mean is to define the parameters by which the goal can be measured. You must establish how to monitor the activities of the team, how to measure them and record them. If you are not able to measure the result in numbers, then your goal is not formulated correctly, and it needs to be reconsidered. For example, if you set the goal to "expand our business", this goal is not measurable, since you have not specified what result you will measure. That is, to achieve a certain level of profit, to reduce staff turnover to a certain level, to come out on top.

Measurement scales.

A scale is a measurement tool, which is a numerical system, where the properties of empirical objects are expressed as properties of a number series. The scale implies the existence of certain rules for its use, for example, establishing a correspondence between numbers and empirical objects.

Scale transformation - renaming of objects of measurement.

Scale type - a group of scales that have the same shape. There are four main types of scales used in sociology.

Scale types:

Nominal scale, scale of names. It is used to measure objects indicated by the name - gender, region of residence, belonging to a political party.

Ordinal scale. Measures the level of agreement with the statement, the degree of satisfaction.

Interval scale. Measures age, income in interval values.

Relationship scale. Measures length of service, age, income.

20. Some concepts of efficiency theory. Efficiency. Criteria and performance indicators. Requirements for performance criteria.

System efficiency

Theory of efficiency. Application area. The theory of effectiveness allows you to evaluate the effectiveness of the use of the management system and choose the best organization for its application under specific circumstances.

Essence. The essence of the theory is to evaluate the effectiveness of achieving the goal by the system and the efforts expended on it. Efficiency theories take into account three groups of process performance indicators that characterize:

The degree of achievement of the goal (target effects);

Resource costs (resource intensity of the process);

Time consumption (process efficiency).

In general, the assessment of operational properties is carried out as an assessment of two aspects:

1. outcome (results) of the operation;

2. an algorithm that provides results.

Efficiency criterion is an indicator that expresses the main measure of the desired result, which is taken into account when considering options for a solution.

The quality of the outcome of the operation and the algorithm that provides the results are evaluated according to the quality indicators of the operation, which include effectiveness, resource intensity and efficiency.

The process of choosing an efficiency criterion, as well as the process of determining the goal, is largely subjective, creative, requiring an individual approach in each individual case.

21. Tasks of efficiency. Method "efficiency - cost" and options for its use.

System efficiency- this is the property of the system to fulfill the goal in the given conditions of use and with a certain quality. Efficiency indicators characterize the degree of adaptability of the system to the fulfillment of the tasks assigned to it and are generalizing indicators of the optimal functioning of the IS.

As an example, let us cite one of the methods for finding compromise solutions, known as "cost-effectiveness" and used in making both important strategic and tactical decisions.

Let us dwell on the main features of the practical application of the "cost-effectiveness" analysis.
Experience shows that the most effective projects are often the most costly. Naturally, if among the considered proposals there was a project whose expected efficiency exceeds the expected efficiency of other projects, and the cost is less than the cost of other projects, then the choice problem would be solved simply. Such a project is the most preferable.

However, in real decision-making practice, this case is extremely rare. Therefore, in order to choose the truly most preferable alternative, additional analysis is needed - an additional multi-criteria, and in this case, two-criteria assessment.
It should be noted that in the cost-effectiveness analysis no attempt is made to find one common measure, the only quantitative assessment that would allow one to compare (rank) alternative project options in terms of preference.

No less often in the practice of decision-making, the so-called "cost-profit" method is used, in which various types of "profit" are considered.

Different types of "profit" here are understood as different criteria that characterize the project, and not necessarily of an economic nature.

One of the main requirements of this method, embedded in the decision-making algorithm, is the ability to add up various types of "profit" with fixed numerical coefficients, obtaining a single composite value - "profit" that characterizes the project.

Similar information.

Stochastic Models

As mentioned above, stochastic models are probabilistic models. At the same time, as a result of the calculations, it is possible to say with a sufficient degree of probability what the value of the analyzed indicator will be when the factor changes. The most common application of stochastic models is forecasting.

Stochastic modeling is, to a certain extent, an addition and extension of deterministic factor analysis. In factor analysis, these models are used for three main reasons:

• it is necessary to study the influence of factors on which it is impossible to build a rigidly determined factorial model (for example, the level of financial leverage);

• it is necessary to study the influence of complex factors that cannot be combined in the same rigidly deterministic model;
• it is necessary to study the influence of complex factors that cannot be expressed in one quantitative indicator (for example, the level of scientific and technological progress).

