Вплив випадкових відхилень у результатах вимірювань на непевність екстремальних спостережень
Date
2018-02-26
Authors
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Видавництво Львівської політехніки
Abstract
Здійснено аналіз та кількісне оцінювання впливу випадкових відхилень у результатах вимірювань на
розширену непевність екстремальних спостережень, які є критичними під час контролю якості багатьох різновидів
продукції. Знайдено значення коефіцієнтів довірчих границь екстремального (мінімального) спостереження, залежно від
комбінацій різних розподілів значень технологічного розкиду досліджуваного параметра (від зразка до зразка) та випадкових
впливів, пов’язаних із самим вимірюванням цих параметрів. Подано результати досліджень для n = 5 кількості спостережень
і таких комбінацій розподілів спостережень і випадкових відхилень: нормальний-рівномірний, рівномірний-рівномірний,
рівномірний-нормальний за різного співвідношення їхніх стандартних відхилень складових. На підставі аналізу одержаних
результатів зроблено висновки, що у разі нестачі інформації про розподіл випадкових впливів коефіцієнти для обчислення
розширеної непевності з достатньою для практики точністю (декілька відсотків) можна взяти такими, як для нормального
розподілу. Результати досліджень можна використовувати для опрацювання результатів вимірювань під час контролю
параметрів якості продукції та виробів у промисловості, сільському господарстві та медицині.
In connection with the set requirements for processing measurement results the attention is considered to analysis of instrumental component of result’s uncertainty. While production control in industry, the mentioned component for concrete measuring instrument with the concrete measurement results has to be determined and analyzed. Important is the question of random impacts effect on to uncertainty of extreme (minimum or maximum) observations. Since in the practice of the test products qualifying there exist the cases when the result of such measurement is an extreme value: the minimal value xmin that is the first one xmin = x1 from ordered observations or the maximal xmax that is the last one xmax = xn from ordered observations. As this question seems to be contemporary, the article is devoted to research of random deviations influence on the uncertainty of extreme observations measurement results. Random deviations in measuring instrument indications are caused by different inside and outside effects. Their avoidance is impossible. To provide the reliable result of the experiment, it is important to take into account the impact of these random deviations in evaluating the uncertainty of processing the measurement results. We estimate here the value of coefficients of the confidence limits of the extreme (minimum) observation and the values of relative errors of the approximate values of the expansion coefficient, depending on the ratio of the standard deviations σr / σx related with two components: (i) instrumental one (σr ) and (ii) dispersion of the parameter of testing samples (σx ), for the number of observations n = 5 and three combinations of the distributions of both components. Research is fulfilled for: (i) the normal distribution of the parameter of testing samples and the uniform distribution of the instrumental component (ii) the uniform distribution of the parameter of testing samples and uniform distribution of the instrumental component and (iii) the uniform distribution of the parameter of testing samples and normal distribution of the instrumental component. With the aim of quality comparison the change in the distribution shape, the histograms of the normalized relative deviation z1,у1 and minimal observation у1 are built. If we measure xi the value of ith tested sample parameter, the random impacts Δri cause the changes of the considered observations yi = xi+Δri. Then standard deviation of registered observation becomes bigger than the standard deviation σx of parameter x. In every measurement these changes are random, and their impact can be described by convolution of the distribution px(x) of the tested parameter values and the random effects distribution pr(Dr). If the distribution of observations and random effects is normal, the its density p1(z1) and other parameters including the expansion coefficient are immutable and remain such as for normal distribution. If the distribution differs from the normal, the resultant distribution is normalized in the next way; due to this the expansion coefficient even less differs from to the expansion coefficient calculated according to the distribution of the observations themselves. I.e., for the number of observations n ≤ 10 the expansion coefficient does not exceed a few percent. Dependences of the expansion coefficients on the random deviation impacts of measuring instruments results at different distributions are investigated byMonte Carlo method. The researches confirm the approximation obtained in the theoretical analysis of the random influences in measuring results. Therefore, in calculation of the expanded uncertainty of extreme observation when distributions of both components in measurements are known, the random impacts can be taken into account directly. In case of shortage of information on distribution of random affects, the coefficients for the calculating the expanded uncertainty with sufficient accuracy (for practice equal to the few percent) are accepted such as for normal distribution.
In connection with the set requirements for processing measurement results the attention is considered to analysis of instrumental component of result’s uncertainty. While production control in industry, the mentioned component for concrete measuring instrument with the concrete measurement results has to be determined and analyzed. Important is the question of random impacts effect on to uncertainty of extreme (minimum or maximum) observations. Since in the practice of the test products qualifying there exist the cases when the result of such measurement is an extreme value: the minimal value xmin that is the first one xmin = x1 from ordered observations or the maximal xmax that is the last one xmax = xn from ordered observations. As this question seems to be contemporary, the article is devoted to research of random deviations influence on the uncertainty of extreme observations measurement results. Random deviations in measuring instrument indications are caused by different inside and outside effects. Their avoidance is impossible. To provide the reliable result of the experiment, it is important to take into account the impact of these random deviations in evaluating the uncertainty of processing the measurement results. We estimate here the value of coefficients of the confidence limits of the extreme (minimum) observation and the values of relative errors of the approximate values of the expansion coefficient, depending on the ratio of the standard deviations σr / σx related with two components: (i) instrumental one (σr ) and (ii) dispersion of the parameter of testing samples (σx ), for the number of observations n = 5 and three combinations of the distributions of both components. Research is fulfilled for: (i) the normal distribution of the parameter of testing samples and the uniform distribution of the instrumental component (ii) the uniform distribution of the parameter of testing samples and uniform distribution of the instrumental component and (iii) the uniform distribution of the parameter of testing samples and normal distribution of the instrumental component. With the aim of quality comparison the change in the distribution shape, the histograms of the normalized relative deviation z1,у1 and minimal observation у1 are built. If we measure xi the value of ith tested sample parameter, the random impacts Δri cause the changes of the considered observations yi = xi+Δri. Then standard deviation of registered observation becomes bigger than the standard deviation σx of parameter x. In every measurement these changes are random, and their impact can be described by convolution of the distribution px(x) of the tested parameter values and the random effects distribution pr(Dr). If the distribution of observations and random effects is normal, the its density p1(z1) and other parameters including the expansion coefficient are immutable and remain such as for normal distribution. If the distribution differs from the normal, the resultant distribution is normalized in the next way; due to this the expansion coefficient even less differs from to the expansion coefficient calculated according to the distribution of the observations themselves. I.e., for the number of observations n ≤ 10 the expansion coefficient does not exceed a few percent. Dependences of the expansion coefficients on the random deviation impacts of measuring instruments results at different distributions are investigated byMonte Carlo method. The researches confirm the approximation obtained in the theoretical analysis of the random influences in measuring results. Therefore, in calculation of the expanded uncertainty of extreme observation when distributions of both components in measurements are known, the random impacts can be taken into account directly. In case of shortage of information on distribution of random affects, the coefficients for the calculating the expanded uncertainty with sufficient accuracy (for practice equal to the few percent) are accepted such as for normal distribution.
Description
Keywords
вимірювання, екстремальні спостереження, непевність результату, випадкові відхилення, метод Монте-Карло, measurement, extreme observations, uncertainty of result, random deviations, Monte Carlo method
Citation
Дорожовець М. М. Вплив випадкових відхилень у результатах вимірювань на непевність екстремальних спостережень / М. М. Дорожовець, І. В. Бубела // Вимірювальна техніка та метрологія : міжвідомчий науково-технічний збірник. — Львів : Видавництво Львівської політехніки, 2018. — Том 79. — № 1. — С. 5–11.