**6. Using rank measure in metrology**

with the experimental data is counted in this series of numerical experiments, formally

for other values of the same parameter, we obtain the required estimate, which looks like a density of probability distribution. If it is required to evaluate number of the factors more than have been considered, then they are evaluated by the same algorithm by performing similar operations.

When using MCD, the estimation algorithm solves the deconvolution problem in the general formulation *M*(*u*(*x*)) → *Pr*= = *D* → *u*(*x*). The model parameters are specified as densities for both the stochastic component and the parameters to be evaluated (as objective function *u*(*x*)). The prediction of the model will be obtained as a certain *n*-dimensional density describing the possible values of the data. It is required to choose both the dimensions and the form of the density of the evaluated parameters so that the metric points out the maximum similarity of the experimental data and the prediction of the model. The natural metric in this approach is the magnitude of the overlap of the prediction density and the actual experimental data, namely, *μ* ≝ ∫±<sup>∞</sup> *Pr*(*x*,*D*(*x*))*dx*

Obviously, the solution in general form, without taking into account the structure of the model and data, is very labour-consuming by both methods. But for simple models and data,

The concept of a rank measure was proposed years ago and analyzed from both the intuitive and the formal points of view. Here, we propose an approach which can be regarded as justi-

*Statement*. For a trivial metrological model, if the source of randomness is described only by its distribution, and the data elements are statistically independent, the implementation of the

*Proof*. From the assumption of data independence, the value of the metric is independent of the permutation of the data elements in the data sample used to identify the trivial model.

In fact, suppose that for two data samples of the same length, all elements are the same. Should the metric distinguish them? It is obvious enough that it is not necessary to distin-

Now, in each sample, one element by element of a different but identical value and in the

Now, in one of the data samples, we change the positions of any two elements. If the data elements are equal, then the samples are indistinguishable. If the data elements are different,

If the data is independent, then any position of each element is equally probable. Thus, the probability of origin is unchanged. The metric must be such that a simple permutation of data elements within one of the samples does not change the value of the metric. Consequently, neither the number nor the step of internal permutations on the value of the metric is affected.

'principle of measuring the probability of origin' leads to a simple 'rank measure'.

same position is replaced. As before, the samples are indistinguishable.

then the samples can be distinguished, but should this be done?

*<sup>N</sup>*|*x*. Repeating a series of experiments

*count*→*C*; sequence metric is *<sup>μ</sup>* <sup>≝</sup> \_\_*<sup>C</sup>*

{*M*(*x*) <sup>→</sup> *Pr*<sup>=</sup> <sup>=</sup> *<sup>D</sup>*}*<sup>N</sup>* ⎯

100 Metrology

**5. Rank measure**

in general and in a case of point data as *μ* ≝ *Pr*(*D*).

fication as rationale in constructive style.

guish and there is no possibility to do this.

the situation is so simplified that it leads to simple algorithms.

In this section we give examples of the application of a rank measure in some basic types of experiments. Let us compare the results obtained by algorithms using a rank measure and the results of normative algorithms. In this section, several varieties of direct measurement experiment and one generalization are considered.

## **6.1. Calibration experiment**

Calibration experiment is main type of experiments in metrology. There is no means of measurement which one way or another would not undergo calibration. The purpose of the calibration experiment is to compare the measuring instrument with the standard, collect the data and describe a correction function that will be used as a priori information in the working measurement experiment.

as the distribution density of the possible values of the measured value. That is, the systematic error is eliminated, and an estimate of the uncertainty of the values of the measurand is given

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On the other hand, the data may already contain a description of the uncertainty, for example, in the form of a probability density *g*(*d*). In this case, the joint probability density distribution of the data and the estimate is calculated *p*(*d*, *x*) = *g*(*d*) ∙ (*f*(̂*d*) = *x*). Note that this is a joint distribution and does not refer to independent distributions because of the large correlation (**Figure 6**). The projection of this joint density will lead to a final evaluation of the measurement result *p*(*x*) = ∫±∞(*g*(*d*) ∙ *f*(̂*d*))*dd*. Thus, a complete and natural synthesis of the available a priori and a posteriori information about this measurement experiment was made without

Measuring the same physical quantity repeatedly, in principle, we get the opportunity to deal with errors and thereby improve the accuracy of the evaluation of the result. The problem of normative statistical tools is that it was far from always possible to use data efficiently, and sometimes efficiency was reduced to zero. From this point of view, since the rank measure uses the form of a specific distribution, it will always be optimal in efficiency with respect to this distribution.

The greatest effect of using the rank measure as statistics for estimating the distribution parameters is observed in a multiple experiment with unknown scattering. According to the principle of probability of origin, the probability of obtaining experimental data from a

to a full evaluation of the experimental result, taking into

*d* ̇).

*6.3.1. The scattering parameter is unknown and is estimated from experimental data*

(**Figure 5**).

any assumptions and approximate calculations.

**Figure 5.** The transition from point experimental data *d* ̇

account a priori information about the property of the measuring instrument *p*(*x*) = *f*(̂

**6.3. Multiple experiment**

In the calibration experiment, the values of the standard and the readings of the measuring instrument are juxtaposed. In this case, the measuring means is used to estimate the value of the standard used. The results are collected and form a data structure, for example, as in **Figure 4** (left).

