6. Monte Carlo simulation applied to metrology

This section presents the limitations of the GUM and shows an alternative methodology based on the propagation of distributions that overcome those limitations. For further details, please refer to the authors' publication that addresses the use of the Monte Carlo methodology applied to uncertainty in measurement [15] or to the JCGM 101:2008 guide [5]. Also, in the field of analytical chemistry, the latest version of EURACHEM/CITAC guide (2012) was updated with procedures to use Monte Carlo simulations [13].

#### 6.1. Limitations of the GUM approach

uyo <sup>¼</sup> Syo ffiffiffiffi

where Syo is the standard deviation of the observations of yo, and Eq. (32) is then expressed as

The methodology presented in the GUM can also be used to evaluate the uncertainty in the calibration of a measuring instrument. Following the steps of the GUM, the measurand for the model used in the calibration must be defined by the quantity that evaluates the systematic error of an instrument over its entire measurement range. Thus, Eq. (36) can be generally used

where e is the systematic error of the instrument for a fixed range, Vind is the value indicated by

From Eq. (36), a basic cause-and-effect diagram can be assembled for the calibration uncer-

The sources of uncertainty in this case involve the repeatability of indicated values, the resolution of the instrument in calibration, and the certificate of calibration of the reference values. Thus, an evaluation of the uncertainty about the systematic error should be done for each nominal value of the instrument in calibration. The combined standard uncertainties uei for

the instrument, and Vref is the reference value corresponding to the indicated value.

each calibrated nominal value are obtained by applying the LPU, as shown in Eq. (37)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>e</sup> yo � <sup>y</sup> � �<sup>2</sup> <sup>b</sup><sup>2</sup> <sup>P</sup>ð Þ xi � <sup>x</sup>

2 vuut (35)

e ¼ Vind � Vref (36)

uxo <sup>¼</sup> <sup>1</sup> b

5. Assessment of uncertainty in instrument calibration

for the evaluation of uncertainty in a calibration process:

tainty assessment of an instrument, as shown in Figure 6.

Figure 6. A general cause-and-effect diagram for the calibration of an instrument.

S2 yo m þ S2 e n þ S2

Eq. (35) [18, 19]:

22 Metrology

<sup>m</sup><sup>p</sup> (34)

As mentioned earlier, the approach to evaluate measurement uncertainties using the LPU as presented by the GUM is based on some approximations that are not valid for every measurement model [5, 20–22]. These approximations comprise (1) the linearization of the measurement model made by the truncation of the Taylor series, (2) the use of a t-distribution as the distribution for the measurand, and (3) the calculation of an effective degrees of freedom for the measurement model based on the Welch-Satterthwaite formula, which is still an unsolved problem [23]. Moreover, the GUM approach usually presents deviated results when one or more of the input uncertainties are relatively much larger than others, or when they have the same order of magnitude than its quantity.

The limitations and approximations of the LPU are overcome when using a methodology that relies on the propagation of distributions. This methodology carries more information than the simple propagation of uncertainties and generally provides results closer to reality. It is


Table 3. A typical format for the result of calibration of an instrument.

described in detail by the JCGM 101:2008 guide (Evaluation of measurement data—Supplement 1 to the "Guide to the expression of uncertainty in measurement"—propagation of distributions using a Monte Carlo method) [5], providing basic guidelines for using Monte Carlo numerical simulations for the propagation of distributions in metrology. This method provides reliable results for a wider range of measurement models as compared to the GUM approach and is presented as a fast and robust alternative method for cases where the GUM approach does not present good results.

#### 6.2. Running Monte Carlo simulations

The propagation of distributions as presented by the JCGM 101:2008 involves the convolution of the probability distributions for the input sources of uncertainty through the measurement model to generate a distribution for the output (the measurand). In this process, no information is lost due to approximations, and the result is much more consistent with reality.

Table 4 resumes the input information for the simulation, which was executed for

Table 4. A summary of sources of uncertainty and their associated distributions for the measurement of torque.

Mass (repeatability) A t-distribution Mean: 35.7653 kg; scale: 9.49 x 10�<sup>5</sup> kg; degrees of freedom: 9

; standard deviation: 0.00001 m/s2

Methods for Evaluation of Measurement Uncertainty http://dx.doi.org/10.5772/intechopen.74873 25

Table 5 summarizes the statistical data of the output distribution, including the upper and

M ¼ 200; 000 trials, generating the output distribution shown in Figure 7.

Statistical data Value (N m) Mean 667.970 Standard deviation 0.096 Lower limit for p = 95% 667.812 Upper limit for p = 95% 668.129

ment of torque.

Table 5. A summary of the statistical data for the output distribution for the measurement of torque.

Figure 7. Output distribution resulting from the Monte Carlo simulation for the evaluation of uncertainty of measure-

Mass (certificate) B Normal Mean: 0 kg; standard deviation: 0.00005 kg

Arm length B Uniform Minimum: 1999.5 mm; maximum: 2000.5 mm

Uncertainty source Type PDF PDF parameters

Local gravity B Normal Mean: 9.80665 m/s<sup>2</sup>

lower limits of a probabilistically symmetric range for a 95% coverage probability.

The main steps of this methodology are similar to those presented in the GUM. The measurand must be defined as a function of the input quantities through a model. Then, for each input, a probability density function (PDF) must be assigned. In this step, the concept of maximum entropy used in the Bayesian statistics should be used to assign a PDF that does not contain more information than that which is known by the analyst. A number of Monte Carlo trials are then chosen and the simulation can be set to run.

