Error analyses

Estimate of errors is fundamental to all branches of natural sciences that deal with experiments. Very frequently, the final result of an experiment cannot be measured directly. Rather, the value of the final result (u) will be calculated from several measured quantities (x, y, Z-., each of which has a mean value and an error)  

We estimate error limits on the target by considering the errors reported for the 

A hypothetical standard deviation A determined for each method for the differences, 298 (cal) - 298 (exp), using different work reactions are given as well. We note that 

5 298 in cal mol K optical isomers number, symmetry number The sum of contributions from 

The greatest error associated with the direct evaluation of chromatograms is in the chromatographic process itself. Errors arise from the application of the initial spots to the plate, from the separation process and from the treatment of the final separated spots prior to evaluation. The spotting error is often the most common cause of poor reproducibility. Creep back , a term describing the tendency for some of the spotted solution to run up the outside of the needle during spotting, is frequently encountered. This quantity of solution may be lost to the spot and could be deposited with the next. The 

Irregularities in the adsorbent layer can lead to significant error by causing poorly reproducible Rp values and irregularities in the size and shape of the spots. Thick areas of the plate may have less of the sample at or near the surface of the layer, causing a reduction in reflectance or fluorescence response when measurements are made on the same side of 

The use of HPLC for quantitative analysis has greatly reduced the sources of error compared to TLC. Usually, the chromatography and detection processes can be better controlled, and reproducible separations are, therefore, carried out more easily. Errors encountered in these processes are systematic and result from either minor defects or failures in the equipment or a poor selection of the operational parameters. Operator errors can result from the sample preparation and the injection on to the column. This is often a matter of technique, and can be easily checked by injecting a number of replicate standards and calculating the reproducibility. Although HPLC is usually preferred for quantitation, 

because of its simplicity, flexibility and cheapness, certainly has its place as a chromatographic method for quantitative and, especially, qualitative analysis. 

The difficult analysis of the sources of error in free energy difference evaluations by computer simulation and the effect of particular implementations of free energy difference techniques has been the subject of a number of recent studies. The error inherent in any computer simulation can formally be categorized by statistical and systematic errors. In practice, these errors are often difficult to separate. 

The most important source of errors stems from the difficulty of adequately sampling phase space in simulations. The correlation between successive steps in molecular dynamics or Monte Carlo simulations makes the adequate sampling of all regions of phase space with their proper Boltzmann probability difficult to achieve within reasonable simulation times. The number of independent samples is always far smaller than the number of generated configurations. It has been estimated that sampling of phase space can be as limited as only one independent sampled configuration per picosecond of simulation.  

Because of the high correlation between successive configurations generated by molecular simulations, an ensemble cannot be treated as a statistically random set of configurations. As a result, the calculation of statistical errors of properties determined as ensemble averages must be evaluated with a method that accounts for this correlation. This requirement applies to any property evaluated as an ensemble average, and several techniques to determine the realistic statistical error have been suggested. -  

An analysis of the effect of sampling errors on free energy difference calculations has demonstrated a definite relation between the calculated error 

Umbrella potentials, employed to increase sampling of certain regions of phase space in, for example, potential of mean force evaluations, may lead to a bias in the determination of thermodynamic properties. Selection criteria to minimize these biases and tests to determine the existence of these biases have 

Subtracting the slope matrix obtained by the multivariate least squares tieatment from that obtained by univariate least squares slope matiix yields the error mahix 

Normally, one does not have hue values of the elements of the slope mah ix M for comparison. It is always possible, however, to obtain y, the vector of predicted y values at each of the known Xi from any of the slope vectors m obtained by the multivariate procedure 

This permits enor analysis of that vector. (Note that the order Xm is necessary for the mahix and vector to be conformable for multiplication.) Repeating the procedure for all m vectors leads to error analysis of the entire matrix M. 

We wish to cany out a proceduie that is the multivariate analog to the analysis in the section on reliability of fitted parameters. A vector multiplied into its hanspose gives a scalar that is the sum of squares of the elements in that vector. The y vector leads to a vector of residuals 

The product e e is the sum of squares of residuals from the vector of residuals. The vai iance is 

When measuring KIE using isotope analysis for the product (Equation 7.17) three quantities are determined experimentally fL, Ros and Rp. For measurements of KIE using substrate (reactant) analysis (Equation 7.18) the corresponding quantities are fL, Ros and Rs. All these measurements, of course, are subject to experimental error. Equation 7.25 expresses the relative error of KIE in terms of the errors in these three experimental quantities  

In Equation 7.25, a is the standard deviation, Rf the isotope ratio corresponding to the fraction of reaction, /, of either substrate (Rs) or product (Rp). Expressions for A and B are given in Table7.3. Ro is the isotope ratio in the starting material. Obviously, for simple reactions the isotope composition of the product after full conversion (Rip) is equal to the isotope composition of the initial substrate (Ros)-The dependence of A and B on the progress of the reaction is illustrated in Fig. 7.6  

Equations 7.17 and 7.18 have been developed assuming that the fraction of reaction for the light isotopomer (fL) is the one monitored. Frequently, however, the chemical (overall) fraction of reaction  

While competitive methods to determine KIE s are free from errors due to differences in reaction conditions (impurities, temperature, pH, etc.) they do require access to equipment that allows high precision measurements of isotope ratios. The selection of an appropriate analytical technique depends on the type of the isotope and its location in the molecule. For studies with stable isotopes the most commonly used technique (and usually the most appropriate) is isotope ratio mass spectrometry (IRMS). 

