Rigorous Method of Weights Calculation in Adjustment of Measurement Data with Different Qualities

: In the traditional measurement theory, precision is defined as the dispersion of measured value, and is used as the basis of weights calculation in the adjustment of measurement data with different qualities, which leads to the trouble that trueness is completely ignored in the weight allocation. In this paper, following the pure concepts of probability theory, the measured value (observed value) is regarded as a constant, the error as a random variable, and the variance is the dispersion of all possible values of an unknown error. Thus, a rigorous formula for weights calculation and variance propagation is derived, which solves the theoretical trouble of determining the weight values in the adjustment of multi-channel observation data with different qualities. The results show that the optimal weights are not only determined by the covariance array of observation errors, but also related to the model of adjustment.


Introduction
The traditional measurement error theory is interpreted following the logic of error classification, in which the measured value (observed value) and random error are regarded as random variables with variance, while the systematic error is regarded as constant without variance [1,2,3,4,5,6] . In this way, when multiple observations with different qualities are obtained from a single measurand, according to this conceptual logic, the weight of each observed value is calculated by its precision, and the final measured value is equal to the weighted average value of all observed values.
However, the biggest trouble faced by this method is that the systematic error of each observation is different from each other. Some observations with large systematic error but small random error get larger weights, which can make the final measured value have better precision, but its trueness will be worse. Although we can explain the existing theory based on the assumption that the systematic error does not exist or can be ignored, this assumption is usually not tenable in actual measurement.
For example, in the field of geodesy, the multiplicative constant error of a geodimeter is considered as systematic error, its measured value given by metrology field is usually used to correct the observed value, and people think that the residual systematic error after correction can be ignored. However, in the field of instrument manufacturing, the multiplicative constant error of geodimeter is originally the residual error after correction, which comes from the residual error of quartz crystal frequency after temperature correction, and still varies with temperature. In the field of instrument, the limited range of this error is also given. Therefore, it is actually meaningless to use a test sample given by metrology field to correct the observed value.
Therefore, this concept of systematic error often makes the measurement practice in trouble. For example, in 2020, the National Bureau of surveying, mapping and geographic information of China announced that the elevation of Mount Everest is 8848.86m, which is actually an adjustment value obtained from the trigonometric leveling data of geodimeter and GNSS satellite survey data according to a certain weight proportion. However, at present, we have not seen the public information about weight calculation, nor the official data about error evaluation.
In recent years, literature [7 , 8 , 9 , 10 , 11 , 12 , 13 ] proposed an interpretation method of measurement theory based new concepts, which clearly pointed out that the traditional measurement theory's understanding of constant and random variable concepts is inconsistent with probability theory. Because the constant in probability theory refers to a numerical value, and the random variable refers to a quantity whose numerical value is unknown, this theory regards the observed value (measured value) as a constant, the error and the true value as random variables, and the variance of error is the dispersion of all possible values of the error. In this conceptual system, errors cannot be classified as systematic error and random error.
Although these documents have been published formally, they do not deal with the problem of weight calculation in multi-channel data adjustment, so this paper will complete the mathematical deduction of this calculation principle.

Basic concepts 2.1. Constant and random variable
In probability theory, a numerical value is regarded as a constant, but a random variable is an unknown quantity without numerical value. To study the random variable, people can only use all its possible values (sample space) to describe its probability interval.

Mathematical expectation and variance
For a random variable with sample space { }, there is ∈ { }, is the probability of occurrence of event { = }, that is, = { = } (the probability density function ( ) is used to express continuous random variable), and its mathematical expectation and variance are respectively defined as: Its meaning is that all possible values of random variable X constitute a density distribution interval (Frequency school) with E(X) as the center and σ 2 (X) as the width evaluation, or the random variable X exists within a probability interval (Bayesian school) described by E(X) and σ 2 (X). In other words, the mathematical expectation E(X) and the variance σ 2 (X) are the evaluation of the probability interval of X.

