Imitation of psychological intuition using artificial neural networks

Introduction

Chapter 1. Psychodiagnostics and neural networks.

1.1 Tasks and methods of modern psychodiagnostics.
1.2 The essence of the intuitive method.
1.3 Mathematical models and psychodiagnostic algorithms.
1.4 Promising algorithms for constructing psychodiagnostic methods.
1.5 Methods of recovering dependencies.
1.6 Algorithms and methods of unconditional optimization.

1.7 Neural networks.
1.7.1 Basic elements.
1.7.2 Network structure.
1.7.3 Direct operation of the network.
1.7.4 Network training.
1.7.5 Reverse operation.

Conclusions of chapter 1.

Chapter 2. Neural networks solving classical problems of psychodiagnostics.

2.1 Classic experiment.
2.2 Assessment of the significance of the test questions.
2.3 Network Contrast on the importance of test questions.
2.4 Results of experiments with contrasted networks.

Conclusions of chapter 2

Chapter 3. Intuitive prediction of relationships by neural networks.

3.1 The problem of evaluating relationships.
3.2 General task of experiments.
3.3 Psychological methods used in experiments.
3.4 Experiments on the prediction of group status.
3.5 Neural network research of the questionnaire structure.
3.6 Estimation of optimization of the task book by the neural network from the standpoint of information theory
3.7 Experiments in predicting pairing relationships.

Conclusions of chapter 3.

LITERATURE

Application. Psychological questionnaire A.G. Kopytov

Introduction

From the very beginning of the information age, the ideas of reproduction of the principles of brain functioning in the work of computers have occupied the minds of scientists. It is known, for example, that Wiener and Rosenblatt worked together on the study of biological neurons, and that from these works the idea of ​​learning Wiener automata and the theory of learning Rosenblatt perceptron networks was born.

The idea of ​​using artificial neural networks in modern computing has taken a firm place in the minds of its developers. Neural networks are used to solve problems of artificial intelligence, in the systems of technical organs of the senses and control of production processes. Hopfield adaptive retinas are used to create interference-resistant communication systems. At the stage of experimental development (for example, in the laboratories of the Siemens company) there are samples of hardware neurocomputers of mass use — neuroprocessors for personal computers.

Neurocomputers are used in many branches of modern science — nuclear physics, geology, meteorology. The study of artificial neural networks constitute significant sections in such sciences as biophysics, computational mathematics, electronics.

Application of artificial neural networks to human sciences would also be attractive. However, the following problem arises: their theory has not yet been formed sufficiently to describe the processes occurring in systems in the form of explicit algorithms that are suitable for modeling on modern computer technology. This is expressed in particular in the fact that the diagnostic apparatus of psychology and medicine in a substantial part is based on approaches connected with the study and systematization of precedents. Simulation of biophysical processes is complicated by the enormous complexity of systems — so, when working with psychological tasks, the functioning of a system consisting of a number of elements of the order of 10 ⁹ (the human brain) is inaccessible for modeling on a computer of any power that can be thought today.

Attempts to use neural network approaches in medicine were undertaken with considerable success by the NeuroComp group (under the guidance of Professor A.N. Gorban). With the help of neural network expert systems, the tasks of predicting the complications of myocardial infarction, early diagnosis and differential diagnosis of malignant vascular membranes of the choroid, simulation of treatment and prediction of its immediate results in patients with obliterating thromboangiitis, differential diagnosis of acute abdomen, and studying immunoreactivity were solved.

In general, on the path of using artificial neural networks to problems from the field of biology, medicine and psychology, we can expect several important results. First, neural networks, working on implicit algorithms and solving problems that do not have an explicit solution, approach the human brain by the mechanism for solving problems, which can provide important material for studying the processes of higher nervous activity. Secondly, neural networks can serve as a mathematical tool for research when searching for relationships and patterns in large information structures, studying the mutual influence of various factors and modeling complex dynamic processes.

Therefore, the development of methods for neural network modeling and information analysis is an important task.

The section of information science, called neuroinformatics, which began in its time by Rosenblatt’s work on the theory of learning of perceptron networks, has experienced several booms and busts. At the moment, the most common ideas about neuroinformatics are:

The principles of operation of neurocomputers resemble the interaction of cells of the nervous system - neurons through special connections - synapses. The basis of the work of self-learning neuroprograms is a neural network, which is a collection of neurons - elements that are interconnected in a certain way.

