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21APPLICATION OF THE LEAST SQUARE METHOD FOR EVALUATION OF UNCERTAINTY OF
MEASUREMENTS
Serikova A.,
2nd year master's student of the educational program "Automation and Control" Almaty University of Power Engineering and Telecommunications named after G.J. Daukeev Almaty, Republic of Kazakhstan Khan S.
PhD, professor
Almaty University of Power Engineering and Telecommunications named after G.J. Daukeev Almaty, Republic of Kazakhstan https://doi.org/10.5281/zenodo.6594891
Abstract
The article studies the least squares method and its application for estimating measurement uncertainty. The article shows how the parameters obtained by fitting, the intercept and the slope of the linear dependence, together with estimates of their variances and covariance, can be used to determine the correction values and its standard uncertainty from the calibration characteristic.
Keywords: measurement uncertainty, least squares method, GUM method, Monte-Carlo method, Kragten method, guidance on expressing measurement uncertainty, combined standard uncertainty, expanded uncertainty.
Introduction. When processing the results of measurements obtained during international comparisons of standards, testing, calibration or verification of measuring instruments for foreign countries, in the study of primary national standards, measurement uncertainty is used as a characteristic of the quality of measurements. In this regard, the task of ensuring international unity in the approach to presenting and estimating the uncertainty of the measurement result is relevant.
Currently, metrological services require a transition from error assessment to measurement uncertainty assessment. A review of various methods for estimating measurement uncertainty has shown that the following methods are available:
1) classical GUM method;
2) Kragten method (spreadsheet method);
3) Monte-Carlo method;
4) least squares method (LSM).
Analytical methods are widely used to estimate measurement uncertainty [1]. Their implementation is based on the law of propagation of uncertainty, which consists in approximating the original model equation by linear terms of the Taylor series, which, in combination with the method of summing variances and covar-iances, makes it possible to obtain an expression for calculating the combined standard uncertainty (classical GUM method).
The Kragten method is a spreadsheet method [2] that can be used to simplify calculations when finding measurement uncertainty using the GUM method. The automated program for calculating measurement uncertainty by the Kragten method developed by the authors is described in the article [3].
The analysis of uncertainty estimation by the classical GUM method shown in the Guide to the Expression of Uncertainty in Measurement [1] is not exhaustive. There are many situations, sometimes quite complex, requiring the use of different statistical methods.
The most universal numerical method is the method of statistical modeling (Monte Carlo) [4],
which is based on generating input values in the form of random numbers with a given distribution law and finding the distribution law of the measured (output) value for the corresponding set of obtained random numbers.
In relatively simple measurement problems, uncertainty components can often be estimated by means of dispersion analysis of the results of hierarchical experiments for a given number of hierarchy levels [5]. Sample estimates of variances and standard deviations of functional dependence parameters, as well as the values predicted from this functional dependence, can be easily calculated using well-known statistical procedures. Therefore, to obtain an estimate Xi of the input value Xh you can use the functional dependence obtained from the experimental data by the least squares method.
Research method. One of the main mathematical methods for processing experimental data is the least squares method. The numerical parameters are selected from the condition of the minimum sum of the squared deviations of the measured values from the calculated values of the approximating function. The meaning of this condition is that the reliability of the approximation results obtained using this condition is the higher, the smaller the sum of the squared deviations.
The example below illustrates the application of the least squares method to construct a linear calibration characteristic for thermometer calibration. The example shows how the parameters obtained by fitting, the intercept and the slope of the linear dependence, together with estimates of their variances and covariance, can be used to determine the correction values and its standard uncertainty from the calibration characteristic [5].