The correction function is constructed as a regression at the calibration data. The obvious representation is the density stretched over the whole measurement range and accumulating all the calibration information [**Figure 4** (right)]. The more calibration data and the more carefully the regression, the more reliable the results. The replacement of the abscissa axis from the value of the reference value to the unknown means that the probability of the value of the standard corresponding to the experimental data is estimated.

The quantity and quality of the information collected in the calibration experiment and the information stored in the correction function largely determine the capabilities of the working measurement experiment. Although modern regulatory documents allow the use of a correction function in this form, for example, IEEE 1451, historically, the systematic error is eliminated separately, and the uncertainty of the measurement tool is described as an interval approximation of the density function in the form of a two-term formula or its simplifications.

#### **6.2. Single experiment**

The correction function is used in a working experiment to fully evaluate the result of the experiment. If the data comes in the form of a point estimate (number), then the corrected measurement result is calculated as cross section of correction function, which is interpreted

**Figure 4.** The structure of the calibration data in graphical form (left) and the correction function (right) as the regression of these data *p*(*x*) = *f* (̂ *d*).

as the distribution density of the possible values of the measured value. That is, the systematic error is eliminated, and an estimate of the uncertainty of the values of the measurand is given (**Figure 5**).

On the other hand, the data may already contain a description of the uncertainty, for example, in the form of a probability density *g*(*d*). In this case, the joint probability density distribution of the data and the estimate is calculated *p*(*d*, *x*) = *g*(*d*) ∙ (*f*(̂*d*) = *x*). Note that this is a joint distribution and does not refer to independent distributions because of the large correlation (**Figure 6**). The projection of this joint density will lead to a final evaluation of the measurement result *p*(*x*) = ∫±∞(*g*(*d*) ∙ *f*(̂*d*))*dd*. Thus, a complete and natural synthesis of the available a priori and a posteriori information about this measurement experiment was made without any assumptions and approximate calculations.

#### **6.3. Multiple experiment**

**Figure 4.** The structure of the calibration data in graphical form (left) and the correction function (right) as the regression

Calibration experiment is main type of experiments in metrology. There is no means of measurement which one way or another would not undergo calibration. The purpose of the calibration experiment is to compare the measuring instrument with the standard, collect the data and describe a correction function that will be used as a priori information in the work-

In the calibration experiment, the values of the standard and the readings of the measuring instrument are juxtaposed. In this case, the measuring means is used to estimate the value of the standard used. The results are collected and form a data structure, for example, as in

The correction function is constructed as a regression at the calibration data. The obvious representation is the density stretched over the whole measurement range and accumulating all the calibration information [**Figure 4** (right)]. The more calibration data and the more carefully the regression, the more reliable the results. The replacement of the abscissa axis from the value of the reference value to the unknown means that the probability of the value of the

The quantity and quality of the information collected in the calibration experiment and the information stored in the correction function largely determine the capabilities of the working measurement experiment. Although modern regulatory documents allow the use of a correction function in this form, for example, IEEE 1451, historically, the systematic error is eliminated separately, and the uncertainty of the measurement tool is described as an interval approximation of the density function in the form of a two-term formula or its simplifications.

The correction function is used in a working experiment to fully evaluate the result of the experiment. If the data comes in the form of a point estimate (number), then the corrected measurement result is calculated as cross section of correction function, which is interpreted

standard corresponding to the experimental data is estimated.

of these data *p*(*x*) = *f* (̂

*d*).

**6.1. Calibration experiment**

ing measurement experiment.

**Figure 4** (left).

102 Metrology

**6.2. Single experiment**

Measuring the same physical quantity repeatedly, in principle, we get the opportunity to deal with errors and thereby improve the accuracy of the evaluation of the result. The problem of normative statistical tools is that it was far from always possible to use data efficiently, and sometimes efficiency was reduced to zero. From this point of view, since the rank measure uses the form of a specific distribution, it will always be optimal in efficiency with respect to this distribution.

#### *6.3.1. The scattering parameter is unknown and is estimated from experimental data*

The greatest effect of using the rank measure as statistics for estimating the distribution parameters is observed in a multiple experiment with unknown scattering. According to the principle of probability of origin, the probability of obtaining experimental data from a

**Figure 5.** The transition from point experimental data *d* ̇ to a full evaluation of the experimental result, taking into account a priori information about the property of the measuring instrument *p*(*x*) = *f*(̂ *d* ̇).

The uncertainty functions for different distributions differ in varying degrees by form but mainly by the scattering estimate. The distributions used in the example are both symmetric

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Now, it became possible to move from a joint estimation of parameters to only an estimate of the shift parameter (usually interpreted as an estimate of the measured quantity). At this stage, it is possible to take into account a priori information about the scattering parameter. This information can be different. One of the polar cases is its complete absence; the scattering

If, for joint uncertainty function, the influence of the form of the model distribution is obvious, then the integral estimates of only the shift parameter differ insignificantly. Small differences can be interpreted as evidence of the prevalent thesis 'if there is a small number of data the form of the distribution is unimportant'. More precisely, when identifying only the shift parameter for a small number of data, the form of the distribution has no important significance and does not introduce significant errors in addition for a wide class of distributions. However, it is possible to construct counterexamples that show that this is not always so, for example, using

The form of the uncertainty function of the result for a number of reasons has heavier tails than the original distribution. Briefly, there are two main reasons. There is still a high probability of obtaining compact data from the distribution with a large value of the scattering parameter, which heavies the tails of the uncertainty function. On the contrary, the probability of compact distributions is concentrated in a small space, which leads to a high probability

