A New Theoretical Interpretation of Measurement Error and Its Uncertainty
The traditional measurement theory interprets the variance as the dispersion of a measured value, which is actually contrary to a general mathematical concept that the variance of a constant is 0. This paper will fully demonstrate that the variance i…
Authors: Huisheng Shi, Xiaoming Ye, Cheng Xing
Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 1 A New Theoretic al Interpretation o f Measurement Error and Its Uncertainty Huisheng Shi 1 , Xiaoming Ye 2 *, Cheng Xing 2 and Shijun Ding 2 1 AVIC Changc heng Institute of Metrology & Measurement, Beijing, 100095, P.R. China . 2 School of Geodesy and Geomatics, Wuhan University, Wuhan, 430079, P.R. China . Correspondence: Xiaoming Ye ; xmye@sgg.whu.edu.cn Abstract The traditional measurement theory interprets the va riance as the dispe rsion of a measured value, which is actuall y contrar y to a general mathematical concept th at the variance o f a constant is 0. This paper will full y demonstrate that the variance in measurement theory is actually the evaluation of probability interval of an error instea d of the dispersion of a measured value, point out the ke y point of mi stake in th e traditional interpretation, and full y int erpret a series of changes in c onceptual logic and proc essing method brought about b y this new concept. 1. Introduction However, in an y measurement, both the measur ed value and every obs erved value are numerical valu es. According to probability theory , the variance of a numerical value (constant) is zero. So, how does a numerical value show dispersion? Next, we illustrate the contradictory expression of variance in traditional theory. For ex ample, the measured value of Mount Eve rest elevation in 2005 is , and its precision is . But in fact, thi s mathematical expression gives a wrong equation , which violates basic mathematical concept, because the equation inevitably leads to the equation , and according to the concept of in probability theory, there must be . Although many other measured values , , ..., can be obtained b y repeatedl y m easuring the height of Mount Eve rest, and there ca n be , the equa tions and still exist. Therefore , t he equation can never be consistent with mathematical concepts. Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 2 It can be seen that in traditional theories, the definition of precision and uncertaint y is the dispersion of all possible mea sured values, but their mathematical expression is the di spersion of a single measured valu e (a sin gle numerical value). This approach is actually a stealth change of concept, and will inevitably lead to a series of conceptual logic tro ubles. A question arise s. All measure d va lues that diverge from e ach other can definitely be used to describe a random variable, but a me asured v alue must be a numerical value and b elong to a constant. So, how should the measurement theory be interpreted? So far, in the measurement industr y, there is no literature questioning the conceptual category of measured values, such as the recent literature [6] . In the references [7,8,9,10 ], the authors proposed some new concepts to reinterpret measurement theory. Reference [7] proposes a new error epistemolog y that all errors follow a random distribution and cannot classified as s ystematic error and random error. Reference [8] points out, the standard deviation (variance) is the evaluation value of the probabilit y interval of error, any error has a variance for evaluating its uncertainty, and so on. Reference [9] points out, the dispersion and deviation of repeated observations are determined by the chang ing rule s of repeated measurement conditions, and it is possible and correct to handle errors according to the func tion model or the random model. Reference [1 0] points out, the mea sured value is a numerical va lue whose variance is zero, a nd the dispersion of a measured value is an incorrect concept. According to these new con cepts, the m athematical expression of the Mount Everest elevation case should be and , where represents the measured value and represents its error. Although these new concepts have been propos ed to re interpret measurement theory, the root of these concepts and the int erpretation process have not bee n fully described mathematically. The refore, in thi s paper, the authors follow strict mathematical concept to point out the misunderstanding of the traditional concepts, give a clear int erpretation for the origin of th ese new measurement concepts, and systematically explain a series of changes in theoretical logic and mathematical processing. 