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Research paper thumbnail of Estimation of the uncertainty bands of regression line for correlated data of variable Y using the GUM rules

Ukrainian Metrological Journal

The estimation of the accuracy of the linear regression method used for measurements based on the... more The estimation of the accuracy of the linear regression method used for measurements based on the GUM recommendations is considered. The impact of correlation and autocorrelation of the variable Y data together with type A and type B uncertainties, not provided in statistical literature about regression methods, is discussed. The theoretical backgrounds are given. The simulated examples of determining the uncertainty bands of the regression line fitted to the measured points with different cases of correlated values and absolute and relative uncertainties of type A and type B of the dependent variable Y are considered.

Research paper thumbnail of Regression and Uncertainty of a Straight-Line for Measurements of x and y Variables with All Correlations

Pomiary Automatyka Robotyka

The work continues the series of publications on the estimation of the parameters of the equation... more The work continues the series of publications on the estimation of the parameters of the equation and the limits of the uncertainty band of the straight-line y = ax + b fitted to the measurement results of both coordinates of the tested points with the use of the linear regression method. A general case was considered when these coordinates have different uncertainties and there are all possible autocorrelations and cross-correlations. Description of matrix equations was used. The results of the coordinate measurements are presented as elements of the X and Y vectors. The propagation of their uncertainty was described by the UZ covariance matrix with four component matrices, i.e., UX and UY – for the uncertainties and autocorrelations of X and of Y, and UXY and its transposition U – for the cross-correlations. The equation of a straight line and of the borders of its uncertainty band are given. Obtained them for the function of parameters a and b satisfying the so-called total crite...

Research paper thumbnail of Dwuwymiarowy model pomiaru dla typowych, założonych rozkładów prawdopodobieństwa wielkości wejściowych

Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki Politechniki Gdańskiej, 2019

Research paper thumbnail of Ocena niepewności punktów charakterystyki z dwu pomiarów kontrolnych

Research paper thumbnail of Extended Method of Estimation Uncertainties of Indirect Multivariable Measurements on the Twoport Example

Ukrainian Metrological Journal, 2020

The new extended mathematical model for evaluation uncertainties of indirect multivariable measur... more The new extended mathematical model for evaluation uncertainties of indirect multivariable measurements, which upgrades the method given in Supplement 2 of guide GUM, is presented. In this model the uncertainties and correlations of parameters of the processing function are taken also into account. This model can be used for multivariable measurements and to describe the accuracy of instruments and systems that perform such measurements. The estimation of uncertainties of voltage and current on the output of a twoport network from indirect measurements on its input with considering influences the uncertainties of twoport elements is included.

Research paper thumbnail of Uncertainty of Measurement in the Linear Regression Method. Part 1. Straight-line and its Uncertainty Bands for Uncorrelated Measurement Data

Pomiary Automatyka Robotyka, 2020

Zezwala się na korzystanie z artykułu na warunkach licencji Creative Commons Uznanie autorstwa 3.... more Zezwala się na korzystanie z artykułu na warunkach licencji Creative Commons Uznanie autorstwa 3.0 1. Wprowadzenie Metoda regresji liniowej jest podstawowym narzędziem matematycznym stosowanym do wyznaczania linii prostej, lub innej zlinearyzowanej funkcji, dopasowanej do danych pomiarowych według określonego kryterium [1-4]. Ocenę dokładności wyników pomiarów w służbach metrologicznych, nauce, przemyśle i wielu innych dziedzinach dokonuje się obecnie według zaleceń Przewodnika Wyznaczania Niepewności GUM [5], opisanych też w literaturze [6-9]. W literaturze o metodzie regresji liniowej dokładność wyznaczania parametrów linii prostej ocenia się statystycznie tylko z niepewności eksperymentu pomiarowego. Wpływu niepewności typu B nie rozpatrywano. Poprawna heurystyczna ocena tej niepewności jako spodziewanego skutku różnych oddziaływań, nieznanych co do wartości