Now, we can write an interval estimate of the measurement result as a quantile of the uncertainty function. For the confidence probability of 0.95 by the normal distribution model, result estimation with uncertainty is 0.153 ± 0.869 and by the uniform distribution model is 0.149 ± 0.94. Uncertainty function has less scattering than the original distribution (at example for normal distribution ±1.96 and for uniform ±2.0), which is actually the goal of increasing the multiplicity of the experiment. The recording of the result by the form is the same as the

**Figure 8.** The uncertainty functions of estimating the shift parameter (left) and their difference (right). The notations on the left figure are a red line for the normal distribution and blue for the uniform, respectively. For a correct comparison, the uncertainty functions are normalized, which is interpreted as an assumption of the validity of both models

for this reason, and the difference in the estimation of the shift parameter is small.

can be any *u*(*μ*) = ∫±<sup>∞</sup> *u*(*μ*, *σ*)*d* (**Figure 8**).

distributions having a significant displacement.

simultaneously.

density near the vertex of the uncertainty function and sharpens it.

**Figure 6.** The transition from experimental data with uncertainty *g*(*d*) to a full evaluation of the experimental result *p* (*x*) <sup>=</sup> <sup>∫</sup>+<sup>∞</sup> *<sup>p</sup>*(*d*, *<sup>x</sup>*)*dd*. In the middle of the figure, an intermediate result is presented *<sup>p</sup>*(*d*, *<sup>x</sup>*) <sup>=</sup> *<sup>g</sup>*(*d*) <sup>∙</sup> *<sup>f</sup>*(̂ *d* ̇).

random process model with a known form of the distribution density is estimated, but the parameters of the shift *μ* and scattering *σ* must be estimated from the experimental data. Note that the form of the distribution can be arbitrary, but it shall be a priori known, for example, obtained from a calibration experiment. We seek a joint distribution of the values of parameters that are estimated *u*(*μ*, *σ*) = *m*({*d*} *n* , *μ*, *σ*), and the distribution density of error source is also described in terms of the values of these parameters *p*(*x*, *μ*, *σ*).

For example, we estimate the shift parameter from the data for normal and uniform distributions {*d*} *<sup>n</sup>* = {−0.125, −0.044, 0.183, 0.349, 0.404}. It is convenient to designate the desired joint distribution as *u*(*μ*, *σ*) ∣ *p* with an explicit indication of the distribution form used in the model and interpret it as a function of the uncertainty of estimates with respect to the distribution used (**Figure 7**).

**Figure 7.** Uncertainty functions *u*(*μ*, *σ*) ∣ *N* for the normal distribution (only the form is used) and *u*(*μ*, *σ*) ∣ *U* for the uniform distribution. For a correct comparison, distribution densities are scaled to the law 2*σ*.

The uncertainty functions for different distributions differ in varying degrees by form but mainly by the scattering estimate. The distributions used in the example are both symmetric for this reason, and the difference in the estimation of the shift parameter is small.

Now, it became possible to move from a joint estimation of parameters to only an estimate of the shift parameter (usually interpreted as an estimate of the measured quantity). At this stage, it is possible to take into account a priori information about the scattering parameter. This information can be different. One of the polar cases is its complete absence; the scattering can be any *u*(*μ*) = ∫±<sup>∞</sup> *u*(*μ*, *σ*)*d* (**Figure 8**).

If, for joint uncertainty function, the influence of the form of the model distribution is obvious, then the integral estimates of only the shift parameter differ insignificantly. Small differences can be interpreted as evidence of the prevalent thesis 'if there is a small number of data the form of the distribution is unimportant'. More precisely, when identifying only the shift parameter for a small number of data, the form of the distribution has no important significance and does not introduce significant errors in addition for a wide class of distributions. However, it is possible to construct counterexamples that show that this is not always so, for example, using distributions having a significant displacement.

The form of the uncertainty function of the result for a number of reasons has heavier tails than the original distribution. Briefly, there are two main reasons. There is still a high probability of obtaining compact data from the distribution with a large value of the scattering parameter, which heavies the tails of the uncertainty function. On the contrary, the probability of compact distributions is concentrated in a small space, which leads to a high probability density near the vertex of the uncertainty function and sharpens it.

Now, we can write an interval estimate of the measurement result as a quantile of the uncertainty function. For the confidence probability of 0.95 by the normal distribution model, result estimation with uncertainty is 0.153 ± 0.869 and by the uniform distribution model is 0.149 ± 0.94. Uncertainty function has less scattering than the original distribution (at example for normal distribution ±1.96 and for uniform ±2.0), which is actually the goal of increasing the multiplicity of the experiment. The recording of the result by the form is the same as the

**Figure 8.** The uncertainty functions of estimating the shift parameter (left) and their difference (right). The notations on the left figure are a red line for the normal distribution and blue for the uniform, respectively. For a correct comparison, the uncertainty functions are normalized, which is interpreted as an assumption of the validity of both models simultaneously.

