Errors of mathematical modeling. Error model in the form of a random elementary function
Production errors can be considered as random variables described by probabilistic (theoretical) and statistical (experimental) methods. An exhaustive characteristic of the error as a random variable is the distribution law with specific values of the corresponding parameters. The description of the distributions of production errors is most consistent with the Gauss law with a probability density calculated by the formula:
where t and σ – mathematical expectation and standard deviation.
The Gaussian distribution has been repeatedly confirmed by experimental data in the range of values corresponding to the range of ±3σ. According to this distribution, the alignment error at a particular point εх in the direction X is perceived as a random variable distributed according to the normal law, with the following characteristics:
(3.16)
where rx– correlation coefficient between the values of displacements of neighboring single sections in the direction X; C2x- the number of combinations of X by 2, calculated from the expression
From relations (3.15) and (3.16) an analytical record of the probability density of the distribution of quantities is derived:
Graphs of the dependence of the alignment errors on the coordinates of points along one axis, which follow from relation (3.18), are shown in Fig. 3.59.
Rice. 3.59. Diagram of layer alignment errors in direction X
In the presence of statistical data, numerical characteristics of the distribution (3.18) can be found for a section of length L with grid spacing h. They are found from the relationships:
(3.19)
where ML, σ L are, respectively, the mathematical expectation and the variance of the deformation of a segment with a length L; - the number of combinations of L/ h by 2.
In general, the error model A 095 (i) can be represented as Up to9 5 (?) = Up to + F(t), where To is the initial error of the SI; F(t) is a random function of time for a set of measuring instruments of this type, due to physical and chemical processes of gradual wear and aging of elements and blocks. Get the exact expression for a function F(t) Based on physical models of aging processes, it is practically not possible. Therefore, based on the data of experimental studies of changes in errors in time, the function F(t) approximated by one or another mathematical dependence.
The simplest model for changing the error is linear:
where v- rate of error change. As studies have shown, this model satisfactorily describes the aging of SI at the age of one to five years. Its use in other time ranges is impossible due to the obvious contradiction between the values determined by this formula and the experimental values of the failure rate.
Metrological failures occur periodically. The mechanism of their periodicity is illustrated in fig. 4.2, a, where a straight line 1 the change in the 95% quantile is shown with a linear law.
In case of a metrological failure, the error D 095 (?) exceeds the value D pr \u003d Do + D 3, where D, is the value of the margin of the normalized error limit necessary to ensure the long-term performance of the MI. With each such failure, the device is repaired, and its error returns to the initial value T? = t ( - - t j _ l failure occurs again (moments t u t 2 , t3 etc.), after which the repair is carried out again. Consequently, the process of changing the MI error is described by broken line 2 in Fig. 4.2, a, which can be represented by the equation
where P - the number of failures (or repairs) of SI. If the number of failures is taken as an integer, then this equation describes discrete points on a straight line 1
(see fig. 4.2, a). If, however, it is conditionally assumed that P can also take fractional values, then formula (4.2) will describe the entire line 1 change in error L 095 (() in the absence of failures.
Metrology failure rate increases with speed v. It also strongly depends on the margin of the normalized error value D 3 in relation to the actual value of the error of the measuring instrument D 0 at the time of manufacture or completion of the repair of the device. Practical opportunities for influencing the rate of change V and margin of error D are completely different. The aging rate is determined by the existing production technology. The margin of error for the first overhaul interval is determined by the decisions made by the MI manufacturer, and for all subsequent overhaul intervals - by the culture level of the user's repair service.
If the metrological service of the enterprise provides during repair the SI error equal to the error D 0 at the time of manufacture, then the frequency of metrological failures will be low. If, during the repair, only the fulfillment of the condition Up to * (0.9-0.95) D pr is ensured, then the error may go beyond the limits of permissible values in the coming months of the operation of the measuring instrument and for most of the calibration interval it will be operated with an error exceeding its class accuracy. Therefore, the main practical means of achieving long-term metrological serviceability of the measuring instrument is to provide a sufficiently large margin D 3 , normalized with respect to the limit D ave.
Gradual continuous consumption of this stock provides for a certain definite period of time a metrologically sound state of the MI. Leading instrument-making factories provide D 3 \u003d (0.4-0.5) D pr, which at an average aging rate V\u003d 0.05 D pr / year allows you to get the overhaul interval T p \u003d A 3 /i= 8-10 years and failure rate co = 1/Gy = 0.1-0.125 year -1 .
When changing the MI error in accordance with formula (4.1), all overhaul intervals T will be equal to each other, and the frequency of metrological failures w = 1 /T will be constant throughout the lifetime.
In the general case, the results of measurements and their errors should be considered as functions that vary in time randomly, i.e. random functions, or, as they say in mathematics, random processes. Therefore, the mathematical description of the results and measurement errors (i.e., their mathematical models) should be based on the theory of random processes. We present the main points of the theory of random functions.
random process X(t) is a process (function) whose value for any fixed value t = tQ is a random variable X(t). A specific type of process (function) obtained as a result of experience is called implementation.
Rice. 4. Type of random functions
Each implementation is a non-random function of time. The family of realizations for some fixed value of time t (Fig. 4) is a random variable called section random function corresponding to time t . Therefore, a random function combines the characteristic features of a random variable and a deterministic function. With a fixed value of the argument, it turns into a random variable, and as a result of each individual experiment, it becomes a deterministic function.
mathematical expectation random function X(t) is a non-random function which, for each value of the argument t, is equal to the mathematical expectation of the corresponding section:
where p(x, t) is the one-dimensional distribution density of the random variable x in the corresponding section of the random process X(t).
dispersion A random function X(t) is a non-random function whose value for each moment of time is equal to the variance of the corresponding section, i.e. the variance characterizes the spread of realizations with respect to m(t).
correlation function- non-random function R(t, t") of two arguments t and t", which for each pair of values of the arguments is equal to the covariance of the corresponding sections of the random process:
The correlation function, sometimes called autocorrelation, describes the statistical relationship between the instantaneous values of a random function separated by a given time value t \u003d t "-t. If the arguments are equal, the correlation function is equal to the variance of the random process. It is always non-negative.
