A New Theoretical Interpretation of Measurement Error and Its Uncertainty
aarXiv:1704.03812 Highlights Variance is the evaluation of probability interval of an error, instead of the dispersion of measured value defined by the existing measurement theory. The dispersion of measured value is 0. Variance is expressed by the dispersion of all possible values of error. All possible values refer to the test values under all possible measurement conditions permitted by measurement specification. Any error has variance used to evaluate its probability range. Error is a unity of regularity and randomness, and it is an incorrect concept to classify errors as systematic error and random error by regularity and randomness. The mathematical expectation of any error is 0, which expresses the mean value of all possible values of error is 0. The basic principle of uncertainty synthesis is covariance propagation law, which is a mathematical process with strict logic. rXiv:1704.03812 The Correct Application of Variance Concept in Measurement Theory
Ye xiaoming School of Geodesy and Geomatics, Wuhan University, Wuhan, Hubei, China, 430079. Key Laboratory of Precise Engineering and Industry Surveying, National Administration of Surveying, Mapping and Geoinformation, Wuhan, Hubei, China, 430079
Abstract : The existing measurement theory interprets the variance as the dispersion of measured value, which is actually contrary to a general mathematical knowledge that the variance of a constant is 0. This paper will fully demonstrate that the variance in measurement theory is actually the evaluation of probability interval of an error instead of the dispersion of a measured value, point out the key point of mistake in the existing interpretation, and fully interpret a series of changes in conceptual logic and processing method brought about by this new concept.
Keywords : measurement error; variance; uncertainty; precision Introduction
In existing measurement theory, because the measured value is viewed as a random variable and variance is interpreted as the dispersion of measured value, both precision and uncertainty are defined as the conceptual connotation of dispersion of measured value [1, 2, 3, 4, 5, 6], so that people can hardly make clear the difference between them. Now, the author uses a case in the 12 th pages of GUM [3] to illustrate the contradictory expression of variance concept in existing theory. Considering that measurement scientists usually believe strongly in their understanding of statistical concepts and are accustomed to accept such contradictory expressions, the author sincerely hopes that scientists can also calmly pay close attention to the strict expression of mathematical concept and read the whole answers given by author. The following is the case study. EXAMPLE A calibration certificate states that the resistance of a standard resistor 𝑅 𝑆 of nominal value ten ohms is at 23°C and that “the quoted uncertainty of defines an interval having a level of confidence of 99 percent”. The standard uncertainty of the resistor may be taken as 𝜎(𝑅 𝑆 ) = (129𝜇Ω)/2,58 = 50𝜇Ω , which corresponds to a relative standard uncertainty 𝜎(𝑅 𝑆 )/𝑅 𝑆 of 5,0 × −6 . The estimated variance is 𝜎 (𝑅 𝑆 ) = (50𝜇Ω) =2,5 × 10 −9 Ω . In this case, the measured value is 𝑅 𝑆 = 10,000742Ω , and its variance is 𝜎 (𝑅 𝑆 ) = 2,5 ×10 −9 Ω . However, because is a constant and there is 𝑅 𝑆 = 10,000742Ω , the measured value 𝑅 𝑆 is a constant instead of a random variable, and according to the definitions of mathematical expectation and variance in probability theory, there must be: ERE S EER S Obviously, 𝜎 (𝑅 𝑆 ) = 0 and 𝜎 (𝑅 𝑆 ) = 2,5 × 10 −9 Ω are contradictory. However, because the equation 𝜎 (𝑅 𝑆 ) = 0 is from the equation 𝑅 𝑆 = 10,000742Ω , the equations 𝑅 𝑆 =10,000742Ω and 𝜎 (𝑅 𝑆 ) = 2,5 × 10 −9 Ω are also contradictory, and the dispersion of a measured value expressed by precision and uncertainty is actually illogical. People usually are difficult to accept this assertion that the measured value 𝑅 𝑆 is a constant, and they usually say that the value is a sample of a Normal variable and that is a 99% confidence interval for such a Normal sampling distribution. Unfortunately, although their understanding is correct, because there is 𝑅 𝑆 = 10,000742Ω , the actual meaning of 𝜎 (𝑅 𝑆 ) is 𝜎 (10,000742Ω) and refers to the dispersion between oneself and oneself of a constant instead of what they mean, while the equation 𝜎 (10,000742Ω) = 2,5 ×10 −9 Ω can never be proved by anyone and is a wrong equation. [email protected] rXiv:1704.03812 Although variance 𝜎 (𝑅 𝑆 ) is also understood as the dispersion of random errors, the expression form of 𝜎 (𝑅 𝑆 ) is also obviously not rigorous logically, because the 𝑅 𝑆 in 𝜎 (𝑅 𝑆 ) is a measured value which is a constant , instead of random errors. Regretfully, besides measurement standard [3], almost all the existing measurement textbooks use the form of 𝜎 (𝑥) or 𝜎 𝑥2 to express variance[7, 8, 9]. That is, in existing measurement theory, there are some troubles in the interpretation of the most basic measurement concepts, and the measurement theory must be reinterpreted. In the literatures [10,11,12], the authors proposed some new concepts of measurement theory, and variance is proposed as the evaluation of probability interval of error instead of the dispersion of measured value. For the above case, it will be expressed as 𝑅 𝑆 = 10,000742Ω and 𝜎 (∆𝑅 𝑆 ) =2,5 × 10 −9 Ω , while the variance 𝜎 (𝑅 𝑆 ) = 0 . Among them, ∆𝑅 𝑆 expresses the error (deviation) of measured value. Although the author has proposed the new variance concept to interpret measurement theory, its concept principle and interpretation process have not been fully described mathematically. Therefore, in this paper, following strict mathematical concept, the author will point out the misunderstanding of existing variance concept, give a clear interpretation for the origin of this new variance concept, and interpret a series of changes in theoretical logic and mathematical processing. Random variable and its probability expression
