Measurement errors, rectification

Romagnoli, J.A. and G. Stephanopoulos, On the Rectification of Measurement Errors for Complex Chemical Plants, Chemical Engineeiing Science, 35, 1980, 1067-1081. 

Rectification accounts for systematic measurement error. During rectification, measurements that are systematically in error are identified and discarded. Rectification can be done either cyclically or simultaneously with reconciliation, and either intuitively or algorithmically. Simple methods such as data validation and complicated methods using various statistical tests can be used to identify the presence of large systematic (gross) errors in the measurements. Coupled with successive elimination and addition, the measurements with the errors can be identified and discarded. No method is completely reliable. Plant-performance analysts must recognize that rectification is approximate, at best. Frequently, systematic errors go unnoticed, and some bias is likely in the adjusted measurements. 

Validation versus Rectification The goal of both rectification and validation is the detecI ion and identification of measurements that contain systematic error. Rectification is typically done simultaneously with reconciliation using the reconciliation resiilts to identify measurements that potentially contain systematic error. Vahdation typically rehes only on other measurements and operating information. Consequently, vahdation is preferred when measurements and their supporting information are hmited. Further, prior screening of measurements limits the possibihty that the systematic errors will go undetected in the rectification step and subsequently be incorporated into any conclusions drawn during the interpretation step. 

It is obvious from these results that measurement /4 is grossly faulty and has to be deleted during the rectification process. Finally, estimates of the measurement error and the process variables when measurement /4 is deleted are given by... 

The reliability of the process measurements1 data is extremely important for good monitoring, control and optimization of chemical process. On-line rectification of a measurement error is possible be it a random error or a gross bias, if additional information is available. Such information is supplied by the extent to which the material and energy balances are satisfied by the recorded data. These balances are simple, involve parameters usually well known, and they should be satisfied independently of the measurements accuracy. 

Define the subset of balance equations which should be used for the rectification of the measurement errors. 

Terry, P.A. and D.M. Himmelhlau, Data Rectification and Gross Error Detection in a Steady-State Process via Artificial Neural Networks, Indushial and Engineeiing Chemistiy Reseaieh, 32, 199.3,. 3020-3028. (Neural networks, measurement test)... 

However, other bias errors are so substantial that their presence will significantly distort any conclusions drawn from the adjusted measurements. Rectification is the detection of the presence of significant bias in a set of measurements, the isolation of the specific measurements containing bias, and the removal of those measurements from subsequent reconcihation and interpretation. Significant bias in measurements is defined as gross error in the literature. 

Relaxation methods for the study of fast electrode processes are recent developments but their origin, except in the case of faradaic rectification, can be traced to older work. The other relaxation methods are subject to errors related directly or indirectly to the internal resistance of the cell and the double-layer capacity of the test electrode. These errors tend to increase as the reaction becomes more and more reversible. None of these methods is suitable for the accurate determination of rate constants larger than 1.0 cm/s. Such errors are eliminated with faradaic rectification, because this method takes advantage of complete linearity of cell resistance and the slight nonlinearity of double-layer capacity. The potentialities of the faradaic rectification method for measurement of rate constants of the order of 10 cm/s are well recognized, and it is hoped that by suitably developing the technique for measurement at frequencies above 20 MHz, it should be possible to measure rate constants even of the order of 100 cm/s. 

Romagnoli, J., and Stephanopoulos, G. (1981). Rectification of process measurement data in the presence of gross errors. Chem. Eng. Sci. 36, 1849-1863. 

The major question related to the rectification of gross errors is the identification of the sources. Quite often this question will be set aside and the gross error will be distributed among several measurements. Although this approach may suffice in the short run, it is unacceptable in the long run. The detection of a gross error requires the construction of an appropiate... 

All nicotinic receptors are somewhat calcium permeable the most permeable are neuronal homomeric receptors (a7, a9) and the least permeable, embryonic muscle receptors. It must be noted that the measurement of relative calcium permeability by the simplest technique (reversal potential shift induced by changes in extracellular calcium concentrations) is error-prone for neuronal nicotinic receptors because their extreme inward rectification makes it difficult to measure reversal potentials accurately. A further technical difficulty (for recombinant receptors) arises from the presence in Xenopus oocytes of a calcium-dependent chloride conductance that has to be suppressed or minimized by either intracellular calcium chelation or chloride depletion. Some of these problems can be overcome by expressing the receptors in mammalian cell lines and using ratiometric measurements of intracellular calcium and coupled with wholecell recording, to obtain a measure of what proportion of the nicotinic current is carried by calcium (a measure that also has the advantage of being physiologically more relevant). 

This approach is equivalent to maximum likelihood rectification for data contaminated by Gaussian errors. The likelihood function is proportional to the probability of realizing the measured data, yj, given the noise-free data, yi. 

Unlike maximum likelihood rectification, Bayesian rectification can remove errors even in the absence of process models. Another useful feature of the Bayesian approach is that if the probability distributions of the prior and noise are Gaussian, the error of approximation between the noise-free and rectified measurements can be estimated before rectifying the data as,... 

Gaussian errors with standard deviations 1, 4, 4, 3, and 1, respectively. The performance of maximum likelihood, single-scale Bayesian, and multiscale Bayesian rectification are compared by Monte-Carlo simulation with 500 realizations of 2048 measurements for each variable. The prior probability distribution is assumed to be Gaussian for the single-scale and multiscale Bayesian methods. The normalized mean-square error of approximation is computed as,... 

The mean and standard deviation of the MSE for 500 realizations of the 2048 measurements per variable are summarized in Table 1, and are similar to those of Johnston and Kramer. The average and standard deviation of the mean-squared errors of single-scale and multiscale Bayesian rectification are comparable, and smaller than those of maximum likelihood rectification. The Bayesian methods perform better than the maximum likelihood approach, since the empirical Bayes prior extracts and utilizes information about the finite range of the measurements. In contrast, the maximum likelihood approach implicitly assumes all values of the measurements to be equally likely. If information about the range of variation of the rectified values is available, it can be used for maximum likelihood rectification, leading to more accurate results. For this example, since the uniformly distributed uncorrelated measurements are scale-invariant in nature, the performance of the single-scale and multiscale Bayesian methods is comparable. 

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