Experiment. The thermometer is calibrated by comparing n = 11 temperature readings tk of the thermometer, having negligible uncertainty, with the corresponding reference temperature values tR k in the range from 21°C to 27°C to obtain correction values bk = tR k - tk to the readings. The measured corrections and
the measured temperatures tk are the input values for the evaluation. The linear calibration characteristic is adjusted to the measurement data of corrections and temperatures using the least squares method:
b(t)=yi+y2(t-t0),
(1)
The two measured (output) values are the parameters yi and y2 - respectively the free term and the slope of the calibration characteristic. The temperature t0 is chosen by agreement as some fixed point, so it is not among the independent parameters to be determined by the least squares method. Once estimates of yt and y2 of their variance and covariance have been obtained, formula (1) can be used to calculate the correction to be applied to the temperature reading t of the thermometer and its standard uncertainty.
Fitting by the least squares method. In view of the above, estimates of the output values y1 and y2, their variances and covariances by the least squares method are obtained by minimizing the sum:
S = ^[bk -yi-y2(tk - to)]2-
(2)
k = 1
which leads to the following formulas for y1 and y2 , their sample variances s2(y1) and s2(y2) and the sample correlation coefficient r(y1,y2) = s(y1,y2)/ s(y1)s(y2), where is the sample covariance:
yi
(Zbk)(Z92k)-(Zbk 9k)(£9k) D '
J 2 =■
nZbk9k-(Zbk)(Z9k)
s2(yi) =
D
s2Z92k D '
(3)
(4)
(5)
correction b(tk) calculated for temperature according to the calibration characteristic (t) = y1 + y2(t — t0). The sample variance s2 is a general measure of the uncertainty of fitting the calibration characteristic to experimental data, and the coefficient l/(n- 2) reflects the fact that, since estimates of two parameters y1 and y2 are obtained based on n observations, the number of degrees of freedom for estimating s2 will be v = n — 2.
If an estimate of one single parameter, the arithmetic mean, is obtained from n independent observations, then the number of degrees of freedom v will be equal to n — l. If n independent observations are used to derive the intercept and slope estimates in the least squares straight line equation, then the number of degrees of freedom to determine the sample standard deviations of these estimates will be v = n — 2. When calculating the least squares method of m curve parameters from n experimental points, the number of degrees of freedom to determine the sample standard deviation of the estimate of each parameter will be v = n
m [3].
Results. The data on which the adjustment is carried out are presented in the second and third columns of Table 1. As a fixed point t0, t0 = 20 °C is taken. Then from formulas (3)-(9) we obtain: y1 = —01712 °C; y2 = —0,00218; s^) = 0,0029 °C; s(y2) = 0,00067;
s = 0,0035 °C; r(yi,y2) = —0,930
The fact that the angular coefficient y2 is more than three times greater than its standard uncertainty indicates the need to use a calibration characteristic, and not a fixed correction, common for the entire temperature range.
After obtaining numerical estimates, the calibration characteristic can be written as:
n
S2(y2) = n — '
r(yi,y2) = -
D Z8k
YXbk-b(tk)}2
n-2
= n^92-(^9k)
= nYj(9k-S)2
= nYj(tk-t)2'
(6)
(7)
(8)
(9)
b = -0,1712(29)°C
+ 0,0021Q(67)(t - 20°C),
(10)
where the numbers in brackets correspond to the least significant digits of the estimates of the free term and the slope of the calibration characteristic and show the numerical values of the standard uncertainties of these parameters. Formula (10) allows you to calculate the correction to the thermometer readings for any temperature t, including the values b(tk) for t = tk. Corrections b(tk) are indicated in the fourth column of Table 1, and in its last column the differences between the calculated and measured values of the correction bk — b(tk) are given.
2
2
s
In the above formulas, summation is carried out over k from 1 to n;9k = tk = t0; 9 = t =
n n
The expression [bk — b(tk)] is the difference between the correction bk measured at temperature tk and the
The analysis of these differences can be used to check the validity of the choice of a linear model as a calibration characteristic using well-known hypotheses testing procedures.
Table 1
Data used to obtain a linear calibration characteristic of a thermometer by the least squares method
Indication Thermometer Measured correction Estimated Difference between measured and
number readings bk = ÍR.k — tk, °C correction calculated corrections
k tk,°C b(tk),°C bk -b(tk),°c.
