Fixed bias calculations

The following is an example of a mathematical/statistical calculation of a calibration curve to test for true slope, residual standard deviation, confidence interval and correlation coefficient of a curve for a fixed or relative bias. A fixed bias means that all measurements are exhibiting an error of constant value. A relative bias means that the systematic error is proportional to the concentration being measured i.e. a constant proportional increase with increasing concentration. 

The equation for the best fit straight line from these calculated values is y = —0.028 + 1.0114x. This equation suggests that a constant error of —0.028 is evident regardless of the true concentration and this is a fixed bias of —0.028 and a relative bias of 1.14%. Using these figures it is possible to calculate the bias at any particular concentration. 

The relative bias gives an error of —0.0223% for 0.5 ppm vanadium and 0.086% for 10.0 ppm vanadium metal and this shows that the relative bias exerts a greater influence on the determination than does the fixed bias (see Table 3.7). The estimated error due to relative bias is calculated by taking differences between 1.0114 and 1.000 of the perfect line and correcting for each concentration is used to calculate each predicted concentration. Table 3.7 gives results obtained along with each residual. 

The calculation above was unable to prove whether a fixed bias exists and the best way to estimate this is to fit the equation y = bx forcing a zero intercept (Table 3.8). 

Estimated values for the slope a and the intercept h are obtained by way of regression calculations as given in Section 6.5. The fixed bias (B)p is equal to h. and the relative bias (B)n is equal to o - I. The composite bias is given by... 

Fig. 23. Typical variation of the ratio n+ / na of the concentrations of H+ and H°, respectively, across ap-n junction, assuming rapid charge-change processes. The dotted, dashed and full curves were calculated assuming no bias, 2.02 V, and 9.88 V reverse bias, respectively, with a distribution of fixed charge in the junction approximately the same as that of the sample used for Fig. 21 before passivation and with the additional (arbitrary) choice of parameters eT+ = em, ed= -0.25 eV, t o/t0+ =. 001, and T = 200°C. 

Although in many laboratories the methods described above remain the methods of choice for determining the proximate analysis of coal, there is also a test method for the proximate analysis of coal by instrumental procedures, assuming that calibration is an integral part of the procedure (ASTM D-5142). This method covers the determination of moisture, volatile matter, and ash and the calculation of fixed carbon in the analysis of coal and coke samples prepared in accordance with standard protocols (ASTM D-2013). The results may require a correction for bias or be corrected for instrument calibration using samples of known proximate... 

Results. The following calculations are examples to determine whether a fixed or relative bias is found in a calibration curve and in an attempt to separate the random variation from any systematic variations the following lines were calculated ... 

Since die FTR will be fixed in size, the criticality, design parameter is fissile enrichment. FTR des models overestimate the required enrichment. The projected bias, in terms of mass of fissile material is 4 %, but, based upon analyses of four FTR criticals, this number is uncertain by 1%. Hence, the calculated enrichment is reduced 0.5% due to the bias and increased 1% to ensure the required excess reactivity. 

The essential feature of PC materials is the ultrafast phase transition between amorphous and crystalline structures that occurs on a nanosecond time scale. In the previous sections, we have discussed extensively the amorphous and crystalline structures of GST and their properties. These correspond to the starting and end points for the actual phase transition, which are crucial to understand the function of PC materials. We now present results for the nucleation-driven crystallization process of GST using DF calculations combined with MD [31], A sample of fl-GST with 460 atoms was studied at 500, 600, and 700 K, and a second sample of 648 atoms was simulated at 600 K. In all cases we used a fixed crystalline seed (58 atoms, 6 vacancies) in order to speed up the crystallization process. More recent experience has shown that the time scale for the crystallization is of the order of several nanoseconds for these system sizes in the absence of a fixed seed, while those here are of the order of 0.3-0.6ns. This means that we cannot discuss the onset of nucleation, but this is also true in the case of smaller systems (<200 atoms) discussed by other groups. In very small systems, periodic boundary conditions bias the process severely. Our larger samples reduce finite-size effects, and we show the effect of choosing different annealing temperatures. Simulations of this scale (up to 648 atoms over 1 ns) are near the limit of present day DF/MD calculations. 

Reptation quantum Monte Carlo (RQMC) [15,16] allows pure sampling to be done directly, albeit in common with DMC, with a bias introduced by the time-step (large, but controllable in DMC e.g. [17]) and the fixed-node approach (small, but not controllable e.g. [18]). Property estimation in this manner is free from population-control bias that plagues calculation of properties in diffusion Monte Carlo (e.g. [19]). Inverse Laplace transforms of the imaginary time correlation functions allow simulation of dynamic structure factors and other properties of physical interest. 

Similar to our SCF calculations we perform our simulations in the grandcanonical ensemble, i.e., we fix the volume V, temperature T, and chemical potentials fis and fip of both species, but the number of particles in the simulation cell flucm-ates. Particle insertions and deletions are implemented via the configuration bias MC method [104, 105,106,107, 108,109]. Additionally, the polymer conformations are updated by local monomer displacements and slithering snake movements [99]. 

Fig. 10. Calculated free energy barrier for homogeneous crystal nucleation of hard-sphere colloids. The results are shown for three values of the volume fraction. The drawn curves are fits to the CNT-expression Eq. (1). For the identification of solid like particles we used the techniques described before. The cutoff for the local environment was set to Vq = 1.4

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