Training of the neural network is achieved by adjusting the parameters - weights of synapses and characteristics of the transducers in order to minimize the error in defining examples of the training sample - pairs of the “required output – obtained output” type.

In the training, an algorithm of ultrafast calculation of the error function gradient with respect to the parameters to be trained using the apparatus of dual functions is used. The presence of methods that allow obtaining the gradient of the error function in highly parallel (with the presence of appropriate hardware) mode allows you to use the extensive apparatus of methods for the unconditional optimization of multidimensional functions for training neural networks.

The experience gained by researchers in the field of neuroinformatics shows that with the help of neural networks, it is possible to satisfy the extremely urgent need of practicing psychologists and researchers in creating psychodiagnostic techniques based on their experience, bypassing the stage of formalization and building a diagnostic model . Thus, this paper is devoted to the study of the possibility of the development of psychological intuition in neural network expert systems.

The purpose of this work was to study the following aspects of the application of neural networks to psychological problems:

study of the functioning of neural networks in solving classical problems of psychodiagnostics;

study of the possibilities and mechanism of the intuitive neural network prediction of relations between people based on their psychological characteristics;

For a more detailed understanding of the mechanism of intuition of artificial neural networks in solving psychological problems characterized by extremely high dimensionality of the space of input signals, it was also required to create a program model of the neuroimulator with optimization of the neural network volume to solve a specific problem.

To achieve these goals, the following tasks were set:

- to assess the fundamental applicability of neural networks for solving psychological problems;

- assess the applicability of the intuitive approach when the neural network makes recommendations, bypassing the creation of descriptive reality;

The first chapter shows a range of tasks related to computer psychodiagnostics and diagnostic intuition. A review of the methods of creating psychodiagnostic methods is performed, the range of mathematical methods and algorithms used in this process is highlighted. In this regard, a detailed survey study of dependency recovery algorithms and unconditional optimization methods was carried out, as well as basic information concerning the apparatus of neural networks.

The second chapter describes a series of experiments aimed at testing the hypothesis of the applicability of neural networks to the problems of psychodiagnostics. The quality (error) of the psychological diagnosis made by the neural network based on the standard LOBI test is examined on the basis of a sliding control test sample of 273 examples.

A study is being carried out on the applicability of neural networks as a psychodiagnostic apparatus - a researcher in determining and optimizing the structure of psychological tests.

The influence of the structure of psychological tests on the diagnostic intuition of an artificial neural network is investigated.

The third chapter analyzes a series of experiments aimed at testing a hypothesis about the possibility of a neural network predicting the relationship between people based on their psychological qualities, which are objectively described by a psychological test. The study was conducted on the material of 48 studied and 474 pairs of mutual elections.

Work was done to determine the optimal structure of the neural network to predict the social status of the subjects based on the questionnaire. The questionnaire was prepared by A.G. Kopytov and published in the annex to the work with his kind permission.

An estimate of the error in the prediction of the status of the subjects studied in the group was made; it was compared with the distance between random examples.

A cross-group intergroup study was performed, as well as a study common to all groups in order to clarify the intragroup locality of the psychological intuition of the neural network.

On the basis of assessments of the significance of the input parameters of the neural network, an assessment of the redundancy of the basic questionnaire was carried out, the influence of the questionnaire minimization on the quality of the prediction status of the subjects studied in the group was investigated. For the first time, work was done to optimize the structure of psychodiagnostic methods based on the study of the mechanism of psychological intuition of software neuroimitators.

An estimate was made of the error in predicting the relationship between the two subjects, and a comparison was made with the distance between random examples.

The results obtained in this paper provide an approach to uncovering the mechanism of intuition of neural networks, which manifests itself in solving psychodiagnostic problems. It also shows the way to use the understanding of the mechanism of psychological intuition of neural network expert systems in a significant simplification of the process of the formation of diagnostic models. The results are of interest for the theory of creation of psychodiagnostic techniques, which allow us to recommend neural networks for use in this area. The intuitive approach to psychodiagnostics, which is non-standard for computer methods, presented in the work, which consists in the exclusion of the construction of the described reality, makes it possible to reduce and simplify work on psychodiagnostic methods. The study of the intuition mechanism of neural networks in the prediction of psychological compatibility in the group and pairwise compatibility provides important material for understanding the mechanism of this phenomenon.