Random processes that proceed uniformly in time, particular implementations of which oscillate around the average function with a constant amplitude, are called stationary. Quantitatively, the properties of stationary processes are characterized by the following conditions:
Mathematical expectation is constant;
The cross-sectional dispersion is a constant value;
The correlation function does not depend on the value of the arguments, but only on the interval.
An important characteristic of a stationary random process is its spectral density S(w), which describes the frequency composition of the random process for w>O and expresses the average power of the random process per unit frequency band:
The spectral density of a stationary random process is a non-negative function of frequency. The correlation function can be expressed in terms of the spectral density
When constructing a mathematical model of measurement error, all information about the measurement and its elements should be taken into account.
Each of them can be due to the action of several different sources of errors and, in turn, also consist of a certain number of components.
Probability theory and mathematical statistics are used to describe the errors, however, a number of essential reservations must be made first:
The application of the methods of mathematical statistics to the processing of measurement results is valid only on the assumption that the individual readings obtained are independent of each other;
Most of the formulas of probability theory used in metrology are valid only for continuous distributions, while the distributions of errors due to the inevitable quantization of readings, strictly speaking, are always discrete, i.e. the error can take only a countable set of values.
Thus, the conditions of continuity and independence for the measurement results and their errors are observed approximately, and sometimes they are not observed. In mathematics, the term "continuous random variable" means a much narrower concept, limited by a number of conditions, than "random error" in metrology.
In metrology, it is customary to distinguish three groups of characteristics and error parameters. The first group is the measurement error characteristics specified as the required or permissible norms (error standards). The second group of characteristics is the errors attributed to the totality of measurements performed according to a certain methodology. The characteristics of these two groups are mainly used for mass technical measurements and represent the probabilistic characteristics of the measurement error. The third group of characteristics - statistical estimates of measurement errors reflect the proximity of a separate, experimentally obtained measurement result to the true value of the measured quantity. They are used in the case of measurements carried out in scientific research and metrological work.
The set of formulas describing the state, motion and interaction of objects obtained within the framework of selected physical models based on the laws of physics will be called mathematical model of an object or process. The process of creating a mathematical model can be divided into a number of stages:
1) drawing up formulas and equations that describe the state, movement and interaction of objects within the framework of the constructed physical model. The stage includes a record in mathematical terms of the formulated properties of objects, processes and relationships between them;
2) the study of mathematical problems, which come at the first stage. The main issue here is the solution of the direct problem, i.e. obtaining numerical data and theoretical consequences. At this stage, an important role is played by the mathematical apparatus and computer technology (computer).
3) finding out whether the results of analysis and calculations or the consequences of them agree with the results of observations within the accuracy of the latter, i.e. whether the accepted physical and (or) mathematical model satisfies the practice, the main criterion for the truth of our ideas about the world around us.
The deviation of the results of calculations from the results of observations indicates either the incorrectness of the applied mathematical methods of analysis and calculation, or the incorrectness of the accepted physical model. Finding out the sources of errors requires great skill and high qualification of the researcher.
Often, when constructing a mathematical model, some of its characteristics or relationships between parameters remain uncertain due to the limited knowledge of the physical properties of an object. For example, it turns out that the number of equations describing the physical properties of an object or process and the relationships between objects is less than the number of physical parameters that characterize an object. In these cases, it is necessary to introduce additional relationships that characterize the object of study and its properties, sometimes even try to guess these properties, so that the problem can be solved and the results correspond to the experimental results within a given error.
Information correction of variable systematic errors of measuring instruments and measuring information systems
Reviewer: Tuz Yu.M.
Director of NII AEI, Doctor of Technical Sciences, Professor, Laureate of the State Prize of Ukraine in the field of science and technology
Introduction
Requirements for accuracy, correctness and convergence of measuring instruments are constantly increasing. The increase in requirements was usually carried out by moving from the used to a new physical principle of measurement, which provided higher quality measurements. At the same time, the methods and techniques of measurements were improved, and the requirements for the set of normal (standard) conditions accompanying the measurement process became more stringent.
Any measuring device, system, channel "responds" not only to the measured value, but also to the external environment, because inevitably associated with it.
A good illustration of this theoretical thesis can be the effect of tidal waves caused by the Moon in the earth's crust on the change in the energy of charged particles obtained at the large ring accelerator at the Center for European Nuclear Research. The tidal wave deforms the 27-kilometer (2.7·10 7 mm) accelerator ring and changes the path length of particles along the ring by approximately 1 mm (!). This leads to a change in the energy of the accelerated particle by almost ten million electron volts. These changes are very small, but they exceed the possible measurement error by about ten times and have already led to a serious error in the measurement of the boson mass.
Formulation of the problem
Metrological provision of radio-electronic measurements can be characterized by the following typical problems. The use of theoretical methods for analyzing the influence of environmental factors on the errors of measuring instruments is difficult. The nature of the influence is complex, unstable, difficult to interpret from the standpoint of logical and professional analysis by a specialist; changeable when moving from instance to instance of the same type of measuring instruments.
It is noted that it is methodologically difficult to obtain dependences of an unknown type on several variables and that “... the possibilities for studying the dependences of the error on environmental factors are very limited and not very reliable, especially in relation to the combined influences of factors and dynamic changes in their values” .