Random variable is an unknown quantity whose actual value cannot be given. Because the random variable is unknown, we can only describe the probability range of its value. In order to study its probability range, it is necessary to study the distribution range of all its possible values, while all possible values refer to the set of test values of random variables under all possible test conditions permitted by people (random test does not have the same conditions). Besides probability density function, mathematical expectation and variance are also two parameters used to describe the probability distribution range of a random variable. Thus, for a random variable 𝐿 with all possible values {𝐿 𝑖 } , its mathematical expectation and variance are defined respectively as nLEL ni i (2-1) )()( ELLEL (2-2) This means that the random variable 𝐿 exists within a probability interval with mathematical expectation 𝐸𝐿 and variance 𝜎 (𝐿) . Mathematical expectation and variance are the evaluation values of probability interval of random variable. Now, suppose that the mathematical expectation of a random variable 𝐿 is 𝐸𝐿 = 𝐶 and its variance is 𝜎 (𝐿) = 0 , then there is: ELLE
By substituting
𝐸𝐿 = 𝐶 , we get: CLE
Therefore, CL That is, when the variance of a random variable 𝐿 is reduced to zero, it becomes constant 𝐶 . In other words, for a constant 𝐶 , because all its possible values are its itself, there are naturally: CEC (2-3) ECCEC (2-4) The mistake of variance concept in existing theory
Traditional measurement textbooks always give such a figure of containing defect, as shown in Figure 1. In this figure, the final measured value is not marked, but the distribution curve of all possible values {𝑥 𝑖 } of measured value under the condition that the “systematic error” is fixed is marked. In this way, for an uncertain measured value 𝑥 , according to definition (2-2), there is rXiv:1704.03812 𝜎 (𝑥) = 𝐸(𝑥 − 𝐸𝑥) , which is the origin of existing variance concept. Therefore, existing textbooks usually use the form of 𝜎 (𝑥) or 𝜎 𝑥2 to express variance. However, in fact, after the measurement is completed, we always have to give a measured value 𝑥 with a definite value. As shown in Figure 2. Although measured value 𝑥 is a member within all possible values {𝑥 𝑖 } , because the measured value 𝑥 is a constant and there is 𝜎 (𝑥 ) = 0 , it is illogical to replace 𝜎 (𝑥 ) with 𝜎 (𝑥) . For example, a student's exam score is 𝑥 , which is known value, and the exam scores of all students in his class is sequence {𝑥 𝑖 } . The 𝑥 is indeed a member within sequence {𝑥 𝑖 } . Although it is reasonable to use the statistic values 𝐸𝑥 and 𝜎 (𝑥) calculated by sequence {𝑥 𝑖 } to express the probability range of an unknown score 𝑥 , because the 𝑥 is a known constant, writing 𝜎 (𝑥) as 𝜎 (𝑥 ) means unjustifiably changing concept. Also, the 𝜎 (𝑥) is obviously not the dispersion of this student’s repeated testing scores. That is, for a measured value 𝑥 , it is a definite constant instead of a random variable, and has no the qualification and need to use the variance concept to describe itself. If we have to view measured value 𝑥 as a random variable, according to formulas (2-3) and (2-4), its mathematical expectation is itself and its variance is 0, that is, 𝐸𝑥 = 𝑥 and 𝜎 (𝑥 ) =0. The equation 𝜎 (10,000742Ω) = 2,5 × 10 −9 Ω is actually to secretly replace 𝜎 (𝑥 ) with 𝜎 (𝑥) . The new interpretation of variance concept
Before interpreting the new variance concept, we need to clarify the concept of error first. The definition of error is the difference between measured value and its true value. That is: T xx (4-1) Because the measured value x is unique, and the true value T x is also unique but unknown, the error must be an unknown constant deviation. In order to evaluate the authenticity of measured value, we can only use the probability theory to evaluate the probability range of this error . As shown in Figure 2. As you can see, the error of final measured value 𝑥 is still a constant deviation, and can be divided into Exx A and TB xEx . However, because the final measured value x is fixed after completing measurement, the two errors A and B are actually constant deviations. Therefore, it is obviously illogical that existing theory interpret A as precision but interpret B as trueness, and the concepts of precision and trueness should not be used to the final unique measured value. Now, the issue we have to solve is to estimate the probability interval of these deviations, and we have no need to be entangled in the question whether error has its classification at all. Random error (precision) Systematic error (trueness) True value Mathematical expectation
Fig 1. Schematic figure containing defect
