Error Propagation Formula Research Articles

A Critical Note on the Guide to the Expression of Uncertainty in Measurement (GUM)

In 1809 Carl Friedrich Gauss' masterpiece “Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium” on the motion of celestial bodies was published. This book contains also a first version of the famous error propagation formula, which is still the basis of modern metrology and of any relevant national and international standard. Gauss' ideas are reflected in the GUM, the “Guide to the Expression of Uncertainty in Measurement”, published in 1993 by ISO, which is a key document for national measurement institutes and industrial calibration laboratories for evaluating uncertainty in the output of a measurement system. Uncertainty constitutes the main problem of any measurement theory. Because uncertainty is investigated in stochastics, any serious measurement theory should necessarily take into account the state of the art in stochastics. Stochastics was initiated by Jakob Bernoulli in his masterpiece “Ars conjectandi”, which appeared posthumously in 1713, i. e., about 100 years before Gauss established the basis of metrology. The difference between Jakob Bernoulli and Carl Friedrich Gauss is the following: Bernoulli aimed at quantifying and measuring uncertainty, while Gauss aimed at quantifying and measuring a measurement error. In this paper a modern measurement theory based on stochastics is briefly outlined. Subsequently, the antiquated concepts and methods as recommended in the GUM, which go back to the early 19th century, are discussed and their weaknesses are listed.

Stochastic Measurement Procedures Based on Stationary Time Series

Stochastic measurement procedures have been proposed and developed in several articles published in EQC. In contrast to the traditional measurement procedures that use the obsolete error propagation formula for taking into account measurement uncertainty, stochastic measurement procedures include the entire uncertainty of the measurement process in a natural and rational way. In this paper the concept of stochastic measurement procedure is extended to the case that the measurement process is given by a stochastic process or more precisely said by a time series. This case appears often for economic phenomena. In this paper, only the feasibility of such measurement procedures shall be demonstrated and the considerations are therefore restricted to some simple and rather unrealistic examples. In a realistic case an appropriate stochastic model must be developed following the rules of stochastic modelling.

Uncertainty calculation in life cycle assessments

Uncertainty is commonly not taken into account in LCA studies, which downgrades their usability for decision support. One often stated reason is a lack of method. The aim of this paper is to develop a method for calculating the uncertainty propagation in LCAs in a fast and reliable manner. The method is developed in a model that reflects the calculation of an LCA. For calculating the uncertainty, the model combines approximation formulas and Monte Carlo Simulation. It is based on virtual data that distinguishes true values and random errors or uncertainty, and that hence allows one to compare the performance of error propagation formulas and simulation results. The model is developed for a linear chain of processes, but extensions for covering also branched and looped product systems are made and described. The paper proposes a combined use of approximation formulas and Monte Carlo simulation for calculating uncertainty in LCAs, developed primarily for the sequential approach. During the calculation, a parameter observation controls the performance of the approximation formulas. Quantitative threshold values are given in the paper. The combination thus transcends drawbacks of simulation and approximation. The uncertainty question is a true jigsaw puzzle for LCAs and the method presented in this paper may serve as one piece in solving it. It may thus foster a sound use of uncertainty assessment in LCAs. Analysing a proper management of the input uncertainty, taking into account suitable sampling and estimation techniques; using the approach for real case studies, implementing it in LCA software for automatically applying the proposed combined uncertainty model and, on the other hand, investigating about how people do decide, and should decide, when their decision relies on explicitly uncertain LCA outcomes-these all are neighbouring puzzle pieces inviting to further work.

New maximum‐likelihood method for reduction of distribution‐coefficient data

AbstractThe sampling distributions for the activity coefficient γ and distribution ratio (or K‐value) in binary fluid mixtures are derived from the assumptions that the pressure errors are negligible, and that the vapor and liquid composition measurements from which they are derived are normally distributed with known mean and variance. The distribution of K is shown to be not normal, with deviations from normality becoming particularly pronounced at the extremes of the composition range. The dispersion of this distribution is considerably greater than that predicted by the classic error‐propagation formula. Deviation from normality invalidates the most important assumption implicit in the use of the least‐squares criterion as the basis for parameter estimation from activity‐coefficient data. This has important implications for the estimation of Henry's law constants for solutes in very dilute solutions, as well as for Raoult's law‐based infinite‐dilution activity coefficients. The use of the exact distributions of γ and K as the basis for a more rigorous maximum‐likelihood method for parameter estimation in excess Gibbs energy models based on either Henry's law or Raoult's law standard states is demonstrated.