As a result of the above reasons and a significant variety of their manifestation, it is concluded that for a group of measuring instruments of the same type, the most adequate description of the errors of measuring instruments from influencing environmental factors should be recognized as an area of uncertainty, the boundaries of which are determined by the extreme dependencies of specimens.
These difficulties in solving the problem of reducing the errors of measuring instruments are a consequence of the system properties of these instruments: emergence, integrity, uncertainty, complexity, stochasticity, etc. . Attempts at a theoretical description at the level of nomographic sciences in the situations under consideration are often ineffective. An experimental-statistical approach is needed, since it allows for an idiographic description of the patterns of specific phenomena in detailed conditions of time and place.
Both in electronic measurements and in ensuring the accuracy of evaluating the results of quantitative chemical analysis, an important feature of errors is noted: the systematic errors of the result for most measuring instruments are significant in the sense that they exceed random, and the error of a given instance of a measuring instrument at each point in the factor space is determined, basically constant.
To further improve the quality of measurements, it is necessary to use not only physical - design, technological, operational - capabilities, but also informational ones. They consist in the implementation of a systematic approach in obtaining information about all types of errors: instrumental, methodological, additional, systematic, progressive (drift), model, and possibly others. Having such information in the form of a multifactorial mathematical model and knowing the values of the factors (conditions) that accompany the process measurements, it is possible to obtain information about the given errors and, therefore, to know the measured value more accurately.
Requirements for the methodology of mathematical modeling of systematic errors of measuring instruments
It is necessary to develop a methodology for multifactorial mathematical modeling of regularly changing systematic errors, taking into account the following requirements.
- A systematic approach to the description of systematic errors, taking into account many factors and, if necessary, many criteria for the quality of a measuring instrument.
- The applied level of obtaining mathematical models, when their structure is not known to the researcher.
- Efficiency (in the statistical sense) of obtaining useful information from the source data and reflecting it in mathematical models.
- The possibility of an accessible and convenient meaningful interpretation of the obtained models in the subject area.
- Efficiency of using mathematical models in the subject area compared to the cost of resources to obtain them.
The main stages of obtaining mathematical models
Let us consider the main stages of obtaining multifactorial mathematical models that meet the above requirements.
Choosing a plan for a multifactorial experiment that provides the necessary properties of the resulting mathematical models
In the considered (metrological) class of ongoing experimental studies, it is possible to use a full and fractional factorial experiment. Under the mathematical model being defined, we mean a model linear with respect to parameters and non-linear in the general case with respect to factors, a model of arbitrarily high, but finite complexity. The extended effects matrix of the full factorial experiment will include a dummy factor column X 0 = 1, columns for all main effects and all possible main effect interactions. If the effects of factors and interactions of factors are expressed as a system of orthogonal normalized contrasts, then the variance-covariance matrix will take the form:
where X
– matrix of effects of the full factorial experiment;
σ y 2 is the dispersion of the reproducibility of the results of the experiments;
N- the number of experiments in the plan of the experiment;
E
is the identity matrix.
The mathematical model obtained by the scheme of a full factorial experiment corresponds to many remarkable properties: the coefficients of the model are orthogonal to each other and are statistically independent; most stable ( cond= 1); each coefficient carries semantic information about the influence of the corresponding effect on the modeled quality criterion; the experimental design meets the criteria D-, A-, E-, G-optimality, as well as the criterion of proportionality of the frequencies of the levels of factors; the mathematical model is adequate at the points of approximation of the response surface. We will consider such a model to be true and “best”.
In cases where the use of a full factorial experiment is impossible due to a large number of experiments, it should be recommended to use multifactorial regular (preferably uniform) experimental designs. With the correct choice of the number of necessary experiments, their properties are as close as possible to the given properties of the full factorial experiment.
Obtaining the structure of a multifactorial mathematical model
The structure of the resulting multifactorial mathematical model, generally unknown to the researcher, must be determined based on the possible set of effects corresponding to the set of effects of the scheme of a complete factorial experiment. It is given by the expression:
where X 1 ,..., X k - factors of the desired mathematical model;
s 1 ,..., s k is the number of factor levels X 1 ,..., X k;
k is the total number of factors;
N n is the number of experiments of a full factorial experiment, equal to the number of structural elements of its scheme.
The search for the necessary effects - main and interactions - in the form of orthogonal contrasts for the desired model is carried out as a multiple statistical testing of hypotheses about the statistical significance of the effects. Statistically significant effects are introduced into the model.
Choosing the number of necessary experiments for a fractional factorial experiment
Usually, the researcher knows (approximately) information about the expected complexity of the influence of factors on the modeled quality criterion. For each factor, the number of levels of its variation is selected, which should be 1 more than the maximum degree of the polynomial required for this factor to adequately describe the response surface. The required number of experiments will be:
where s i is the number of factor levels X i ; 1 ≤ i ≤ k.
A coefficient of 1.5 is chosen for the case when the number of necessary experiments is significant (of the order of 50...64 or more). With a smaller required number of experiments, a factor of 2 should be chosen.
Choosing the Structure of a Multifactorial Mathematical Model
To select the structure of the resulting mathematical model, it is necessary to use the developed algorithm. The algorithm implements a sequential scheme for selecting the necessary structure based on the results of a planned multifactorial experiment.