**Figure 7.** Uncertainty functions *u*(*μ*, *σ*) ∣ *N* for the normal distribution (only the form is used) and *u*(*μ*, *σ*) ∣ *U* for the

random process model with a known form of the distribution density is estimated, but the parameters of the shift *μ* and scattering *σ* must be estimated from the experimental data. Note that the form of the distribution can be arbitrary, but it shall be a priori known, for example, obtained from a calibration experiment. We seek a joint distribution of the values of param-

**Figure 6.** The transition from experimental data with uncertainty *g*(*d*) to a full evaluation of the experimental result *p*

For example, we estimate the shift parameter from the data for normal and uniform distri-

joint distribution as *u*(*μ*, *σ*) ∣ *p* with an explicit indication of the distribution form used in the model and interpret it as a function of the uncertainty of estimates with respect to the

*<sup>n</sup>* = {−0.125, −0.044, 0.183, 0.349, 0.404}. It is convenient to designate the desired

, *μ*, *σ*), and the distribution density of error source is

*d* ̇).

*n*

(*x*) <sup>=</sup> <sup>∫</sup>+<sup>∞</sup> *<sup>p</sup>*(*d*, *<sup>x</sup>*)*dd*. In the middle of the figure, an intermediate result is presented *<sup>p</sup>*(*d*, *<sup>x</sup>*) <sup>=</sup> *<sup>g</sup>*(*d*) <sup>∙</sup> *<sup>f</sup>*(̂

also described in terms of the values of these parameters *p*(*x*, *μ*, *σ*).

eters that are estimated *u*(*μ*, *σ*) = *m*({*d*}

distribution used (**Figure 7**).

butions {*d*}

104 Metrology

uniform distribution. For a correct comparison, distribution densities are scaled to the law 2*σ*.

normative one, but in fact it has a more rigorous meaning. Tails of joint distributions (as well as clouds of estimates) are cut vertically, but not by the sector as in the normative case.

#### *6.3.2. The scattering parameter is known fully or partially*

There are many cases when the scattering parameter is known a priori with greater or lesser accuracy. The direct way to take into account information about the value of the scattering parameter is to solve the estimation problem for an unknown parameter and only then to use a priori information *u*(*μ*, *<sup>σ</sup>*)⟶*σ*? <sup>→</sup> *<sup>u</sup>*(*μ*). The algorithm for solving the problem formally depends on the form of the representation of this information but at the heart of all algorithms lies an integral that somehow projects the joint uncertainty function to the shift parameter uncertainty.

The most often known is the range of possible values of the scattering parameter *σ*¯. The solution reduces to a simple integral *u*(*μ*) <sup>=</sup> <sup>∫</sup>*σ*¯ *u*(*μ*, *σ*)*d*. Two polar variants are evident. It is complete ignorance *<sup>σ</sup>*¯ <sup>=</sup> <sup>±</sup>∞ and exact knowledge *σ*¯ <sup>=</sup> *<sup>σ</sup>*̇<sup>±</sup> <sup>0</sup> which are solved analogically. The case of a known density distribution *g*(*σ*) of the scattering parameter *u*(*μ*) = ∫±<sup>∞</sup> *g*(*σ*)*u*(*μ*, *σ*)*d* is slightly more complicated (**Figure 9**).

respect to the shift parameter is calculated directly as the product of these functions *u*(*x*) =

**Figure 10.** Illustration of the use of the set of ordinal correction functions. On the left is a set of correction functions for a triple experiment, and on the right is an example of estimating the value of the measured parameter for data {0.4, 0.41,

0.44} each of the three ordinal estimations (color lines) and resultant estimation (black line).

This tool is more refined because it can take into account the change in the form of the distribution of the correction function for different elements from the data set. But it is more vulnerable because it does not provide for any additional sources of randomness that cannot be the

The situation where the scattering parameter is known sufficiently accurately is not so rare, although it is hidden inside the measuring instrument. At best, the user can adjust the 'accumulation time'. If the accumulation of information is made in digital form, then this is a direct analogue to the number of repeated measurements, but in the analogue form, the accumula-

The abstraction of point data is very useful from a practical point of view. Its application seriously simplifies both calculations and their interpretation, and the results are of quite satisfactory quality. In most cases, it should be used. However, in the strict approach, each data element must be assigned to its own individual uncertainty. For many applications, including the case of multiple measurement experiments, an adequate form of describing the uncertainty of the experimental data is the probability density of the obtained value *D* ≔ {*gk*

interpreted as the reliability of the point fragment of this estimate. It is because the basis is the

Normative documents including GUM solve this problem taking into account uncertainties apart, for example, preliminarily dividing the uncertainties into type A and type B and then

(*δ*) }*n*

tion is not fundamentally different from the effect of repeated measurements.

(*dk*). The formula is interpreted as uncertainty of a repetitive measurement of the same physical quantity by an imperfect means of measurement but with a known scattering parameter of it stochastic model. It is assumed that new sources of randomness, not accounted for by the calibration experiment, are not added. This is what distinguishes the repeated experiment

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∏*<sup>k</sup>*=1 *<sup>n</sup> fk* <sup>⁄</sup>*<sup>n</sup>*

from a multiple experiment.

taken into account in the calibration experiment.

*6.3.4. The uncertainty of the experimental data is known*

single experiments in which the initial data are obtained.

