A solution is proposed for the problem of statistical analysis of a sum of random measurement errors or the

result of indirect measurements based on the Monte-Carlo method using an algorithm that generates

correlated samples of random quantities with various distributions.

Fairly often measurement practice and instrument design require evaluation of a resultant error from known esti-

mates of its components: additive summation of random measurement errors or finding indirect-measurement errors, when it

is also necessary to take into account the transformation of the distribution when random quantities are multiplied and divid-

ed. As has been pointed out [1], “error summation is one of the main problems in both instrument design and evaluation of

the errors of the measurements themselves.”

There exist several approaches to the solution of this problem. One approach is based on the description of each

error component by its corresponding distribution and the analytic construction of the resultant multivariate distribution,

which describes their combined effect. This approach provides the most complete and accurate result. In this formulation,

however, the problem of error summation of transformation becomes practically insoluble for as few as three or four com-

ponents (not to mention the tens of components in the analysis of complex devices), since operations with multivariate dis-

tributions are very complicated.

A second approach involves the description of each error component by some numerical estimate: the standard devi-

ation (SD) σ, the entropy error γe, or the confidence error γc. Summation of these estimates by particular rules provides a

similar numerical estimate for the resultant additive error. This approach requires allowance for the deformation of the dis-

tributions of the random quantities when they are summed, and also for the correlations between individual error components.

These difficulties make it necessary to employ approximate formulas or graphs, which can sometimes greatly affect the accu-

racy of the result.

In a third approach, the resultant univariate distribution is determined by statistical modeling. Each component of

the resultant additive error (or the results of direct measurements in indirect calculations) is represented by a corresponding

distribution and coefficients of correlation with the other components. Modeling yields a univariate distribution that can be

identified by an analytic expression or its corresponding numerical estimate can be calculated without constraints or simpli-

fications. The algorithm is the basis of a program developed by the authors, which can be represented as follows.

1. Theoretical or empirical determination of the probability density function p(x) of the individual error components

or the results of direct measurements and their analytic approximation in the class of distributions most often used.This

problem can be solved by various software products for statistical analysis of data [3, 4]. However, these products have lim-

ited sets of distributions (usually 10–15, or a maximum of 26 in [4]), which does not always permit a good analytic approx-

imation for the experimental data. We therefore developed a program for identification of the histogram of a random quan-

tity [2]. The program seeks the best description of the histogram by an analytic formula from 30 classes of symmetric and

asymmetric unimodal distributions that include practically all unimodal distributions known in the literature. The conver-

Measurement Techniques, Vol. 43, No. 11, 2000

SUMMATION OF RANDOM MEASUREMENT ERRORS

AND ANALYSIS OF INDIRECT-MEASUREMENT

ERRORS BY MONTE-CARLO METHOD

S. A. Labutin and M. V. Pugin UDC 621.317.08+519.22

Translated from Izmeritel’naya Tekhnika, No. 11, pp. 6–9, November, 2000. Original article submitted January 12, 2000.

0543-1972/00/4311-0918$25.00 ©2000 Plenum Publishing Corporation918

gence of the empirical and theoretical distributions is evaluated quantitatively by the distance between these functions using

Pearson’s test (for the probability density function) or Kolmogorov’s test (for the distribution function).

2. Experimental determination of the coefficients of correlation between individual components of the resultant

error or the results of direct measurements.For random errors x and y with different distribution functions F1(x) and F2(y),

mathematical expectations mx and my, and standard deviations σx and σy, it is advisable to move to random quantities

and uniformly distributed on the interval [0; 1] and then determine the coefficient of

linear correlation by the formula [3]

Determination of the correlation coefficient in this way simplifies the problem of generating correlated random quantities

with different (in general) distributions and eliminates the dependence of p on the form of the distributions.

If the correlation coefficient for two samples of random quantities x and y with different distributions was determined

in the usual way [3], the program calculates the corresponding correlation coefficient after the random quantities x and y have

been converted to uniformly distributed random quantities Z1 and Z2.

3. Generation of samples of random numbers with given distributions and correlation coefficients. This problem is

solved by a software-implemented inversion method [5] that produces random numbers Zi that are uniformly distributed on

the interval [0; 1] and then converts the Zi values by the formula xi = F –1(Zi), where F –1(Z) is the inverse of the distribution

function of random quantity xi.

4. Element-by-element algebraic summation of the samples for additive summation of the errors or nonlinear con-

version of the elements of the samples in the case of indirect measurements to form a resultant sample.

5. Identification of the univariate distribution of the total error and evaluation of the parameters of that distribu-

tion. In particular, determination of the entropy coefficient K and SD σ of the resultant sample for calculation of the entropy

value γe = Kσ or confidence value γc = tσ of the resultant error, where t is the quantile of the resultant distribution for a given

fiducial probability [1].

