Stochastic line sampling

Coupling this to the stochastic evolution computed from solving the OU parts in both and p yields an alternative to Langevin dynamics that is 2nd order accurate in its steady state distribution and long term averages. Let us refer to this as the stochastic line sampling with leapfrog method. 

The direct MCS described above is the basic version of the method and is often used in the literature as an exact (reference) approach for validating the results of other methods (Adhikari 2011). Several variants of this approach (e.g., importance sampling, subset simulation, line sampling) have been developed in the last twenty years especially for the efficient solution of reliability problems where the calculation of small failure probabilities requires a very large number of samples. These variants will be examined in section Reliability of Stochastic Systems. ... 

The category of algebraic equation models is quite general and it encompasses many types of engineering models. For example, any discrete dsmamic model described by a set of difference equations falls in this category for parameter estimation purposes. These models could be either deterministic or stochastic in nature or even combinations of the two. Although on-line techniques are available for the estimation of parameters in sampled data systems, off-line techniques... 

Figure 3 The collapse of the peptide Ace-Nle30-Nme under deeply quenched poor solvent conditions monitored by both radius of gyration (Panel A) and energy relaxation (Panel B). MC simulations were performed in dihedral space 81% of moves attempted to change angles, 9% sampled the w angles, and 10% the side chains. For the randomized case (solid line), all angles were uniformly sampled from the interval —180° to 180° each time. For the stepwise case (dashed line), dihedral angles were perturbed uniformly by a maximum of 10° for 4>/ / moves, 2° for w moves, and 30° for side-chain moves. In the mixed case (dash-dotted line), the stepwise protocol was modified to include nonlocal moves with fractions of 20% for 4>/ J/ moves, 10% for to moves, and 30% for side-chain moves. For each of the three cases, data from 20 independent runs were combined to yield the traces shown. CPU times are approximate, since stochastic variations in runtime were observed for the independent runs. Each run comprised of 3 x 107 steps. Error estimates are not shown in the interest of clarity, but indicated the results to be robust.

In the more practical case where the impedance is sampled at a finite number of frequencies, r, x) represents the error between an interpolated function and the "true" impedance value at frequency x. This error is seen in Figure 22.3, where a region of Figure 22.2 was expanded to demonstrate the discrepancy between a straight-line interpolation between data points and the model that conforms to the interpolation of the data. This error is composed of contributions from the quadrature and/or interpolation errors and from the stochastic noise at the measurement frequency (v. Effectively, equation (22.75) represents a constraint on the integration procedure. In the limit that quadrature and interpolation errors are negligible, the residual errors r( c) should be of the same magnitude as the stochastic noise r(o ). 

FIGURE 33.2 Flowchart of a PK/PD simulation at an individual level. Shaded boxes denote stochastic elements. Arrows denote the flow of information and inputs/outputs from a model component. Lines with solid circles denote sampling components. 

Higle, J.L. and Sen, S., 1996, Stochastic decomposition, Kluwer Academic Publisher. Kalgnanam, J.R. and Diwekar, U.M., 1997, An efficient sampling technique for off-line quality control. Technometrics, 39(3), 308. 

When complex classification problems arise (e.g. the different classes of sample overlap or distribute in a non-linearly separable shape) one can have resource either to ANNs (which implies that one must be aware of their stochastic nature and of the optimisation tasks that will be required) or increase the dimensionality of the data (i.e. the variables that describe the samples) in the hope that this will allow a better separation of the classes. How can this be possible Let us consider a trivial example where the samples were drawn/ projected into a two-dimensional subspace (e.g. two original variables, two principal components, etc.) and the groups could not be separated by a linear border (in the straight line sense. Figure 6.9a). However, if three variables were considered instead, the groups would be separated easily (Figure 6.9b). How to get this(these) additional dimension(s) is what SVM addresses. 

The anisotropy of the g and hyperfine tensor leads to a dependence of the spectral line shape of nitroxides on the reorientation rate in soft matter or liquid solution. In the simplest case, nitroxide motion can be considered as isotropic Brownian rotational diffusion and can then be characterized by a single rotational correlation time Zr. To understand Zr, one can consider the reorientation of the molecular z axis caused by stochastic molecular motion. With the angle 0 between the orientation of this axis at zero time and the orientation at time t, the correlation fimction (cos0) exhibits exponential decay with time constant (the brackets () denote the average over a large ensemble of nitroxide molecules). Starting from the rigid limit, exemplified by a solid sample at very low... 

Another approach to obtain subdifiraction resolution can be classified under RESOLFT. This technique also relies on switching molecules between two states but in this case, the molecules are not switched on or off stochastically. Instead, the sample is illuminated with patterned laser excitation light. For instance, let us assume we have a sample labeled with dye molecules which can be switched between a fluorescent (on) and a nonfluorescent state (off). The sample can now be illuminated with a spatial intensity distribution with zero-intensity points or lines to selectively illuminate certain parts of the sample If this excitation light will convert the molecules to the off state, only the molecules that were not illuminated will still show fluorescence. Despite the theoretical diffraction limit, the size of the zero-intensity points or lines can be made infinitely small, by adjusting the intensity and the duration of the laser pulse. An example of such an intensity distribution is a donuf mode, which is characterized by a zero-intensity point in the center surrounded by high intensity. This means that when exciting with this donut-shaped laser light, all molecules except the one in the center will be switched to the off state. 

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