#### *6.3.3. Repetitive experiment*

Under favorable conditions, instead of the joint uncertainty function of the parameters, one can use the fact that the correction function itself is a distribution. Consequently, one complete correction function can be replaced by a set of ordinal correction functions with the same external characteristics. This is done either experimentally in a calibration experiment or analytically from the formulas of the densities of ordinal distributions for each value of the measured quantity in the entire measurement range. We obtain a family of correction functions passing along and partially overlapping {*fk* <sup>⁄</sup>*<sup>n</sup>* (*d*) }*n* . Each of these functions is used to correct its data element from an ordered data sample [**Figure 10** (left)]. The uncertainty function with

**Figure 9.** Illustration of the use of a priori information on the scattering parameter in order to convert joint uncertainty to uncertainty function of the shift parameter. Legend on the right picture field: The green line is the exact knowledge of the scattering parameter *σ* = 1.0; black line corresponds to the normal distribution *g*(*σ*) = *N*(*σ*, 1,0.2) (graph on the left); blue line corresponds to the interval number *<sup>σ</sup>*¯ <sup>=</sup> [0.5,1.5]; and red line for complete ignorance *σ*¯ <sup>=</sup> <sup>±</sup>∞. Legend on the Centre picture field: The joint uncertainty distribution with respect to both parameters is the same as in **figure 7** on the left; the green line is the exact knowledge of the scattering parameter; and blue wideband is the image of the scattering parameter interval.

**Figure 10.** Illustration of the use of the set of ordinal correction functions. On the left is a set of correction functions for a triple experiment, and on the right is an example of estimating the value of the measured parameter for data {0.4, 0.41, 0.44} each of the three ordinal estimations (color lines) and resultant estimation (black line).

respect to the shift parameter is calculated directly as the product of these functions *u*(*x*) = ∏*<sup>k</sup>*=1 *<sup>n</sup> fk* <sup>⁄</sup>*<sup>n</sup>* (*dk*). The formula is interpreted as uncertainty of a repetitive measurement of the same physical quantity by an imperfect means of measurement but with a known scattering parameter of it stochastic model. It is assumed that new sources of randomness, not accounted for by the calibration experiment, are not added. This is what distinguishes the repeated experiment from a multiple experiment.

This tool is more refined because it can take into account the change in the form of the distribution of the correction function for different elements from the data set. But it is more vulnerable because it does not provide for any additional sources of randomness that cannot be the taken into account in the calibration experiment.

The situation where the scattering parameter is known sufficiently accurately is not so rare, although it is hidden inside the measuring instrument. At best, the user can adjust the 'accumulation time'. If the accumulation of information is made in digital form, then this is a direct analogue to the number of repeated measurements, but in the analogue form, the accumulation is not fundamentally different from the effect of repeated measurements.

#### *6.3.4. The uncertainty of the experimental data is known*

**Figure 9.** Illustration of the use of a priori information on the scattering parameter in order to convert joint uncertainty to uncertainty function of the shift parameter. Legend on the right picture field: The green line is the exact knowledge of the scattering parameter *σ* = 1.0; black line corresponds to the normal distribution *g*(*σ*) = *N*(*σ*, 1,0.2) (graph on the left); blue line corresponds to the interval number *<sup>σ</sup>*¯ <sup>=</sup> [0.5,1.5]; and red line for complete ignorance *σ*¯ <sup>=</sup> <sup>±</sup>∞. Legend on the Centre picture field: The joint uncertainty distribution with respect to both parameters is the same as in **figure 7** on the left; the green line is the exact knowledge of the scattering parameter; and blue wideband is the image of the scattering

normative one, but in fact it has a more rigorous meaning. Tails of joint distributions (as well as clouds of estimates) are cut vertically, but not by the sector as in the normative case.

There are many cases when the scattering parameter is known a priori with greater or lesser accuracy. The direct way to take into account information about the value of the scattering parameter is to solve the estimation problem for an unknown parameter and only then to use a priori information *u*(*μ*, *<sup>σ</sup>*)⟶*σ*? <sup>→</sup> *<sup>u</sup>*(*μ*). The algorithm for solving the problem formally depends on the form of the representation of this information but at the heart of all algorithms lies an integral that somehow projects the joint uncertainty function to the shift parameter uncertainty. The most often known is the range of possible values of the scattering parameter *σ*¯. The solu-

plete ignorance *<sup>σ</sup>*¯ <sup>=</sup> <sup>±</sup>∞ and exact knowledge *σ*¯ <sup>=</sup> *<sup>σ</sup>*̇<sup>±</sup> <sup>0</sup> which are solved analogically. The case of a known density distribution *g*(*σ*) of the scattering parameter *u*(*μ*) = ∫±<sup>∞</sup> *g*(*σ*)*u*(*μ*, *σ*)*d* is

Under favorable conditions, instead of the joint uncertainty function of the parameters, one can use the fact that the correction function itself is a distribution. Consequently, one complete correction function can be replaced by a set of ordinal correction functions with the same external characteristics. This is done either experimentally in a calibration experiment or analytically from the formulas of the densities of ordinal distributions for each value of the measured quantity in the entire measurement range. We obtain a family of correction functions

> (*d*) }*n*

data element from an ordered data sample [**Figure 10** (left)]. The uncertainty function with

*u*(*μ*, *σ*)*d*. Two polar variants are evident. It is com-

. Each of these functions is used to correct its

*6.3.2. The scattering parameter is known fully or partially*

tion reduces to a simple integral *u*(*μ*) <sup>=</sup> <sup>∫</sup>*σ*¯

passing along and partially overlapping {*fk* <sup>⁄</sup>*<sup>n</sup>*

slightly more complicated (**Figure 9**).

*6.3.3. Repetitive experiment*

106 Metrology

parameter interval.

