Features of Аpplication of a Сombined Аpproach to the Еvaluation of the Мeasurement Results Uncertainty

The article exposes description of features of the combined approach application to the evaluation of measurement results uncertainty. The aim of this work is the justification and development of new science-driven approaches to achieve maximum efficacy of measurements on the criteria "accuracy/costs" at the stated level of confidence. It provides theoretical base for correctness of combined approach to assess measurement results uncertainty. There is proposition to conventionally divide measurement process into fragments – combining objects, each from shall be considered as individual element for evaluation. It is well known that combining objects can be formed by grouping individual components (resources) of the measurement process either via separate stages of the measurement process. Correctness of such approach is based on the application of "resource" and "process" approaches as regards identification of the factors that affect the measurement results uncertainty. This article provides recommendations on selection of model or empiric approach for evaluating of particular contributions from combining objects of different types into total uncertainty of the final measurement result. In order to improve the validity of empiric approach of the criteria of sufficiency of measurement method uncertainty examination was formulated. It is recommended to evaluate the total uncertainty of the final measurement result by complexation of evaluations of particular total uncertainty of the results for all fragments according to the uncertainties distribution law. It is determined two typical cases of effective application of the combined approach to evaluation of measurement results uncertainty: method of direct measurements and method of indirect measurements. This article considers features of effective application of the combined approach for both situations providing corresponding examples. Special attention is paid to the application of the combined approach to assessing the test results uncertainty. As distinct from the measurement process realized under normal conditions, testing process includes additional external influence factors that are determined by test conditions.

Until recently the main method for evaluation of measurement results uncertainty was considered as model approach, stated in GUM 1 . Herewith this approach validity in fact is not guaranteed due to several reasons [1].
Technical reports of EUROLAB and other metrological organizations in last several years emphasize the use of empiric approach to evaluate the measurement results uncertainty as an alternative against the strict mathematic modeling [2][3][4][5][6][7]. The main argument for empiric approaches -considerable improvement of the efficacy of examination of the measurement process accuracy. Empiric approach is based on a quite trivial idea: full uncertainty evaluation can be reached in parallel with carrying out procedures that are mandatory for accredited laboratories, for example, for measurement method validation by intra-and interlaboratory examinations of method accuracy characteristics and quality control through laboratory participation in qualification check programs. As for empiric approach disadvantages we shall mention impossibility of the analysis of influence factors contributions into overall uncertainty of the measurement result, that does not allow to correct and improve the method.
The methodological base for empiric approach are standards 2,3 . From legitimacy point of view, this approach has the same high status as the model one.
From efficacy point of view, the most rational is combined approach to estimating the measurement results uncertainty that envisages the participation of elementary as well as complex factors (grouped data) in the final integrated model [2][3][4][5][6][7] laboratories on the base of their experience prefer combined approach [2]. However, despite of obvious advantages, combined approach is rarely used in the measuring laboratories practice. The aim of this article is to provide scientific and methodical justification for combined approach correctness and analysis of its application possibilities in metrological practice.

Theoretical base for combined approach
Taking into account obvious equivalency of the empiric and model approaches it is logic to suppose their possible combination within a single measurement method.
The key aspect of the combined approach to estimating overall measurement result uncertainty is grouping of individual components of the measurement process ( Figure 1а  Identification and grouping of individual measurement process components as a variety of input quantities x i , realize so-called "resource" approach to the evaluation of the overall uncertainty u c (Y) (Figure 1а).
Identification and grouping of the measurement process individual stages as a variety of input quantities x i realize so-called "process" approach to the evaluation of the total uncertainty u c (Y) (Figure 1b). The "process" approach for revealing factors as sources of uncertainty in the model of measurement task can be rationally applied in cases of complex measurement methods from the point of view of the quantity and expressiveness of process stages.
Complex methods are those, for example, like measurement methods in analytical chemistry, that suppose the presence of such typical, relatively individual stages as reference materials preparation, measuring equipment calibration, sample preparation and carrying out measurements [5][6][7].
