Sum, error

The two last points, of course, apply to any numerical method to compute energy or forces for the Coulomb or dipolar sum. Error estimates can tell us if we might see artifacts in simulations due to too small cutoffs, or if our observations have some other origin. They can tell us how the algorithm scales at its optimal point and they can help us save a lot of expensive computer time. Unfortunately, some of the free or commercially available computer programs choose parameter combinations automatically, according to some more or less known rules. Often, the user is not aware of the applied approximations, which is a very dangerous route, since, after all, one needs to interpret the data. Therefore, we stress here the point that for all our implemented routines where we have error estimates, we make use of them. 

GammaVision provides a Umited means for performing a comparative analysis in that there is an option to provide an interpolative efficiency curve. If the efficiency calibration data is provided for each gamma-ray of each nuchde to be measured, then each request for an efficiency value would return the actual calibration data derived from the standard spectrum. If true coincidence summing were a problem, then as long as the standard spectra were of the same nuclides measured under the same conditions as the sample spectra, the summing errors would cancel out. The procedure is not elegant, but as far as I can see, should work satisfactorily. 

Set A spectra contained only °Co, Sr and Cs, while Set B spectra contained nuclides expected to present a more difficult analysis problem with trae coincidence summing errors likely Sb, Cs, Cs, Eu, Eu and, as an un-certified impurity, Eu. The intention is that the user should analyse the spectra using only the basic information provided with the spectra - essentially blind . The results can then be checked against the reference values provided in a sealed envelope. 

It is clear from the reported results that true coincidence summing errors are present, as expected. The report comments on the fact that none of the software packages... 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

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... 

Abstract. A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the Ca atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Angstrom. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic progrmnming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has local minima within 1.3-4.7A of the PDB geometry, with one exception that has an error of S.SA. 

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

Deserno M and C Holm 1998b. How to Mesh Up Ewald Sums. II. An Accurate Error Estimate for the Particle-Particle-Particle-Mesh Algorithm. Journal of Chemical Physics 109 7694-7701. 

After application of the 6 time-stepping method (see Chapter 2, Section 2.5) and following the procedure outlined in Chapter 2, Section 2.4, a functional representing the sum of the squares of the approximation error generated by the finite element discretization of Equation (4.118) is formulated as... 

In so doing, we obtain the condition of maximum probability (or, more properly, minimum probable prediction error) for the entire distribution of events, that is, the most probable distribution. The minimization condition [condition (3-4)] requires that the sum of squares of the differences between p and all of the values xi be simultaneously as small as possible. We cannot change the xi, which are experimental measurements, so the problem becomes one of selecting the value of p that best satisfies condition (3-4). It is reasonable to suppose that p, subject to the minimization condition, will be the arithmetic mean, x = )/ > provided that... 

If the experimental error is random, the method of least squares applies to analysis of the set. Minimize the sum of squares of the deviations by differentiating with respect to m. 

The term on the left-hand side is a constant and depends only on the constituent values provided by the reference laboratory and does not depend in any way upon the calibration. The two terms on the right-hand side of the equation show how this constant value is apportioned between the two quantities that are themselves summations, and are referred to as the sum of squares due to regression and the sum of squares due to error. The latter will be the smallest possible value that it can possibly be for the given data. 

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