2. Constant a nd rando m variable In probability theo ry, a constant is a numerica l value, such as 100, 150, x =100, x =8844.43, and so on. Unlike constants, rando m variable is an unknow n quantit y whose actual value cannot be given. Because the random variable is unknown or uncerta in, we can only describe the probability range of its value. I n order to study its probability range , it is necessary to study the distribution rang e of all its possible values (sample space), while all possible values refer to the set of test values o f random variables und er all permitted possible test conditions (random test doe s not have the same conditions). Mathematical expectation and va riance are the numerical expression of probability range of random variable. For a random variable with all possible values , there is , is the probabilit y that each is (Continuous random variables correspond to the probability densit y functio n ), and its mathematical expectation is defined as follows: n i i i L P L E 1 ) ( or dL L LP L E ) ( ) ( (2 -1) And its variance is defined as the dispersion of its possible values : 2 2 )] ( [ ) ( L E L E L σ (2-2) This means that the r andom variable exists withi n a probability i nterval with mathematical expectation and variance , or that mathematical expectation and variance are the ev aluation values of its probability interval. Note that de scribing a random Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 3 variable requires two par ameters: mathematical expectation and varia nce, both of which are indispensable. Now, suppose that there is a constant , and there are and , then: 0 )] ( [ 2 L E L E By substituting , we get: 0 ) ( 2 C L E Therefore, C L That is, when the variance of a random varia ble is reduc ed to zero, it becomes a consta nt . In other words, for a constant , because all its possible values are itself, we get : C C E ) ( (2-3) 0 )] ( [ ) ( 2 2 C E C E C σ (2 - 4) That is to sa y, a constant is a special random variable, and its mathematical ex pectation and variance a re itself and 0 respec tively. Of c ourse, a constant is a known qua ntity, and usually does not need to be expressed in terms of probability. It can be seen that bo th constant and random variable have their own mathematical expectation and variance. Therefore, if we can give the variance of a quanti ty but cannot give its mathematical expectation, there must be a conceptual mistake. It should be noted that for the random variable , its basic feature is that its value is unknown; but for a sample , because it i s a numer ical value, it is still a constant rather than a random variable, and there are and . Obviously, and . That is, constant and random va riable are distinguished by whether they have a numerical value, and the sample is a numerical value, which is a constant a nd cannot be described by the entire set . The conc eptual differences b etween random variable and sample is shown in Table 1. Table 1: T he conceptual differences between the rando m variable and the sample . Random variable Sample 2 2 )] ( [ ) ( ) ( L E L E L σ L P L E i i 0 ) ( ) ( 2 k k k L σ L L E ) ( ) ( ) ( ) ( 2 2 k k L σ L σ L E L E L Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 4 92 . 2 ) ( 0 ) 6 ( 0 ) 2 ( 0 ) 1 ( 5 . 3 ) ( 6 ) 6 ( 2 ) 2 ( 1 ) 1 ( 2 2 2 2 L σ σ σ σ L E E E E C C L E L L E L Δ ) ( ) ( C C E ) ( 0 ) ( 2 C σ 0 ) ( ) ( )] ( [ ) ( L E L E L E L E C E Δ ) ( )] ( [ ) ( )] ( [ ) ( 2 2 2 2 2 L σ L E L E C E C E C E C σ Δ Δ Δ Δ 3. The origin of co nceptual troubles in t raditional th eory Figure 1 is the schematic diag ram of the measurement c oncept in traditional measurement theory. Because people n otice that measu red value is in a stat e of random change in rep eated measurement, the measured value and the random err or are considered as random variables, and the v ariance is the di spersion of measured value or random error . Besides, the s ystematic error