We will study the following methods Classical Method 

New Method with Vanished Phase-Lag and its First Derivative (developed in paragraph 3.1) 

The radial time independent Schr—dinger equation is of the form 

Based on the paper of Ixaru and Rizea [25], the function f(x) can be written in the form  

We express the derivatives w 1 = 2,3,4,which are terms of the local truncation error formulae, in terms of the equation (23). The expressions are presented as polynomials of G 

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Clifford, A. A., "Multivariate Error Analysis," Halsted Press, Division of John Wiley Sons, New York (1973). 

CLIFFORD, A. A., MULT IVARI ATE ERROR ANALYSIS, HALSTED PRESS, N.Y., 1973. 

Bevington P R 1969 Data Reduction and Error Analysis for the Physical Sciences (New York McGraw Hill) pp 36-43... 

E. Hairer and Ch. Lubich. The life-span of backward error analysis for numerical integrators. Numer. Math. 76 (1997) 441-462... 

T. R. Littell, R. D. Skeel, and M. Zhang. Error analysis of symplectic multiple time stepping. SIAM J. Numer. Anal., 34 1792-1807, 1997. 

Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vec tors in iT and for matrices that are analogous to the notion of length of a geometric vector. Let R denote the set of all vec tors with n components, x = x, . . . , x ). In dealing with matrices it is convenient to treat vectors in R as columns, and so x = (x, , xj however, we shall here write them simply as row vectors. 

Rasmussen, J. 1979. Notes on human error analysis and prediction. In G. Apostalakis and G. Volta (Eds.), Synthesis and Analysis Methods for Safety and Reliability Studies, Plenum, New York. 

Equation 4.7 is referred to as the variance equation and is eommonly used in error analysis (Fraser and Milne, 1990), variational design (Morrison, 1998), reliability... 

ATHEA A Technique for Human Error Analysis NUREG/CR-6350... 

CREAM Cognitive Reliability and Error Analysis Method Hollnagel, 1993... 

The probability density of the normal distribution f x) is not very useful in error analysis. It is better to use the integral of the probability density, which is the cumulative distribution function... 

Chapter 7, Case Studies, uses examples that illustrate the application of the various error analysis and reduction techniques to real world process industry cases. 

The third category of methods addressed in this chapter are error analysis and reduction methodologies. Error analysis techniques can either be applied in a proactive or retrospective mode. In the proactive mode they are used to predict possible errors when tasks are being analyzed during chemical process quantitative risk assessment and design evaluations. When applied retrospectively, they are used to identify the underlying causes of errors giving rise to accidents. Very often the distinction between task analysis and error analysis is blurred, since the process of error analysis always has to proceed from a comprehensive description of a task, usually derived from a task analysis. 

To provide data for error analysis methods by pinpointing error-likely situations... 

In addition to their descriptive fimctions, TA techniques provide a wide variety of information about the task that can be useful for error prediction and prevention. To this extent, there is a considerable overlap between Task Analysis and Human Error Analysis (HEA) techniques described later in this chapter. HEA methods generally take the result of TA as their starting point and examine what aspects of the task can contribute to human error, hr the context of human error reduction in the CPI, a combination of TA and HEA methods will be the most suitable form of analysis. 

HTA can be used as a starting point for using various error analysis methods to examine the error potential in the performance of the required operations. 

The application of human error analysis (HEA) techniques is to predict possible errors that may occur in a task. The next stage of error analysis is to identify error recovery possibilities implicit within the task, and to specify possible... 

The other main application area for predictive error analysis is in chemical process quantitative risk assessment (CPQRA) as a means of identifying human errors with significant risk consequences. In most cases, the generation of error modes in CPQRA is a somewhat unsystematic process, since it only considers errors that involve the failure to perform some pre-specified function, usually in an emergency (e.g., responding to an alarm within a time interval). The fact that errors of commission can arise as a result of diagnostic failures, or that poor interface design or procedures can also induce errors is rarely considered as part of CPQRA. However, this may be due to the fact that HEA techniques are not widely known in the chemical industry. The application of error analysis in CPQRA will be discussed further in Chapter 5. 

Error analysis techniques can be used in accident analysis to identify the events and contributory factors that led to an accident, to represent this information in a clear and simple manner and to suggest suitable error reduction strategies. This is achieved in practice by identification of the causal event sequence that led to the accident and the analysis of this sequence to identify the root causes of the system malfunction. A discussion of accident analysis techniques is included in Chapter 6. 

Predictive human error analysis can be performed manually or by means of a computer software package. Three types of analysis are possible within PHEA. 

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