Δ=Δ A +Δ B Δ B Δ A True value 𝑥 𝑇 Mathematical expectation 𝐸𝑥 Fig 2. Figure 1 after correcting defect
Measured value 𝑥 x rXiv:1704.03812 As shown in Figure 2. Although the measured value 𝑥 is not a random variable, its error ∆ 𝐴 = 𝑥 − 𝐸𝑥 is unknown and is a random variable, and the error sequence {𝑥 𝑖 − 𝐸𝑥} are all possible values of error ∆ 𝐴 . In this way, we can view the deviation ∆ 𝐴 as any one unknown deviation 𝑥 − 𝐸𝑥 , that is ∆ 𝐴 = 𝑥 − 𝐸𝑥 , then, according to variance’s definition (2-2), there is )()( AAA EE Because 𝐸∆ 𝐴 = 𝐸(𝑥 − 𝐸𝑥) = 𝐸𝑥 − 𝐸𝑥 = 0 , there is )( )()( ExxEE AA (4-2) In formula (4-2), variance 𝜎 (∆ 𝐴 ) expresses the dispersion of all possible values of an error ∆ 𝐴 . More importantly, because error ∆ 𝐴 is a member within its all possible values, so the variance 𝜎 (∆ 𝐴 ) is also the evaluation of probability interval of single error ∆ 𝐴 . That is to say, variance is the variance of a single error, and has no direct relation with measured value 𝑥 , which is completely different from that existing theory uses formula 𝜎 (𝑥) = 𝐸(𝑥 − 𝐸𝑥) to interpret variance as the dispersion of measured value 𝑥 . Taking normal distribution as an example, standard deviation )( A expresses that the deviation ∆ 𝐴 is within the interval of [ )( A , )( A ] under the confidence probability of 68%. Standard deviation is actually a concept of error range with probability meaning, and expresses an error’s possible deviation degree. Because ∆ 𝐴 exists within a symmetric probability interval [ )( A , )( A ] centered with 0, naturally, there is: A E (4-3) In fact, there was already ExExExxEE A . In formula (4-2), single deviation A is a member within its all possible values, and the dispersion interval of all possible values is naturally the probability interval of this single deviation A . Note that, an unknown deviation follows a random distribution, which means that all possible values of the deviation follow a random distribution; it is because the value of deviation is unknown that we use all its possible values to study its probability distribution. This principle obviously can be extended to the error B . In fact, when we trace back to the upstream measurement of forming error B , we will find that the formation principle of error B is similar to that of current error A , that the error B is also a member within all its possible values, and that there is also a standard deviation )( B to evaluate the probability interval of error B . For example, the multiplicative constant error R of geodimeter[14, 15] comes from the frequency error of quartz crystal, and is always viewed as a systematic error by the existing measurement theory. However, in document [12], the authors demonstrated the process of obtaining its standard deviation from upstream instrument manufacturer. Thus, a more generalized definition of variance is expressed as below: )( )( xEx (4-4) That is to say, the x in formulas of (4-4) not only expresses the deviation A between measured value and expectation, but also can expresses the deviation B between expectation and true value. It can even expresses the deviation BA between measured value and true value. Please notice, because the final measured value is unique and constant, both A and B are constant unknown deviations. In addition, both A and B have their own variance. Hence, it is incorrect that the existing measurement theory considers A as random error and considers B as systematic error. And, corresponding precision and trueness concepts are also incorrect. rXiv:1704.03812 Any error exists within a probability distribution centered with 0, and naturally its mathematical expectation is 0. That is xE (4-5) This law (4-5) can also be proved as follows. Suppose the error of a measured value 𝑥 is ∆𝑥 and there is CxE , and make xExC , then there is,
CC xExxEx (4-6) For the constant 𝐶 in equation (4-6), there are CEC (4-7) ECCEC (4-8) For the error ∆𝐶 in equation (4-6), there are xExE xExECE (4-9) )( )( )( )()( x xExE CE CECEC (4-10) It can be seen, the constant C is a known error, belongs to the measured value (the measured value of error) instead of error, and is bound to be used for the correction of final measured value 𝑥 , while the remaining residual error xExC takes 0 as its mathematical expectation. In other words, an error, whose mathematical expectation isn’t 0, cannot exist in measurement practice. That is, there is always 𝐸∆𝑥 = 0 . It should be emphasized that, the law (4-5) means that the mean value of all possible values of an unknown error is 0, which expresses the probability of positive and negative value of an unknown error is equal in our subjective cognition. From a statistical perspective, all possible values of an error refer to the set of all error values under all possible measurement conditions permitted by measurement specification, so the existing concept of "repeated measurement under the same conditions" must be abandoned [11,12], otherwise an unique error value obtained under a particular condition is only one sample within all possible values and does not represent all possible values, which is very easy to cause the illusion of