Theoretical base for combined approach is linked to the second consequence of central limit theorem of the probability theory and mathematic statistics relating to complexation of random quantities dispersions.
The interpretation of this consequence in relation to the measurement process is provided in GUM 1 : the total uncertainty u c (Y) of the measurement result Y, obtained by complexation of constituents -standard uncertainties u (x i ) of the factors x i (i = 1, …, N ), involved in measurement process, according to the measurement model f is expressed as: Note. Quantities x i are random, normally distributed and not inter-correlated.
Let's suppose that measurement process for the quantity Y (see Figure 1а) includes five not inter-correlated influence factors x 1 , x 2 , x 3 , x 4 , x 5 . According to the second consequence formula (1) is valid, where N = 5. It is obvious, that we can consider as mathematically correct the following expression (1): where u (1+2+3) is the intermediate total uncertainty of the measurement result Y, evaluated, for example, using empiric approach according to the results of measuring method validation via its intra-laboratory examination in accordance with the standard 5 . At that, as the result of measuring experiment we have determined standard deviation in intermediate conditions of precision S R (1+2+3) from three varying factors: measuring instrument - Standard uncertainties u 4 and u 5 for condition factors (x 4 ) and sample (x 5 ) correspondingly in the Formula (2) are assessed using the model approach. At that, standard uncertainty u 4 , for example, is evaluated by type В on the base of a priori laboratory knowledge, and standard uncertainty u 5 is assessed by type А by realization of one-factor experiment according to the GUM 1 approach.
The combined approach essence can be formulated as follows: during the process of evaluating the uncertainty of the result, the measurement process is conventionally divided into fragments, each of them is considered as individual object of evaluation. To determine the total uncertainty of each fragment result, model or empiric approach can be applied. Assessment of the total result uncertainty for entire measurement process u c (Y ) is performed by complexation of assessments of individual total uncertainties of measurement results from all fragments according to the "law of uncertainties distribution" from the GUM approach 1 .

Fields of combined approach rational application
It is obvious that the combined approach due to limits of theoretical, material, technical and economical nature, has specific rational fields of preferred application.
From our point of view, we can determine at least two typical cases of effective application of the combined approach to measurement results uncertainty evaluation: method of direct measurements and method of indirect measurements.

Case 1. Using a combined approach for the direct measurement method
Using of combined approach for the direct measurement method in most cases is not rational. As a rule, model of direct measurement according to GUM 1 is as follows: Uncertainty of the results of direct measurements can be evaluated by model or by empiric approaches.
Here we shall exclude situations where combined approach is the rational one: -measurements that involve variety of consecutive operations, for example, in the field of analytic chemistry; -measurements within the test framework.
Tests have a special place here. As distinct from measurement process realized under normal conditions, testing process includes additional external factors determined by test conditions. Accordingly, test results uncertainty will be determined taking into account two main groups of standard uncertainty sources: 1) attributed to the measurement process under normal conditions; 2) attributed to additional factors of external influences. The last are reproduced and controlled (measured) within the framework of the measurement process. Therefore, we can see obvious availability of objective conditions for the combined approach application.
Let's consider features of combined approach application in such cases using an example.
Laboratory participates in inter-laboratory examinations within the framework of validation of the method of measurements of the UHFsignals power for UHF-equipment testing. During examinations, the laboratory used an instrument for measuring the absorbed power E4417A with a power sensor E9300A (manufacturer -"Keysinght Technologies Microwave Products (M) Snd Bhd", Malaysia). Measurements where performed in coaxial transmission line at the frequency 50 MHz at the points 1 mW and 10 mW, as well as within frequency range from 10 MHz to 18 GHz at the point 1 mW. Reference value was realized using a reference power calibrator F1130 and a thermistor bridge 1806А (manufacturer -"Tegam", USA). Multimeter 3458А (manufacturer -"Agilent Technologies Inc.", USA) was used as measuring block.