The origin of the conceptual trouble in traditional measurement theory
In Section 4.2 of literature [2] and Section 5.17 of literature [3] , a method process is described as follows: When obtaining n observed values , the final measured value q is The variance of each observed value is The variance of the final measured value q is Obviously, in formula (2-3), q and each k q respectively represent a numerical value; but in formulas (2)(3)(4) and (2)(3)(4)(5), these numerical values q and k q are given a variance that is not zero, which obviously violates the formula (2-2). For example, a DC voltage is repeatedly measured 100 times, and there are 50 observed values of 5.0V and 50 observed values of 5.1V. According to formula (2-3) (2-4) (2-5), we can get: That is, in formula (2-1), all numerical values i x or x are used to describe variable X , but the formulas (2-4) and (2)(3)(4)(5) do not follow this principle and imposes the concept of variable on numerical values itself.
Because the numerical values (observed value and measured value) are regarded as random variable, the traditional measurement theory uses Fig 1 to interpret its basic measurement concepts, which further leads to the philosophy of error classification and misleads the conceptual difference between trueness and precision. Formulas (2)(3)(4) and (2)(3)(4)(5) are the mathematical expressions of the so-called precision concept.
In a word, the concept of error classification in traditional measurement theory is obtained by misinterpreting the concept of random variable, so the classification concept of precision and trueness is actually not correct [7][8] [9][10] [11][12] [13] .

The probability expressions of measured value, error and true value
In references [7,8,9,10,11,12,13] , Fig 2 is used to interpret its basic measurement concepts, which is different from the traditional measurement theory. Among them, the final measured value is an observed value 0 which is a sample within all possible observed values { }, and { } is the sample space of random variable .

Fig 1. Schematic diagram in traditional measurement theory
Because 0 is a numerical value, according to the formula (2-2), there must be: For the unknown deviation ∆ = 0 − ( ) , because it has the same sample space { − ( )} as the random variable ∆ = − ( ), we can use ∆ = − ( ) to express the deviation ∆ , that is, ∆ = − ( ). Therefore, there are Its concept meaning is that variance 2 (∆ ) is the dispersion (Frequency School) of all possible values { − ( )} of deviation ∆ , and is also the evaluation value of the probability interval (Bayesian School) of deviation ∆ .
For the unknown deviation ∆ = ( ) − , because it comes from the previous measurement, its formation principle is similar to that of deviation ∆ , and it also has all its possible values, so there are For example, the field of geodesy generally considers that the multiplicative constant error of geodimeter is a systematic error without variance, but from the perspective of instrument manufacturing, this error is the random error, and the instrument manufacturer also uses all its possible values to count its probability interval (usually expressed by the maximum permissible error MPE). That is, the multiplicative constant error of geodimeter also has its variance like the output deviation in the field of geodesy.
That is to say, both ∆ and ∆ are unknown deviations, have their own variances, and cannot be classified as random error and systematic error respectively. That is, the concepts of precision and trueness were abolished.
In this way, there are In conclusion, the above concepts are summarized in Table 2 [10][11] [12] .
The above conclusion is derived by using an observed value 0 as the final measured value. When obtaining multiple observations , the submission method of the best measured value is shown in Section 3.

Regularity and randomness of error
Any error has variance to evaluate its probability range, and the regular error is of course the same, because the variance of error is the dispersion of all its possible values. Fig 3 is a conceptual diagram describing the regularity and randomness of the periodic error of the geodimeter. When observing the corresponding relationship between the error value and the range, we can see the regularity. When observing the density distribution of the error values, we can see the randomness.

The contribution form of error to repeated measurement
Because errors vary with various measurement conditions, the variation of measurement conditions in repeated measurement determines the contribution form of errors to repeated observations (deviation, dispersion, outlier or both). It is precisely because errors can lead to the dispersion of repeated observations, so the variance of any error can be obtained through experimental statistics, and these data exist in the instrument specifications or instructions. The contribution form of error to the final measured value is only to make it deviate from the true value. This is because, after the measurement adjustment, the final measured value is unique, and the difference between it and the true value is a unique unknown deviation.

The law of covariance propagation
Because every error has its variance, the concept of variance can be extended to any two errors, which is called covariance. It comes from the common component of the two errors.
Thus, for an error array , the definition of variance is as follows: Therefore, according to the definition of variance (2-16), when there is an error propagation equation This is the law of covariance propagation, which is the propagation relation of probability interval between errors, or the propagation relation of the dispersion of all possible values of errors.
After the data processing, the difference between the measured value and the true value is an unknown deviation. Therefore, the task of measurement is to adjust the measured value to minimize the variance of the deviation and submit its value.