Processing the results of experiments
For complex processing of the results of experiments and obtaining the necessary information for interpreting the results in the subject area, the software tool "Planning, regression and analysis of models" (PS PRIAM) has been developed. The developer is the Laboratory of Experimental and Statistical Methods of the Department of Mechanical Engineering Technology of the National Technical University of Ukraine "Kyiv Polytechnic Institute". The evaluation of the quality of the resulting mathematical models includes the following criteria:
- obtaining an informative subset of the main effects and interactions of factors to be adopted as the structure of the desired multi-factor mathematical model;
- ensuring the highest possible theoretical efficiency (up to 100%) of extracting useful information from the source data;
- testing for statistical significance of a potential mathematical model;
- testing various assumptions of multiple regression analysis;
- verification of the adequacy of the resulting model;
- checking for informativeness, i.e. the presence in the mathematical model of useful information and its statistical significance;
- checking for the stability of the coefficients of the mathematical model;
- verification of the actual efficiency of extracting useful information from the source data;
- assessment of semanticity (information) according to the obtained coefficients of the mathematical model;
- checking the properties of residues;
- a general assessment of the properties of the obtained mathematical model and the possibility of its use to achieve the goal.
Interpretation of the results
It is carried out by a specialist (or specialists) who understand well both the formal results in the obtained models and the applied goals for which the models should be used.
The mathematical method for obtaining useful information about the systematic errors that accompany the process of measuring a physical quantity, and the measuring instrument create a supersystem with interaction (in other words, emergence) with each other. The effect of interaction - a higher accuracy of the measured value - in principle cannot be obtained only at the expense of individual subsystems. This follows from the structure of the mathematical model Ŷ (ŷ 1 ,..., ŷ p) = f j (SI, MM) for experiment 2 2 //4 (the absence of a subsystem is set by “–1”, and the presence of “1”) the indicated subsystems:
where Ŷ (ŷ 1 ,..., ŷ p) is the vector of efficiency of the measuring instrument, 1 ≤ j ≤ p;
1 - symbol of the average value of the result (conditional reference point);
SI - measurement result obtained only from the measuring instrument;
MM - information obtained by a multifactorial mathematical model about the systematic errors of the measuring instrument used with knowledge of the internal and external measurement conditions relative to its conditions;
SI · MM - the effect of interaction (emergence) of the measuring instrument and the mathematical model, provided that they are used together.
Improving the measurement accuracy is achieved by obtaining more information about the measurement conditions and the properties of the measuring instrument in interaction with the internal and external environment relative to it.
The combination of physical and information principles in practice means the intellectualization of known systems, in particular, the creation of intelligent measuring instruments. Combining physical and informational principles into a single integrated system makes it possible to solve old problems in a fundamentally new way.
An example of increasing the accuracy of measuring digital scales
Let's consider the possibilities of the proposed approach on the example of increasing the accuracy of digital scales with a weighing range of 0...100 kgf. Capacitive type weighing sensor self-powered from a portable voltage source. Scales are intended for operation in the range of temperature of environment (air) 0...60 °C. The voltage from an autonomous voltage source during the operation of the balance can vary in the range of 12.3 ... 11.7 V at a calculated (nominal) value of 12 V.
A preliminary study of digital scales showed that changes in ambient temperature and supply voltage in the above ranges have relatively little effect on the readings of the capacitive sensor and, consequently, on the weighing results. However, it was not possible to stabilize these external and internal conditions with the required accuracy and maintain them during the operation of the balance, due to the fact that the balance should be operated not under stationary (laboratory) conditions, but on board a moving object.
A study of the accuracy of the scales without taking into account the influence of changes in temperature and supply voltage showed that the average absolute approximation error is 0.16%, and the root-mean-square error of the remainder (in units of measurement of the weighing output value) is 53.92.
To obtain a multifactorial mathematical model, the following designations of factors and the values of their levels were adopted.
X 1 - hysteresis. Levels: 0 (load); 1 (unloading). Quality factor.
X 2 – ambient temperature. Levels: 0; 22; 60°C.
X 4 - measured weight. Levels: 0; 20; 40; 60; 80; 100 kgs.
Taking into account the accepted levels of factor variation and the relatively inexpensive amount of testing, it was decided to conduct a full factorial experiment, i.e. 2 3 2 6//108. Initial test data were provided by prof. P.V. Novitsky. Each experiment was repeated only once, which cannot be considered a good solution. It is advisable to repeat each experiment twice. A preliminary analysis of the initial data showed that they contain gross errors with a significant probability. These experiments were repeated and their results were corrected.
The natural values of the levels of factor variation were converted into orthogonal contrasts, otherwise into a system of orthogonal Chebyshev polynomials.
Using a system of orthogonal contrasts, the structure of a complete factorial experiment will look like this:
(1 + x 1) (1 + x 2 + z 2) (1 + x 3 + z 3) (1 + x 4 + z 4 + u 4 + v 4 + ω 4) → N 108
where x 1 ,..., x 4 ; z 2 ,..., z 4 ; u 4 , v 4 , ω 4 - respectively linear, quadratic, cubic, fourth and fifth degree contrast factors X 1 ,..., X 4 ;
N 108 is the number of structural elements for the scheme of a complete factorial experiment.
All effects (principal and interactions) were normalized
where x iu (p) is the value p th orthogonal contrast i-th factor for the uth row of the planning matrix, 1 ≤ u ≤ 108, 1 ≤ p ≤ s i - 1; 1 ≤ i ≤ 4.
A preliminary calculation of the mathematical model showed that the (approximately) value of 20.1 can be chosen as an estimate of the reproducibility variance.
Number of degrees of freedom (conditionally) accepted V 2 = 108.
The variance was used to determine the standard error of the regression equation coefficients.