Research paper thumbnail of Wyznaczanie niepewności w pomiarach wieloparametrowych wielkości o skorelowanych składowych typu A oraz B

Research paper thumbnail of Estimation of vector uncertainties of multivariable indirect instrumental measurement systems on the star circuit example

Journal of Physics: Conference Series, 2018

Signal processing in a multi-variable indirect measurement system and its uncertainties is consid... more Signal processing in a multi-variable indirect measurement system and its uncertainties is considered. It was proposed to extend the vector method of estimating uncertainties, given in Supplement 2 to GUM for the use to describe the accuracy of instrumental systems for indirect multivariable measurements. A formula for the covariance matrix of relative uncertainties is also given. As the example, the covariance matrix for indirect measurements of the star form circuit resistances from its terminals was determined and influence of of the measurement channels uncertainties are analyzed.

Research paper thumbnail of Numerical determination of continuous ray tracing: the four-component method

Applied Optics, 1994

Two interpolants are presented for Runge-Kutta ray tracing using the four-component method. Such ... more Two interpolants are presented for Runge-Kutta ray tracing using the four-component method. Such a continuous interpolant of ray trajectory connects only two adjacent Runge-Kutta points and requires only the information needed to find the position and ray vector for interpolant of ray position and ray vector to the fourth-order error. However Runge-Kutta schemes in contrast to other schemes are not based on approximating the continuous ray path with a polynomial. For this reason an optimal interpolant is shown that gives the same accuracy for calculating ray position and ray vector as a single-step integration in the Runge-Kutta scheme.

Research paper thumbnail of Computing optical-path length in inhomogeneous media—an alternative method

OSA Annual Meeting

In this paper a new method for OPL computation for gradient-index media has been presented. This ... more In this paper a new method for OPL computation for gradient-index media has been presented. This method is suitable for use with the fast and accurate ray-trace procedures which were developed earlier.

Research paper thumbnail of Method of estimation uncertainties of indirect multivariable measurements including the accuracy of processing function as extension of GUM-S2

Series on Advances in Mathematics for Applied Sciences, 2022

Research paper thumbnail of Estimation of Uncertainties in Indirect Multivariable Measurements: Part 2. Influence of the Processing Function Accuracy

Advances in Intelligent Systems and Computing, 2020

This two-part paper presents an improved version of the method of evaluation uncertainties in the... more This two-part paper presents an improved version of the method of evaluation uncertainties in the multivariable indirect measurements. It has an extended scope of application compared to the method given in Supplement 2 of the international guide with the acronym GUM [1], which assumes a perfectly accurate parameters multidimensional function of processing input measurement data. Part 1 discussed the cases of various correlations of measured values. The law of variance propagation is described in the form of dependencies between input and output covariance matrices. General formulas for uncertainty and correlation coefficients at the output are given as well as formulas for several characteristic cases. It has been shown that for pairs of all measured quantities, the effect of correlation between sets of deviations with uncertainty for each of types A and B should be considered separately. Proposed is to calculate output covariance matrices of correlated variables separately for uncertainties type A and type B An example of associated two-parameter (2D) measurements is also discussed. In this Part 2, the effect of uncertainties of the processing function parameters on the elements of output multimeasurand covariance matrix has been determined. General formulas for full correlation and its absence are given. An example of the evaluation uncertainties of indirect current and voltage measurements on the inaccessible input of the T-type impedance twoport network based on measurements at its output is considered in detail. Final conclusions are given. The proposed upgraded method is versatile, because it allows both the estimation of the accuracy of indirect multivariable measurements as well a description of the accuracy of instruments and measuring systems performing such measurements. Keywords: Multivariable measurements Á Multi-measurand Á Propagation of uncertainties Á Covariance matrix Á Correlation coefficient Á Correlations of deviations of type A or type B Á Uncertainties of processing function