𝐸∆𝑥 ≠ 0 . For example, if a rule 99 cm long is marked wrongly with 100 cm, every measurement done with such rule will lead to errors that expected to be 1 cm, which causes the illusion of 𝐸∆ 𝐵 ≠ 0 . The root of this illusion is actually from the traditional concept of "repeated measurement under the same conditions", and the key question is whether we can determine that all possible values of error ∆ 𝐵 are 1cm. In actual applied measurements, if we can determine ∆ 𝐵 = 1𝑐𝑚 , that is, all possible values of ∆ 𝐵 are 1cm, then this 1 cm belongs to a constant 𝐶 in equation (4-6), and after the 1 cm is corrected, the remaining residual error still takes 0 as its mathematical expectation; if we cannot determine ∆ 𝐵 = 1𝑐𝑚 under a new measurement condition, that is, 1cm does not represent all possible values of error ∆ 𝐵 , then this 1 cm belongs to one sample within all possible values of ∆ 𝐵 , while we should use the maximum permissible error (MPE) given by ruler manufacturer as the dispersion evaluation of all possible values of ∆ 𝐵 , and like the constant 𝐶 in equation (4-6), the mathematical expectation of all possible values of ∆ 𝐵 has been corrected by manufacturer, which is also the reason why the measuring instrument is only annotated MPE but never annotated mathematical expectation. Besides, all the instrumental error test values, which are given by the field of metrological testing, should be viewed as a sample within all possible values of error, and should be not an obstacle to our understanding of 𝐸∆𝑥 = 0 . In short, being different from measured value, the error is unknown; because the error is unknown, we can only study its probability range; because of studying its probability range, we must study all possible values of error; because of studying all possible values of error, error samples rXiv:1704.03812 must come from all the possible measurement conditions permitted by measurement specification, and the traditional concept of "repeated measurement under the same measurement conditions" must be abandoned. Now, we only need to study the variance of error. Because the number of error samples is always a finite value in measurement practice, the formula (4-4) can be approximated as nxx ni i )( )( (4-11) Notice that formula (4-11) is also the source of least squares principle. That is to say, under the new concept logic, only the concept of error evaluation has changed, while the principle of least square method used to obtain the best measured values has not changed. Considering that two correlated errors have common component, the formula (4-4) is extended to any two errors. That is )()( jiji xxExx (4-12) Thus, for the error sequence Tt xxx X , the definition of variance is: T E ))(()( XXXD (4-13) That is tt xxxxxxE )( XD Obviously, the true definition of variance is actually formula (4-13). Formula (4-4) is only a special case of formula (4-13) when t =1, and formula (4-3) is only a special case of formula (4-4) when interpreting x as Exx - . Any error has variance, including the regular error, because regular error also has all its possible values. For example, the cyclic error of phase photoelectric distance meter [13, 14] is sine regularity, and its function model is )2sin( DA . However, when we only observe the density distribution of all its possible values, this cyclic error’s probability density function )( f can be derived as: AAAf See Figure 3. Further, its variance can be derived as A , and its mathematical expectation can be derived as E . Another example, the rounding error is a sawtooth cycle regularity function of true value w . However, when we only observe the density distribution of all its possible values, the error is also to follow a random distribution, as shown in Figure 4, and its probability density function is: δ D f(δ) Fig. 3 The random distribution of sine error rXiv:1704.03812 aaaf Its variance can be derived as a , and its mathematical expectation can be derived as E . That is to say, regularity and randomness are the effect of observing all possible values of error from different perspectives, there is no opposition between them, and there is actually no need to dwell on the error’s regularity in the discussion of error evaluation. In other words, when a regular error is unknown, we can still use mathematical expectation and variance to describe its probability range. Furthermore, existing measurement theories use regularity and randomness to classify errors into systematic errors and random errors, which is also proved to be incorrect. Because any error has all its possible value and has variance, the law of covariance propagation is extended to any error. In addition, the law of covariance propagation can only be interpreted as the propagation law of error’s probability interval, and cannot be interpreted as propagation law of measured value’s dispersion. Now, there is a linear function: KKXZ ( ) In equation (4-14), t ZZZ Z , tntt nn kkk kkk kkk
21 22221 11211 K , t kkk K Take total differential for (4-14) equation:
XKZ ( ) According to formula (4-13), the covariance matrix of the error sequence Z is TT TT )( KXXK XKXK ZZZD
EEE
That is, T )()( KXKDZD (4-16) Please note the relationship between equations (4-15) and (4-16): 1. In the error equation (4-15), the direct participants of synthesis is the error itself, each error is a deviation, and error synthesis always follows algebra rule. 2. In variance equation (4-16), the participants of synthesis are all possible values of each error instead of each error itself. It expresses the dispersion’s propagation relation between error groups, and is also the probability interval’s propagation relation between errors.