According to the results of the measurements carried out under strictly approved conditions, the laboratory received participation certificate from provider, where method accuracy characteristics are stated. According to EUROLAB recommendations [2] the laboratory accepted the model of the total method uncertainty assessment: where S RW -the value of the standard deviation of inter-laboratory method reproducibility. And now this laboratory should have the full right to use this assessment in routine tests assigning this to the results of one-time measurements carried out. However, in this case, this solution is not correct, because the uncertainty of the measurement results depends on the factor "mistuning in measuring line", the value of what is determined via reflection factors of UHFsignal source and absorbed power meter.
Difference between the reflection coefficients of UHV-signal source applied during inter-laboratory examinations (power calibrator F1130 with measuring bridge 1806А) and the UHF-signal source used for routine measurements will be considered as an additional factor implying on the measurement result uncertainty [10][11][12].
I. e., in cases where conditions or objects under measurement in fact differ from those applied in inter-laboratory validation, additional examinations of measurement results uncertainty are required. For such cases, it is reasonable to use the standard 3 , that, according to the expanded statistical model of the measurement result, recommends revealing additional influencing factors by applying expert method. These factors should be evaluated and unified with the standard deviation of the inter-laboratory reproducibility S RW . For this purpose, the standard 3 recommends to use, instead of the model (3), expanded model of uncertainty assessment: where c i u i -contributions of additional influence factors, i = 1, …, N, that were not included in interlaboratory comparisons process.  , vol. 11, no. 1, pp. 60-69 P.S. Serenkov et al. Приборы и методы измерений 2020 Examples of additional effects that were not taken into account in inter-laboratory comparisons but can significantly imply on the uncertainty of the results achieved by the laboratory using standardized method, can be like sampling and preparation conditions, measurement conditions and so on.
Assessments of standard uncertainties of the factors due to additional effects, as a rule, are determined using a model approach (by type А or В) or an empiric approach (by the method of intralaboratory examinations [3]).

Case 2. Using a combined approach for indirect measurements
It is just the case where in order to assess the measurement results uncertainty it is reasonable to apply combined approach.
Algorithm for combined approach application is considered on the example of a method of measurement of the valid value of the direct current rate 10 А using multimeter 34401А (manufacturer -"Hewlett Packard", Germany). As the top multimeter limit for the current is 3 А, measurement method supposes using measuring shunt В3 (manufacturer -"Excelsiorwerk Rudolf Kiesewetter Messtechnik mbH", Germany) with nominal electrical resistance 10 mOhm. According to multimeter indications in measuring mode for direct current voltage and data on the valid value of the shunt resistance (provided in the calibration certificate for the shunt of type В3) by indirect method we examined the direct current rate. Measurement conditions were normal.
The model of indirect measurement of the direct current rate is as follows: where I -valid value for the direct current rate, А; U -valid value of the voltage measured, V; R SHvalid value of the shut resistance, Ohm.
To evaluate the uncertainty of the result of the measurement of direct current rate I we propose to use two-level algorithm to resolve this task taking into account hierarchic structure of the indirect measurement model (Figure 2). First level of the task of evaluating the uncertainty u c (I) for the results of indirect measurements of I supposes model approach application. This selection is due to availability of the measurement model I, which has the status of a physical law. It shall be noted, that the proposed two-level algorithm for estimating the uncertainty of indirect measurements results, allows for the first level of the task, not to include into measurement model (4) factors affecting the direct measurement results U and R SH . They will be taken into account at the second level of solving the assessment task.
Model for evaluating the overall measurement result uncertainty (model М1 on the Figure 2) is formed according to the algorithm "8 steps" GUM 1 .