Research paper thumbnail of Uncertainty Bands of the Regression Line for Data with Type A and Type B Uncertainties of Dependent Variable Y

Advances in Intelligent Systems and Computing, 2021

Research paper thumbnail of Uncertainties of Multivariable Indirect Measurements of DC Electrical Circuits

In this paper three examples of processing uncertainties of a indirect multi-variable measurement... more In this paper three examples of processing uncertainties of a indirect multi-variable measurement system are considered. It was proposed to extend the vector method of estimating measurement uncertainties, given in Supplement 2 to GUM, on the statistical description of the accuracy of whole ranges of indirect multivariable measurement system. Formula for the covariance matrix of relative uncertainties of the vector measurand is given. The covariance matrixes of uncertainties of few DC electrical measurement circuits are presented, i.e.: for indirect measurement of three resistances with using them in three variants of balanced Wheatstone bridge or without disconnection this circuit but with apply unconventional current supplies; the measurement of three internal resistances of the star circuit from its terminals and estimation of uncertainty of powers of two currents if two other currents are measured and their uncertainties are known. Formulas for absolute and relative uncertainties and their correlation coefficients are given. The general conclusion is that in the description accuracy of multivariable measurement systems the relative uncertainties are sometimes preferable than the absolute ones, and uncertainties of their main measurement functions have been also considered.

Research paper thumbnail of Estimation of Uncertainties in Indirect Multiparameter Measurements of Correlated Quantities

2019 12th International Conference on Measurement, 2019

The paper presents an upgraded version of the vector method of evaluation of multiparameter measu... more The paper presents an upgraded version of the vector method of evaluation of multiparameter measurement uncertainties stated in the Supplement 2 to GUM guide. This was done on the example of two-parameter jointed measurements. It consists of the correlation of individual components of the type A and/or type B uncertainty of input measurands. The general formulas for the covariance matrix, final uncertainties, and correlation coefficient were determined. The 3D graph shows the correlation coefficients of the output quantities as a function of the type B contributions in the uncertainty of two input quantities. It has been demonstrated that the inclusion of correlations of uncertainty components makes the uncertainty evaluations more reliable and accurate.

Research paper thumbnail of Estimation of the Extended Uncertainty of Points of the Characteristic from Two Control Measurements

Pomiary Automatyka Robotyka, 2020

Research paper thumbnail of Extended Method for Evaluation Uncertainties of Indirect Multi-Parameter Measurements and of Circuits for these Measurements. Part 1. Influence of Correlation and Uncertainties of Processing Function – Basic Formulas

Pomiary Automatyka Robotyka, 2019

Zezwala się na korzystanie z artykułu na warunkach licencji Creative Commons Uznanie autorstwa 3.... more Zezwala się na korzystanie z artykułu na warunkach licencji Creative Commons Uznanie autorstwa 3.0 1. Wprowadzenie W 1993 r. w międzynarodowym przewodniku wyznaczania niepewności pomiarów o angielskim akronimie GUM [1] wprowadzono nowe pojęcie "niepewność" do oceny dokładności pomiarów. Pojawiła się wówczas niejednolitość z dotychczas stosowanym opisem dokładności przyrządów, urządzeń i systemów pomiarowych przez błędy pomiarowe. Producenci aparatury pomiarowej podają nadal maksymalny dopuszczalny błąd przyrządu MPE (ang. maximum permissible error) [3]. Przy założeniu równomiernego rozkładu odchy

Research paper thumbnail of Upgraded Method of the Estimation of Uncertainties in Multiparameter Measurements. Part 1. Theoretical Basis for Correlated Measured Variables

Pomiary Automatyka Robotyka, 2019

Research paper thumbnail of Estimation of the uncertainty of function values from measurements at control points

Pomiary Automatyka Robotyka, 2018

Research paper thumbnail of Estimation of uncertainties in multivariable indirect measurements on the example of two resistance circuits 3D