The calculation of the mathematical model and all its quality criteria was carried out using the PS PRIAM. The resulting mathematical model has the form
| ŷ = 28968,9 – 3715,13x 4 + 45,2083x 3 – 37,5229z 2 + 23,1658x 2 – 19,0708z 4 – 19,6574z 3 – 9,0094x 2 z 3 – 9,27434z 2 x 4 + 1,43465x 1 x 2 + 1,65431z 2 x 3 , | (2) |
x 1 = 2 (X 1 – 0,5);
x 2 = 0,0306122 (X 2 – 27,3333);
z 2 = 1,96006 (x 2 2 – 0,237337x 2 – 0,575594);
x 3 = 3.33333 (X 3 – 12);
z 3 = 1,5 (x 2 3 – 0,666667);
x 4 = 0,02 (X 4 – 50);
z 4 = 1,875 (x 2 4 – 0,466667);
u 4 = 3,72024 (x 3 4 – 0,808x 4);
v 4 = 7,59549 (x 4 4 – 1,08571x 2 4 + 0,1296).
Table 1
Criteria for the quality of the obtained mathematical model
| Analysis of model adequacy | |
| Residual dispersion | 21,1084 |
| Reproducibility dispersion | 20,1 |
| Estimated value F- criteria | 1,05017 |
| Significance level F- criterion for adequacy 0.05 for degrees of freedom V 1 = 97; V 2 = 108 | |
| Table value F-criteria for adequacy | 1,3844 |
| Table value F-criteria (in the absence of repeated experiments) | 1,02681 |
| Standard error of estimation | 4,59439 |
| Correct. taking into account degrees of freedom | 4,80072 |
| Model | adequate |
| Note: Reproducibility variance is user defined |
|
| Analysis of the informativeness of the model | |
| Fraction of dispersion explained by the model | 0,999997 |
| Introduced regressors (effects) | 11 |
| Multiple correlation coefficient | 0,999999 |
| (corrected for degrees of freedom) | 0,999998 |
| F attitude for R | 3.29697 10 6 |
| Significance level F-criterion for informativeness 0.01 for degrees of freedom V 1 = 10; V 2 = 97 | |
| Table value F-criteria for informativeness | 2,50915 |
| Model | informative |
| Box and Wetz criterion for informativeness | over 49 |
| Informativeness of the model | very high |
table 2
Statistical characteristics of regression coefficients
| Name of the main effect or interaction of the main effects | Regression coefficient | Standard error of the regression coefficient | Computed value t-Crete. | Share of participation in explaining the dispersion of the modeled value |
| x 4 | b 1 = –3715,13 | 0,431406 | 5882,9 | 0,999557 |
| x 3 | b 2 = 45,2083 | 0,431406 | 85,5631 | 0,000211445 |
| z 2 | b 3 = –37,5229 | 0,431406 | 62,2275 | 0,000111838 |
| x 2 | b 4 = 23,1658 | 0,431406 | 40,7398 | 4.79362 10 -5 |
| z 4 | b 5 = –19,0708 | 0,431406 | 33,0808 | 3.16065 10 -5 |
| z 3 | b 6 = –19,6574 | 0,431406 | 32,22 | 2.9983 10 -5 |
| x 2 z 3 | b 7 = –9,0094 | 0,431406 | 11,2035 | 3.62519 10 -6 |
| z 2 x 4 | b 8 = –9,27434 | 0,431406 | 10,5069 | 3.18838 10 -6 |
| x 1 x 2 | b 9 = 1,43465 | 0,431406 | 2,523 | 1.83848 10 -7 |
| z 2 x 3 | b 10 = 1,65431 | 0,431406 | 2,24004 | 1.44923 10 -7 |
b 0 = 28968,9
Significance level for t-criterion - 0.05
For degrees of freedom V 1 = 108. Table value t-criteria - 1.9821
In table. 1 shows a printout of the quality criteria for the resulting multifactorial mathematical model. The model is adequate. The proportion of dispersion explained by the model is very high, because the model is highly accurate, the variability of the response function is large, and its random variability is relatively small. Multiple correlation coefficient R is very close to 1 and is stable, since, when adjusted for degrees of freedom, it practically does not change. Statistical Significance R is very large, i.e. the model is very informative. The high information content of the model is also confirmed by the value of the Box and Wetz criterion. The coefficients of the model are maximally stable: the condition number cond= 1. The resulting model is semantic in the informational sense, since all its coefficients are orthonormal: they are statistically independent and can be compared in absolute value with each other. The sign of the coefficient shows the nature of the influence, and its absolute value - the strength of the influence. The resulting model is most convenient for interpretation in the subject area.
Taking into account the semantic properties of the obtained mathematical model and the participation share of each of the model effects in the total dispersion share explained by the model, it is possible to carry out a meaningful informational analysis of the formation of the measurement result of the studied digital weights.
The prevailing share in the simulation results, equal to 0.999557, is created by a linear main effect x 4 (with coefficient b 1 = -3715.13), i.e. measured weight (Table 2). Nonlinearity z 4 (with coefficient b 5 = –19.07) is relatively small (3.16 10 –5) and its inclusion in the model improves the measurement accuracy. Line effect x 4 relatively weakly (3.19 10 -6) interacts with the quadratic effect z 2 ambient temperatures: interaction z 2 x 4 (b 8 = -9.27). Therefore, the mathematical model only depends on the factor measured weight X 4 should also include the effect of ambient temperature
ŷ 1 = 28968,90 – 3715,13x 4 – 19,07z 4 – 9,27z 2 x 4 ,
whose factor X 2 is unmanaged.
The supply voltage changes the weighing results as a linear effect x 3 (b 2 = 45.21) and quadratic effect z 3 (b 6 = -19.66). Their total share of participation is 2.41·10 -4 .
Ambient temperature affects as a quadratic z 2 (b 3 = -37.52) and linear x 2 (b 4 \u003d 23.17) effects with a total participation share of 1.60 10 -4.
Ambient temperature and supply voltage form a pair interaction x 2 z 3 (b 7 \u003d -9.01) with a participation share of 3.63 10 -6.