Results are expressed in terms of the average value for the output PDF, its standard deviation, and the end points that cover a chosen probability p.

Example: Returning once more to the torque measurement example, one can consider the following PDFs for the input sources:

Mass (m). For repeated indications, the JCGM 101:2008 suggests the use of a scaled and shifted t-distribution. Thus, the distribution should use 35.7653 kg as its average, a scale value of s= ffiffiffi <sup>n</sup> <sup>p</sup> <sup>¼</sup> <sup>0</sup>:3 g<sup>=</sup> ffiffiffiffiffi <sup>10</sup> <sup>p</sup> <sup>¼</sup> <sup>9</sup>:<sup>49</sup> � <sup>10</sup>�<sup>5</sup> kg, and <sup>n</sup> � <sup>1</sup> <sup>¼</sup> 9 degrees of freedom.

For the calibration component, the supplement 1 recommends the use of a normal distribution if the number of degrees of freedom is not available. In this case, the mass value of 35.7653 kg is taken as the mean and a standard deviation of Um=k ¼ 0:1 g=2 ¼ 0:00005 kg should be used. However, to facilitate the calculation of the final mean value of the measurand, the mean should be shifted to zero, since both values for the mass sources will be added together.

Local gravity acceleration (g). This case is similar to the case of the balance certificate, for which we have values of expanded uncertainty and coverage factor without information on the number of effective degrees of freedom. Thus, a normal distribution with a mean of 9.80665 m/s<sup>2</sup> and a standard deviation of Ug=<sup>k</sup> <sup>¼</sup> ð Þ <sup>0</sup>:00002 m=s2 <sup>=</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:00001 m/s2 are assumed.

Length of the arm (L). In this case, as poor information about the interval is available (�0.5 mm), an uniform distribution is assumed with a minimum value of 1999.5 mm and a maximum value of 2000.5 mm.


described in detail by the JCGM 101:2008 guide (Evaluation of measurement data—Supplement 1 to the "Guide to the expression of uncertainty in measurement"—propagation of distributions using a Monte Carlo method) [5], providing basic guidelines for using Monte Carlo numerical simulations for the propagation of distributions in metrology. This method provides reliable results for a wider range of measurement models as compared to the GUM approach and is presented as a fast and robust alternative method for cases where the GUM

The propagation of distributions as presented by the JCGM 101:2008 involves the convolution of the probability distributions for the input sources of uncertainty through the measurement model to generate a distribution for the output (the measurand). In this process, no informa-

The main steps of this methodology are similar to those presented in the GUM. The measurand must be defined as a function of the input quantities through a model. Then, for each input, a probability density function (PDF) must be assigned. In this step, the concept of maximum entropy used in the Bayesian statistics should be used to assign a PDF that does not contain more information than that which is known by the analyst. A number of Monte Carlo trials are

Results are expressed in terms of the average value for the output PDF, its standard deviation,

Example: Returning once more to the torque measurement example, one can consider the

Mass (m). For repeated indications, the JCGM 101:2008 suggests the use of a scaled and shifted t-distribution. Thus, the distribution should use 35.7653 kg as its average, a scale value of

For the calibration component, the supplement 1 recommends the use of a normal distribution if the number of degrees of freedom is not available. In this case, the mass value of 35.7653 kg is taken as the mean and a standard deviation of Um=k ¼ 0:1 g=2 ¼ 0:00005 kg should be used. However, to facilitate the calculation of the final mean value of the measurand, the mean should be shifted to zero, since both values for the mass sources will be added together.

Local gravity acceleration (g). This case is similar to the case of the balance certificate, for which we have values of expanded uncertainty and coverage factor without information on the number of effective degrees of freedom. Thus, a normal distribution with a mean of 9.80665 m/s<sup>2</sup> and a standard deviation of Ug=<sup>k</sup> <sup>¼</sup> ð Þ <sup>0</sup>:00002 m=s2 <sup>=</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>:00001 m/s2 are

Length of the arm (L). In this case, as poor information about the interval is available (�0.5 mm), an uniform distribution is assumed with a minimum value of 1999.5 mm and a

<sup>p</sup> <sup>¼</sup> <sup>9</sup>:<sup>49</sup> � <sup>10</sup>�<sup>5</sup> kg, and <sup>n</sup> � <sup>1</sup> <sup>¼</sup> 9 degrees of freedom.

tion is lost due to approximations, and the result is much more consistent with reality.

approach does not present good results.

6.2. Running Monte Carlo simulations

then chosen and the simulation can be set to run.

following PDFs for the input sources:

10

maximum value of 2000.5 mm.

s= ffiffiffi

24 Metrology

assumed.

<sup>n</sup> <sup>p</sup> <sup>¼</sup> <sup>0</sup>:3 g<sup>=</sup> ffiffiffiffiffi

and the end points that cover a chosen probability p.

Table 4. A summary of sources of uncertainty and their associated distributions for the measurement of torque.

Table 4 resumes the input information for the simulation, which was executed for M ¼ 200; 000 trials, generating the output distribution shown in Figure 7.

Table 5 summarizes the statistical data of the output distribution, including the upper and lower limits of a probabilistically symmetric range for a 95% coverage probability.

Figure 7. Output distribution resulting from the Monte Carlo simulation for the evaluation of uncertainty of measurement of torque.


Table 5. A summary of the statistical data for the output distribution for the measurement of torque.
