Statistics combining errors

The statistical prediction error does not account for biases in concentration or pathlei h changes. The (SS ) matrix depends only on the pure spectra, and the re ual spectra only depends on how well dS can find a linear combination of pures to fit the sample spectrum. In other words, the statistical prediction error is a measure of precision, not accuracy. 

Fig. 17.15. Ratio F /F. Inner errors are statistical, outer systematic and statistical combined.

It is hoped that the more advanced reader will also find this book valuable as a review and summary of the literature on the subject. Of necessity, compromises have been made between depth, breadth of coverage, and reasonable size. Many of the subjects such as mathematical fundamentals, statistical and error analysis, and a number of topics on electrochemical kinetics and the method theory have been exceptionally well covered in the previous manuscripts dedicated to the impedance spectroscopy. Similarly the book has not been able to accommodate discussions on many techniques that are useful but not widely practiced. While certainly not nearly covering the whole breadth of the impedance analysis universe, the manuscript attempts to provide both a convenient source of EK theory and applications, as well as illustrations of applications in areas possibly u amiliar to the reader. The approach is first to review the fundamentals of electrochemical and material transport processes as they are related to the material properties analysis by impedance / modulus / dielectric spectroscopy (Chapter 1), discuss the data representation (Chapter 2) and modeling (Chapter 3) with relevant examples (Chapter 4). Chapter 5 discusses separate components of the impedance circuit, and Chapters 6 and 7 present several typical examples of combining these components into practically encountered complex distributed systems. Chapter 8 is dedicated to the EIS equipment and experimental design. Chapters 9 through 12... 

Statistical and algebraic methods, too, can be classed as either rugged or not they are rugged when algorithms are chosen that on repetition of the experiment do not get derailed by the random analytical error inherent in every measurement,i° 433 is, when similar coefficients are found for the mathematical model, and equivalent conclusions are drawn. Obviously, the choice of the fitted model plays a pivotal role. If a model is to be fitted by means of an iterative algorithm, the initial guess for the coefficients should not be too critical. In a simple calculation a combination of numbers and truncation errors might lead to a division by zero and crash the computer. If the data evaluation scheme is such that errors of this type could occur, the validation plan must make provisions to test this aspect. 

The application of optimisation techniques for parameter estimation requires a useful statistical criterion (e.g., least-squares). A very important criterion in non-linear parameter estimation is the likelihood or probability density function. This can be combined with an error model which allows the errors to be a function of the measured value. A simple but flexible and useful error model is used in SIMUSOLV (Steiner et al., 1986 Burt, 1989). 

Traditionally, analytical chemists and physicists have treated uncertainties of measurements in slightly different ways. Whereas chemists have oriented towards classical error theory and used their statistics (Kaiser [ 1936] Kaiser and Specker [1956]), physicists commonly use empirical uncertainties (from knowledge and experience) which are consequently added according to the law of error propagation. Both ways are combined in the modern uncertainty concept. Uncertainty of measurement is defined as Parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand (ISO 3534-1 [1993] EURACHEM [1995]). 

The minimization of the statistical error can be done by examining the variance of cxp(—/ zL4) directly. Bennett [55] has studied this problem by combining the forward and reverse FEP simulations, and the same analysis can be followed for the OS case. In Bennett s analysis, the weighting function w is placed to balance the forward and reverse FEP contributions [7, 55]... 

Following Bennett, Crooks proposed the generalized acceptance ratio (GAR) method to combine the forward and reverse NEW calculations to minimize the statistical error of the relative free energy [56]... 

Based on previous recommendations [31], a combination of graphical techniques and error index statistics was used for evaluating the goodness-of-fit between the simulated and observed streamflow values, both during the calibration and validation period. The used statistics were the mean error (ME), the percent bias (PBIAS, [32]) and the Nash-Sutcliffe efficiency (NSeff, [33]) ... 

The presence of gross errors invalidates the statistical basis of the common data reconciliation procedures, so they must be identified and removed. Gross error detection has received considerable attention in the past 20 years. Statistical tests in combination with an identification strategy have been used for this purpose. A good survey of the available methodologies can be found in Mah (1990) and Crowe (1996). 

This section briefly discusses an approach that combines statistical tests with simultaneous gross error identification and estimation. The strategy is called SEGE (Simultaneous Estimation of Gross Error Method). It was proposed by Sanchez and Romagnoli (1994). 

If H0 is rejected, a two-stage procedure is initiated. First, a list of candidate biases and leaks is constructed by means of the recursive search scheme outlined by Romagnoli (1983). All possible combinations of gross errors (measurement biases and/or process leaks) from this subset are analyzed in the second stage. Gross error magnitudes are estimated simultaneously for each combination and chi-square test statistic calculations are performed to identify the suspicious combinations. We will now explain the stages of the procedure. 

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