Measurement error estimates

V covariance matrix of measurement error estimates W covariance matrix of d... 

The method used here is based on a general application of the maximum-likelihood principle. A rigorous discussion is given by Bard (1974) on nonlinear-parameter estimation based on the maximum-likelihood principle. The most important feature of this method is that it attempts properly to account for all measurement errors. A discussion of the background of this method and details of its implementation are given by Anderson et al. (1978). 

Evaluating Indeterminate Error Although it is impossible to eliminate indeterminate error, its effect can be minimized if the sources and relative magnitudes of the indeterminate error are known. Indeterminate errors may be estimated by an appropriate measure of spread. Typically, a standard deviation is used, although in some cases estimated values are used. The contribution from analytical instruments and equipment are easily measured or estimated. Indeterminate errors introduced by the analyst, such as inconsistencies in the treatment of individual samples, are more difficult to estimate. 

Variations in measurable properties existing in the bulk material being sampled are the underlying basis for samphng theory. For samples that correctly lead to valid analysis results (of chemical composition, ash, or moisture as examples), a fundamental theoiy of sampling is applied. The fundamental theoiy as developed by Gy (see references) employs descriptive terms reflecting material properties to calculate a minimum quantity to achieve specified sampling error. Estimates of minimum quantity assumes completely mixed material. Each quantity of equal mass withdrawn provides equivalent representation of the bulk. 

Systematic Operating Errors Fifth, systematic operating errors may be unknown at the time of measurements. Wriile not intended as part of daily operations, leaky or open valves frequently result in bypasses, leaks, and alternative feeds that will add hidden bias. Consequently, constraints assumed to hold and used to reconcile the data, identify systematic errors, estimate parameters, and build models are in error. The constraint bias propagates to the resultant models. 

The percentage error in the temperature difference translates directly to the percentage error in the estimate Q. As temperature-measurement error increases, so does the heat transfer coefficient error. 

Model Development PreHminary modeling of the unit should be done during the familiarization stage. Interactions between database uncertainties and parameter estimates and between measurement errors and parameter estimates coiJd lead to erroneous parameter estimates. Attempting to develop parameter estimates when the model is systematically in error will lead to systematic error in the parameter estimates. Systematic errors in models arise from not properly accounting for the fundamentals and for the equipment boundaries. Consequently, the resultant model does not properly represent the unit and is unusable for design, control, and optimization. Cropley (1987) describes the erroneous parameter estimates obtained from a reactor study when the fundamental mechanism was not properly described within the model. 

Verneuil et al. (Verneuil, V.S., P. Yan, and F. Madron, Banish Bad Plant Data, Chemical Engineering Progress, October 1992, 45-51) emphasize the importance of proper model development. Systematic errors result not only from the measurements but also from the model used to analyze the measurements. Advanced methods of measurement processing will not substitute for accurate measurements. If highly nonlinear models (e.g., Cropley s kinetic model or typical distillation models) are used to analyze unit measurements and estimate parameters, the Hkelihood for arriving at erroneous models increases. Consequently, resultant models should be treated as approximations. 

Increa.se the number of mea.surements included in the mea.sure-ment. set by using mea.surements from repeated. sampling. Including repeated measurements at the same operating conditions reduces the impact of the measurement error on the parameter estimates. The result is a tighter confidence interval on the estimates. 

For sources having a large component of emissions from low-level sources, the simple Gifford-Hanna model given previously as Eq. (20-19), X = Cqju, works well, especially for long-term concentrations, such as annual ones. Using the derived coefficients of 225 for particulate matter and 50 for SO2, an analysis of residuals (measured minus estimated) of the dependent data sets (those used to determine the values of the coefficient C) of 29 cities for particulate matter and 20 cities for SOj and an independent data set of 15 cities for particulate matter is summarized in Table 20-1. For the dependent data sets, overestimates result. The standard deviations of the residuals and the mean absolute errors are about equal for particulates and sulfur dioxide. For the independent data set the mean residual shows... 

Flow Low mass flow indicated. Mass flow error. Transmitter zero shift. Measurement is high. Measurement error. Liquid droplets in gas. Static pressure change in gas. Free water in fluid. Pulsation in flow. Non-standard pipe runs. Install demister upstream heat gas upstream of sensor. Add pressure recording pen. Mount transmitter above taps. Add process pulsation damper. Estimate limits of error. 