and the tru e value are constant in repeated measurement, so the systematic error and the true value are considered a s consta nts which have no variances (or the variance is zero). I n this way, according to formula (2 -2), there is . Therefore, traditional Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 5 textbooks [11,12,13] usually use the form of o r to ex press the variance. However, these are obviousl y inconsis tent with the meanings of random variabl es and constants described in Section 2. Besides, in actu al measurement, we alwa ys have t o give a numerical value as the final measured value. Therefore, the actual schematic diagram is shown in Figure 2. According to the conc epts in section 2, although measured value is a sample within a random distribution , because the measured value is a numer ical value and there a re and , it is illogical to replace with . In addition, simply replacin g with c annot express a complet e mathematical meaning, because the traditional theory cannot submit the mathematical expectation . On the other h and, the systematic error and the tru e value are unkno wn and are re garded as constants by traditional theory . Ho wever, a ccording to formula （ 2-3 ） , the mathematic al expectation of a constant is itself, so it is impossible to give the numerical values of mathematical expectation s of sy stematic error and true value. Therefore, th is so -called const ant is obvious ly not the same concept as the constant in the probabilit y theory, and it is also a conceptual trouble in the traditional measurement theory. In sho rt, in the traditional theory, ex cept for the conceptual trouble of viol ating the c oncept that the variance of a constant is zero, the conceptual trouble of missing mathematical expectations is shown in Table 2. Table 2: T he conceptual trouble of missing mathematical exp ectations. Random error (precision) Systematic error (trueness) True value Mathematical expectation Figure 1. Schematic diagram in traditional m easurement theory Δ= Δ A + Δ B Δ Δ A True value Mathematical expectation Figure 2. Schematic diagram in the new concept t heory Measured value Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 6 Measur ed value Random err or Systemat ic err or T rue value Because ra ndom errors have variance but s ystematic err ors cannot be quantitatively evaluated, traditional theories believe that truene ss and acc uracy are both qualitative c oncepts, and the relationship between the uncertainty concept and them is of course very difficult to explain. 4. Probability e xpression of basic m easurem ent concepts It can be seen that be cause of the wrong understanding of the concept of random variable, the conceptual logic of tr aditional theory actuall y has systematic troubles. Therefore, we need to reorganize the basic measurement concept log ic according to the concepts in section 2. 4.1 Measured value 0 0 ) ( x x E 0 ) ( 0 2 x σ 4.2 Error As shown in Figure 2, the true value of measura nd is , and the error of the final measured value , whic h is a unknown deviati on T x x 0 Δ , can be divided into ) ( 0 x E x A Δ and T B x x E ) ( Δ . First of all, the error is a ra ndom variable, a nd there is . Moreove r, the sample spa ce of error is also , so error can be used to re present , that is, . According to the formulas (2 -1) and (2 - 2) , there are: 0 )] ( [ ) ( x E x E E A Δ (4 -3) ) ( )] ( [ )]} ( [ ) ( { ) ( 2 2 2 2 A A E x E x E x E x E x E x E σ Δ Δ (4 -4) That is, although deviation is unknown, it exists within a proba bility interval with 0 as center and as width eva luation. I n other words, is the eva luation of probabilit y interval of deviation , which expresses the degree that surveyor cannot determine the value of deviation . Taking the norm al distribution a s an example, the vari ance ex presses that the deviation is withi n the interval of [ ) ( A , ) ( A ] under the confidence probabilit y of Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 7 68%. Variance is actually a concept of error r ange with probability mea ning, and expresses an error s possible deviation degree. In formula (4-4), the single deviation A is a member within all its poss ible values, and the dispersion interval of all its possible values is the probability inte rval of this