In contrast to the rigidly deterministic approach, the stochastic approach for implementation requires a number of prerequisites:

1. the presence of a population;
2. sufficient volume of observations;
3. randomness and independence of observations;
4. homogeneity;
5. the presence of a distribution of signs close to normal;
6. the presence of a special mathematical apparatus.

The construction of a stochastic model is carried out in several stages:

• qualitative analysis (setting the goal of the analysis, determining the population, determining the effective and factor signs, choosing the period for which the analysis is carried out, choosing the analysis method);
• preliminary analysis of the simulated population (checking the homogeneity of the population, excluding anomalous observations, clarifying the required sample size, establishing the laws of distribution of the studied indicators);
• building a stochastic (regression) model (refinement of the list of factors, calculation of estimates of the parameters of the regression equation, enumeration of competing models);
• assessing the adequacy of the model (checking the statistical significance of the equation as a whole and its individual parameters, checking the correspondence of the formal properties of the estimates to the objectives of the study);
• economic interpretation and practical use of the model (determination of the spatio-temporal stability of the constructed dependence, assessment of the practical properties of the model).

Basic concepts of correlation and regression analysis

Correlation analysis - a set of methods of mathematical statistics that allow estimating the coefficients that characterize the correlation between random variables and testing hypotheses about their values ​​based on the calculation of their sample counterparts.

Correlation analysis called the method of processing statistical data, which consists in studying the coefficients (correlations) between variables.

correlation(which is also called incomplete, or statistical) appears on average, for mass observations, when the given values ​​of the dependent variable correspond to a certain number of probable values ​​of the independent variable. The explanation for this is the complexity of the relationships between the analyzed factors, the interaction of which is influenced by unaccounted random variables. Therefore, the relationship between the signs is manifested only on average, in the mass of cases. With a correlation, each value of the argument corresponds to randomly distributed values ​​of the function in a certain interval.

In the most general form, the task of statistics (and, accordingly, economic analysis) in the field of studying relationships is to quantify their presence and direction, as well as to characterize the strength and form of influence of some factors on others. To solve it, two groups of methods are used, one of which includes the methods of correlation analysis, and the other - regression analysis. At the same time, a number of researchers combine these methods into a correlation-regression analysis, which has some grounds: the presence of a number of common computational procedures, complementarity in interpreting the results, etc.

Therefore, in this context, we can talk about correlation analysis in the broad sense - when the relationship is comprehensively characterized. At the same time, there are correlation analysis in the narrow sense - when the strength of the connection is studied - and regression analysis, during which its form and the impact of some factors on others are evaluated.

Tasks proper correlation analysis are reduced to measuring the closeness of the relationship between varying traits, identifying unknown causal relationships and assessing the factors that have the greatest impact on the resulting trait.

Tasks regression analysis lie in the field of establishing the form of dependence, determining the regression function, using an equation to estimate unknown values ​​of the dependent variable.

The solution of these problems is based on appropriate techniques, algorithms, indicators, which gives grounds to talk about the statistical study of relationships.

It should be noted that the traditional methods of correlation and regression are widely represented in various statistical software packages for computers. The only thing left for the researcher is to properly prepare the information, choose a software package that satisfies the requirements of the analysis, and be ready to interpret the results. There are many algorithms for calculating communication parameters, and at present it is hardly advisable to carry out such a complex type of analysis manually. Computational procedures are of independent interest, but knowledge of the principles of studying the relationships, possibilities and limitations of certain methods of interpreting the results is a prerequisite for research.

Methods for assessing the tightness of the connection are divided into correlation (parametric) and non-parametric. Parametric methods are based on the use, as a rule, of normal distribution estimates and are used in cases where the population under study consists of quantities that obey the normal distribution law. In practice, this position is most often taken a priori. Actually, these methods are parametric and are commonly called correlation methods.

Nonparametric methods do not impose restrictions on the law of distribution of the studied quantities. Their advantage is also the simplicity of calculations.

autocorrelation- statistical relationship between random variables from the same series, but taken with a shift, for example, for a random process - with a shift in time.