The principal difficulty is that of generating samples of random quantities with arbitrary distributions and given

coefficients of correlation between those random quantities. Let us examine the sequence in which a pair of correlated ran-

dom numbers with different distributions is obtained. Three uncorrelated random numbers A, B, and C with uniform distri-

butions on the interval [0; 1] are generated in the first step. And from them a pair of correlated numbers Y1 and Y2 are gen-

erated by the following formulas:

(1)

For expressions (1),it can be rigorously demonstrated that ρ* is the coefficient of correlation between random quan-

tities Y1 and Y2 if ρ* is defined in the usual way [3]. In general, numbers Y1 and Y2 have symmetrical trapezoidal distribu-

tions with large and small bases. Then the trapezoidal distribution of numbers

Y1 and Y2 is converted to a uniform distribution on the interval [0; 1] by the following formulas:

If Yi > S1/2, then for p = S2 – Yi

Z

p

S Sp

S S

S S p

S Sp

S Si =

−−

− >

−− +

+− ≤

12

2 2

14

2 2 2

2

12

22

1 2

2 1

1 2

1 2

, ,

( ), ;

S SS2 1

11

2= − ρ*S1 1= + −ρ ρ* *

Y A B Y A C1 21 1= + − = + −ρ ρ ρ ρ* * * * .and

ρσ σ

=−

−

−

=

−− −

− −

= =∑ ∑12

1

1

2

1

2

12

1

1 2 1 21 2

11 2

1N

Z ZN

Fx m

Fy m

i ii

Ni x

x

i y

yi

N/ /

.

Z Fx my

y2 2=

−

σ

Z Fx mx

x1 1=

−

σ

919

and if Yi ≤ S1/2, then for p = Yi

As a result,we obtain random numbers Z1 and Z2, which are uniformly distributed on the interval [0; 1] and have,

as a computer check shows,a correlation coefficient ρ that is related to ρ * as

ρ * = ρ + 0.005086 + 0.01739sin(6.3986ρ + 5.9575).

Thus,the value of ρ * calculated by the above relation must be specified when random numbers Y1 and Y2 are

formed to find the required value of the correlation coefficient ρ for random quantities Z1 and Z2. The absolute systematic

error of ρ in this method of obtaining two correlated samples does not exceed 0.0025.

Random numbers with a given distribution are obtained by the inversion method, in which a uniform distribution is

converted to the given distribution by the formula: q = F–1(Z) or q = F–1(1 – Z) (to obtain a correlation coefficient equal to

–ρ ), where F–1 = x(F) is the inverse of the distribution function,whose discrete values are found by numerical solution of

the equation for F ∈ [0; 1]. When the two pairs of parameters A and B and A and C are transposed in for-

mulas (1),random quantities Z1 and Z2 will have a correlation coefficient of 1 – ρ , which makes it possible to form sam-

ples of two random quantities with correlation coefficients of ±(1 – ρ ) and arbitrary distributions.

Thus,the errors to be summed can be divided into uncorrelated groups of three types:those that include any num-

ber of uncorrelated error; strictly correlated errors with ρ = ±1 with respect to one error of that group that is taken as a base;

and errors with pair correlation coefficients with respect to a single base error of ±ρ and ±(1 – ρ ).

It should be noted that there can be several groups of the second and third types,and ρ can have an arbitrary value

in each of them. If the number of errors in a group is greater than 2,there are objective constraints on the cross-correlation

p z dz F

x

( )

−∞∫ =

Z

p

S Sp

S S

S S p

S Sp

S Si =

−− >

− ++

− ≤

2

2 2

4

2 2 2

2

12

22

1 2

2 1

1 2

1 2

, ,

( ), .

920

0 0.5 1.0ρ

1.075

0.975

1.025

0.925

0.875

γe, %

Fig. 1. Estimate of total entropy error γe for end of measurement

range versus correlation coefficient ρ for two errors with triangular

probability density functions.

coefficients. For example, ρ12 = 1 and ρ13 = 1 cannot be specified simultaneously if the correlation coefficient for the sec-

ond and third random quantities ρ23 ≠ 1.