deviation A . A n unknown deviation follo ws a random dist ribution, which means that all possible values of the deviation follow a random distribution. This principle obviously can be extended to the error B . I n fact, when we tr ace 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 , a nd that the e rror B Δ is a lso a member within all its possible values. Therefore, there is also a variance ) ( 2 B σ Δ to evaluate the probability interval of error B , and there is also 0 ) ( B E Δ . For example, the multiplicative constant error R of a geodimeter [1 4,15] comes from th e frequency error of the quartz c rystal, a nd is alw ays viewed as a systematic error without variance by traditional measurement theory. Ho wever, it is the output error in the field of instrument manufacturing, and the submission pr ocess of its variance will be demonstrated in the case in Section 6. Obviously, acc ording to the principle of formula (2-5) ~ (2-9), if the mathematical expectation of an error is C rather than 0, then C must be co rrected to the final measured value, and the mathematical ex pectation of the remaini ng un known error is still 0 . That is, for any unknown error x Δ , there is always 0 ) ( x E Δ (4-5) Thus, variance is expressed as below: ) ( )] ( [ ) ( 2 2 2 x E x E x E x σ Δ Δ Δ (4-6) The x in formulas (4 -5) and (4-6 ) can express not onl y the deviation A between the measured v alue and the mathematical expectati on, but also the deviation B between the mathematical expectation and the true value. I t c an even ex press the deviation B A between the measured value and the true value. In this way, according to formulas (2-1) and (2-2), there are: B A Δ Δ Δ (4 -7) 0 ) ( ) ( ) ( B A E E E Δ Δ Δ (4 -8) ) ( ) ( ) ( ) ( )] ( [ ) ( 2 2 2 2 2 2 B A B A σ σ E E E E σ Δ Δ Δ Δ Δ Δ Δ Δ (4 -9) Because the fina l measured value is unique and constant, both A and B are unknown deviations. In addition, both of them have their own variance, hence, it is incorrect that the traditional measurement theory considers A as random error and considers B as s ystema tic error. Moreover, the corresponding concepts of precision and trueness are also incorrect. It should be emphasized that, the formula (4 -5) m eans that the mean v alue of all possible values of an unknown error is 0, which expresses the probabilit ies that an unknown error takes positive and negative value are equal in our subjective c ognition. F rom a statistical pe rspective, all possible values of an error refer to the set of all error values under a ll possible mea surement conditions permitted by measurement specification, so the traditional concept of "repeated measurement under the same conditions" must be abandoned [ 8,9] , otherwise an unique error Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 8 value obtained under a p articular condition is only on e sample withi n all possible values and does not represent all possible values, which is very eas y to ca use the illusion of . In short, being diffe rent from measured value, th e error is unknown; beca use the error is unknown, we c an onl y study its probability range; beca use o f stud ying it s probability r ange, we must stud y all possible values of error; because of stud ying all possible values of error, error sample s must come fr om all the possi ble measurement conditions permitted b y measurement specificati on, and the traditional concept of "repeated measurement under the same measurement conditions" must be abandoned. 4.3 Va riance of regular error ) 2 s i n( D A ) ( f A A A f 0 1 ) ( 2 2 2 ) ( 2 2 A 0 ) ( δ E w a a a f 0 2 1 ) ( δ D f( δ) Figure 3. Regularity and randomness of periodic error -a a f(δ) δ w Figure 4. Regularity and randomness of rounding error Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 9 3 ) ( 2 2 a 0 ) ( δ E 4.4 Probability expression of true value T x x 0 0 x x T 0 0 0 ) ( ) ( ) ( ) ( x E x E x E x E T Δ Δ ) ( ) ( ) ( ) ( 2 2 0 0 2 T 2 x x E Ex x E x T T Table 3: T he probability expression of true value, measured value and er ror . The above is the case where an observed value is used as the final measured value. I f the mean value o f n obse rvations is taken as the final measured v alue, it can b e inferred that the variance will decrease by n times. Please see section 6.1. 