According to Figure 2, the total error of final measured value is BA (4-17) Among them, A is the difference between measured value and expectation, and B is the difference between expectation and true value. Because the two errors are usually irrelevant, according the law of covariance propagation(4-16), there is: )()()( BA (4-18) -a a f(δ) δ w Fig. 4 The random distribution of sawtooth error rXiv:1704.03812 This total standard deviation )( is the uncertainty, expresses the probability range of the total error of final measured value. Thus, )( A and )( B are referred as the uncertainty of Type A and the uncertainty of Type B respectively. The current )( B is actually the )( of historical upstream measurement, and the current )( can also be used as the )( B in future downstream measurement. The traceability chain of quantity value is a chain of uncertainty transmission. It can be seen, the uncertainty is actually a rigorous concept instead of a loose concept as said by VIM[4]. Note that formula (4-18) comes from the covariance propagation law (4-16). That is to say, the basic principle of uncertainty synthesis is covariance propagation law (4-16), and the uncertainty synthesis does not need to always apply formula (4-18) mechanically.
The principle of uncertainty synthesis is actually the propagation law of covariance. However, because of the existence of covariance between different errors, there is naturally a concept of co-uncertainty, which is another focus in uncertainty concept system. The mean square synthesis of uncertainty is based on the premise that the errors are independent of each other. But in fact, because the problem of error correlation is universal, the co-uncertainty is an unavoidable issue in uncertainty synthesis. So, what is the meaning of co-uncertainty? It is assumed that the errors k , p and q are uncorrelated with each other, and that their standard deviations are pk , and q respectively. Now, there are two errors pk and qk , that is, both and contain a communal error component 𝑘 , therefore, there are: pk ( ) qk ( ) According to the definition of covariance: )()()( ))(( )( pqEkqEkpEEk qkpkE E ( ) Because the errors k , p and q have been assumed to be irrelevant from each other, there are kpE , kqE and pqE , and the equation (4-21) becomes into: k Ek ( ) Covariance is the co-uncertainty between 𝛿 and 𝜀 , which is actually the variance of their communal error component 𝑘 . That is to say, the mathematical meaning of co-uncertainty is the probability evaluation of the communal error component among two errors. As long as there are communal error component among different errors, there must be a co-uncertainty between them. Of course, the symbol and coefficient of communal error component should be considered in the actual measurement. For example, the two measured value’s errors measured by the same instrument have correlation, and the errors of two instruments calibrated by the same benchmark have correlation, too. Moreover, like the above principle, when two errors are associated with the same measurement condition, there is also a covariance between them. For example, both the error of light speed in atmosphere and the thermal expansion error of metal are functions of temperature, and there is a correlation between all possible values of the two errors. The uncertainty synthesis relies on historical data. But in historical data, there are very few data about the co-uncertainty (covariance) between different instruments and between different measured values. Naturally, the concept of co-uncertainty is also a new topic in the future to improve the measurement theory and metrology management system. rXiv:1704.03812 Now, we already know, for the measured value 𝑥 , there are 𝐸𝑥 = 𝑥 and 𝜎 (𝑥 ) = 0 ; for the error ∆ , there are 𝐸∆= 0 and 𝜎 (∆) = 𝐸(∆) . Further, because the error is the difference between measured value and true value, that is T xx , so, there are: xx T (4-23) )( x EExxEEx T (4-24) )( )( )()( xxE ExxEx TT (4-25) Thus, the probability expressions of true value 𝑥 𝑇 , measured value 𝑥 and error ∆ are summarized as Table 1. It can be seen that uncertainty is also the possible degree that the true value deviates from the measured value. Uncertainty is not only the uncertainty of error but also the uncertainty of true value, but is not the uncertainty of measured value. As a constant, the measured value has no uncertainty. Statistical calculation of variance
In actual measurement, in order to achieve the reduction and evaluation of measurement error, a large number of observations are usually carried out, and the observation values are obtained. Because errors make a large number of observations contradict from each other, the optimal measured values must be given by adjustment process, while the errors of measured values are also evaluated. Here, we only discuss the case of the least square adjustment.