This formula leaves as unknown standard uncertainties u (U) and u (R SH ), that are obviously represent complex assessment of the uncertainty of the results of direct measurements of the values U and R SH . Second level of the task of estimating the uncertainty of the result of indirect measurements of the valid value of the direct current rate I supposes self-sustained resolving of the tasks of estimating the uncertainties u (U) and u (R SH ) of the results of direct measurements of quantities U and R SH , included in the Formula (5). Task 1 (model М1.1, Figure 2). Assessment of the uncertainty u (U) of the results of measurements by direct method as for direct current voltage value U.
Laboratory has already performed validation of the direct current voltage U method of measurement. Validation method is an intra-laboratory examination of the method accuracy characteristics (bias and precision) according to the standards ISO 5725 series (Accuracy (trueness and precision) of measurement methods and results, Part 1 -Part 6). Thus, to estimate the total uncertainty u (U), it is reasonable to apply an empiric approach, where the assessment model is as follows [3]: where b U -assessment characterizing uncertainty fraction associated with the bias of the results of direct measurement of the direct current voltage [3]; SRW U -standard deviation of the results of method examination under conditions of intralaboratory reproducibility, assessed according to the standard 5 .
Evaluation of the constituent b U (6). In this example, b U was evaluated at the point 100 mV on the basis of experimental data obtained during intra-laboratory method examinations using, as standard, multimeter 3458А (manufacturer "Agilent Technologies Inc.", USA). Value of b U was expressed as follows [3]: where ∆ st -average deviation of repeated measurement results from the corresponding standard value minus correction; U ref -assessment of the standard value uncertainty (obtained from the calibration certificate of the standardmultimeter 3458А); S rw -standard deviation of the repeated measurement's results n, determined within the framework of the method validation according to the standard 5 .
Note. ∆ st can be considered as an analogue of non-excluded systematic constituent of the measurement error, that is assessed as follows: where ∆ -method bias determined within the framework of validation method, ∆ corr -method correction significant for the stated level of confidence probability. In theory, ∆ st ≈ 0.
Evaluation of the constituent SRW U (6). Criteria for the correctness of the SRW U assessment is its representativeness, i. е. the value of SRW U must include influence effects from all significantly implying factors in laboratory that a priori are not known if full.
At the first stage, in order to examine and assess SRW U , we recommend to accept the base statistic model of the direct measurements result U (M1.1 on the Figure 2) according to the standard 5 : where m U -overall average value of the measurement result (expectancy); B U -aboratory bias constituent; e -bias constituent, arising at each measurement under repeatability conditions. Summand B U is a fixed complex of influencing factors-corrections: different operators -B 1 , different measuring instruments -B 2 , different measure- Devices and Methods of Measurements 2020, vol. 11, no. 1, pp. 60-69 P.S. Serenkov et al. Приборы и методы измерений 2020 ment time -B 3 , repeated calibration (validation) of measuring instruments -B 4 . Variation of these factors forms conditions for intra-laboratory method repeatability. According to the results of a measurement experiment carried out within the framework of method validation in strict accordance with hierarchical plan (Figure 3), that takes into account laboratory factors (В 1 -В 4 ) , method standard deviation value S R (B 1 -B 4 ) under repeatability conditions was obtained. It is reasonable to carry out the analysis of the correctness of the results of a measuring experiment using the method of dispersion analysis (software package STATISTICA can be applied) [13,14]. During this analysis, the following can be determined: -Required dispersion of measurement results SR (B 1 − B 4 ) 2 associated with conditions of intralaboratory reproducibility (В 1 -В 4 ); -Residual dispersion Se 2 associated with factors not taken into account in the model (7) (summand е).
Decision on the representativeness of the obtained value SR (B 1 − B 4 ) is supposed to be accepted taking into account the criteria of fullness and nonredundancy: -If SR (B 1 − B 4 ) ˃ S e -accepted statistical measurement model (7) is adequate, assessment SR (B 1 − B 4 ) is representative and can be used in model (6) for the assessment of the uncertainty u (U) as summand SRW U . Further method examination is inappropriate.