5. Covariance pr opagation Measur ed value Err or T rue value Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 10 5.1 Covariance ) ( ) ( j i j i x x E x x T t x x x 2 1 X T E ) )( ( ) ( X X X D t t x x x x x x E Δ Δ Δ Δ Δ Δ Δ 2 1 2 1 ) ( X D 2 tt t2 t1 2t 2 22 21 1t 12 2 11 σ σ σ σ σ σ σ σ σ t x ) ( - x E x It is assumed that the errors k , p and q are uncorrelated with each othe r, and that their variance are 2 2 , p k σ σ and 2 q σ respectively. Now, there are two errors p k and q k , and they contain a communal error component . Therefore, we can get: 2 2 2 p k （ ） 2 2 2 q k （ ） ) ( ) ( ) ( ) ( )] )( [( ) ( 2 pq E kq E kp E k E q k p k E δε E σ δε （ ） k p q 0 ) ( kp E 0 ) ( kq E 0 ) ( pq E 2 2 ) ( k δε σ k E σ （ ） Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 11 5.2 The law of covariance propagation ) ( X Z F X K Z t Z Z Z 2 1 Z tn t t n n k k k k k k k k k 2 1 2 22 21 1 12 11 K n x x x 2 1 X According to formula (5-2), the covariance matrix of the error sequence Z is T T T ) ( ) ( K X D K X K X K Z Z Z D Δ Δ Δ Δ Δ Δ E E (5-9) Equation (5-9) is the l aw of cov ariance propagation. The relationship between equations (5-8) and (5-9) is: 1. In the e rror equation ( 5-8), the dire ct participants of s ynthesis is the e rror itself, each error is a deviation, and error synthesis alway s follows the algebra rule. 2. I n the variance equation ( 5-9), the participants of s y nthesis are all possible values o f each error instead of each error itself. It expresses the propagation relation of dispersion of all possible values of errors , and also the propagation relation of probability intervals between errors. 6. Statistical calcu lation of variance n x x n i i 1 2 2 ) ( ) ( Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 12 6.1 Direct measurement for single measurand A measurand is direct ly measured b y n times, and n observations are obtained. In thi s way, using to represent the best measured value, the err or equations are: 0 0 2 2 0 1 1 y x v y x v y x v n n (6-2) According to the least square method, the final measured value is n i i n x y 1 0 (6-3) The measurement model of this measurement method is Y X V , , and are samples of random variables V , X and Y respectively. Taking the total differential of equation (6-3), the error propagation e quation is n i i n x y 1 Δ Δ (6-4) Now we onl y dis cuss the variances of the error components ) ( X E x i and ) ( - 0 Y E y , and make ) ( X E x x i i Δ and ) ( - 0 Y E y y Δ . Because ) ( X E x x i i Δ is unknown, is a random variable, and has the same sa mple space ) ( X E x i as ) ( X E X , so ) ( X E x x i i Δ can be represen ted by ) ( X E X x Δ , that is, ) ( X E X x x i Δ Δ . Similarly, there is ) ( - Y E Y y Δ . By applying the law of covariance propagation to formula (6-4), there is, n x σ y σ ) ( ) ( 2 2 Δ Δ (6 -5) According to the measurement model Y X V , there is y V Y V E Y V X E X x Δ Δ ) ( ) ( (6 -6) Therefore, according to the definition of varia nce, there is ) ( ) ( ) ( ) ( ] [ ] ) [( ) ( 2 2 2 2 2 2 2 y σ V E y E V E y V E x E x σ Δ Δ Δ Δ Δ (6-7) Substituting n i i v n V E 1 2 2 1 ) ( and formula (6-5) into equation (6-7), we get: n x σ n v x σ n i i ) ( ) ( 2 1 2 2 Δ Δ (6-8) Therefore 1 ) ( 1 2 n v x σ n i i Δ (6-9) Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 13 6.2 Indirect measurement for single measurand Different from the direct measure ment, each observation in the indirec t measurement is the measured data of times of measura nd. The error equations of the repeated measurement are: 0 0 2 2 2 0 1 1 1 y a x v y a x v y a x v n n n (6- 10 ) According to the least square method, the final mea sured value is: n i i n i i i a x a y 1 2 1 0 (6-11) The measurement model is Y a X V i i i , , and are samples of random variables i i X V , and Y respectively. Similarly, for the errors ) ( i i X E X x Δ and ) ( - Y E Y y Δ , the covariance propagation relationship is n i i a x σ y σ 1 2 2 2 ) ( ) ( Δ Δ (6 -12) Similarly, according to the measurement model Y a X V i i i , there is y a V Y a V E Y a V X E X i i i i i i i i Δ ) ( ) ( (6-13) According to the definition of variance, there is n x σ v n y σ n a v n y a V y a V n X E X X E X n x E x σ n i i n i i n i i n n ) ( 1 ) ( 1 ] ) ( ) [( 1 lim } )] ( [ )] ( {[ 1 lim ] [ ) ( 2 1 2 2 1 2 1 2 2 2 2 2 1 1 2 2 2 2 1 1 2 2 Δ Δ Δ Δ Δ Δ (6 -14) Therefore 1 ) ( 1 2 n v x σ n i i Δ (6 -15) 6.3 Indirect measurement for multiple me asurands In this measure ment mode, there are t different measurands, a nd e ach obse rvation value is obtained by measuring the linear superposition value of multi ple measurands. The error equations of the repeated measurement are: Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 14 t nt n n t t n n y y y a a a a a a a a a x x x v v v 2 1 2 1 2 22 21 1 12 11 2 1 2 1 (6- 16 ) That is: AY X V (6-17) According to the principle of the lea st squares, its measured values are: X A A A Y T 1 T (6-18) The measurement model is t it i i i i Y Y Y a a a X V 2 1 2 1 , , and are samples of random variables i i X V , and j Y respectively. The error propagation equation is: X A A A Y T 1 T (6 -19) Similarly, for the errors ) ( i i X E X x Δ and ) ( - j j j Y E Y y Δ , the covariance propagation relationship is: 1 2 ) ( A A Y D T x (6 -20) Similarly, according to the measurement model, there is t it i i i i i y y y a a a V X E X Δ Δ Δ 2 1 2 1 ) ( (6 -21) According to the definition of variance, there is n i t T t n i i n i t it i i i n n y y y y y y n v n y y y a a a V n X E X X E X n x E x σ 1 2 1 2 1 1 2 1 2 2 1 2 1 2 2 2 2 1 1 2 2 1 1 ) ... ( 1 lim } )] ( [ )] ( {[ 1 lim ) ( ) ( Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ A A (6 -22) Omitting the tedious algebraic calculation process, the final result is t n v x n i i 1 2 ) ( (6 -23) that is written as , and represents the dispersion of all possible values of the deviation . Also, the standard deviation or , which is given b y the formula (6-5), (6-12 ) or (6-20 ), is also the evaluation of p robability inte rval of th e deviation or Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 15 . Obviousl y , it is incorrec t to express Bessel's formula as n j j k x x n x σ 1 2 2 1 1 ) ( in the literatures [1,4] , because is a numerical value. For example: the measured frequenc y values of a quartz crystal at different temperatures are shown in Table 4, and the nominal value of frequency is 5.000050MHz. Now, we need to give a temperature correction model for the freque nc y error and evaluate the standard deviation of the residual error after correction. Table 4: Ob served values T emperature ˚ C Fr equency MHz Err or value ) 10 (1 / -6 0 f f R i i -40° 4.999900 -30 -30° 4.999975 -15 -20° 5.000040 -2 -10° 5.000085 7 0° 5.000115 13 10° 5.000110 12 20° 5.000070 4 30° 5.000035 -3 40° 5.000010 -8 50° 4.999995 -11 60° 4.999995 -11 70° 5.000010 -8 80° 5.000045 -1 90° 5.000125 15 100° 5.000235 37 We use the first 4 terms of the Taylor series as the temperature model of the frequenc y error, that is 3 2 dT cT bT a R . In this way, the error equation set is: d c b a T T T T T T T T T R R R v v v n n n n n 3 2 3 2 2 2 2 3 1 2 1 1 2 1 2 1 1 1 1 - According to the least square method, there is 3 2 6 5 4 3 5 4 3 2 4 3 2 3 2 i i i i i i i i i i i i i i i i i i i i i i T R T R T R R d c b a T T T T T T T T T T T T T T T n Substituting the values in Table 4 in to above equation, there are: 42713000 304500 4610 1 000 1983295000 0 2195250000 256870000 292500 0 2195250000 256870000 2925000 41500 256870000 2925000 41500 450 292500 41500 450 15 d c b a Solving the equations, get: . 0 . 0 0 0 2 1 4 , 0 . 0 1 8 6 01 0 .0 13 5 18 , , 1 9. 9 8 3 2 5 d c b a Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 16 Therefore, t s function model is fitted as: 3 2 0 .0 0 02 1 4 0 .0 1 86 0 1 0. 0 1 3 5 1 8 9 .9 8 32 5 1 T T T R Fig5 is the comparison curve between the model and the actual error. According the formula (6-23), the standard deviation of residual error is 6 1 2 10 3 . 2 4 ) ( n v R σ n i i Δ Finally, the frequency of quartz crystal is given as follows: ) 10 1 ( 6 0 R f f That is, temperature- frequency e rror can be corrected b y the measured value s o f temperature sensor, and a more accurate frequency value can be calculated. Residual error (as shown in Fig6), which is still a regular error, is also processed b y statistical rules, and the standard deviation of the residual error is ± 2.3 × 10 -6 . This error processing method has been widely used in the manufacture of photoelectric geodimeter [14,15] . 