Evidence for the statistical significance of the last two effects x 1 x 2 and z 2 x 3 cannot be substantiated, since they are significantly less than the effects x 2 z 3 and z 2 x 4, and, unfortunately, there was no reasonable value for the dispersion of reproducibility based on the results of repeated experiments in the presented initial data.
In table. 2 shows the statistical characteristics of the regression coefficients. Note that the values of the regression coefficients are divided into normalization coefficients of orthogonal contrasts, which are not included in the orthogonal contrast formulas given. This explains the fact that when dividing the values of the regression coefficients by their standard error, the obtained values t-criteria differ from the given correctly calculated values of this criterion in Table. 2.
Rice. one. Histogram of Residuals
On fig. 1 shows a histogram of residuals 
. It is relatively close to the normal distribution law. In table. Figure 3 shows the numerical values of the residuals and their deviation percentages. The time graph of the residuals (Fig. 2) indicates the random nature of the change in the residuals from the time (sequence) of the experiments. Further increase in the accuracy of the model is not possible. Analysis of the dependence of residuals on ŷ
(calculated value) shows that the largest residual scatter is observed for X 4 = 0 kgf ( y= 32581...32730) and X 4 = 100 kgf ( y= 25124...25309). The smallest spread at X 4 = 40 kgf. However, the statistical significance of such a conclusion requires knowledge of the reasonable value of the reproducibility variance.
Rice. 2. Residue timeline
Taking into account various systematic errors, nonlinearities, interactions of uncontrolled factors in the mathematical model made it possible to increase the accuracy of the measuring instrument by the criterion of the average absolute approximation error up to 0.012% - by 13.3 times, and by the criterion of the root-mean-square approximation error up to 4.80 (Table 1) - 11.2 times.
Experiment plan 2 2 //4 for the average absolute approximation error in % and the results obtained using only measuring instruments and measuring instruments with a mathematical model of systematic errors are presented in Table. 4.
The mathematical model for the average absolute error of approximation, obtained by experiment 2 2 //4, with the structure of the model (1) and the results of the functioning of the measuring instrument without the mathematical model and with its use, has the form
ŷ = 0,043 + 0,043x 1 ...0,037x 2 ...0,037x 1 x 2
where x 1 - orthogonal contrast factor X 1 (SI) - measuring instrument;
x 2 - orthogonal contrast factor X 2 (MM) - mathematical model of systematic errors of the used measuring instrument;
x 1 x 2 - interaction of factors X 1 (SI) and X 2 (MM).
Table 3
Residuals and their percentage deviations
1
– Experience number; 2
– Response to the experiment; 3
– Model response; 4
- Remainder;
5
– Deviation percentage; 6
– Experience number; 7
– Response to the experiment;
8
– Model response; 9
- Remainder; 10
– Deviation percentage
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 32581 | 32574,2 | 6,832 | 0,0210 | 55 | 32581 | 32576,6 | 4,431 | 0,0136 |
| 2 | 31115 | 31108,7 | 6,349 | 0,0204 | 56 | 31115 | 31111,1 | 3,948 | 0,0127 |
| 3 | 29635 | 29631,7 | 3,308 | 0,0112 | 57 | 29633 | 29634,1 | –1,092 | –0,0037 |
| 4 | 28144 | 28143,3 | 0,710 | 0,0025 | 58 | 28141 | 28145,7 | –4,691 | –0,0167 |
| 5 | 26640 | 26643,4 | –3,445 | –0,0129 | 59 | 26637 | 26645,8 | –8,846 | –0,0332 |
| 6 | 25128 | 25132,2 | –4,159 | –0,0165 | 60 | 25124 | 25134,6 | –10,559 | –0,0420 |
| 7 | 32625 | 32638,6 | –13,602 | –0,0417 | 61 | 32649 | 32641 | 7,997 | 0,0245 |
| 8 | 31175 | 31173,1 | 1,915 | 0,0061 | 62 | 31179 | 31175,5 | 3,514 | 0,0113 |
| 9 | 29694 | 29696,1 | –2,126 | –0,0072 | 63 | 29699 | 29698,5 | 0,473 | 0,0016 |
| 10 | 28208 | 28207,7 | 0,276 | 0,0010 | 64 | 28209 | 28210,1 | –1,125 | –0,0040 |
| 11 | 26709 | 26707,9 | 1,120 | 0,0042 | 65 | 26711 | 26710,3 | 0,719 | 0,0027 |
| 12 | 25198 | 25196,6 | 1,407 | 0,0056 | 66 | 25199 | 25199 | 0,006 | 0,0000 |
