Rejection algorithm

Locatelli, M. and F. Schoen. Random Linkage A Family of Acceptance/Rejection Algorithms for Global Optimization. Math Prog 85 379-396 (1999). 

We can show that the full conditional distributions in (2.29) through (2.31) are log-concave in each of these parameters. Therefore, we can use the adaptive rejection algorithm of Gilks and Wild (1992) or the localized Metropolis algorithm discussed in Chen et al. (2000) to sample (y, 0, Oq). 

FIGURE 40.10 Typical result of comparing an unknown sample with 460 members of a database. The horizontal dotted hne, at 100 is the set threshold value. The black triangles indicate selected samples crosses indicate samples that were selected but were then rejected by the outlier rejection algorithm. The solid vertical line indicates the predicted value for the test sample while the dashed vertical line indicates the actual database value for the test sample. One outlier was automatically rejected. 

Two different types of calibration marks are used in our experiments, planar circles and circular balls. The accuracy of the calibration procedure depends on the accuracy of the feature detection algorithms used to detect the calibration marks in the images. To take this in account, a special feature detection procedure based on accurate ellipses fitting has been developed. Detected calibration marks are rejected, if the feature detection procedure indicates a low reliability. 

FIGURE 11.9 Outliers, (a) Dose-response curve fit to all of the data points. The potential outlier value raises the fit maximal asymptote, (b) Iterative least squares algorithm weighting of the data points (Equation 11.25) rejects the outlier and a refit without this point shows a lower-fit maximal asymptote. 

Step 1. Separate the initial data into two sets, corresponding to temperatures above and below Tb. Step 2. Make an initial selection from the low temperature set by rejecting all points with zero uncertainty and all points with uncertainties above a limit determined by the data selection algorithm described in section 1.5.2. Zero uncertainties are assigned to values that are not experimental and are included for comparison only (these are most often values recommended in other compilations). 

The number z2 / z, is a weighted mean of all points in the set other than the y-th point. The weighting factor decreases exponentially with the difference in temperature of the /-th point from the y -th point. The parameter gj determines the rate of decrease. This procedure compares the uncertainty of they -th point to the weighted mean of other points. The parameter g2 determines the rejection level from this comparison for the y -th point. Larger values of g2 are less selective. Values for gi and g2 are supplied to the algorithm. For all cases gj is in the range of 1 to 2 (usually 1.8). The value of g2 is in the range of 2 to 3 (kg m3) (usually 2.5). 

Thus, the way the algorithm works is to set an initial value for the control parameter. At this setting of the control parameter, a series of random moves are made. Equation 3.11 dictates whether an individual move is accepted or rejected. The control parameter (annealing temperature) is lowered and a new series of random moves is made, and so on. As the control parameter (annealing temperature) is lowered, the probability of accepting deterioration in the objective function, as dictated by Equation 3.11, decreases. In this way, the acceptability for the search to move uphill in a minimization or downhill during maximization is gradually withdrawn. 

After the momenta are selected from the distribution (8.39), the dynamics is propagated by a standard leapfrog algorithm (any symplectic and time-reversible integrator is suitable). The move is then accepted or rejected according to a criterion based on the detailed balance condition... 

In principle, in the absence of noise, the PLS factor should completely reject the nonlinear data by rotating the first factor into orthogonality with the dimensions of the x-data space which are spawned by the nonlinearity. The PLS algorithm is supposed to find the (first) factor which maximizes the linear relationship between the x-block scores and the y-block scores. So clearly, in the absence of noise, a good implementation of PLS should completely reject all of the nonlinearity and return a factor which is exactly linearly related to the y-block variances. (Richard Kramer)... 

Some of this variance was indeed rejected by the PLS algorithm, but the amount, compared to the Principal Component algorithm, seems to have been rather minuscule, rather than providing a nearly exact fit. 

Step 4 rejects the new point and decreases the step bounds if ratiok < 0. This step can only be repeated a finite number of times because, as the step bounds approach zero, the ratio approaches 1.0. Step 6 decreases the size of the trust region if the ratio is too small, and increases it if the ratio is close to 1.0. Zhang et al. (1986) proved that a similar SLP algorithm converges to a stationary point of P from any initial point. 

Liu, J., Luijten, E. Rejection-free geometric cluster algorithm for complex fluids. Phys. Rev. Lett. 2004, 92, 035504. 

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