Statistical noise

As modern one- or two-dimensional detectors are used, every pixel of the detector is enforcedly receiving the same exposure (time). Only by means of an old-fashioned zero-dimensional detector the scattering curve can be scanned in such a manner that every pixel receives the same number of counts with the consequence that the statistical noise is constant at least in a linear plot of the SAXS curve. The cost of this procedure is a recording time of one day per scattering curve. 

By means of this procedure our problem is not only reduced from three to two dimensions, but also is the statistical noise in the scattering data considerably reduced. Multiplication by —4ns2 is equivalent to the 2D Laplacian89 in physical space. It is applied for the purpose of edge enhancement. Thereafter the 2D background is eliminated by spatial frequency filtering, and an interference function G(s 2,s ) is finally received. The process is demonstrated in Fig. 8.27. 2D Fourier transform of the interference function... 

These considerations raise a question how can we determine the optimal value of n and the coefficients i < n in (2.54) and (2.56) Clearly, if the expansion is truncated too early, some terms that contribute importantly to Po(AU) will be lost. On the other hand, terms above some threshold carry no information, and, instead, only add statistical noise to the probability distribution. One solution to this problem is to use physical intuition [40]. Perhaps a better approach is that based on the maximum likelihood (ML) method, in which we determine the maximum number of terms supported by the provided information. For the expansion in (2.54), calculating the number of Gaussian functions, their mean values and variances using ML is a standard problem solved in many textbooks on Bayesian inference [43]. For the expansion in (2.56), the ML solution for n and o, also exists, lust like in the case of the multistate Gaussian model, this equation appears to improve the free energy estimates considerably when P0(AU) is a broad function. 

Since the mean velocity and Reynolds-stress fields are known given the joint velocity PDF /u(V x, t), the right-hand side of this expression is closed. Thus, in theory, a standard Poisson solver could be employed to find (p)(x, t). However, in practice, (U)(x, t) and (u,Uj)(x, t) must be estimated from a finite-sample Lagrangian particle simulation (Pope 2000), and therefore are subject to considerable statistical noise. The spatial derivatives on the right-hand side of (6.61) are consequently even noisier, and therefore are of no practical use when solving for the mean pressure field. The development of numerical methods to overcome this difficulty has been one of the key areas of research in the development of stand-alone transported PDF codes.38... 

The principal advantage of using (6.184) to determine u is that the feedback of statistical noise through the particle-pressure field in (6.178) will be minimized by solving (6.185) for (U). Indeed, for homogeneous turbulence, (6.184) is independent of X ... 

On the other hand, as was pointed out above, all MC simulation codes suffer from statistical noise that must be minimized (or at least understood) before valid comparisons can be made with experimental data (or other CFD methods). 

For stationary flows, the time-averaged values should be used in place of X, y in the central-difference formula in order to improve the smoothness of the estimated fields. For non-stationary flows, it may be necessary to filter out excess statistical noise in u Uj X, iy before applying (7.71). In either case, the estimated divergence fields are given by... 

Kalman Filter Process model, (including process kinetics, but it is possible to estimate some kinetic parameters on-line), process inputs, statistical noise properties. Well known approach. It takes into account the measuring noise as well as process inputs noise. Model hnearization Inputs knowledge Stabihty and convergence are only locally vahd. [6p... 

Figure 3.S Two different autocorrelation functions. The solid curve is for a property that shows no significant statistical noise and appears to be well characterized by a single decay time. The dashed curve is quite noisy and, at least initially, shows a slower decay behavior. In the absence of a very long sample, decay times can depend on the total time sampled as well...

Fig. 5.15. Power-law approach to the equilibrium concentration in excited dye molecules for different proton concentrations in solution (mM) [85] 1 - 0.01, 2 - 4 3 - 1.5 and 4 - 15. Full lines show smoothed experimental curves, dashed lines correspond to large statistical noise.

Under some conditions, feelings of sleepiness may be great and protocols that must be run when subjects are extremely sleepy encounter the problem of floor effects. For example, physiological sleepiness may be so great because of a drag effect that sleep latencies are too short to be above statistical noise. In these circumstances, it is difficult to detect subjects with abnormal sleepiness or to detect the effect of any remedial intervention. 

The data in the above figures may be improved by using more intense excitation. A new oscillator rod and a new amplifier rod have recently been installed and this should improve intensity and reduce the statistical noise. It should also be possible to achieve better imaging from the spectrograph and lens coupling systems. For example, the rounded effect at the peak of the curves in Fig. 7 is caused by rather wide linewidths (about 10 channels FWHM). This will be reduced in future studies partly by better imaging and partly by deconvolution. 

We should mention here that a better way to extract the activation energies for vacancy-mediated diffusion would be to plot the tracer diffusion coefficient of the embedded atoms vs. 1/kT, rather than their jump rate. As we discussed in Section 4, the mean square jump length depends on the proximity of steps, and so does the average jump frequency. This adds non-statistical noise to the two plots in Fig. 12. However, it can be shown easily that these effects on jump length and jump rate cancel in the resulting tracer diffusion coefficient, which thus becomes independent of the distance to steps. In this way, a more accurate value for the activation energy has been obtained for the case of In/Cu(0 0 1) of 717 30meV [23]. 

Because of numerical problems emanating from the inherent statistical noise in Gci(t), it is impossible to evaluate the Fourier transform in Equation (19) at the required oxygen frequency of >o/2jrc = 1552.5 cm-1. We have used several methods to extrapolate the numerical results to high frequency, including the use of the ansatz in Equation (33) (4,23). Here, however, we simply assume that the rate would follow an exponential energy gap law (2), k], 0 oc e F" and therefore we perform a linear extrapolation on a log plot (6). 

A sharp peak at about 6 ns, occurs when backscattered positrons pass the grid and reach the CEMA and trigger timing pulses without the secondary electron time of flight. This 6 ns peak vanishes in statistical noise in the case of samples that cause longer lifetimes. In the lifetime analysis, the data in the 6 ns peak region are ignored in the present discussion. 

As can be seen in Figure 7.15 very different lifetimes occur and each dataset shows different lifetimes. In principle the sample material and the size distribution of pores and the open porosity component each generate an annihilation rate (f ) with some relative intensity (I ). The spectrum is then convoluted with the experimental response function (R). Random statistical noise and a certain background (B) level are added. The measured time spectmm M(t)... 

Even though the lifetime distributions appear to be quite different, the recreated data are almost identical except for a small deviation near 200 ns and less obvious ones at shorter times. If the relative differences are plotted, systematic differences beyond the statistical noise are noticeable up to 200 ns, particularly when several channels are binned together. Given sufficient statistics, in principle, one can tell the difference between a bimodal and a monomodal distribution. The shown simulated spectra are based on 108 counts, five to 10 times the amount collected for the data discussed here. 

The detector noise of a spectrometer operating in the visible range can be very low (as for a grating Raman spectrometer with a cooled photomultiplier). In this case, the main source of noise is the statistical noise of the analyzed radiation d>s, according to Poisson statistics — v - The signal-to-noise ratio, SNR, is therefore given by ... 

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