The following is an observation equation set based on the error model tj jijii yaxv . tntnn ttnn yyyaaa aaa aaaxxxvvv That is
AYXV (5-1) Among them, 𝑥 , 𝑥 , ⋯ 𝑥 𝑛 are the observed values, and 𝑦 , 𝑦 , ⋯ 𝑦 𝑡 are measured values, and there is 𝑛 > 𝑡 . Now, the task we want to accomplish is to solve the calculation method of variance matrix of deviations XXX E and YYY E . According to the principle of least squares, its normal equation is: XAAYA T T (5-2) Its measured values are: XAAAY T1 T (5-3) The error propagation equation is: XAAAY T1 T (5-4) Now, we only discuss the variance of ∆ 𝐴 , so we only see the error in formula (5-4) as XXX E and YYY E . According to the covariance propagation law (4-16), the Table 1. The probability expression of true value, measured value and error
Measured value 𝑥 Error ∆ True value 𝑥 𝑇 Mathematical expectation 𝑥 𝑥 Variance 0 𝜎 (∆) 𝜎 (∆) rXiv:1704.03812 covariance matrix )( YD is: )()( XDAAYD T (5-5) Now, the covariance matrix )( XD is: xxx
00 00 00)( XD Therefore, )( AAYD T x (5-6) Here, the AA T is called the co factor matrix. According to equation (4-5) and AYXV , there is AYXV
EEE
Therefore
AYX EE (5-7) )( YYAV YAAYAYX AYX XXX
E EEE (5-8) Make
YYY E , that is jjj Eyyy , there is
YAVX
That is tntnn ttnn yyyaaa aaa aaavvvxxx (5-9) Now, the measured values Y have been given by equation (5-3), and the numerical value of every error 𝑣 𝑖 also has been given by equation (5-1). According to the formula (4-11), there is: )( )( tt tTtni ini i ayayayv yyyyyyvxxn AA (5-10) Take mathematical expectation from both sides of the equation (5-10), and be aware of jjjjj EyEyEyyEyE . Then get: tTtni i yyyyyyEvxn )( AA ttttt tttni i yyyAAA AAA AAAyyyEv tj jjtttj jjtj jjni i yAyEyAyEyAyEv
11 221 111 2 (5-11) Now, let's see the matrix )( YADA T , and please note that AA T is a symmetric matrix. rXiv:1704.03812
221 22221 1122121 22212 11211 )( ttt tttttt ttT AAA AAA AAA
YADA tttt tttttt tt yyyyyy yyyyyy yyyyyyAAA AAA AAAE
21 22212 1211121 22212 11211 tj jittj jjttj jjt tj jitj jjtj jj tj jitj jjtj jj yAyEyAyEyAyE yAyEyAyEyAyE yAyEyAyEyAyE (5-12) However, according to the formula (5-6), there is: )(00 0)(0 00)()()( xxx T XDYADA (5-13) Make equal substitution for equation (5-12) and equation (5-13), and there is: )(
211 221 11 xyAyEyAyEyAyE tj jjtttj jjtj jj (5-14) Replace equation (5-14) into the equation (5-11): )( )()()()(
21 2 2221 22 xtv xxxvxn ni ini i (5-15) Therefore, tn vx ni i )( (5-16) This is the Bessel formula. The uncertainty of Type A, which is the evaluation of the probability interval of error YY E , is obtained by formula (5-6). In multivariate model tj jijii yaxv in section 5.1, we make t =1 to obtain the single variable model as yaxv iii . The error equations of repeated measurement are: yaxv yaxv yaxv nnn
222 111 (5-17) According to formula (5-3), the final measured value given by least square method is: rXiv:1704.03812 ni ini ii axay (5-18) According to formula (5-16), the Bessel formula is: n vx ni i (5-19) According to formula (5-6), for the final measured value y , the uncertainty of Type A, which is the evaluation of the probability interval of error Eyyy , is: ni i axy )()( (5-20) In the model yaxv iii in section 5.2, we make 𝑎 𝑖 ≡ 1 to obtain the direct measurement model as yxv ii . A measurand was directly measured by n times, and the error equations are: yxv yxv yxv nn
22 11 (5-21) According to formula (5-18), the final measured value given by least square method is ni i nxy . (5-22) According to formula (5-19), the Bessel formula is still: n vx ni i (5-23) The standard deviation )( x is the evaluation of the probability interval of any an error Exx . Then, according to the equation (5-20), the uncertainty of Type A, which is the evaluation of probability interval of error Eyyy , is: nxy )()( (5-24) It can be seen, in the sense of forms, the conclusion of above three models is the same as the conclusion of existing measurement theory, but the difference is only that 𝜎(𝑥) is written as 𝜎(∆𝑥) ! It must be noted that, the standard deviation, which is given by the formula (5-6), (5-20) and (5-24), is the evaluation of probability interval of constant deviation y or j y between final measured value and its mathematical expectation, and belong to uncertainty of Type A. Uncertainty synthesis
The observational errors i x discussed in section 5 are completely uncorrelated, and thus contribute to dispersion of i v , which cause the error A in final result’s error. But in fact, there are always some special regular error i , which exactly adapts to the change rule of repeated measurement conditions and don’t contribute dispersion of sequence i v . Also, its existence does not cause any effect on the calculation and analysis process of standard deviation )( A , therefore, rXiv:1704.03812 there is no need to consider its existence in section 5. However, it also causes a deviation to final measured value, which is the error B . Therefore, analyzing this special regular error i is a very important task. 1. For the single variable direct measurement, the error model is yxv ii , and the final measured value is given by equation (5-22). That is ni i nxy (6-1) When every observed value i x contains the same error Bi , according to equation (6-1), the error of final result y is added an error component B , but the dispersion of i v isn’t affected. That is, the uncertainty of Type A isn’t affected. 2. For the single variable indirect measurement, the error model is yaxv iii , and the final measured value is given by equation (5-18). That is ni ini ii a xay (6-2) When each observed value i x contains an error component Bii a respectively, according to equation (6-2), the error of final result y is added an error component B , but the dispersion of i v isn’t affected. That is, the uncertainty of Type A isn’t affected. 3. For the multivariate indirect measurement, the error model tj jijii yaxv , and the final measured value is given by equation (5-3). That is XAAAY TT (6-3) When each observed error i x contains an error component tj Bjiji a respectively, according to equation (6-3), the error of each final result j y is added an error component Bj respectively, but the dispersion of i v isn’t affected. That is, the uncertainty of Type A isn’t affected. Therefore, it is an important task to analyze the influencing characteristics of error sources, and analyzing the influence characteristics of error sources in the current measurement method has become an essential skill for surveyors. Because standard deviation )( B cannot be obtained by current measurement, it can only be obtained by tracing its source and looking up historical measurement data. After obtaining the standard deviation )( B , the total uncertainty of final measured value can be obtained by the law of covariance propagation. Obviously, the way of A/B classification is difficult to understand and apply. However, as mentioned earlier, the fundamental principle of uncertainty synthesis is covariance propagation law, and we do not need to always copy this A/B classification thinking mechanically.
Especially for the indirect measurement principle mentioned above, directly using covariance propagation law will make the problem simple and easy. Here's a simple example to illustrate this principle, which is also a comparison with the traditional practice. For example, the weights of three objects of A, B and C are measured with a steelyard, and the observed values are obtained as shown in Table 2. How do you obtain the final measured values and uncertainties? Assuming that the final measured values of A, B and C are y , y and y respectively, the error Table 2. Observation values under different conditions Measuring method Observation value Measure A separately 𝑥 Measure B separately 𝑥 Measure C separately 𝑥 A and B were measured together 𝑥 B and C were measured together 𝑥 A and C were measured together 𝑥 rXiv:1704.03812 equations are as follows:
101 110 011 100 010 001 yyyxxxxxxvvvvvv (7-1) According to the formula (6-3), the final measured values are: xxxxxx xxxxxxyyy (7-2) For the existing measurement theory, the next step is to substitute (7-2) into (7-1), and six residual 𝑣 𝑖 are obtained. Then )( x is obtained by Bessel formula tn vx ni i )( , and )( y 、 )( y and )( y are obtained by covariance propagation law. Among them, )( y 、 )( y and )( y are called as precision. However, from the perspective of the new variance concept, there are three serious problems in the above variance submission process: 1. The degree of freedom 𝑛 − 𝑡 is too small, so it is meaningless to apply Bessel formula. 2. Both the observed value 𝑥 𝑖 and the measured value 𝑦 𝑗 are constants, and their variances should be 0. The variances submitted actually belong to some errors. 3. The contribution of covariance between the errors of each observation value 𝑥 𝑖 has not been taken into account at all (uncertainty synthesis issue). The following is the variance submission process based on the new variance concept for this case. Taking the total differential of equation (7-2), the error propagation equation is obtained as follows: xxxxxxyyy (7-3)
Applying covariance propagation law (4-16) to equation (7-3), the covariance propagation equation is obtained as follows: rXiv:1704.03812
323 332 233 411 141 114)(332411 233141 3231140101)( ij xy DD (7-4) Among them,