Organic carbon flux in Shiraho coral reef (Ishigaki Island, Japan)

Organic carbon flux and community production rates were estimated on Shiraho coral reef (Ishigaki Island, Japan) from 19 to 26 September 1998. The daily net community production (P n ) and respiration rate (R) during the study period were 3 to 79 and 596 mmolC m -2 d -1 , respectively. This resulted in a daily gross community production (P g ) of 599 to 675 mmol C m -2 d -1 , The variation of P n associated with the uncertainty of the curve fitting parameters of light response curves for photosynthesis was estimated using an error propagation formula. The averaged P n ± SE was 36 ± 12 mmol C m -2 d -1 (n = 23), indicating that the P n was significantly positive (t-test, p < 0.05). The apparent fluxes of dissolved organic carbon (DOC) and particulate organic carbon (POC) on the reef were estimated as 30 to 36 and 5 to 7 mmol C m -2 d -1 , respectively; The sum of which was comparable with the P n during the study period. The sediment trap study conducted at 1 km off the reef and 40 m depth showed that the vertical flux of POC was 1.0 mmol C m -2 d -1 . The results indicated that 6 to 7 % of the P g was exported to offshore and about 14 to 20% of the POC exported from the reef flat and 0.2 % of the Pg reached 1 km off and 40 m depth.

Precision of prediction in second‐order calibration, with focus on bilinear regression methods

AbstractWe consider calibration of hyphenated instruments with particular focus on determination of the unknown concentrations of new specimens. A hyphenated instrument generates for each specimen a two‐way array of data. These are assumed to depend on the concentrations through a bilinear regression model, where each constituent is characterized by a pair of profiles to be determined in the calibration. We discuss the problem of predicting the unknown concentrations in a new specimen, after calibration. We formulate three different predictor construction methods, a ‘naive’ method, a least squares method, and a refined version of the latter that takes account of the calibration uncertainty. We give formulae for the uncertainty of the predictors under white noise, when calibration can be seen as precise. We refine these formulae to allow for calibration uncertainty, in particular when calibration is carried out by the bilinear least squares (BLLS) method or the singular value decomposition (SVD) method proposed by Linder and Sundberg (Chemometrics Intell. Lab. Syst. 1998; 42: 159–178). By error propagation formulae and previous results on the precision of $\widehat{A}$ and $\widehat{B}$ we can obtain approximate standard errors for the predicted concentrations, according to each of the two estimation methods. The performance of the predictors and the precision formulae is illustrated on both real (fluorescence) and simulated data. Copyright © 2002 John Wiley &amp; Sons, Ltd.

ON THE PAPER “INVERSE METHODS FOR ANALYZING AEROSOL SPECTROMETER MEASUREMENTS: A CRITICAL REVIEW”

In this paper several statements of Kandlikar and Ramachandran concerning the one-to-one correspondence between the spectrometric channels and aerosol size classes, overlaps of the kernels of integral equation, errors of the algebraic systems’ solutions and some inversion algorithms are discussed. It is shown that expression A −1 e ( A -linear algebraic system matrix, e -vector of the experimental errors) cannot be interpreted as the standard deviation of random errors and its adequate interpretation is given. The expressions for standard deviation of the system's solution are derived using the error propagation formula. It is indicated that known inequality for the errors of linear systems’ solutions including condition number of matrix cond is not applicable to the random errors, and the new inequality is given. The approach of smooth continuous size distribution function concept is shown to meet with difficulties since its actual realization is discrete, stochastic and has only two values: 0 and 1.

ERROR ANALYSIS OF DUAL-ENERGY GAMMA RADIATION MEASUREMENTS

Dual-energy gamma radiation systems are currently used to simultaneously determine either the dry bulk density of a porous medium and the volumetric content of one fluid, the volumetric fluid contents of two fluids, or the salt concentration and volumetric water content. In this paper an error analysis based on the well-known error propagation formula is presented for all three types of measurements. For each application, formulae are derived that calculate the variance of a measured variable as a function of the observed count rates, attenuation coefficients, and path length. These formulae can be implemented in data acquisition or postprocessing programs. In addition, formulae are presented that relate the variance to the values of the measured variables, measurement counting time, calibration counting time, path lengths, source intensities, and attenuation coefficients. The latter formulae benefit experimental design. The effect of counting and calibration time, attenuation coefficients, source strengths, and path lengths on the probable error are demonstrated. For each type of measurement, experiments are conducted to test the validity of the presented theory. It was shown that, in general, the difference between the measured value and the true value of a variable is less than 1 standard deviation. 14 refs., 5 figs.,more » 3 tabs.« less

The effect of neglecting correlations when propagating uncertainty and estimating the population distribution of risk.