Pomiary Automatyka Robotyka, 2018

Research paper thumbnail of Estimation of the uncertainty bands of regression line for correlated data of variable Y using the GUM rules

Ukrainian Metrological Journal

The estimation of the accuracy of the linear regression method used for measurements based on the... more The estimation of the accuracy of the linear regression method used for measurements based on the GUM recommendations is considered. The impact of correlation and autocorrelation of the variable Y data together with type A and type B uncertainties, not provided in statistical literature about regression methods, is discussed. The theoretical backgrounds are given. The simulated examples of determining the uncertainty bands of the regression line fitted to the measured points with different cases of correlated values and absolute and relative uncertainties of type A and type B of the dependent variable Y are considered.

Research paper thumbnail of Regression and Uncertainty of a Straight-Line for Measurements of x and y Variables with All Correlations

Pomiary Automatyka Robotyka

The work continues the series of publications on the estimation of the parameters of the equation... more The work continues the series of publications on the estimation of the parameters of the equation and the limits of the uncertainty band of the straight-line y = ax + b fitted to the measurement results of both coordinates of the tested points with the use of the linear regression method. A general case was considered when these coordinates have different uncertainties and there are all possible autocorrelations and cross-correlations. Description of matrix equations was used. The results of the coordinate measurements are presented as elements of the X and Y vectors. The propagation of their uncertainty was described by the UZ covariance matrix with four component matrices, i.e., UX and UY – for the uncertainties and autocorrelations of X and of Y, and UXY and its transposition U – for the cross-correlations. The equation of a straight line and of the borders of its uncertainty band are given. Obtained them for the function of parameters a and b satisfying the so-called total crite...

Research paper thumbnail of Dwuwymiarowy model pomiaru dla typowych, założonych rozkładów prawdopodobieństwa wielkości wejściowych

Zeszyty Naukowe Wydziału Elektrotechniki i Automatyki Politechniki Gdańskiej, 2019

Research paper thumbnail of Ocena niepewności punktów charakterystyki z dwu pomiarów kontrolnych

Research paper thumbnail of Extended Method of Estimation Uncertainties of Indirect Multivariable Measurements on the Twoport Example

Ukrainian Metrological Journal, 2020

The new extended mathematical model for evaluation uncertainties of indirect multivariable measur... more The new extended mathematical model for evaluation uncertainties of indirect multivariable measurements, which upgrades the method given in Supplement 2 of guide GUM, is presented. In this model the uncertainties and correlations of parameters of the processing function are taken also into account. This model can be used for multivariable measurements and to describe the accuracy of instruments and systems that perform such measurements. The estimation of uncertainties of voltage and current on the output of a twoport network from indirect measurements on its input with considering influences the uncertainties of twoport elements is included.

Research paper thumbnail of Uncertainty of Measurement in the Linear Regression Method. Part 1. Straight-line and its Uncertainty Bands for Uncorrelated Measurement Data

Pomiary Automatyka Robotyka, 2020

Zezwala się na korzystanie z artykułu na warunkach licencji Creative Commons Uznanie autorstwa 3.... more Zezwala się na korzystanie z artykułu na warunkach licencji Creative Commons Uznanie autorstwa 3.0 1. Wprowadzenie Metoda regresji liniowej jest podstawowym narzędziem matematycznym stosowanym do wyznaczania linii prostej, lub innej zlinearyzowanej funkcji, dopasowanej do danych pomiarowych według określonego kryterium [1-4]. Ocenę dokładności wyników pomiarów w służbach metrologicznych, nauce, przemyśle i wielu innych dziedzinach dokonuje się obecnie według zaleceń Przewodnika Wyznaczania Niepewności GUM [5], opisanych też w literaturze [6-9]. W literaturze o metodzie regresji liniowej dokładność wyznaczania parametrów linii prostej ocenia się statystycznie tylko z niepewności eksperymentu pomiarowego. Wpływu niepewności typu B nie rozpatrywano. Poprawna heurystyczna ocena tej niepewności jako spodziewanego skutku różnych oddziaływań, nieznanych co do wartości