The solution of weight values
The smaller the error is, the better the quality of the measured value is. However, we don't know the error value, and can only use the variance of the error to describe its probability interval, so we can only think that the quality of the measured value is the best when the variance of the error is the smallest. Therefore, the mathematical problem we are faced with is to minimize the variance of the error of the final measured value by assigning the weights of each observed value.

Direct measurement model for single measurand
As shown in Fig 2, when n different observed values are obtained by repeated observation of a measurand, assuming that the measured value is y , the error equation is as follows: The error sample set { 1 , 2 , ⋯ , } constitutes a sample space of error V. Therefore, according to the formula in Table 2, there are: The best measured value 0 y is the value when ) and this is a minimum problem of quadratic function, so making 0 )) ( ( , the best measured value is obtained as follows: This is the least square method. As you can see, in the above deduction process, each and do not represent a random variable, which is consistent with the conceptual principle in Section 2. Of course, formula (3-3) is actually an equal weight processing method, that is, the weight of each observation is 1 . However, if n observation errors ∆ have different variances and covariances, that is, the quality of each observation is different from each other, then the equal weight processing is not scientific. At this time, different weights must be given to each observed value to ensure that the quality of the final measured value is the best. Therefore, in order to study each weight takes the minimum value, the equation (3-1) is modified as: Similarly, by using the least square method, there is Please note that in the above derivation process, it is impossible to obtain the best weights , because the best weight 0 actually depends on the evaluation 2 (∆ ) of error ∆ = − rather than the evaluation 2 ( ) of error = − .
For the convenience of calculation, making Now, using the quality of each , we begin to solve the general calculation method of each weight .
, so according to the formula (2-11), there is: For any two observation errors ∆ and ∆ , according to the formula (2-15), there is: Therefore, according to the formula (3-11), there is:  . Moreover, it can be seen that the traditional measurement theory classifies ∆ as systematic error, which is actually to negate the correlation between ∆ and ∆ .  Table 3. Please give the best measured value of diameter and the evaluation of uncertainty.
The four measured values are as follows: The error ∆ of the observed value consists of three components: fixed error , proportional error and non-uniform dividing error . That is:

Assuming
= ±0.01 , = ±1 × 10 −5 and = ±0.02 , there is: As you can see, 01 is a negative value, which is beyond our inherent concept. In the past, we used to follow the conceptual logic of traditional theory to determine weights according to precision, ignoring the correlation between errors, so that (∆ ) is a diagonal matrix, and 0 is always positive. Now, variance is the dispersion of all possible values of error, and any error has variance and covariance, so (∆ ) is no longer a diagonal matrix, and 0 can be negative.
However, this negative weight is also a positive contribution to the final measured value. In this case, 1 is the smallest, and the fixed error K is too harmful to the observed value 1 . Therefore, Lagrange algorithm can only choose negative weights to correct this harm, which is exactly the scientific point of this algorithm.

Indirect measurement model for single measurand
The observation error equation is as follows.  According to the least square method, there is:
And because according to , there is:

Indirect measurement model for multiple measurands
The observation error equation is as follows. According to the least square method, there is:

X Y A P
The error propagation equation is obtained by taking the total differential of formula (3-32), that is: Applying the law of covariance propagation to formula (3-33), we get: However, in the error propagation equation , the propagation law of each error j y 0  is different from each other, so it is difficult to ensure that the variance of all errors can be minimized at the same time. This is also a topic worthy of further study by readers, and the authors consider this topic as follows.
There are t output errors  . Therefore, this is still a question that needs further research.

Conclusion
Following the pure concept of probability theory, both the observed value and the measured value are numerical values and belong to constant, while the error and true value are unknown and belong to random variable; the variance of error is the dispersion of all possible values of the error, and is the evaluation value of the probability interval where the error exists; any error is a deviation and has its variance, so it can't be classified as systematic error or random error. Based on these concepts, and analysing the covariance propagation relationship between the observation error and the final error of measured value, it is a pure mathematical problem to give the weight assignment with the minimum variance, and the dilemma of traditional measurement theory has been solved.
In an adjustment model, the weight allocation of observations is not only determined by the covariance array of observation error, but also related to the parameters in the adjustment model, which is particularly important for a single measurand model. For the adjustment model with multiple measurands, because the errors of multiple measured values come from different propagation laws, the variances of all errors cannot be minimized at the same time, which is a question that needs further research.