The abstraction of point data is very useful from a practical point of view. Its application seriously simplifies both calculations and their interpretation, and the results are of quite satisfactory quality. In most cases, it should be used. However, in the strict approach, each data element must be assigned to its own individual uncertainty. For many applications, including the case of multiple measurement experiments, an adequate form of describing the uncertainty of the experimental data is the probability density of the obtained value *D* ≔ {*gk* (*δ*) }*n* interpreted as the reliability of the point fragment of this estimate. It is because the basis is the single experiments in which the initial data are obtained.

Normative documents including GUM solve this problem taking into account uncertainties apart, for example, preliminarily dividing the uncertainties into type A and type B and then combining them in a specific way. The method is simple but strictly adequate only for normal distribution and simple models. For distributions similar to normal distribution, the deterioration in the result still is quite acceptable.

regular grid (MCD) (in example used 12 grid knots for each data distribution). Each frame

are uniform and equal, then probabilities of frames are equal. At the end of the algorithm, all frames (in example 123 = 1728 frames) are summed according to their probability. The result

The uncertainty is large compared to the distance between data; hence, the probability of accidental coincidence of data is large, which leads to a touch of the uncertainty function of the estimates to the abscissa axis [**Figure 11** (centre)]. The uncertainty of the data, as it was, 'smears out' the uncertainty function of the estimate. Uncertainty is greater in all respects but especially strongly affects the top of the uncertainty function of the estimate of the measured

This allows us to build a logical chain from the interpretation of data by interpreting possible estimates to the final estimate of the uncertainty of the measurand. For example, *<sup>D</sup> <sup>f</sup>* (̂*d*) ⎯→{*d*(*x*)}⎯

tion obtained from a calibration experiment, {*d*(*x*)} is the data in the form of densities that have adjusted by the calibration experiment, {*p*(*x*)} is the set of admissible types of distributions in measurement model, {*u*(*x*)} is the set of densities of estimates measurand and *x*¯ is the final evaluation,

The purpose of the multifactorial experiment is to estimate the value of several quantities in the form of a joint uncertainty function by factors. The number of factors considered varies easily, so in the examples we confine ourselves to two. And so, *u*(*ξ*, *υ*) is estimated by the data structure ΞΥ for each of the factors. The result of the evaluation and the complexity of the algorithm are essentially determined by the relationships within the data structure. The simplest solution is obtained when the data for different factors are not related to each other. For example, a data structure is simply a list of independent data differing only belonging to its factor ΞΥ ≔ Ξ, Υ. The solution consists of a multiplication of uncertainty functions for each of the factors calculated independently *u*(*ξ*, *υ*) = *u*(*ξ*)*u*(*υ*). The number of data for each factor can

Another solution is obtained if the experimental data are obtained synchronously ΞΥ ≔ {*ξ*, *υ*}

<sup>|</sup><sup>ξ</sup> <sup>∈</sup> <sup>Ξ</sup>, <sup>υ</sup> <sup>∈</sup> <sup>Υ</sup>. If the statistical relationships between the factors do not manifest themselves *<sup>r</sup>*

For example, if the multiplicity of experiment is 3, the number of factors is 2, *p*(*ξ*) is a normal distribution and *g*(*υ*) is the uniform distribution, then the figure of the set of ordinal distribu-

The rank measure is constructed as follows. The data structure (in the example this is three data pairs) is ordered by one of the factors, for example, by *ξ*. This predefines the selection of the columns of the set of ordinal distributions. The rows are selected in accordance with the order of the data for the second factor. As a result, for each experiment of the nine distributions, three will be chosen. Using them as a function of the data values, we get three probabilities for each

(*ξ*, *υ*) = *p*(*ξ*) ∙ *g*(*υ*), then it is possible to express the densities of order distribution *rk*

¯, where *D* is the initial experimental data given in point form, *f*(

*<sup>n</sup> gk* (*δk*

). When all data distributions

A New Statistical Tool Focused on Metrological Tasks http://dx.doi.org/10.5772/intechopen.74872

̂

{*p*(*x*)} → 109

*n*

*ξ* ,*k υ*⁄*n* (*ξ*, *υ*) = *pk ξ*⁄*n* (*ξ*) ∙ *gk υ*⁄*n* (*υ*).

*d*) is the correction func-

corresponds a probability that is calculated by formula ∏*<sup>k</sup>*=1

parameter and often changes the form of the evaluation function.

is shown in **Figure 11** in the centre.

for example, obtained for the worst case.

*6.3.5. Multifactor multiple experiment*

tions will look like in **Figure 12**.

{*u*(*x*)} → *x*

be different.

To strictly take into account the uncertainty of the measuring instrument, it is sufficient to slightly upgrade the rank measure to.

$$m(D) = \prescript{u}{}{\int\_{\r^\*}} \left( m\left(sort(\{\delta\}\\_)\right) \cdot \prod\_{k=1}^n \mathbb{g}\_k(\bullet) \right) d\delta^u.$$

The formula is interpreted as an n-fold integral of a rank measure from deviation to point data with their joint probability. The complexity of applying the formula is the multiplicity of the integral and the need to constantly check the order of the data if the density of the data distribution overlaps. When the distribution density of data is reduced to the delta function, the upgraded measure reduces to the original measure. The delta function is the model of point data. From this point of view, uncertainty function for point data is the most likely, but for data deviations it is a less likely alternative.

In a more general case, all sources of uncertainty are taken into account in a natural way when calculating the model's predictions and when a comparison of the prediction and an adequate data model is made.