Research paper thumbnail of Wyznaczanie niepewności w pomiarach wieloparametrowych wielkości o skorelowanych składowych typu A oraz B

Research paper thumbnail of Estimation of vector uncertainties of multivariable indirect instrumental measurement systems on the star circuit example

Journal of Physics: Conference Series, 2018

Signal processing in a multi-variable indirect measurement system and its uncertainties is consid... more Signal processing in a multi-variable indirect measurement system and its uncertainties is considered. It was proposed to extend the vector method of estimating uncertainties, given in Supplement 2 to GUM for the use to describe the accuracy of instrumental systems for indirect multivariable measurements. A formula for the covariance matrix of relative uncertainties is also given. As the example, the covariance matrix for indirect measurements of the star form circuit resistances from its terminals was determined and influence of of the measurement channels uncertainties are analyzed.

Research paper thumbnail of Numerical determination of continuous ray tracing: the four-component method

Applied Optics, 1994

Two interpolants are presented for Runge-Kutta ray tracing using the four-component method. Such ... more Two interpolants are presented for Runge-Kutta ray tracing using the four-component method. Such a continuous interpolant of ray trajectory connects only two adjacent Runge-Kutta points and requires only the information needed to find the position and ray vector for interpolant of ray position and ray vector to the fourth-order error. However Runge-Kutta schemes in contrast to other schemes are not based on approximating the continuous ray path with a polynomial. For this reason an optimal interpolant is shown that gives the same accuracy for calculating ray position and ray vector as a single-step integration in the Runge-Kutta scheme.

Research paper thumbnail of Computing optical-path length in inhomogeneous media—an alternative method

OSA Annual Meeting

In this paper a new method for OPL computation for gradient-index media has been presented. This ... more In this paper a new method for OPL computation for gradient-index media has been presented. This method is suitable for use with the fast and accurate ray-trace procedures which were developed earlier.

Research paper thumbnail of Method of estimation uncertainties of indirect multivariable measurements including the accuracy of processing function as extension of GUM-S2

Series on Advances in Mathematics for Applied Sciences, 2022

Research paper thumbnail of Estimation of Uncertainties in Indirect Multivariable Measurements: Part 2. Influence of the Processing Function Accuracy

Advances in Intelligent Systems and Computing, 2020

This two-part paper presents an improved version of the method of evaluation uncertainties in the... more This two-part paper presents an improved version of the method of evaluation uncertainties in the multivariable indirect measurements. It has an extended scope of application compared to the method given in Supplement 2 of the international guide with the acronym GUM [1], which assumes a perfectly accurate parameters multidimensional function of processing input measurement data. Part 1 discussed the cases of various correlations of measured values. The law of variance propagation is described in the form of dependencies between input and output covariance matrices. General formulas for uncertainty and correlation coefficients at the output are given as well as formulas for several characteristic cases. It has been shown that for pairs of all measured quantities, the effect of correlation between sets of deviations with uncertainty for each of types A and B should be considered separately. Proposed is to calculate output covariance matrices of correlated variables separately for uncertainties type A and type B An example of associated two-parameter (2D) measurements is also discussed. In this Part 2, the effect of uncertainties of the processing function parameters on the elements of output multimeasurand covariance matrix has been determined. General formulas for full correlation and its absence are given. An example of the evaluation uncertainties of indirect current and voltage measurements on the inaccessible input of the T-type impedance twoport network based on measurements at its output is considered in detail. Final conclusions are given. The proposed upgraded method is versatile, because it allows both the estimation of the accuracy of indirect multivariable measurements as well a description of the accuracy of instruments and measuring systems performing such measurements. Keywords: Multivariable measurements Á Multi-measurand Á Propagation of uncertainties Á Covariance matrix Á Correlation coefficient Á Correlations of deviations of type A or type B Á Uncertainties of processing function