𝐃(∆y 𝑗 ) is the covariance matrix of error sequence {∆𝑦 𝑖 } and 𝐃(∆𝑥 𝑖 ) is the covariance matrix of error sequence {∆𝑥 𝑖 } . The acquisition process of covariance matrix 𝐃(∆𝑥 𝑖 ) is as follows. For the i th observation value 𝑥 𝑖 , its error ∆𝑥 𝑖 can be regarded as the composition of three parts: zero-point error a , proportional error 𝑏𝑥 𝑖 and scale non-uniformity error 𝑐 𝑖 . The error combination relationship is as follows: iii cxbax ( ) Its variance is cbiaxi x ( ) Among them, 𝜎 𝑎 , 𝜎 b and 𝜎 𝑐 can be obtained by consulting instrument (steelyard) instructions or the tolerance standard in instrument specification. Furthermore, according to (4-13), 𝐃(∆𝑥 𝑖 ) can be deduced as follows: )( cbababa bacbaba babacbai xxxxx xxxxx xxxxx xxxxxxxxxxEx D (7-7) Finally, substituting the covariance matrix 𝐃(∆𝑥 𝑖 ) into equation (7-4), the covariance matrix 𝐃(∆y 𝑗 ) is obtained. Among them, the 𝜎(∆y 𝑗 ) is called as uncertainty. Conclusion
In short, we can summarize the main point as follows. Variance (standard deviation) or uncertainty is the evaluation of probability interval of a single error (deviation) instead of dispersion of a measured value, and any regular error’s size degree can be evaluated by them. Any error is a deviation, follows random distribution, has standard deviation which can be used to evaluate its size, and cannot be classified according to systematic and random way. The error synthesis follows the algebraic rule, and the synthesis of variance follows the probability principle. In the interpretation of measurement theory based on the new concepts, there is no need to reuse those old concepts such as systematic error, random error, precision, trueness and accuracy. rXiv:1704.03812 REFERENCES [1] Guide to the Expression of Uncertainty in Measurement, International Organization for Standard, First edition corrected and reprinted,1995,ISBN92-67-10188-9 [2] JJF1059-1999, Evaluation and Expression of Uncertainty in Measurement [3] JCGM 100:2008, Guide to the Expression of Uncertainty in Measurement(GUM) [4] JCGM 200:2012, International vocabulary of metrology — Basic and general concepts and associated terms (VIM) [5] JJF1001-2011, General Terms in Metrology and Their Definitions [6] GB/T14911-2008, Basic Terms of Surveying and Mapping [7] School of Geodesy and Geomatics, Wuhan University. Error Theory and Foundation of Surveying Adjustment[M]. Wuhan University Press, 2016 6. [8] Fei Yetai. Error Theory and Data Processing[M]. Machinery Industry Press, 2016 3 [9]Ye Depei. Understanding, Evaluation and Application of Measurement Uncertainty[M]. China Metrology Publishing House, 2007 10 [10] Ye Xiao-ming, Ling Mo, Zhou Qiang, Wang Wei-nong, Xiao Xue-bin. The New Philosophical View about Measurement Error Theory. Acta Metrologica Sinica, 2015, 36(6): 666-670. [11] Ye Xiao-ming, Xiao Xue-bin, Shi Jun-bo, Ling Mo. The new concepts of measurement error theory, Measurement, Volume 83, April 2016, Pages 96-105 [12] Ye Xiao-ming, Liu Hai-bo, Ling Mo, Xiao Xue-bin. The new concepts of measurement error's regularities and effect characteristics, Measurement, Volume 126, October 2018, Pages 65-71 [13]. JJG703-2003, Electro-optical Distance Meter (EDM instruments) [14]. ISO 17123-4:2012,Optics and optical instruments -- Field procedures for testing geodetic and surveying instruments -- Part 4: Electro-optical distance meters (EDM measurements to reflectors)[1] Guide to the Expression of Uncertainty in Measurement, International Organization for Standard, First edition corrected and reprinted,1995,ISBN92-67-10188-9 [2] JJF1059-1999, Evaluation and Expression of Uncertainty in Measurement [3] JCGM 100:2008, Guide to the Expression of Uncertainty in Measurement(GUM) [4] JCGM 200:2012, International vocabulary of metrology — Basic and general concepts and associated terms (VIM) [5] JJF1001-2011, General Terms in Metrology and Their Definitions [6] GB/T14911-2008, Basic Terms of Surveying and Mapping [7] School of Geodesy and Geomatics, Wuhan University. Error Theory and Foundation of Surveying Adjustment[M]. Wuhan University Press, 2016 6. [8] Fei Yetai. Error Theory and Data Processing[M]. Machinery Industry Press, 2016 3 [9]Ye Depei. Understanding, Evaluation and Application of Measurement Uncertainty[M]. China Metrology Publishing House, 2007 10 [10] Ye Xiao-ming, Ling Mo, Zhou Qiang, Wang Wei-nong, Xiao Xue-bin. The New Philosophical View about Measurement Error Theory. Acta Metrologica Sinica, 2015, 36(6): 666-670. [11] Ye Xiao-ming, Xiao Xue-bin, Shi Jun-bo, Ling Mo. The new concepts of measurement error theory, Measurement, Volume 83, April 2016, Pages 96-105 [12] Ye Xiao-ming, Liu Hai-bo, Ling Mo, Xiao Xue-bin. The new concepts of measurement error's regularities and effect characteristics, Measurement, Volume 126, October 2018, Pages 65-71 [13]. JJG703-2003, Electro-optical Distance Meter (EDM instruments) [14]. ISO 17123-4:2012,Optics and optical instruments -- Field procedures for testing geodetic and surveying instruments -- Part 4: Electro-optical distance meters (EDM measurements to reflectors)