Let us explain this with an example (**Figure 11**). The data is the same as for **Figure 10**. We will supplement the data with uncertainty ±0.05. The uncertainty is the same for all data elements, but it can also have an individual value. The law of distribution of uncertainty will be assumed to be uniform. The model of the measurement experiment being studied differs from the trivial model only in the presence of two sources of randomness. One source has a normal distribution law, for example, the error of manufacturing samples from the same material whose property is being investigated. Another source has a uniform distribution of, for example, uncertainty of a digit measuring instrument.

The work of the algorithm can be interpreted as the creation of a film. Each frame is an estimate of the parameters from a given set of point data {*δk*}*<sup>n</sup>* . Each frame is similar to the one in **Figure 11** (left). The difference between frames is a consequence of the differences in the data. Data is selected from specified distributions either randomly (MCM) or according to a

**Figure 11.** Illustration of identification of a trivial model with two different sources of randomness. The left figure is obtained for point data, and the central figure is obtained for data with uncertainty. The right figure shows the uncertainty functions of the estimates for the two preceding figures: *p*, without data uncertainties, and *g*, with uncertainties.

regular grid (MCD) (in example used 12 grid knots for each data distribution). Each frame corresponds a probability that is calculated by formula ∏*<sup>k</sup>*=1 *<sup>n</sup> gk* (*δk* ). When all data distributions are uniform and equal, then probabilities of frames are equal. At the end of the algorithm, all frames (in example 123 = 1728 frames) are summed according to their probability. The result is shown in **Figure 11** in the centre.

The uncertainty is large compared to the distance between data; hence, the probability of accidental coincidence of data is large, which leads to a touch of the uncertainty function of the estimates to the abscissa axis [**Figure 11** (centre)]. The uncertainty of the data, as it was, 'smears out' the uncertainty function of the estimate. Uncertainty is greater in all respects but especially strongly affects the top of the uncertainty function of the estimate of the measured parameter and often changes the form of the evaluation function.

This allows us to build a logical chain from the interpretation of data by interpreting possible estimates to the final estimate of the uncertainty of the measurand. For example, *<sup>D</sup> <sup>f</sup>* (̂*d*) ⎯→{*d*(*x*)}⎯ {*p*(*x*)} → {*u*(*x*)} → *x* ¯, where *D* is the initial experimental data given in point form, *f*( ̂ *d*) is the correction function obtained from a calibration experiment, {*d*(*x*)} is the data in the form of densities that have adjusted by the calibration experiment, {*p*(*x*)} is the set of admissible types of distributions in measurement model, {*u*(*x*)} is the set of densities of estimates measurand and *x*¯ is the final evaluation, for example, obtained for the worst case.

### *6.3.5. Multifactor multiple experiment*

combining them in a specific way. The method is simple but strictly adequate only for normal distribution and simple models. For distributions similar to normal distribution, the deterio-

To strictly take into account the uncertainty of the measuring instrument, it is sufficient to

The formula is interpreted as an n-fold integral of a rank measure from deviation to point data with their joint probability. The complexity of applying the formula is the multiplicity of the integral and the need to constantly check the order of the data if the density of the data distribution overlaps. When the distribution density of data is reduced to the delta function, the upgraded measure reduces to the original measure. The delta function is the model of point data. From this point of view, uncertainty function for point data is the most likely, but

In a more general case, all sources of uncertainty are taken into account in a natural way when calculating the model's predictions and when a comparison of the prediction and an adequate

Let us explain this with an example (**Figure 11**). The data is the same as for **Figure 10**. We will supplement the data with uncertainty ±0.05. The uncertainty is the same for all data elements, but it can also have an individual value. The law of distribution of uncertainty will be assumed to be uniform. The model of the measurement experiment being studied differs from the trivial model only in the presence of two sources of randomness. One source has a normal distribution law, for example, the error of manufacturing samples from the same material whose property is being investigated. Another source has a uniform distribution of,

The work of the algorithm can be interpreted as the creation of a film. Each frame is an esti-

in **Figure 11** (left). The difference between frames is a consequence of the differences in the data. Data is selected from specified distributions either randomly (MCM) or according to a

**Figure 11.** Illustration of identification of a trivial model with two different sources of randomness. The left figure is obtained for point data, and the central figure is obtained for data with uncertainty. The right figure shows the uncertainty functions of the estimates for the two preceding figures: *p*, without data uncertainties, and *g*, with uncertainties.

. Each frame is similar to the one

*<sup>n</sup>*)) ∙ ∏*<sup>k</sup>*=1 *<sup>n</sup> gk* (*δ*) ) *<sup>d</sup><sup>n</sup>* .

<sup>∫</sup>±∞(*m*(*sort*({*δ*}

*<sup>m</sup>*(*D*) <sup>=</sup> *<sup>n</sup>*

ration in the result still is quite acceptable.

for data deviations it is a less likely alternative.

for example, uncertainty of a digit measuring instrument.

mate of the parameters from a given set of point data {*δk*}*<sup>n</sup>*

data model is made.

108 Metrology

slightly upgrade the rank measure to.

The purpose of the multifactorial experiment is to estimate the value of several quantities in the form of a joint uncertainty function by factors. The number of factors considered varies easily, so in the examples we confine ourselves to two. And so, *u*(*ξ*, *υ*) is estimated by the data structure ΞΥ for each of the factors. The result of the evaluation and the complexity of the algorithm are essentially determined by the relationships within the data structure. The simplest solution is obtained when the data for different factors are not related to each other. For example, a data structure is simply a list of independent data differing only belonging to its factor ΞΥ ≔ Ξ, Υ. The solution consists of a multiplication of uncertainty functions for each of the factors calculated independently *u*(*ξ*, *υ*) = *u*(*ξ*)*u*(*υ*). The number of data for each factor can be different.

