Propagation error

Sensitivity analysis also can be applied to examinations of error propagations that arise from the use of nonoptimal potential constants in empirical 

To understand the insights that each principal component provided, one could use Eq. [25] to write dC) and dO in the following form  

Principal component analysis can also be carried out by using simulation data obtained at different conditions, such as at different temperatures, so that more observables can be used to construct a larger sensitivity matrix. This has been done for the evaluation of serine and threonine dipeptides in methano , but the key findings were essentially the same as those described above when the results from only two simulations were used in the analysis. 

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Following the law of error propagation for independant values (root sum square ) in the worst case for the parameters mentioned above, the overall reduction of the visibility level will be Rvl = -0.55, that means VLmin = 0.45 VL om... 

A kinetics text with a strong theoreticai bent that overviews transient kinetic methods and discusses data anaiysis issues such as error propagation and sensitivity anaiysis. 

Forward Analysis In this type of analysis, we are interested in the propagation of initial perturbations Sxq along the flow of (1), i.e., in the growth of the perturbations 5x t xo) = (xo -h Sxq) — xq. The condition number K,(t) may be defined as the worst case error propagation factor (cf. textbook [4]), so that, in first order perturbation analysis and with a suitable norm j ... 

In experiments, the two symplectic methods ROT and SPL performed very similarly in terms of error propagation and long term stability. The ex-... 

Researchers must be particularly cautious when using one estimated property as the input for another estimation technique. This is because possible error can increase significantly when two approximate techniques are combined. Unfortunately, there are some cases in which this is the only available method for computing a property. In this case, researchers are advised to work out the error propagation to determine an estimated error in the final answer. 

Critica.1 Properties. Several methods have been developed to estimate critical pressure, temperature, and volume, U). Many other properties can be estimated from these properties. Error propagation can be large for physical property estimations based on critical properties from group contribution methods. Thus sensitivity analyses are recommended. The Ambrose method (185) was found to be more accurate (186) than the Lyderson (187) method, although it is computationally more complex. The Joback and Reid method (188) is only slightly less accurate overall than the Ambrose method, and is more accurate for some specific substances. Other methods of lesser overall accuracy are also available (189,190) (T, (191,192) (T, P ),... 

Rumelhart, D.E., Hinton, G.E. and Williams, R.J. (1986) Learning internal representations by error propagation. In Parallel Distributed Processing, Rumelhart, D.E. and McClelland, J.L. (eds.), M.I.T. Press, Cambridge, Mass. 

Control and alignment of segmented-mirror telescopes matrices, modes, and error propagation Chanan, G., MacMartin, D., Nelson, J., Mast, T., 2004, Applied Optics 43, 1223... 

Of all the requirements that have to be fulfilled by a manufacturer, starting with responsibilities and reporting relationships, warehousing practices, service contract policies, airhandUng equipment, etc., only a few of those will be touched upon here that directly relate to the analytical laboratory. Key phrases are underlined or are in italics Acceptance Criteria, Accuracy, Baseline, Calibration, Concentration range. Control samples. Data Clean-Up, Deviation, Error propagation. Error recovery. Interference, Linearity, Noise, Numerical artifact. Precision, Recovery, Reliability, Repeatability, Reproducibility, Ruggedness, Selectivity, Specifications, System Suitability, Validation. 

Classical error propagation (2) must not be overlooked if the final result R is arrived at by way of an algebraic function... 

Reinhart and Rippin (1986) proposed two methods for design under uncertainty (1) introduction of a penalty function for the probability of exceeding the available production time, whereby the probability can be generated by standard error propagation techniques for technical or commercial uncertainties, and (2) the Here and Now method. 

Finally, Chapter 16 provides information about the handling of U-series data, with a particular focus on the appropriate propagation of errors. Such error propagation can be complex, especially in the multi-dimensional space required for U- " U- °Th- Th chronology. All too often, short cuts are taken during data analysis which are not statistically justified and this chapter sets out some more appropriate ways of handling U-series data. 

Error correlations are often ignored both mathematically and graphically in the treatment of °Th/U data. Nonetheless, error correlations can be as important a parameter in error propagation and evaluation as the errors themselves, and must be considered in all cases. Though sometimes approached with trepidation, the significance of error correlations is easily understood at an intuitive level and demonstrated visually (Fig. 1). Basic examples of procedures for calculating error correlations are given in Appendix I. 

A worked example of the detrital-component correction and error propagation for the data of Figure 4 is given in Appendix 11. 

Albarede (1995), p. 294-307 gives a modem summary of EWLS isochron mathematics and error propagation. Bevington (1969), chapter 6.3 presents the classical approach, limited to cases withy-error only. 

Ignore the problem completely and simply quote the errors propagated from the analytical errors of the data points. 

Anderson GM (1976) Error propagation by the Monte Carlo method in geochemical calculations. Geochim Cosmochim Acta 40 1533-1538... 

WORKED EXAMPLE OF DETRITAL CORRECTION AND ERROR PROPAGATION... 

If we assume that the residuals in Equation 2.35 (e,) are normally distributed, their covariance matrix ( ,) can be related to the covariance matrix of the measured variables (COV(sy.,)= LyJ through the error propagation law. Hence, if for example we consider the case of independent measurements with a constant variance, i.e. 

Having an estimate (through the error propagation law) of the covariance matrix L, we can obtain the ML parameter estimates by minimizing the objective function,... 

It is not always possible to tell strictly the difference between random and systematic deviations, especially as the latter are defined by random errors. The total deviation of an analytical measurement, frequently called the total analytical error , is, according to the law of error propagation, composed of deviations resulting from the measurement as well as from other steps of the analytical process (see Chap. 2). These uncertainties include both random and systematic deviations, as a rule. 

By careful proceeding of measurements random variations can be minimized, but fundamentally not eliminated. The appearance of random errors follow a natural law (often called the Gauss law ). Therefore, random variations may be characterized by mathematical statistics, namely, by the laws of probability and error propagation. 

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]). 

Jochum C, Jochum P, Kowalski BR (1981) Error propagation and optimal performance in multicomponent analysis. Anal Chem 53 85... 

A realistic uncertainty interval has to be estimated, namely by considering the statistical deviations as well as the non-statistical uncertainties appearing in all steps of the analytical process. All the significant deviations have to be summarized by means of the law of error propagation see Sect. 4.2. 

The variance characterizes the spread of AA if an infinite number of independent simulations are carried out, each with a finite sample of size N. In practice, usually only one estimate (or a small number of repeats) of free energy differences are taken, and the variance in free energy must be estimated. One way to compute the variance is to use the error propagation formula (for a forward calculation)... 

The variance of the free energy calculation is (with error propagation)... 

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