The discovery of the intuition mechanism of neural networks using the apparatus for calculating the importance of input parameters allows us to simplify psychodiagnostic models, reducing the dimension of the feature space.

The results of the work are summarized in the thesis: Dorrer M. G., Psychological intuition of artificial neural networks, thesis, ... 1998.

Chapter 1. Psychodiagnostics and Neural Networks

1.1. Tasks and methods of modern psychodiagnostics

Psychodiagnostics occupies an important place among the tasks of modern psychology - making a decision about the present psychological state of a person as a whole or in relation to any particular human property. According to [26], the purpose of psychodiagnostics according to modern concepts is to describe individually - psychological features, personality traits in the interests of theory and practice.

According to one of the most commonly used interpretations [71], psychodiagnostics is a science, in the course of which the following questions are solved:

1. What is the nature of psychological phenomena and the fundamental possibility of their scientific evaluation?

2. What are currently the general scientific grounds for the principle knowability and quantitative assessment of psychological phenomena?

3. To what extent do psychodiagnostic tools used conform to accepted general scientific, methodological requirements?

4. What are the main methodological requirements for various psychodiagnostic tools?

5. What are the grounds for the reliability of the results presented for the conditions of psychodiagnostics, the means of processing the results obtained and the ways of its interpretation?

6. What are the main procedures for constructing and testing the scientific methods of psychodiagnostics, including tests?

Accurate psychodiagnosis in any psychological experiment involves the assessment of the psychological properties of the subject.

One of the key in modern psychodiagnostics is the concept of diagnosis, which in [61] is interpreted as follows: “The concept of“ diagnosis ”is a kind of expression and concretization of the general scientific concept of“ state ”, reflecting the dominant way of changing and developing systems in these relationships, in a certain place and time.

According to [21], diagnostics as a practical activity is carried out in order to transform the real state of an object. Diagnostic knowledge as a whole is such a type of knowledge in which the subject, based on his practical needs, sets a definite goal - to use the laws of the functioning of the object being diagnosed to intervene in the system, that is, bring it to a state of normal functioning by management methods.

However, the psychodiagnostic method according to [7], [26] has its own characteristics. His analysis allows to identify specific motives that determine the activity of the subject, a special strategy of his behavior, the specifics of the situation - both social (interaction of the psychologist and the researched) and stimulus (for example, with varying degrees of structure) - etc.

The paradox of theoretical and psychodiagnostic description of the same reality, the essence of which lies in the gnoseological difference between the “theoretical” and the “measured” personality, which differs in turn from the real person, constitutes a significant difficulty in psychodiagnostics. The consequence of this complexity is that attempts to identify the “theoretical” and “measured” personality turn out to be, in the end, unproductive, artificial.

The scope of psychodiagnostics according to [71] is very wide. It includes:

- Testing hypotheses, tested in experiments;

- applied research in which you want to check the result of the introduction of certain innovations;

- psychological counseling, for which the psychologist must have the correct diagnosis of the subject, see the essence of his problem;

- practical psychocorrectional work;

- medical psychology;

- pathopsychology;

- engineering psychology;

- the psychology of labor.

It can be argued that psychodiagnostics can be applied wherever exact knowledge of the degree of development of certain properties of a person is required.

According to [47], psychodiagnostics is characterized by a wide range of methodological approaches. This variety determines the existence of various classification systems of psychodiagnostic experiment depending on the attributes that are important for classification. For computer psychodiagnostics, the formalizability of the psychodiagnostic method, which allows determining the possibility of using computer information technology in the psychodiagnostic experiment, can serve as such a significant attribute.

The concept of “formalizability” is specified by dividing into independently systematizing elements: the impact on the test subject during the experiment (stimuli), the responses (responses) of the test subject to this effect, and operations with information generated by the test person’s response to the stimuli.

1.2. Essence of the intuitive method

According to [81], intuition is knowledge that arises without awareness of the ways and conditions for obtaining it, by virtue of which the subject has it as a result of “immediate discretion”. Intuition is interpreted both as a specific ability (for example, artistic and scientific intuition) and as “holistic embracing” of the conditions of a problem situation (sensory intuition, intellectual intuition) and as a mechanism of creative activity (creative intuition).

Scientific psychology considers intuition as a necessary, internally determined by the nature of creativity, the moment when the stereotypes of behavior and, in particular, logical programs for finding a solution to the problem go beyond the boundaries.