Interest in examining both the uncertainty and variability in environmental health risk assessments has led to increased use of methods for propagating uncertainty. While a variety of approaches have been described, the advent of both powerful personal computers and commercially available simulation software have led to increased use of Monte Carlo simulation. Although most analysts and regulators are encouraged by these developments, some are concerned that Monte Carlo analysis is being applied uncritically. The validity of any analysis is contingent on the validity of the inputs to the analysis. In the propagation of uncertainty or variability, it is essential that the statistical distribution of input variables are properly specified. Furthermore, any dependencies among the input variables must be considered in the analysis. In light of the potential difficulty in specifying dependencies among input variables, it is useful to consider whether there exist rules of thumb as to when correlations can be safely ignored (i.e., when little overall precision is gained by an additional effort to improve upon an estimation of correlation). We make use of well-known error propagation formulas to develop expressions intended to aid the analyst in situations wherein normally and lognormally distributed variables are linearly correlated.

The Effect of Neglecting Correlations When Propagating Uncertainty and Estimating the Population Distribution of Risk

Interest in examining both the uncertainty and variability in environmental health risk assessments has led to increased use of methods for propagating uncertainty. While a variety of approaches have been described, the advent of both powerful personal computers and commercially available simulation software have led to increased use of Monte Carlo simulation. Although most analysts and regulators are encouraged by these developments, some are concerned that Monte Carlo analysis is being applied uncritically. The validity of any analysis is contingent on the validity of the inputs to the analysis. In the propagation of uncertainty or variability, it is essential that the statistical distribution of input variables are properly specified. Furthermore, any dependencies among the input variables must be considered in the analysis. In light of the potential difficulty in specifying dependencies among input variables, it is useful to consider whether there exist rules of thumb as to when correlations can be safely ignored (i.e., when little overall precision is gained by an additional effort to improve upon an estimation of correlation). We make use of well-known error propagation formulas to develop expressions intended to aid the analyst in situations wherein normally and lognormally distributed variables are linearly correlated.

A statistical approach for predicting accuracies of soil properties measured by single, double and dual gamma beams

SUMMARYA statistical analysis is developed to describe the accuracies of soil properties measured through single, double and dual gamma‐beam methods. The gamma sources are 137Cs for the high‐energy beam (662 keV) and 241Am for the low‐energy beam (60 keV). The theoretical confidence intervals of the thicknesses or water contents of samples are determined using a modification of the method of Gardner el al. (1972), who derived theoretical variances using standard error propagation formulae. The finding of Gardner et al. that, in the double‐beam method, errors in the soil bulk density and water content measurements are primarily caused by random emission from the sources, is confirmed by the theoretical confidence intervals and is applied to the single‐beam and dual‐beam methods. The effects of interferences between gamma rays in the dual‐beam method on the accuracies of estimations are examined. The minimum measurement time to attain a required accuracy can be predicted from the theoretical confidence intervals.

A Family of Highly Accurate Interpolation Functions for Magnitude-Mode Fourier Transform Spectroscopy