7. Uncertainty According to Figure 2, the total error of the final measured value is B A (7-1) Where A is the deviation between mea sured value and expectation, and B is the deviation between expectation and true value. Because the two errors are usua lly irrelevant, according the law of covariance propagation (5 -9), there is: ) ( ) ( ) ( 2 2 B A (7-2) -40 -30 -20 -10 0 10 20 30 40 50 -40 -20 0 20 40 60 80 100 -4 -3 -2 -1 0 1 2 3 4 5 -40 -20 0 20 40 60 80 100 T R Figure5.The function model fitting of frequency err or Figure6.The residual error ’ s curve R T Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 17 This tot al standard deviation is the evaluation of probabilit y interval of total error (the dispersion of all possible values of total error ). It can be seen that formula (7-2) is consistent with the traditional uncertainty evalu ation (But th e e xpression is changed fr om to ). Therefore, this total standard deviation is actuall y the uncertaint y , which expresses the probability range o f the total error of final measu red value. A nd t he uncertaint y concept, whi ch is interpreted as the dispersion o f measured v alue (constant) in the tr aditional measurement theory, is also proved to be incorrect. It can be se en from Tabl e 3 that the uncerta inty is also the possible deg ree that the true value deviates from the measured value. That is, the uncertainty is not only th e uncertaint y of the error but also the unc ertainty of the true value, but is not the uncertainty of the measured value. The measured value, which is a numerical value, has no uncertainty. According to the interpretation of the traditional theory, ) ( A and ) ( B are re ferred as the uncertainty of Type A and the uncerta int y of Type B re spectively. However, the current ) ( B is actually the ) ( of historical upstream measurement, and the current ) ( can also be used as the ) ( B in future down stream measurement. T h is kind of int erpretation with A/B classification of the uncertainty evaluation is obviously too rigid. Moreover, the currently widely used formula (7-2) is only applicable to the direct repeated measurement model in section 6.1, but h as no use at all fo r the indirect repeated measurement in se ctions 6.2 and 6.3, because in indirect re peated measurement, there are usually some error sources which not only contribute to dispersion of repeated observations but also c ontribute to their deviation, and it is diff icult to distinguish them with A/B classification method. Therefore, A/B classification method is not universal in practice. Formula (7 - 2) comes from the covariance propagation law (5 -9). Thus, the basic principle of uncertaint y s ynthesis is covariance propagation law (5 - 9) , and the uncertainty synthesis do es not need to apply the interpretation of A/B classification mechanically. Here is a simple example to illustrate this principle, which is also a comparison with the traditional practice. For ex ample, four points A, B, C and D are lo cated on a strai ght li ne (Fig 7), and the observation data of distances obtained b y geodimeter [1 4,15] are shown in Table 5. Please solve the final measured values of each line segment and the uncertainty of eac h error. Table 5: Observed values Line segment Observ ed values 1 2 3 4 5 6 Using 1 y , 2 y and 3 y to express the final measured values of AB, BC and CD respectively, and using k to express the measured value of the additive constant error of geodimeter, the observation error equation is A B C D Figure 7. Distances measurement Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 18 k y y y x x x x x x v v v v v v 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 1 0 1 0 1 0 0 1 (7 -3) According to the least square method, there are: 6 2 1 1 T 3 2 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 x x x k y y y 6 2 1 2 - 0 0 2 2 2 2 1 1 - 1 2 - 1 - 1 1 1 2 - 1 2 - 2 1 - 1 1 - 2 - 1 4 1 x x x (7 -4) Substituting the numerical values of a ll observed values into equation (7-4 ), we get: ) ( 0013 . 0 - 0030 . 320 9583 . 159 8549 . 