Pair correlation

The simplest technique for identifying a relationship between two features is to build correlation table:

\Y\X\	Y 1	Y2	...	Yz	Total	Y i
x1	f 11		...	f 1z
x1	f 21		...	f2z
...	...	...	...	...	...	...
X r	f k1	k2	...	fkz
Total			...		n
			...			-

The grouping is based on two traits studied in the relationship - X and Y. Frequencies f ij show the number of corresponding combinations of X and Y.

If f ij are arranged randomly in the table, we can talk about the absence of a relationship between the variables. In the case of the formation of any characteristic combination f ij, it is permissible to assert a connection between X and Y. In this case, if f ij is concentrated near one of the two diagonals, there is a direct or reverse linear relationship.

A visual representation of the correlation table is correlation field. It is a graph where X values ​​are plotted on the abscissa axis, Y values ​​are plotted along the ordinate axis, and the combination of X and Y is shown by dots. By the location of the points, their concentration in a certain direction, one can judge the presence of a connection.

correlation field the set of points (X i , Y i ) on the XY plane is called (Figures 6.1 - 6.2).

If the points of the correlation field form an ellipse whose main diagonal has a positive slope (/), then there is a positive correlation (an example of such a situation can be seen in Figure 6.1).

If the points of the correlation field form an ellipse, the main diagonal of which has a negative slope angle (\), then there is a negative correlation (an example is shown in Figure 6.2).

If there is no regularity in the location of the points, then they say that in this case there is a zero correlation.

In the results of the correlation table for rows and columns, two distributions are given - one for X, the other for Y. Let's calculate for each X i the average value of Y, i.e. , how

The sequence of points (X i , ) gives a graph that illustrates the dependence of the average value of the effective feature Y on the factor X, - empirical regression line, showing how Y changes as X changes.

In essence, both the correlation table, and the correlation field, and the empirical regression line already characterize the relationship beforehand, when the factorial and resulting features are selected and it is required to formulate assumptions about the form and direction of the relationship. At the same time, a quantitative assessment of the closeness of the connection requires additional calculations.

Any real process peculiar random fluctuations caused by the physical variability of any factors over time. In addition, there may be random external influences on the system. Therefore, with an equal average value of the input parameters at different times, the output parameters will be different. Therefore, if random effects on the system under study are significant, it is necessary to develop probabilistic (stochastic) object model, taking into account the statistical laws of distribution of system parameters and choosing the appropriate mathematical apparatus.

When building deterministic models random factors are neglected, taking into account only the specific conditions of the problem being solved, the properties and internal connections of the object (almost all sections of classical physics are built on this principle)

The idea behind deterministic methods- in using the own dynamics of the model during the evolution of the system.

In our course, these methods are: molecular dynamics method, the advantages of which are: the accuracy and certainty of the numerical algorithm; the disadvantage is the complexity due to the calculation of the forces of interaction between particles (for a system of N particles, at each step it is necessary to perform
operations for calculating these forces).

At deterministic approach are given, and the equations of motion are integrated with respect to time. We will consider systems of many particles. The positions of the particles give the potential energy contribution to the total energy of the system, and their velocities determine the contribution of the kinetic energy. The system moves along a trajectory with constant energy in the phase space (further explanations). For deterministic methods, a microcanonical ensemble is natural, the energy of which is the integral of motion. In addition, it is possible to study systems for which the integral of motion is temperature and (or) pressure. In this case, the system is not closed, and it can be represented in contact with a thermal reservoir (canonical ensemble). To model it, we can use an approach in which we limit a number of degrees of freedom of the system (for example, we set the condition
).

As we have already noted, in the case when the processes in the system occur unpredictably, such events and the quantities associated with them are called random, and algorithms for modeling processes in the system - probabilistic (stochastic). Greek stoohastikos- literally means "he who can guess".

Stochastic methods use a slightly different approach than deterministic ones: it is required to calculate only the configuration part of the problem. The equations for the momentum of the system can always be integrated. The problem that then arises is how to conduct transitions from one configuration to another, which in the deterministic approach are determined by the impulse. Such transitions in stochastic methods are carried out with a probabilistic evolution in Markov process. The Markov process is a probabilistic analog of the model's own dynamics.

This approach has the advantage of being able to model systems that do not have any intrinsic dynamics.

Unlike deterministic methods, stochastic methods on a PC are easier and faster to implement, however, to obtain close to true values, good statistics are required, which requires modeling a large ensemble of particles.