-If SR (B 1 − B 4 ) ≤ S e -statistical measurement model (7) is non-adequate, assessment SR (B 1 − B 4 ) is not representative, i. е., it does not reflect contributions of all significant factors of the laboratory and thus can not be used in model (6) for the assessment of the uncertainty u (U) as summand SRW U . Additional examinations of the direct measurement process U are required.
In the last case, statistical measurement model (7) shall be revised. For this, it is necessary to organize the second stage of examinations. It is reasonable to take as a base an expanded statistical measurement model according to the standard 3 : where, in addition to (7) the summand presents additional factors reveled during detailed examination of the direct measurement method U.
As the most rational ways of revealing all influencing factors x i in (8) [8]. Next, taking into account new revealed factors c i x i we shall implement changes into the hierarchic plan of the measurement experiment and realize it at the points according to the new plan positions. Figure 3 provides an example of correction of the plan of validation experiment due to revealing of an additional influence factor B5 -P (sample).
Dispersion analysis of the results of cumulative measurement experiment (factors B1 -B5) will allow to determine the corrected values S R (B1 − B5) and S e .
For the final decision on representativeness of the estimate S R (B1−B5) we shall apply the criteria of fullness and non-redundancy once again. If the U m B cx e (8) is still not correct, in order to reveal additional influence factors we can recommend more complex searching methods, for example, functional modeling of the measurement process according to the methodology IDEF0 [9]. Finally, verified according to representativeness condition assessment S R (B1 − B5) can be used in Formula (6) as a summand SRW U . Assessment of the uncertainty u (U) for direct current voltage measurements U , obtained from Formula (6) according to method validation results, can be considered as correct. Task 2 (model М1.2, Figure 2). Assessment of the uncertainty u (R SH ) of the valid value of the shunt resistance R SH .
To evaluate u (R SH ) it is reasonable to use model approach. To create an adequate measurement model, the laboratory realized an approach based on the application of cause-and-effect diagrams. For identification of included quantities x i the laboratory used classification of the constituents of the total measurement uncertainty from the classical error theory 4 : 1) instrumental, 2) methodological, 3) subjective, 4) measurement conditions. Further expert analysis of these constituents allowed to state the following: -possible methodological ∆ method and subjective ∆ operator factors, that influence the shunt accuracy, are negligible; -among the factors of measurement conditions ∆ condition potentially significant factor "temperature" is identified. However, taking into account the fact, that routine measurements will be carried out under normal conditions, the factor "temperature" is recognized as only slightly influencing shunt accuracy characteristics.
As the result, the final measurement model is as follows: where R SH calib -valid value of the shunt resistance R SH obtained from calibration certificate. Hence, correct model of uncertainty of the resistance value reproduced by the shunt according to GUM 1 is as follows: where the value u (R SH calib ) is also obtained from the calibration certificate (by type В).
After obtaining reliable uncertainty estimates for direct measurements results u (U ) and u (R SH ) it is necessary to return to the first stage of the resolution of the task of the evaluating the indirect measurements results uncertainty (model М1, Figure 2). Values u (U ) and u (R SH ) are substituted in the Formula (5), that then gives assessment of u с (I ) looked for. Therefore, the task is solved with maximum level of rationality and correctness.

Conclusion
Thus it can be stipulated, that a correct estimation of the measurements results uncertainty by a specific method can be carried out using a combined approach, efficacy and universality of which is described in this article. Any measurement process can be conventionally divided into fragmentscombination objects (resource components of the measurement process or its individual operations), each from shall be considered as individual evaluation element. In order to assess intermediate total uncertainty of each object of combining, we can equivalently select a model approach or an empiric approach. This selection is mainly based on efficacy of solving the problem. To improve empiric approach validity the sufficiency criteria of the measurement method uncertainty examination is recommended. Evaluation of the total uncertainty of the final measurement result should be done by complexation of assessments of individual total uncertainties of the results of all fragments according to the "law of errors distribution" of the GUM approach.