BazTech - Yadda

1

Evaluation of the uncertainty type A of the random stationary signal component from its autocorrelated observations

Warsza Z. L., Dorozhovets M.

Measurement Automation Monitoring

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2015

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Vol. 61, No. 8

399--402

EN

A proposal of evaluation of the uncertainty type A of the stationary random component of measured signal from its regularly sampled observations when they are auto-correlated is described. In the first step the regularly variable components of the signal are identified and removed from the raw sample data. Then upgreaded formulas for standard uncertainty type A of the sample and of the mean value are expressed with use the correction coefficients or the so-called "effective number" of observations. These quantities depend on number of observations and on the autocorrelation function of the sample cleaned from regular components. Two methods of finding and estimating the autocorrelation function for the sample data are also described. Some numerical examples are included.

2

Niepewność typu A pomiaru o obserwacjach samoskorelowanych

Warsza Z. L., Zięba A.

Pomiary Automatyka Kontrola

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2012

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R. 58, nr 2

157-162

PL

Omówiono ograniczenia zalecanej w Przewodniku GUM metody wyznaczania niepewności pomiarów typu A. Opisano rozszerzenie jej na pomiary o równomiernym próbkowaniu menzurandu z uwzględnieniem wpływu funkcji autokorelacji wartości obserwacji. Przedstawiono poprzedzającą niezbędną identyfikację i usunięcie składowych regularnie zmiennych z surowych danych pomiarowych. Podano wzory dla równoważnej, tzw. efektywnej liczby nieskorelowanych obserwacji ηeff, zależnej od funkcji autokorelacji ρk próbki. Umożliwia ona poprawne wyznaczenie niepewności pomiarów według dotychczasowej procedury GUM. Omówiono sposób oszacowania estymaty funkcji autokorelacji τk z danych pomiarowych. Rozważania zilustrowano przykładami liczbowymi.

EN

Expanding of the application range of the present formalism of GUM to the case of regularly sampled mutually correlated observations is proposed. First, the obvious previous identification and cleaning of the raw sample data from regularly variable components is discussed briefly. The formulae for standard deviation and standard deviation of the mean are expressed with the use of the so-called effective number of observation ηeff. This quantity depends of real number of observation n and elements of the autocorrelation function ρk. The another parameter named effective degree of freedom νeff describes the dispersion of both estimators of standard deviation and can be used to calculate the expanded uncertainty. We also show how to adopt this formalism if only an estimate τk of the ACF derived from a sample is available. A novel method is introduced based on truncation of the τk function at the point of its first transit through zero (FTZ). This method can be applied to non-negative ACFs which occurs most often in practice. Considerations are illustrated by the numerical example.

3

Szacowanie niepewności w pomiarach z autokorelacją obserwacji

Warsza Z. L.

Elektronika : konstrukcje, technologie, zastosowania

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2012

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Vol. 53, nr 8

108-113

PL

Opisano rozszerzenie podanej w Przewodniku GUM metody wyznaczania niepewności typu A na pomiary o równomiernym próbkowaniu mezurandu w funkcji czasu z uwzględnieniem wpływu autokorelacji obserwacji. Obejmuje ona wyodrębnienie składowej losowej w surowych danych pomiarowych przez identyfikację i usunięcie z nich estymatora sygnału menzurandu oraz innych składowych regularnie zmiennych i outlierów. Podano wzory do obliczania odchylenia standardowego dla pojedynczej obserwacji i dla wartości średniej próbki. Uwzględniają one autokorelację przez zastosowanie tzw. „efektywnej liczby” równoważnych nieskorelowanych obserwacji. Dzięki temu można oszacować niepewność rozszerzoną wg dotychczasowej procedury GUM. Podano też metodę wyznaczania estymaty funkcji autokorelacji z danych próbki. Rozważania zilustrowano przykładami.

EN

Expanding of the application range of the GUM uncertainty evaluation method to the case of autocorrelated observations of the regularly sampled in time measurand is described. As the first step the random components of the sample are selected by identification estimate of the measured signal and as well other regularly variable components and removing them from the raw data. Then for the remainder random component with autocorrelated data, given are formulas for standard deviation of the single observation and of the mean value of sample. Formulas are expressed by means of an equivalent so-called „effective number” of uncorrelated observations, which depends on the sample autocorrelation function. Calculation of the expanded uncertainty due GUM recommendations is now possible. As proposal for use in practice given is also method of the estimation of autocorrelation function from sample data. Considerations are illustrated by two numerical examples.

4

Standard Deviation of the Mean of Autocorrelated Observations Estimated with the Use of the Autocorrelation Function Estimated From the Data

Zięba A., Ramza P.

Metrology and Measurement Systems

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2011

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Vol. 18, nr 4

529-542

EN

Prior knowledge of the autocorrelation function (ACF) enables an application of analytical formalism for the unbiased estimators of variance s²a and variance of the mean s²a(x‾). Both can be expressed with the use of so-called effective number of observations neff. We show how to adopt this formalism if only an estimate {rk} of the ACF derived from a sample is available. A novel method is introduced based on truncation of the {rk} function at the point of its first transit through zero (FTZ). It can be applied to non-negative ACFs with a correlation range smaller than the sample size. Contrary to the other methods described in literature, the FTZ method assures the finite range 1 < nˆeff ≤ n for any data. The effect of replacement of the standard estimator of the ACF by three alternative estimators is also investigated. Monte Carlo simulations, concerning the bias and dispersion of resulting estimators sa and sa(x‾), suggest that the presented formalism can be effectively used to determine a measurement uncertainty. The described method is illustrated with the exemplary analysis of autocorrelated variations of the intensity of an X-ray beam diffracted from a powder sample, known as the particle statistics effect.

5

Effective number of observations and unbiased estimators of variance for autocorrelated data - an overview

Zięba A.

Metrology and Measurement Systems

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2010

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Vol. 17, nr 1

3-16

EN

When observations are autocorrelated, standard formulae for the estimators of variance, s², and variance of the mean, s²(x), are no longer adequate. They should be replaced by suitably defined estimators, sa² and sa²(x), which are unbiased given that the autocorrelation function is known. The formula for sa² was given by Bayley and Hammersley in 1946, this work provides its simple derivation. The quantity named effective number of observations neff is thoroughly discussed. It replaces the real number of observations n when describing the relationship between the variance and variance of the mean, and can be used to express sa² and sa²(x) in a simple manner. The dispersion of both estimators depends on another effective number called the effective degrees of freedom veff. Most of the formulae discussed in this paper are scattered throughout the literature and not very well known, this work aims to promote their more widespread use. The presented algorithms represent a natural extension of the GUM formulation of type-A uncertainty for the case of autocorrelated observations.