According to [80], intuition is a heuristic process consisting in finding a solution to a problem on the basis of search orientations that are not connected logically or insufficient for obtaining a logical conclusion. The intuition is characterized by the speed (sometimes instantaneous) of the formulation of hypotheses and decision-making, as well as the lack of awareness of its logical bases.

Intuition manifests itself in conditions of subjectively or objectively incomplete information and organically enters the ability to extrapolate the intrinsic thinking of a person.

The mechanism of intuition consists in the simultaneous integration of several informative features of different modalities into complex guidelines that guide the search for a solution. In such a simultaneous consideration of information that is different in its quality, intuitive processes are different from discursive processes, in which only a single modification of the attributes of a problem that are interconnected can be taken into account in a single thinking act (logical step).

Search guidelines in intuitive and discursive processes do not have a fundamental difference in the composition of the information contained in them. Logical principles, including formal ones, are included in the intuitively formed informative complex and, being in themselves insufficient for obtaining a solution, in combination with other information links determine the direction of the search.

The main role in intuition is played by semantic generalizations relating to a given problem domain. Such is the intuition of a doctor or a scientist.

1.3. Mathematical models and psychodiagnostic algorithms

In the work of a psychodiagnostic test design researcher, it is customary to distinguish three stages [20], [47].

At the first stage, a “draft” version of the test is constructed. It includes tasks, the answers to which, in the opinion of the experimenter, should reflect the individual psychological differences of the subjects on a given construct.

At the second stage, the researcher chooses a diagnostic model and determines its parameters. A diagnostic model is understood as a way of arranging (transforming, aggregating) the initial diagnostic features (variants of answers to test tasks) into a diagnostic indicator.

At the third stage, standardization and testing of the constructed diagnostic model is carried out.

The most commonly used in psychodiagnostics is a linear diagnostic model. No serious attempt to construct or adapt tests can do without the use of empirico-statistical analysis [97]. The initial material for such an analysis is the results of an experimental survey of a representative sample of subjects using a “draft” version of the psychodiagnostic test. A table of experimental data is formed from the obtained data (see Table 1).

Table 1. The structure of the table of experimental data

In tab. ... N is the total number of objects (subjects), p is the total number of signs, x _j is the j-th sign, x _ij is the value of the j-th sign, measured at the i-th object, X = (x ₁ , .. ., x _p ) ^T is the feature vector, X _i = (x _i1 , ..., x _ip ) ^T is the i-th object, X = {X _i } is the set of objects.

Initial characteristics x _j , as a rule, are measured in nominal and ordinal (ordinal) scales [18], [82], [89]. For most objective methods, neither quantitative relations nor order relations can be established a priori, since their attributes are nominal dimensions. Often, during the formalization of test methods, “dichotomization” [65] is used — a procedure for transforming the initial indicators into a set of attributes with two gradations.

For ordinal features, only the gradation order on the scale is significant, and for them any monotonic transformations that do not violate this order are considered acceptable. Methodically rigorous is the application to ordinal features of processing methods, the result of which is invariant with respect to admissible transformations of the ordinal scale [49].

Further, after forming a table of experimental data, a diagnostic model is built. It is considered that the model should in a certain form express the relationship between the vector of input features and the property being tested (the value of the property severity will be denoted further by y). The model should reflect the transformation mechanism y = y (x).

The preliminary step in the construction of diagnostic models is usually the elucidation of the structure of the experimental data table. At this stage, the correlation between the factors and the proximity between the objects is evaluated. The set of mathematical models and algorithms used for this is determined on the basis of the specifics of the experimental data in psychodiagnostics.

To determine the degree of connection between the signs are used [48], [65], [73]:

- Pearson's correlation coefficient, which is a measure of the linear relationship of two variables: and designed to measure the relationship of two dichotomous traits [73]. The coefficient is calculated on the basis of the tables of conjugacy of signs (see Table 2) using the formula

Table 2. Contingency table of dichotomous features

- Kendell's rank correlation coefficient based on counting the number of discrepancies in the ranking of objects by matching variables. This coefficient is based on the task of interpreting the process of measuring the relationship between variables without the aid of the principle of the production of moments. Two signs are considered. and , on each of which N objects are displayed in N consecutive ranks. Of N objects formed par. Then the coefficient is calculated by the formula where P is the number of order matches on a sign with order on the tag , Q is the number of mismatches.

The degree of connection between features can be used to assess the redundancy of the set of features of the “draft” model, for mutual control of scales, etc.