An example of a fully stochastic method is Monte Carlo method. Stochastic methods use the important concept of a Markov process (Markov chain). The Markov process is a probabilistic analogue of the process in classical mechanics. The Markov chain is characterized by a lack of memory, i.e., the statistical characteristics of the near future are determined only by the present, without regard to the past.

Practically busy 2.

Random Walk Model

Example(formal)

Let us assume that particles are placed at arbitrary positions at the nodes of a two-dimensional lattice. At each time step, the particle “jumps” to one of the lucky positions. This means that the particle has the ability to choose the direction of the jump to any of the four nearest places. After the jump, the particle "does not remember" where it jumped from. This case corresponds to a random walk and is a Markov chain. The result at each step is a new state of the particle system. The transition from one state to another depends only on the previous state, i.e. the probability of the system being in state i depends only on state i-1.

What physical processes in a solid remind us of the (similarity) of the described formal random walk model?

Of course, diffusion, i.e., the most, processes, the mechanisms of which we considered in the course of heat and mass transfer (3 course). As an example, let us recall the usual classical self-diffusion in a crystal, when, without changing their visible properties, atoms periodically change their places of temporary residence and wander around the lattice, using the so-called “vacancy” mechanism. It is also one of the most important mechanisms of diffusion in alloys. The phenomenon of atomic migration in solids plays a decisive role in many traditional and non-traditional technologies - metallurgy, metalworking, the creation of semiconductors and superconductors, protective coatings and thin films.

It was discovered by Robert Austen in 1896, observing the diffusion of gold and lead. Diffusion- the process of redistribution of the concentrations of atoms in space by chaotic (thermal) migration. The reasons, from the point of view of thermodynamics, there can be two: entropy (always) and energy (sometimes). The entropy cause is the increase in chaos when the atoms of the carved variety are stirred. Energy - promotes the formation of an alloy, when it is more profitable to be near an atom of a different sort, and promotes diffusion decay, when the energy gain is ensured by placing atoms of the same sort together.

The most common diffusion mechanisms are:

vacancy

internodal

displacement mechanism

To implement the vacancy mechanism, at least one vacancy is required. Migration of vacancies is carried out by moving to an unoccupied site of one of the neighboring atoms. An atom, on the other hand, can carry out a diffusion jump if there is a vacancy next to it. Vacancy cm, with a period of thermal vibrations of an atom in a lattice sitec, at a temperature T = 1330 K (by 6 K< точки плавления), число скачков, которое совершает вакансия в 1с, путь за одну секунду-см=3 м (=10 км/ч). По прямой же путь, проходимый вакансиейсм, т. е. в 300 раз короче пути по ломаной.

Nature needed it. so that the vacancy changes its place of residence within 1s time, passes along a broken line of 3m, and shifts along a straight line by only 10 μm. Atoms behave more calmly than vacancies. But they also change their place of residence a million times per second and move at a speed of about 1 m / h.

So. that one vacancy per several thousand atoms is enough to move atoms at the micro level at a temperature close to melting.

Let us now form a random walk model for the phenomenon of diffusion in a crystal. The wandering process of an atom is chaotic and unpredictable. However, for an ensemble of wandering atoms, statistical regularities should appear. We will consider uncorrelated jumps.

This means that if
and
is the movement of atoms at the i and jth jumps, then after averaging over the ensemble of wandering atoms:

(mean product = product of means. If the walks are completely random, all directions are equal and
=0.)

let each particle of the ensemble make N elementary jumps. Then its total displacement is:

;

and the mean square of the displacement

Since there is no correlation, the second term =0.

Let each jump have the same length h and a random direction, and the average number of jumps per unit time v. Then

It's obvious that

Let's call the quantity
- diffusion coefficient of wandering atoms. Then
;

For the 3D case -
.

We got parabolic diffusion law- the average square of the displacement is proportional to the wandering time.

It is this task that we have to solve in the next laboratory work - modeling random one-dimensional walks.

Numerical model.

We define an ensemble of M particles, each of which makes N steps, independently of each other, to the right or to the left with the same probability. stride length = h.

For each particle, we calculate the square of the displacement
in N steps. Then we average over the ensemble -
. Value
, if
, i.e., the mean square of the bias is proportional to the random walk time
- average time of one step) - parabolic law of diffusion.