A new family of highly efficient interpolating functions, the KCe functions, KCe( ω) = ( aω2 + bω + c) e, where e is the exponent, is developed for three-point frequency interpolation of discrete, magnitude-mode, apodized Fourier transform spectra. The family is characterized by high interpolation accuracy and ease of implementation. Various members of the family can be generated by varying the exponent. Prior work from this laboratory indicated that the parabola is the interpolating function of choice for interpolation of discrete, apodized magnitude spectra. We show here that, compared to parabolic interpolation, KCe interpolation typically gives residual systematic errors which are lower by between one and two orders of magnitude. These systematic errors are analytically derived and the efficacy of interpolation is rigorously examined as a function of the KCe exponent, the number of zero-fillings, the amount of damping in the transient, and the window function used to apodize the spectrum. For Hanning-apodized spectra, the KC5.5 function gives the lowest residual systematic errors, which are typically 15 times less than those remaining after parabolic interpolation. Similarly, the KC6.6 function is optimal for Hamming-apodized spectra (22 times better than parabolic interpolation) and the KC9.5 function is optimal for Blackman-Harris-apodized spectra (80 times better than parabolic interpolation). By extrapolation from other optimal KCe functions, we estimate that the optimal KCe function for interpolation of Kaiser-Bessel-apodized spectra is KC12.5. Analytical formulae for propagation of random errors in spectral intensity into random errors in interpolated frequency are derived for parabolic interpolation and for KCe interpolation. These error propagation formulae give random errors which are inversely proportional to the SNR of the spectrum. These formulae are evaluated with the appropriate KCe exponent for each of the Hanning, the Hamming, and the Blackman-Harris windows. In all cases we find that the random error is essentially independent of both window type and interpolation scheme. While zero-filling prior to interpolation reduces the residual systematic frequency interpolation error, it increases the random frequency error. The increase in random error with higher levels of zero-filling is explained. Because the random errors are proportional to noise level, the optimal number of zero-fillings varies with SNR. If the apodizing window is chosen to match the dynamic range of the spectrum, as we have previously recommended, then the systematic error for KCe interpolation of non-zero-filled spectra is so low that the overall error is dominated by the random error. In this case, KCe interpolation is, for all intents and purposes, exact. Since the random error is minimized by no zero-filling, the lowest overall error will be achieved by a combination of no zero-filling and KCe interpolation. In constrast, the minimum total error for parabolic interpolation is achieved by interpolation of the once-zero-filled spectrum. A further advantage of KCe interpolation, over and above its lower total error, is that KCe interpolation obviates the need for zero-filling.

Explanation, verification and application of helical-axis error propagation formulas

Finite helical axes (screw axes or axes of rotation) are often used for the quantitative description of joint movement. The helical-axis parameters can be calculated from the co-ordinates of markers in the body segments comprising the joint, where the segments are modelled as rigid bodies which are observed in discrete positions. It appears from the biomechanical literature that the helical parameters are highly sensitive to landmark measurement errors. In the present paper, the mathematical results of Woltring et al. (1984) are explained from a geometrical point of view, and various recommendations in the literature for optimizing experimental procedures in the study of joint kinematics are discussed in the light of these findings. Furthermore, the error propagation formulas are verified through a simulation study, and applied in an experiment on in-vitro, tarsal joint movement.

Error analysis for the determination of tectonic rotation from paleomagnetic data

Appendix B is available with entire article on microfiche. Order from American Geophysical Union, 2000 Florida Avenue, N.W., Washington, D.C. 20009. Document B83‐001; $2.50. Payment must accompany order.Recent determinations of tectonic rotation and flattening from paleomagnetic data have exaggerated the size of the confidence limits or error bars by about 25%. This has been caused by the misuse of conventional error propagation formulas to derive one degree of freedom error bars from a two dimensional cone of confidence (α95). An analytical calculation, which is valid when error bars are small, shows that the correct error bars are 75% to 80% of the values derived in previous studies. I have used numerical integration to check on the applicability of the analytical solution and to calculate error bars when the analytical approximation cannot be used. The analytical approximation may be used whenever α95 is less than 10° and the mean inclination is not too close to vertical. Confidence limits given for tectonic rotation and flattening in recent literature are too large and should be reduced by 20–25%. Narrowing the error bars in this way permits a more accurate interpretation of paleomagnetic data.

A Taylor's Theorem—Central Limit Theorem Approximation: Its Use in Obtaining the Probability Distribution of Long-Range Profit

Although the functional relationship between long-range profit (LRP) and its various components is usually well defined, the difficulty of determining the probability distribution of LRP prevents an immediate answer to a question such as “What are the chances that the LRP associated with decision D will exceed X dollars?” In a situation such as this, the first thought is to use Monte Carlo (i.e., random sampling) in order to provide an approximation to the distribution of LRP. An alternative approach, the use of error propagation formulas based on a Taylor series expansion coupled with a Central Limit Theorem approximation, is discussed.

Practical Tests of Error Propagation Formulae and Additional Notes on they‐parallax Method

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