To determine the proximity of objects, various distance measures are used:

- Euclidean distance

- Weighted Euclidean distance .

- Distance Mahalanobis where S is the covariance matrix of the population from which the objects are extracted and .

- Distance Minkowski (city metric), used to measure the distance between objects described by ordinal features. equal to the difference of numbers of gradations by the k-th attribute of the compared objects and .

- Hamming distance which is used to determine differences between objects defined by dichotomous features and is interpreted as the number of mismatches between the values ​​of features of the considered objects and

Information on the proximity of objects obtained on the basis of a metric (for more details - [25], [48], [50]) can be used to single out their groupings.

Presentation of information on the structure of experimental data serves as an intermediate in the construction of a diagnostic model. Regardless of the type of model, its creation may be based on two approaches:

1. A strategy based on autoinformativeness of experimental data.

The high degree of closeness between a group of signs may indicate that the signs included in the group reflect an empirical factor corresponding to the diagnostic construct.

Selection of geometric groupings in the space of objects may indicate the difference of the studied objects according to the property being tested, which allows us to construct a diagnostic algorithm.

For strategies based on the auto-informativeness of experimental data, the consistency of the test assignments is an important category.

The consistency of measurable responses of test subjects to test stimuli means that they must have a statistical focus on the expression of the general, main trend of the test.

The strategy based on the autoinformativeness of experimental data is used to construct a diagnostic algorithm using the principal component method [17], [18], [19], factor analysis [66] and the contrast group method [97].

2. Strategy based on the criteria of external information content. External information can be represented as binding to objects of values ​​of the “dependent” variable, measured on a quantitative scale, as a number of a class that is homogeneous by the tested property of a class, as a serial number (rank) of an object among all objects, ordered by the degree of manifestation of the property being diagnosed or в виде совокупности значений набора внешних (не включенных в таблицу экспериментальных данных) признаков, характеризующих тестируемый психологический феномен.

Методы, основанные на внешней информативности признаков, принято подразделять на экспертные, экспериментальные и жизненные.

Among the expert criteria include assessments, judgments, conclusions about the subjects, made by one expert or their group.

Experimental criteria are the results of a simultaneous and independent study of the subject by another test, which is considered to be approved and measure the same property as the constructed test.

Objective socio-demographic and biographical data are used as life criteria.

The strategy, based on the external information content of experimental data, is used to construct a diagnostic algorithm using a regression analysis, a discriminant analysis [49] and a typological approach [60], [99].

The most widely used at present linear diagnostic models. However, in conditions of heterogeneity of the training set, they have practical success not higher than 70–80% [60].

The constructed diagnostic model can be considered a psychodiagnostic test only after it has passed tests to check the psychometric properties - reliability and validity [20], [27].

The reliability of the test is a characteristic of the technique, reflecting the accuracy of psychodiagnostic measurements, as well as the stability of the test results to the effects of extraneous random factors [27].

Validity - a measure of the conformity of test scores with the ideas about the nature of properties or their role in a given activity [60].

1.4. Promising algorithms for constructing psychodiagnostic methods

A promising direction in the construction of psychodiagnostic methods is now considered the use of the apparatus of pattern recognition theory [2], [13], [47].

The classification of pattern recognition methods is diverse. Parametric, non-parametric and heuristic methods are distinguished, there are classifications based on the terminology of established scientific schools. In [52], pattern recognition methods are classified as follows:

- methods based on the principle of separation;

- statistical methods;

- methods like "potential functions";

- methods for calculating ratings (voting);

- methods based on the apparatus of propositional calculus.

In addition, the method of knowledge representation may be essential for a method based on the pattern recognition theory. Currently, there are two main ways [78]:

1. Intensional representations - schemes of relations between attributes (attributes)

2. Extensional representations - concrete facts (objects, examples).

The group of intensional pattern recognition methods includes the following subclasses:

1) Methods based on estimates of the densities of the distribution of attribute values ​​(methods of non-parametric statistics) [18].

2) Methods based on assumptions about the class of decision functions (methods that use minimization of the risk functional or error functional), [6], [15], [36], [41], [94].

3) Logical methods based on the apparatus of the algebra of logic and allowing to operate with information contained not only in individual features, but also in the combination of their values ​​[49].

4) Linguistic (structural) methods based on the use of special grammars that generate languages, with the help of which a set of properties of recognizable objects can be described [93].