The stochastic model describes the situation when there is uncertainty. In other words, the process is characterized by some degree of randomness. The adjective "stochastic" itself comes from the Greek word "guess". Since uncertainty is a key characteristic of everyday life, such a model can describe anything.

However, each time we apply it, the result will be different. Therefore, deterministic models are more often used. Although they are not as close as possible to the real state of affairs, they always give the same result and make it easier to understand the situation, simplify it by introducing a set of mathematical equations.

Main features

A stochastic model always includes one or more random variables. She seeks to reflect real life in all its manifestations. Unlike stochastic, it does not aim to simplify everything and reduce it to known values. Therefore, uncertainty is its key characteristic. Stochastic models are suitable for describing anything, but they all have the following common features:

• Any stochastic model reflects all aspects of the problem for which it was created.
• The outcome of each of the phenomena is uncertain. Therefore, the model includes probabilities. The correctness of the overall results depends on the accuracy of their calculation.
• These probabilities can be used to predict or describe the processes themselves.

Deterministic and stochastic models

For some, life seems to be a succession for others - processes in which the cause determines the effect. In fact, it is characterized by uncertainty, but not always and not in everything. Therefore, it is sometimes difficult to find clear differences between stochastic and deterministic models. Probabilities are quite subjective.

For example, consider a coin toss situation. At first glance, it looks like there is a 50% chance of getting tails. Therefore, a deterministic model must be used. However, in reality, it turns out that much depends on the dexterity of the hands of the players and the perfection of the balancing of the coin. This means that a stochastic model must be used. There are always parameters that we do not know. In real life, the cause always determines the effect, but there is also a certain degree of uncertainty. The choice between using deterministic and stochastic models depends on what we are willing to give up - simplicity of analysis or realism.

In chaos theory

Recently, the concept of which model is called stochastic has become even more blurred. This is due to the development of the so-called chaos theory. It describes deterministic models that can give different results with a slight change in the initial parameters. This is like an introduction to the calculation of uncertainty. Many scientists have even admitted that this is already a stochastic model.

Lothar Breuer elegantly explained everything with the help of poetic images. He wrote: “A mountain stream, a beating heart, an epidemic of smallpox, a column of rising smoke - all this is an example of a dynamic phenomenon, which, as it seems, is sometimes characterized by chance. In reality, such processes are always subject to a certain order, which scientists and engineers are only just beginning to understand. This is the so-called deterministic chaos.” The new theory sounds very plausible, which is why many modern scientists are its supporters. However, it still remains little developed, and it is rather difficult to apply it in statistical calculations. Therefore, stochastic or deterministic models are often used.

Building

Stochastic begins with the choice of the space of elementary outcomes. So in statistics they call the list of possible results of the process or event being studied. The researcher then determines the probability of each of the elementary outcomes. Usually this is done on the basis of a certain technique.

However, the probabilities are still quite a subjective parameter. The researcher then determines which events are most interesting for solving the problem. After that, it simply determines their probability.

Example

Consider the process of building the simplest stochastic model. Suppose we roll a die. If "six" or "one" falls out, then our winnings will be ten dollars. The process of building a stochastic model in this case will look like this:

• Let us define the space of elementary outcomes. The die has six sides, so one, two, three, four, five, and six can come up.
• The probability of each of the outcomes will be equal to 1/6, no matter how much we roll the die.
• Now we need to determine the outcomes of interest to us. This is the loss of a face with the number "six" or "one".
• Finally, we can determine the probability of the event of interest to us. It is 1/3. We sum up the probabilities of both elementary events of interest to us: 1/6 + 1/6 = 2/6 = 1/3.

Concept and result

Stochastic simulation is often used in gambling. But it is also indispensable in economic forecasting, as it allows you to understand the situation deeper than deterministic ones. Stochastic models in economics are often used in making investment decisions. They allow you to make assumptions about the profitability of investments in certain assets or their groups.

Modeling makes financial planning more efficient. With its help, investors and traders optimize the distribution of their assets. Using stochastic modeling always has advantages in the long run. In some industries, refusal or inability to apply it can even lead to the bankruptcy of the enterprise. This is due to the fact that in real life new important parameters appear daily, and if they are not, it can have disastrous consequences.