39 3 2 1 m k y y y (7 -5) For the traditional measurement theor y, the next step is to subst itute (7-5) into (7-3), and six residual are obtained. Then ) ( x σ is obtained by Bessel formula t n v x σ n i i 1 2 ) ( , and ) ( 1 y , ) ( 2 y , ) ( 3 y σ and ) ( k σ are obtained b y covariance propagation law. Finally, ) ( 1 y 、 ) ( 2 y ) ( 3 y σ and ) ( k σ are called as p recision or uncertaint y of T ype A, but the uncertainty of Type B is almost impossible to discuss. However, from the perspective of the new conceptual theory , ther e are three conceptual troubles in the above v ariance submission process : 1. The degree of the fre edom is too small, so it is meaningless to apply Bessel formul a. 2. In Table 5, each observed va lue is a numerical v alue, and acc ording to equation (7 -5), each m easured value is also a nume rical value, so, their variances should be 0. 3. The contribution of the covariance betwee n the err ors of each observation value has not been taken into account at all (uncertainty sy nthesis iss ue). The following is the variance submission process of the new conceptual theory for this case. Taking the total differential of equation (7 -5), the error prop agation equation is obtained as follows: 6 2 1 3 2 1 2 - 0 0 2 2 2 2 1 1 - 1 2 - 1 - 1 1 1 2 - 1 2 - 2 1 - 1 1 - 2 - 1 4 1 x x x k y y y Δ Δ Δ Δ Δ Δ Δ (7 -6) Applying covariance propagation law (5 -9) to equation (7-6), the covariance propagation equation is obtained as follows: Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 19 2 - 2 1 2 0 1 1 1 - 0 1 - 1 1 2 1 2 - 1 - 2 2 - 1 2 - 2 1 - 2 - 1 ) ( 2 - 0 0 2 2 2 2 1 1 - 1 2 - 1 - 1 1 1 2 - 1 2 - 2 1 - 1 1 - 2 - 1 16 1 2 2 2 2 3 2 1 3 3 2 3 1 3 2 3 2 2 1 2 1 3 1 2 1 1 X D Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ k y k y k y k k y y y y y y k y y y y y y k y y y y y y σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ (7 -7) The acquisition process of the covariance matrix is as follows. For the observ ed value , its error is composed of three parts: additive constant error , multiplication constant error and uneven indexing error . That is: i i i c x R K x Δ (7-8) Its variance is 2 2 2 2 2 c R i K xi σ σ x σ σ Δ (7-9) The , and are obtain ed by consulting instrument instructions or the tolerance standard in instrument specification. F urthermore, according to formula (5-2), can be deduced as follows: 6 2 1 6 2 1 ) ( x x x x x x E Δ Δ Δ Δ Δ Δ Δ X D 2 2 2 6 2 2 6 2 2 2 6 1 2 2 2 6 2 2 2 2 2 2 2 2 1 2 2 1 6 2 2 1 2 2 2 2 2 1 2 c R K R K R K R K c R K R K R K R K c R K σ σ x σ σ x x σ σ x x σ σ x x σ σ σ x σ σ x x σ σ x x σ σ x x σ σ σ x σ (7 -10) Finally, the covariance matrix is obtained b y su bstituting the equation (7 -10) into equation (7-7), where is called as uncertainty. Assuming that there are , and , we can get: ) ( 00 . 5 50 . 0 50 . 0 50 . 0 50 . 0 85 . 0 05 . 0 26 . 0 50 . 0 05 . 0 78 . 0 01 . 0 50 . 0 26 . 0 01 . 0 75 . 0 2 2 2 2 2 3 2 1 3 3 2 3 1 3 2 3 2 2 1 2 1 3 1 2 1 1 mm σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ k y k y k y k k y y y y y y k y y y y y y k y y y y y y Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Therefore, the uncertainties are: mm σ mm σ mm σ mm σ k y y y 2 . 2 9 . 0 9 . 0 9 . 0 3 2 1 Δ Δ Δ Δ It can be seen that ther e is . Besides, it can be seen that the indi cation errors not only lead to the dispersion of repeated observations , but also lead to their overall deviation. If we e ntangle in the influence c haracteristics (A/B c lassification) of indication error on repeated observations, i t will not onl y be unable to express, but also will not help to solve the problem. Moreover, whether errors or error are all deviations, a nd have variances used to evaluate their probability intervals, while the s ystematic error without variance does not exist. 8. Conclusions Discrete Dynamics in Nature and Society https://doi.org/ 10.1155/2 020/386 4578 20 Table 6: Concep tual logic difference bet ween the two theories . References [1] JCGM 100 :2008, Guide to the Expression of U ncertainty in Measure ment (GUM). [2] JJF1059 -2012, Evaluation and Expressio n of Uncertainty in Meas urement. [3] JCGM 200:2 012, International vocabulary of metrolog y Basic and general co ncepts and associated terms (VIM). [4] JJF1001 -2011, General Ter ms in Metrology and Th eir Definitions. [5] GB /T14911- 2008 , Basic Terms of Surveying and Mapping . [6] Huang He ning. A uni fied theory o f measurement errors and uncertai nties. Measure ment Science and Technology, 2018 , 29(12). [7 ] Xiao-ming Y e, Mo Ling, Qian g Z hou, Wei-nong W ang, Xue-bin Xiao . 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