1 21,521 -0,171 -0,1679 -0,0031
2 22,012 -0,169 -0,1668 -0,0022
3 22,512 -0,166 -0,1657 -0,0003
4 23,003 -0,159 -0,1646 +0,0056
5 23,507 -0,164 -0,1635 -0,0005
6 23,999 -0,165 -0,1625 -0,0025
7 24,513 -0,156 -0,1614 +0,0054
8 25,002 -0,157 -0,1603 +0,0033
9 25,503 -0,159 -0,1592 +0,0002
10 26,010 -0,161 -0,1581 -0,0029
11 26,511 -0,160 -0,1570 -0,0030
Uncertainty of the calculated correction. The expression for the combined standard uncertainty of the calculated correction is obtained by taking the functional dependence from formula (1), b(t) = f(y-i, y2) and taking u(y±) = s(y±) u u(y2) = s(y2):
u2c[b(t)]
= u2(y1) + (t1-t0)2u2(y2) (11)
+ 2(t- t0)u(y1)u(yc)r(yi,yc).
The variance estimate uc [¿(t)] has a minimum at the temperature tmin = t0- u(yi)r(yi,yc)/u(yc), which in this case is tmin = 24,0085 °C.
As an example of the use of formula (11), suppose that it is necessary to find a correction to the readings of a thermometer and its uncertainty at a temperature t = 30 °C, which is outside the calibration range. Substituting t = 30 °C into formula (10) gives:
b(30 °C) = -0,1494 °C,
and formula (11) after substituting the same value takes the form
uc[b(30°C)] = (0,0029°C)c + (10°C)c(0,00067)c +
+2(10°C)(0,0029°C)(0,0067)(-0,930) = 17,1 • 10-6 °Cc
or
uc[b(30°C)] = 0,0041 °C.
Thus, the correction at 30 °C is - 0,1494 °C with a combined standard uncertainty uc = 0,0041°C for v = n - 2 = 9 degrees of freedom.
Assume that the output is normally distributed and estimate the expanded uncertainty with probability P=95%, for which the coverage factor is approximately k=2. Then the expanded uncertainty is:
U = 2 • uc = 2 • 0,0041 = 0,0082°C « 0,008°C.
Measurement result:
t= (30 ± 0,008)°° k=2,P = 95%
Conclusion. The least squares method is used for calibration purposes to assess the uncertainties associated with short-term and long-term random changes in the results of comparisons of material standards with unknown sizes of units of quantities (for example, gage blocks, mass standards) with comparison standards with known transmitted sizes of units of quantities.
In the course of the results of the measurement uncertainty estimation by the least squares method, it was concluded that the least squares method is a practical alternative to the GUM uncertainty estimation method, in many cases it is easier to use and also provides a high reliability of measurement uncertainty estimation.
The combination of the classical GUM method, the Kragten method, the Monte Carlo method, and the least squares method is useful for developing an appropriate strategy, since each of the four approaches illuminates different sides of the problem.
REFERENCES:
1. Guide to the Expression of Uncertainty in Measurement: First edition. ISO, Geneva, 1993.
2. EURACHEM/CITAC Guide Quantifying Uncertainty in Analytical Measurements, Third Edition, 2012.
3. Khan S.G., Serikova A.A. Development of an automated program for calculating measurement uncertainty by the Kragten method, "The Europe and the Turkic World: Science, Engineering and Technology": Materials of the VII International Scientific-Practical Conference. In three volumes. Volume I - Mersin, Turkey: Regional Academy of Management, 2022. - 363 p.
4. Cox M., Harris P., Siebert B.R.-L. Estimation of measurement uncertainty based on the transformation of distributions using Monte Carlo simulation // Izmeritelnaya tekhnika. - 2003. - No 9. - P. 9-14.
5. ISO/IEC Guide 98-3:2008 «Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995)»