| 13 | 32659 | 32666,7 | –7,680 | –0,0235 | 67 | 32660 | 32669,1 | –9,081 | –0,0278 |
| 14 | 31199 | 31201,2 | –2,163 | –0,0069 | 68 | 31200 | 31203,6 | –3,564 | –0,0114 |
| 15 | 29723 | 29724,2 | –1,204 | –0,0040 | 69 | 29726 | 29726,6 | –0,605 | –0,0020 |
| 16 | 28241 | 28235,8 | 5,198 | 0,0184 | 70 | 28242 | 28238,2 | 3,797 | 0,0134 |
| 17 | 26741 | 26736 | 5,042 | 0,0189 | 71 | 26742 | 26738,4 | 3,642 | 0,0136 |
| 18 | 25232 | 25224,7 | 7,329 | 0,0290 | 72 | 25233 | 25227,1 | 5,928 | 0,0235 |
| 19 | 32632 | 32636,5 | –4,543 | –0,0139 | 73 | 32630 | 32637 | –7,012 | –0,0215 |
| 20 | 31175 | 31177,1 | –2,086 | –0,0067 | 74 | 31173 | 31177,6 | –4,554 | –0,0146 |
| 21 | 29705 | 29706,2 | –1,185 | –0,0040 | 75 | 29703 | 29706,7 | –3,654 | –0,0123 |
| 22 | 28225 | 28223,8 | 1,157 | 0,0041 | 76 | 28223 | 28224,3 | –1,311 | –0,0046 |
| 23 | 26734 | 26730,1 | 3,942 | 0,0147 | 77 | 26733 | 26730,5 | 2,474 | 0,0093 |
| 24 | 25233 | 25224,8 | 8,170 | 0,0324 | 78 | 25233 | 25225,3 | 7,702 | 0,0305 |
| 25 | 32710 | 32707,4 | 2,623 | 0,0080 | 79 | 32710 | 32707,8 | 2,155 | 0,0066 |
| 26 | 31251 | 31247,9 | 3,081 | 0,0099 | 80 | 31249 | 31248,4 | 0,612 | 0,0020 |
| 27 | 29777 | 29777 | –0,019 | –0,0001 | 81 | 29775 | 29777,5 | –2,488 | –0,0084 |
| 28 | 28294 | 28294,7 | –0,676 | –0,0024 | 82 | 28292 | 28295,1 | –3,145 | –0,0111 |
| 29 | 26799 | 26800,9 | –1,891 | –0,0071 | 83 | 26799 | 26801,4 | –2,360 | –0,0088 |
| 30 | 25297 | 25295,7 | 1,336 | 0,0053 | 84 | 25296 | 25296,1 | –0,132 | –0,0005 |
| 31 | 32730 | 32723,7 | 6,349 | 0,0194 | 85 | 32729 | 32724,1 | 4,880 | 0,0149 |
| 32 | 31269 | 31264,2 | 4,806 | 0,0154 | 86 | 31267 | 31264,7 | 2,338 | 0,0075 |
| 33 | 29794 | 29793,3 | 0,707 | 0,0024 | 87 | 29793 | 29793,8 | –0,762 | –0,0026 |
| 34 | 28310 | 28311 | –0,951 | –0,0034 | 88 | 28309 | 28311,4 | –2,419 | –0,0085 |
| 35 | 26814 | 26817,2 | –3,166 | –0,0118 | 89 | 26814 | 26817,6 | –3,634 | –0,0136 |
| 36 | 25309 | 25311,9 | –2,938 | –0,0116 | 90 | 25309 | 25312,4 | –3,407 | –0,0135 |
| 37 | 32616 | 32619,1 | –3,053 | –0,0094 | 91 | 32608 | 32616,2 | –8,183 | –0,0251 |
| 38 | 31152 | 31154,5 | –2,525 | –0,0081 | 92 | 31148 | 31151,7 | –3,656 | –0,0117 |
| 39 | 29677 | 29678,6 | –1,555 | –0,0052 | 93 | 29675 | 29675,7 | –0,686 | –0,0023 |
| 40 | 28192 | 28191,1 | 0,858 | 0,0030 | 94 | 28192 | 28188,3 | 3,727 | 0,0132 |
| 41 | 26696 | 26692,3 | 3,713 | 0,0139 | 95 | 26692 | 26689,4 | 2,582 | 0,0097 |
| 42 | 25189 | 25182 | 7,010 | 0,0278 | 96 | 25189 | 25179,1 | 9,880 | 0,0392 |
| 43 | 32713 | 32707,9 | 5,132 | 0,0157 | 97 | 32704 | 32705 | –0,998 | –0,0031 |
| 44 | 31244 | 31243,3 | 0,660 | 0,0021 | 98 | 31240 | 31240,5 | –0,471 | –0,0015 |
| 45 | 29770 | 29767,4 | 2,630 | 0,0088 | 99 | 29764 | 29764,5 | –0,501 | –0,0017 |
| 46 | 28285 | 28280 | 5,043 | 0,0178 | 100 | 28278 | 28277,1 | 0,912 | 0,0032 |
| 47 | 26784 | 26781,1 | 2,898 | 0,0108 | 101 | 26778 | 26778,2 | –0,233 | –0,0009 |
| 48 | 25262 | 25270,8 | –8,805 | –0,0349 | 102 | 25262 | 25267,9 | –5,935 | –0,0235 |
| 49 | 32717 | 32710,7 | 6,318 | 0,0193 | 103 | 32710 | 32707,8 | 2,187 | 0,0067 |
| 50 | 31249 | 31246,2 | 2,845 | 0,0091 | 104 | 31245 | 31243,3 | 1,715 | 0,0055 |
| 51 | 29770 | 29770,2 | –0,185 | –0,0006 | 105 | 29767 | 29767,3 | –0,315 | –0,0011 |
| 52 | 28280 | 28282,8 | –2,772 | –0,0098 | 106 | 28279 | 28279,9 | –0,903 | –0,0032 |
| 53 | 26779 | 26783,9 | –4,917 | –0,0184 | 107 | 26779 | 26781 | –2,048 | –0,0076 |
| 54 | 25267 | 25273,6 | –6,619 | –0,0262 | 108 | 25267 | 25270,8 | –3,750 | –0,0148 |
| The average absolute relative error in percent is 0.0119. | |||||||||
Table 4
Experiment plan 2 2 //4
Analysis of the coefficients of the model shows that the factor X 2 (MM) reduces the systematic error not only in the form of the main effect x 2 (coefficient b 2 = -0.037), but also due to the interaction (emergence) of the factors X 1 (SI) X 2 ( MM) (coefficient b 12 = -0.037).
A similar model can also be obtained for the criterion of the root-mean-square approximation error.
For the actual implementation of the obtained model (2), it is necessary to measure and use information about the ambient temperature and supply voltage using sensors and calculate the result using a microprocessor.