The group of extensional methods includes:

1) The method of comparison with the prototype, used when recognizable classes are displayed in the space of features by compact geometric groupings.

2) The method of k-closest neighbors, in which the decision to classify an object to a class is made on the basis of information on the membership of k of its nearest neighbors.

3) Algorithm for calculating estimates (voting), consisting in calculating priorities (similarity scores) characterizing the “proximity” of recognizable and reference objects according to the system of ensembles of signs, which is a system of subsets of a given set of signs [51], [52], [53].

When comparing extensional and intensional pattern recognition methods in [47], the following analogy is used: intensional methods correspond to the left-hemisphere way of thinking, based on knowledge of the static and dynamic patterns of the structure of perceived information; extensional methods correspond to the right-hemisphere way of thinking, based on the holistic mapping of the objects of the world.

1.5. Dependency Recovery Techniques

Most widely in this paper will be considered methods for constructing psychodiagnostic methods based on intensional methods based on assumptions about the class of decision functions. Therefore, we consider them in more detail.

The main advantage of methods based on the assumption of a class of decision functions is the clarity of the mathematical formulation of the recognition problem as an extremum search. The variety of methods of this group is explained by a wide range of used functionals of the quality of a decision rule and extremum search algorithms. The generalization of this class of algorithms is the method of stochastic approximation [94].

In this class of pattern recognition algorithms, a meaningful statement of the problem according to [29] is put as follows:

There are a number of observations that belong to p different classes. It is required, using information about these observations and their classifications, to find such a rule by which it would be possible to classify the newly emerging observations with a minimum number of errors.

The observation is given by the vector x, and its classification by the number ( ).

Thus, it is required, having a sequence of l observations and classifications build such a decisive rule which, with as few errors as possible, would classify new observations.

For the formalization of the term “error”, it is assumed that there is some rule defining a classification for each vector x which is called "true." By classifying a vector x with a rule called a classification in which and do not match.

Further it is assumed that in the space of vectors x there exists an unknown probability measure (denoted by the density ). In accordance with Randomly and independently situations arise x , which are classified by the rule . This determines the training sequence.

The quality of the decision rule written as where

The problem therefore is to build a decision rule so as to minimize the functionality

Similar to the problem of pattern recognition is the task of restoring regression, the prerequisites for which are formulated as follows:

Two sets of elements are related by functional dependence, if each element x can be associated with element y . This dependence is called a function if the set x is vectors, and the set y - scalars. However, there are dependencies where each vector x is assigned a number y , obtained by random testing, according to the conditional density . In other words, each x is assigned a law , according to which the choice of y is realized in a random test .

The existence of such relationships reflects the presence of stochastic dependencies between the vector x and the scalar and scalar y . Complete knowledge of stochastic dependence requires conditional density recovery. However, this task is very difficult and in practice (for example, in the tasks of processing measurement results) can be narrowed down to the task of determining the conditional expectation function. This narrowed problem is formulated as follows: define a conditional expectation function, that is, a function that assigns to each x a number y (x) equal to the expectation of a scalar y : . The function y (x) is called the regression function, and the task of restoring the conditional expectation function is the task of restoring the regression.

The strict formulation of the problem is as follows:

In a medium characterized by a probability distribution density P (x) , situations x appear randomly and independently. In this environment, the converter functions, which each vector x associates with the number y , obtained as a result of the implementation of a random test, according to the law . Environment properties P (x) and law unknown, however it is known that there is a regression . Required by a random, independent sample of pairs restore regression, that is, in the class of functions find function closest to the regression

The task of regression restoration is one of the main tasks of applied statistics. It addresses the problem of interpreting direct experiments.

The problem is solved in the following assumptions:

-

- The purpose of the study is to determine the dependence in a situation where, at any point x , a direct experiment can be conducted to determine this dependence, that is, direct measurements of . However, due to the imperfection of the experiment, the measurement result will determine the true value with some random error, that is, at each point x it is not possible to determine the value , and value where - experiment error,

- At no point x, the experimental conditions do not allow systematic errors, that is, the expectation of measurement functions at each fixed point is equal to the value of the function at this point:

- Random values and are independent.

Under these conditions it is necessary to restore the function by a finite number of direct experiments. . The required relationship is regression, and the essence of the problem is finding the regression by the sequence of pairs

The problem of regression restoration is usually reduced to the problem of minimizing the functional on set (integrable with square as