After correlated samples of random quantities,statistical modeling is used to construct a random sample of values

of the resultant measurement error or the results of indirect measurement with a size of from 104 to 5·105 values. In partic-

ular, the relative error of SD calculation can be evaluated by the formula [1] , where n is the sample size

and ε is the kurtosis of the distribution. It follows from this formula that the error δσ = 0.007 for n = 104 and ε = 3 (normal

distribution),while δσ = 0.001 for 5·105. The obtained sample of values of the resultant error or indirect-measurement result

makes it possible to find numerical estimates that characterize the distribution (mathematical expectation, standard deviation,

asymmetry factor, kurtosis,and entropy coefficients),construct a histogram,and find the best analytic description of the prob-

ability density function or distribution (according to Pearson’s or Kolmogorov’s test,respectively). In conclusion,we calcu-

late the confidence γc or entropy γe value of the error.

The program was tested in a calculation of the resultant error of a measurement channel [1]. Eight random-error

components were taken into account:four with uniform probability density functions; two triangular; one normal; and one

arcsine. The correlation between the two errors with triangular probability density functions (ρ = 1) and between one of

the pairs of errors with uniform probability density functions (ρ = –1) was taken into account. The remaining errors were

uncorrelated (ρ = 0).

Note that the analytic procedure for calculation of γe or γc presented in [1] does not permit solution of the problem

in question when errors with different distributions are correlated or ρ ≠ 0 or ±1. The problem of determining the entropy

coefficient of a composition of uncorrelated errors from the entropy coefficients and relative weights of the variances of each

of them in the total variance is today solved analytically or numerically only for certain cases of summation of pairs of ran-

dom quantities with normal,uniform, exponential,arcsine, or similar distributions. It should also be noted that the given ana-

lytic approach is poorly suited for determination of the confidence error γc, since in general numerical estimates of the resul-

tant distribution are unknown (except for the standard deviation and kurtosis),and in some cases even the shape of the dis-

tribution can be found only approximately. All of this greatly limits the application of the analytic approach to calculation of

γe or γc recommended in [1].

As a result of numerical calculations by statistical modeling using the described program,we obtained entropy-error

estimates γe ≈ 0.489 and 1.069%,respectively, for the beginning and end of the measurement range of the channel with an

analog recorder, which practically coincide with the values γe ≈ 0.475 and 1.066% given in [1]. However, the program per-

mits not only more accurate estimation of the entropy error γe and the entropy coefficient of the distribution of the resultant

error for arbitrary values of the correlation coefficient ρ for two errors with triangular probability functions (Fig. 1),but also

determination of its distribution,as well as numerical estimates of other parameters of that distribution. When a digital volt-

meter is used instead of an analog recorder, the parameter γe takes a value of 0.644% at the beginning of the range, which

practically coincides with the value of 0.61% given in [1]; for the end of the range, these values are 1.208 and 1.16%,respec-

tively. The size of the total-error sample n = 5·105; the calculation time for one value of γe on a computer with an Intel P-166

MMX processor is 8 sec.

The possibilities of the program are retained in analysis of a resultant distribution and evaluation of its parameters

for indirect measurements. An example of calculation of the heat-transfer coefficient of the surface of the cylinder of a pis-

ton machine from the results of direct measurements of three temperatures was examined in [1]; values of γe = 4.6,5.7,and

14% were found by the approximate analytic method.

The difference from the problem of additive summation of measurement errors lies only in the formula for the resul-

tant random quantity, which in indirect measurements can also include the operations of multiplication and division of ran-

dom quantities. The probability density function of a random quantity can be described by nonunimodal functions or uni-

modal functions that have statistical moments above the zeroth order, when it is impossible to use the authors’ program for

histogram identification. However, modeling yields a random sample of values of indirect-measurement results that can be

analyzed by other programs for statistical data processing [3,4].

Conclusion. The proposed approach to solution of problems of summation of random errors and analysis of the

errors of indirect measurements is fairly universal in comparison with the analytic method [1],is more accurate and infor-

δσ ε= −1 2/( )n

921

mative, and when implemented as a finished program requires minimal computation time, which is especially important in

the optimization of measurement apparatus (including SAPR).

REFERENCES

1. P. V. Novitskii and I. A. Zograf, Evaluation of Measurement-Result Errors [in Russian],Énergoizdat, Leningrad

(1991).

2. S. A. Labutin and M. V. Pugin, in: Abstracts of Proceedings of Third All-Russia Scientific-Technical Conference

“Methods and Instruments for Measurement of Physical Quantities” [in Russian],Part 10, Nizhnyi Novgorod

(1998),p. 3.

3. Yu. N. Tyurin and A. A. Makarov, Statistical Analysis of Data by Computer[in Russian],IFNRA-M, Moscow

(1998).

4. B. Yu. Lemeshko, Statistical Analysis of Univariate Observations of Random Quantities. Program System[in

Russian],NGU, Novosibirsk (1995).

5. B. Ya. Sovetov and S. A. Yakovlev, System Modeling[in Russian],Vysshaya Shkola,Moscow (1998).

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