Statistical scattering matrix

W - pooled within-class scatter matrix, see Eq. 5-33, under the assumption that the covariance matrices of the classes are equal (in the statistical sense) ... 

The true values of X and jx are usually estimated from a small sample of size n. When n is very large, the estimates x and S are very good however, n is usually small, and thus the estimates x and S have a lot of uncertainty. In this case it is necessary to make an adjustment for the confidence interval, 100%(1 - a), of the sample mean and scatter matrix by use of Hotelling s T2 statistic. 

This is based on the statistical approximation to the scattering matrix [77, 78]... 

The sputter rate (in units of A/nA.s or nm/nA.s) is then defined by dividing the crater depth (in units A or nanometer), by the sum of the sputter time (in units of seconds) and the primary ion current used in forming the respective crater (in units of nanoampere). This definition assumes a uniform sputter rate over the region (depth) sputtered, hence is only applicable to a specific matrix type. Such measurements should be carried out for each crater to ensure utmost in precision (averaging procedures are often employed to reduce statistical scatter). In the case of multilayered structures, surface profilometry measurements should be carried out once each subsequent layer is sputtered, such that sputter rates pertaining to each of the different layers can be de-convoluted (recall from Section 3.2.2 that sputter rates are dependent on many parameters including those defining the matrix). 

We shall derive in this section the fundamental equations for the kinetic quantities in the adiabatic channel model from the point of view of the statistical S-matrix in scattering theory, which may seem to be the most logical approach following Refs. 2 and 17. In theoretical quantum dynamics we start from the time-dependent Schrodinger equation (3) ... 

The raw data in the more comprehensive study (61) were conversions, determined in duplicate, when each of 104 coals selected from three geological provinces was heated with tetralin under standard conditions, together with the results of 14 commonly made analytical determinations for each coal. An early observation in this study was that when data for all 104 samples were plotted against volatile matter, a steady decrease of conversion with decreasing volatile matter was apparent. But there was a great deal of scatter (r=0.85). In any case, the formal requirements that make possible the employment of valid statistical analyses were not met by the data matrix, as evidenced by skewed and bimodal relationships between the variables the sample set was heterogeneous. ... 

Dependencies may be detected using statistical tests and graphical analysis. Scatter plots may be particularly helpful. Some software for statistical graphics will plot scatter plots for all pairs of variables in a data set in the form of a scatter-plot matrix. For tests of independence, nonparametric tests such as Kendall s x are available, as well as tests based on the normal distribution. However, with limited data, there will be low power for tests of independence, so an assumption of independence should be scientifically plausible. 

We also did linear regression analysis, checking whether there were any statistically significant relationships between pairs of features. When these correlations were displayed in a matrix, we noted that there was a small cluster of pairs that seemed to be highly correlated (above. 75, or closest to 1.00). Making a command decision, we decided that these were statistically significant correlations—and the more scatter-plotted, less-well-corre-lated pairs were not. 

Here the authors consider the possibility of inferring such statistical characteristics from the spectral features of probe photons or particles that are scattered by the density fluctuations of trapped atoms, notably in optical lattices, in two hitherto unexplored scenarios, (a) The probe is weakly (perturbatively) scattered by the local atomic density corresponding to the random occupancy of different lattice sites, (b) The probe is multiply scattered by an arbitrary (possibly unknown a priori) multi-atom distribution in the lattice. The highlight of the analysis, which is based on this random matrix approach, is the prediction of a semicircular spectral lineshape of the probe scattering in the large-fluctuation limit of trapped atomic ensembles. Thus far, the only known case of quasi-semicircular lineshapes in optical scattering has been predicted [Akulin 1993] and experimentally verified [Ngo 1994] in dielectric microspheres with randomly distributed internal scatterers. 

Figure 14. Extinction and scattering spectra for ballistic RF aggregates with different conjugate numbers N = 1-100. All data are averaged over random orientations (T-matrix method) without statistical averaging. Parameters of conjugates are the shell thickness and refractive index s = 2.5 nm, =1.40, respectively, the gold core diameter <7 =15 nm (a,b) and 60 nm (c,d).

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