The precondition for the use of the normal distribution in estimating the random error is that adequate reliable estimates are available for the parame-rcrs ju. and cr. In case of a repeated measurement, the estimates are calculated using Eqs. (12.1) and (12,3). When the sample size iiicrease.s, the estimates m and s approach the parameters /c and cr. A rule of rhumb is that when s 30. the normal distribution can be osecl,... 

Every measured quantity or component in the main equations, Eqs. (12.30) and (12.31), influence the accuracy of the final flow rate. Usually a brief description of the estimation of the confidence limits is included in each standard. The principles more or less follow those presented earlier in Treatment of Measurement Uncertainties. There are also more comprehensive error estimation procedures available.These usually include, beyond the estimation procedure itself, some basics and worked examples. 

Consider now a situation in which the bias limits in the temperature measurements are uncorrelated and are estimated as 0.5 °C, and the bias limit on the specific heat value is 0.5%. The estimated bias error of the mass flow meter system is specified as 0.25% of reading from 10 to 90% of full scale. According to the manufacturer, this is a fixed error estimate (it cannot be reduced by taking the average of multiple readings and is, thus, a true bias error), and B is taken as 0.0025 times the value of m. For AT = 20 °C, Eq. (2.9) gives ... 

Another important insight obtained from this example is related to the number of Monte Carlo trials which must be averaged to obtain a comparison value to the experimentally observed quantities. In order to produce a reasonable estimate of the distribution a suitable ratio of shimmer to measurement error must be achieved. A reasonable value based on experience only was found to be 0.2. In this example 100 Monte Carlo trials were required. With such a large number of trials computer logistics are an important concern. The details of the computer run and of the mapping procedure are discussed by Duever (7 ). ... 

The experimental errors on the %DE measurements are estimated to be between 1 and 2 %, taking into account a relative long time span and the involvement of different lab-workers. As indicated by Table 2 the best models converge to an RMSEP of 1.5 % to refine the models further the experimental chemical errors have to be thoroughly investigated. 

Compared with the experimental values for which was noted a high level of measurement error, a level of agreement was found that is not worse than the disparities found for a lot of compounds, which were the subject of independent measurement. Note in particular the good estimates obtained with two compounds that have relatively complex structures, such as styrene oxide and glycidyl acrylate. Nevertheless, there are two estimates that seem sufficiently different from the experimental values to require explanation. 

The sequence of the innovation, gain vector, variance-covariance matrix and estimated parameters of the calibration lines is shown in Figs. 41.1-41.4. We can clearly see that after four measurements the innovation is stabilized at the measurement error, which is 0.005 absorbance units. The gain vector decreases monotonously and the estimates of the two parameters stabilize after four measurements. It should be remarked that the design of the measurements fully defines the variance-covariance matrix and the gain vector in eqs. (41.3) and (41.4), as is the case in ordinary regression. Thus, once the design of the experiments is chosen... 

Automatic process control involves the maintenance of a desired value of a measured or estimated quantity (controlled variable) within prescribed limits (deviations, errors), without the direct action of an operator. Generally, this involves three steps ... 

In all the above cases we presented confidence intervals for the mean expected response rather than a future observation (future measurement) of the response variable, y0. In this case, besides the uncertainty in the estimated parameters, we must include the uncertainty due to the measurement error (so). 

We also use a linearized covariance analysis [34, 36] to evaluate the accuracy of estimates and take the measurement errors to be normally distributed with a zero mean and covariance matrix Assuming that the mathematical model is correct and that our selected partitions can represent the true multiphase flow functions, the mean of the error in the estimates is zero and the parameter covariance matrix of the errors in the parameter estimates is ... 

The MaxEnt method will always deflate deformation features by the (<80 ) ,1S corresponding to measurements error [39]. To obtain an empirical estimate of this intrinsic spread allowed by the noise, twenty noisy data sets were generated as in formula (31), and fitted with BUSTER using the fragment and NUP already described in the previous paragraph. 

Faber K, Kowalski BR (1997a) Improved prediction error estimates for multivariate calibration by correcting for the measurement error in the reference values. Appl Spectrosc 51 660... 

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