Another solution is obtained if the experimental data are obtained synchronously ΞΥ ≔ {*ξ*, *υ*} *n* <sup>|</sup><sup>ξ</sup> <sup>∈</sup> <sup>Ξ</sup>, <sup>υ</sup> <sup>∈</sup> <sup>Υ</sup>. If the statistical relationships between the factors do not manifest themselves *<sup>r</sup>* (*ξ*, *υ*) = *p*(*ξ*) ∙ *g*(*υ*), then it is possible to express the densities of order distribution *rk ξ* ,*k υ*⁄*n* (*ξ*, *υ*) = *pk ξ*⁄*n* (*ξ*) ∙ *gk υ*⁄*n* (*υ*).

For example, if the multiplicity of experiment is 3, the number of factors is 2, *p*(*ξ*) is a normal distribution and *g*(*υ*) is the uniform distribution, then the figure of the set of ordinal distributions will look like in **Figure 12**.

The rank measure is constructed as follows. The data structure (in the example this is three data pairs) is ordered by one of the factors, for example, by *ξ*. This predefines the selection of the columns of the set of ordinal distributions. The rows are selected in accordance with the order of the data for the second factor. As a result, for each experiment of the nine distributions, three will be chosen. Using them as a function of the data values, we get three probabilities for each

Although in the natural sciences and in technology one can find very complex principal models of the experiment, metrology strives to avoid indirect experiments. This is achieved through the creation of new standards and the construction of suitable calibration schemes (calibration hierarchy). Even if the measurement tool uses inside the complex indirect model but being calibrated in the target units, then it realizes direct experiment. All that metrology can afford is the use of an indirect experiment as a temporary means in cases where a direct reference to the standard is not yet possible. Of course, one can complicate the formulation of the problem of indirect experiment in different ways, for example, in the analogy of Section 6.3.5, complicating the data structure, but it is unlikely that metrologists will be

A New Statistical Tool Focused on Metrological Tasks http://dx.doi.org/10.5772/intechopen.74872 111

The tools that metrology now uses have been created by statisticians at the beginning of the last century. By the middle of the century, metrology had mastered them. Over the years, the goals and circumstances of their creation and some of the properties have been forgotten. This creates some misunderstandings when interpreting the results of their application. Attempting to implement the GUM has been useful by simplifying and standardizing their

As a result of the application of new tools, a direct and obvious chain of information gathering and use is built up in the performance of metrology tasks from calibration to the final result.

A. V. Luikov Heat and Mass Transfer Institute, National Academy of Sciences of Belarus,

[1] Uncertainty in Measurement (GUM). https://www.bipm.org/en/publications/guides/

[2] Chernukho EV. Estimation of arbitrary-distribution parameters from the data of a repetitive experiment. Journal of Engineering Physics and Thermophysics. 2010;**83**(2):431-437

[3] Chernukho EV. Substantiation of the rank measure as an efficient statistic for estimating distribution parameters of arbitrary form. Journal of Engineering Physics and

application, but the tools themselves remained the same.

Address all correspondence to: charnukha@tut.by

Thermophysics. 2012;**85**(1):239-248

interested in this.

**7. Conclusion**

**Author details**

Eugene Charnukha

Minsk, Belarus

**References**

**Figure 12.** Initial joint distribution and set of ordinal joint distributions.

rank. Multiplying them we get the value of a rank measure. You can take advantage of this in a working experiment when the data about the same physical quantity comes in completely different ways.

For example, let's use the model whose distribution is shown in **Figure 12**. The received data is {(−0.5,0.5), (0.1, −0.4), (0.5,0.7)}. Their order is {(1, 2), (2, 1), (3, 3)}. The values of probabilities from the densities of ordinal distributions are {(0.578), (0.841), (1.114)}. Hence the value of a rank measure is 0.542. Next, it is possible to identify the shear and scattering parameters of the model in the usual way.

In the event that the statistical links between the factors are significant, the task is solved only numerically. For MCM, this is a direct numerical experiment. MCD is a search for direct and inverse transformations of such that make the distribution of the model independent by factors.

#### **6.4. Indirect experiment**

In order to pass from the model of direct measurement to the model of the indirect measurement experiment, it is necessary to replace the measurand of trivial model by a more complex measurement principle model *x* = *f*(*ξ*, *υ*), where *ξ* and *ν* are the quantities measured from the direct measurement experiment. In a simple formulation, the problem of an indirect experiment consists in calculating the uncertainty function of the new measurand *u*(*x*), starting from the uncertainties obtained from the experimental data *u*(*ξ*) and *u*(*υ*). As a rule, the problem is easily solved by both MCM and MCD.

Although in the natural sciences and in technology one can find very complex principal models of the experiment, metrology strives to avoid indirect experiments. This is achieved through the creation of new standards and the construction of suitable calibration schemes (calibration hierarchy). Even if the measurement tool uses inside the complex indirect model but being calibrated in the target units, then it realizes direct experiment. All that metrology can afford is the use of an indirect experiment as a temporary means in cases where a direct reference to the standard is not yet possible. Of course, one can complicate the formulation of the problem of indirect experiment in different ways, for example, in the analogy of Section 6.3.5, complicating the data structure, but it is unlikely that metrologists will be interested in this.