Results of mathematical modeling of six-component tensometric measuring systems
Mathematical modeling of six-component tensometric measuring systems is considered. The proposed method was introduced at the Kiev Mechanical Plant (now the Aviation Scientific and Technical Complex named after O.K. Antonov). For the first time in the practice of carrying out similar measurements, this method to a large extent made it possible to exclude the consequences of physical imperfections of measuring systems, which manifest themselves in the form of interaction between channels, the influence of other channels on the channel under consideration, nonlinearities, and to study the structural relationships of various channels.
The use of the method of mathematical modeling in the real conditions of the enterprise showed that the time of the experiments is reduced by 10...15 times; significantly (up to 60 times) increases the efficiency of processing measurement information; the number of performers involved in measurement experiments is reduced by 2...3 times.
The final conclusion about the advisability of using the above approach depends on the economic efficiency of the following compared options.
A high-precision measuring instrument and, therefore, more expensive, used in normalized (standard) conditions that need to be created and maintained.
Means of measurement of less high accuracy, used in non-standardized (non-standard) conditions using the obtained mathematical model.
Main conclusions
1) A successfully implemented systematic approach in the mathematical modeling of the measuring instrument made it possible to take into account the influence of external factors - ambient temperature - and the internal environment - supply voltage. The efficiency of extracting useful information from the original data was 100%.
2) In the resulting multi-factor mathematical model, the structure of which was not a priori known to the researcher, the non-linearity of the measuring instrument and the systemic influence of factors (emergence) of the external and internal environment are disclosed in a form convenient for interpretation in the subject area. Under real operating conditions, the stabilization of these factors with the required accuracy is not possible.
3) Taking into account the mathematical model of systematic errors made it possible to increase the accuracy of measurements by the criterion of the average absolute error by 13.3 times and by the criterion of the root-mean-square error by 11.2 times.
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Literature:
- Rybakov I.N. Fundamentals of accuracy and metrological support of radio-electronic measurements. - M.: Publishing house of standards, 1990. - 180 p.
- Radchenko S.G. Mathematical modeling of technological processes in mechanical engineering. - K .: CJSC "Ukrspetsmontazhproekt", 1998. - 274 p.
- Alimov Yu.I., Shaevich A.B. Methodological features of evaluating the results of quantitative chemical analysis // Journal of Analytical Chemistry. - 1988. - Issue. 10. - T. XLIII. - S. 1893 ... 1916.
- Planning, regression and analysis of PRIAM models (PRIAM). SCMC-90; 325, 660, 668 // Catalog. Software products of Ukraine. Catalog. Software of Ukraine. - K .: JV "Teknor". - 1993. - C. 24 ... 27.
- Zinchenko V.P., Radchenko S.G. Method for modeling multicomponent tensometric measuring systems. - K.: 1993. - 17 p. (Prepr. / Academy of Sciences of Ukraine. Institute of Cybernetics named after V.M. Glushkov; 93 ... 31).
Requirements for models describing measurement errors
Models of measurement errors
Requirements:
1.should reflect the essential metrological properties of the measuring instrument or measurement procedure,
2. provide solutions to practical problems that use measurement results;
3. quantitative assessment of the error;
5.correct the readings of the measuring instrument and make corrections to the measurement results to reduce errors;
6. determine the probability of failure-free operation of the measuring instrument for a certain period of time;
7. must take into account the production and operational tolerances for the values of metrological characteristics.
The more stringent requirements are imposed on the model, the more detailed conclusions should be drawn from the measurement results, the more complex the structure of the error model should be.
The type of mathematical model of errors is chosen based on:
Theoretical or experimental study of methods and measuring instruments;
Analysis of statistical data on the quantities influencing the results, taking into account the measurement conditions.
When solving practical metrological problems, one and the same model can be used both to describe and evaluate the measurement results and their errors.
The most commonly used models describing errors are:
Measurement error is a function of time. With a monotonic change in the error, the simplest description of the nature of its change is the approximation of the error by a monotonic function of time
Where is a monotone non-random function of time;
Z- random value.
If this model is used to estimate the errors of the same type of measuring instruments, then
the random component makes it possible to take into account the difference in errors for each individual measuring instrument, and the spread of errors under the influence of various conditions.
If the model is used to describe the errors of the same measuring instrument, the random component makes it possible to take into account that the errors take on different values for different combinations of influencing factors.
The most convenient monotonic random functions that allow describing errors are
LINEAR!!!
Linear-uniform;
And linear-fan functions (Fig. 30).
Linear-uniform functions of the form 
include a random part, i.e. individual implementations of the quantity a and a monotone non-random component .
In linear-fan functions 
magnitude a is non-random, and the term is a separate realization of the random component.
The generalized error model in the form of a linear function can be the expression 
, wherein BUT is the initial value of the error; AT is the rate of error change.
The components of the model are random, usually mutually uncorrelated quantities.
NONLINEAR!!!
Also, monotone elementary random functions are non-linear fan-shaped random functions of time (Fig. 31), for example, exponential or power functions. In Fig.31, a an error model is presented that takes into account the decrease in the rate of change of the error over time and its gradual approach to some practically unchanged value. In Fig.31, b the model used in the case when the rate of change of the error increases and tends to some stationary value is given.
Such models can be used, for example, when the error is caused by two oppositely influencing factors, while one of them is valid for a limited time. Even at a constant rate of change in the error for the same type of devices, due to the difference in dynamic technological, physical and mechanical properties (wear rate, aging, changes in external factors), the model is represented by an ensemble of implementations.
In the above models, the argument can be not only time, but also other monotonically changing parameters.
The monotonic component in the error model can take into account:
Changing the parameters of the power source that feeds the measuring circuit of the device;
Aging of measuring circuit elements;
Monotonically changing in time external influencing factors;
Gradual wear of the elements of the measuring instrument, etc.
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