Research paper thumbnail of Uncertainty Bands of the Regression Line for Data with Type A and Type B Uncertainties of Dependent Variable Y

Advances in Intelligent Systems and Computing, 2021

Research paper thumbnail of Uncertainties of Multivariable Indirect Measurements of DC Electrical Circuits

In this paper three examples of processing uncertainties of a indirect multi-variable measurement... more In this paper three examples of processing uncertainties of a indirect multi-variable measurement system are considered. It was proposed to extend the vector method of estimating measurement uncertainties, given in Supplement 2 to GUM, on the statistical description of the accuracy of whole ranges of indirect multivariable measurement system. Formula for the covariance matrix of relative uncertainties of the vector measurand is given. The covariance matrixes of uncertainties of few DC electrical measurement circuits are presented, i.e.: for indirect measurement of three resistances with using them in three variants of balanced Wheatstone bridge or without disconnection this circuit but with apply unconventional current supplies; the measurement of three internal resistances of the star circuit from its terminals and estimation of uncertainty of powers of two currents if two other currents are measured and their uncertainties are known. Formulas for absolute and relative uncertainties and their correlation coefficients are given. The general conclusion is that in the description accuracy of multivariable measurement systems the relative uncertainties are sometimes preferable than the absolute ones, and uncertainties of their main measurement functions have been also considered.

Research paper thumbnail of Estimation of Uncertainties in Indirect Multiparameter Measurements of Correlated Quantities

2019 12th International Conference on Measurement, 2019

The paper presents an upgraded version of the vector method of evaluation of multiparameter measu... more The paper presents an upgraded version of the vector method of evaluation of multiparameter measurement uncertainties stated in the Supplement 2 to GUM guide. This was done on the example of two-parameter jointed measurements. It consists of the correlation of individual components of the type A and/or type B uncertainty of input measurands. The general formulas for the covariance matrix, final uncertainties, and correlation coefficient were determined. The 3D graph shows the correlation coefficients of the output quantities as a function of the type B contributions in the uncertainty of two input quantities. It has been demonstrated that the inclusion of correlations of uncertainty components makes the uncertainty evaluations more reliable and accurate.

Research paper thumbnail of Estimation of the Extended Uncertainty of Points of the Characteristic from Two Control Measurements

Pomiary Automatyka Robotyka, 2020

Research paper thumbnail of Extended Method for Evaluation Uncertainties of Indirect Multi-Parameter Measurements and of Circuits for these Measurements. Part 1. Influence of Correlation and Uncertainties of Processing Function – Basic Formulas

Pomiary Automatyka Robotyka, 2019

Zezwala się na korzystanie z artykułu na warunkach licencji Creative Commons Uznanie autorstwa 3.... more Zezwala się na korzystanie z artykułu na warunkach licencji Creative Commons Uznanie autorstwa 3.0 1. Wprowadzenie W 1993 r. w międzynarodowym przewodniku wyznaczania niepewności pomiarów o angielskim akronimie GUM [1] wprowadzono nowe pojęcie "niepewność" do oceny dokładności pomiarów. Pojawiła się wówczas niejednolitość z dotychczas stosowanym opisem dokładności przyrządów, urządzeń i systemów pomiarowych przez błędy pomiarowe. Producenci aparatury pomiarowej podają nadal maksymalny dopuszczalny błąd przyrządu MPE (ang. maximum permissible error) [3]. Przy założeniu równomiernego rozkładu odchy

Research paper thumbnail of Upgraded Method of the Estimation of Uncertainties in Multiparameter Measurements. Part 1. Theoretical Basis for Correlated Measured Variables

Pomiary Automatyka Robotyka, 2019

Research paper thumbnail of Estimation of the uncertainty of function values from measurements at control points

Pomiary Automatyka Robotyka, 2018

Research paper thumbnail of Estimation of uncertainties in multivariable indirect measurements on the example of two resistance circuits 3D

Pomiary Automatyka Robotyka, 2018