Bootstrapping process

The following sections describe the crystallographic bootstrapping process in more detail. 

Inspection of Equation (2) shows that in the limit of CRE ages long compared to 1/X, the concentration of a cosmogenic radionuclide approaches the constant value P/X. Hence, in this case, the activity at the time of a meteorite fall (or the collection of a lunar sample) is equal to the production rate. Once the production rate is known for one meteorite, one may analyze for the same radionuclide in a second meteorite with, perhaps, a shorter CRE age. If P was the same in the second meteorite (an important assumption), then we can solve Equation (2) for the CRE age. In a kind of bootstrapping process, measurements of other cosmogenic nuclides in the second meteorite then can serve as the basis for calculating their production rates, and so on. 

The bootstrap process can be continued indefinitely, but clearly becomes increasingly cumbersome. The cases for five and six identical substituents are... 

Figure 35 The bootstrapping process (reproduced from Figure 2 of ref. [61 i] with permission from the copyright owner).

This implies that a cut-off limit must be established for functional depths, so that the observations incorrectly classified as outliers do not exceed 1% [9], A bootstrap process is made to determine the cut-off limit [10], which calculates this cut-off parameter by determining functions with depths below the calculated value as functional outliers. This is proposed below for the first time as a valid tool for the analysis of quality characteristics of electrical power systems and particularly for the analysis of harmonics. 

Expression (7-9) formalizes the bootstrapping process described in Chapter 3. Essentially, the n-yeai discount factor is computed using the discount factors for years 1 to -l and the -year swap or zero-coupon rate. Given the discount factor for any period, that periods zero-coupon, or spot, rate can be derived using (7.9) rearranged as (7.10). 

Note that all the information needed to calculate the first-order energy and wave function is available from the zeroth order problem. For each higher order, the corrections to the energy and wave function only depend on the (n — l)th order information so, although tedious, each order of corrections can be obtained by a sort of mathematical bootstrap process. Perhaps perturbation theory is conceptually useful in that we now know that we can look at an unsolved problem and by mental modeling see that the answer is like a solvable problem with some modification. 

Larger steam turbine or bootstrap required Expander and generator part of process Addition of expander requires prior planning Induction M/G decreases power factor Loss of production on loss of generator or expander... 

One might also anticipate that the influence of bootstrap effects (Section 8.3.1.2) would be quite different in living and non-living processes. 68 A comprehensive study of reactivity ratios in living and conventional radical polymerization may provide a test of the various hypotheses for the origin of this effect. 

CV or bootstrap is used to split the data set into different calibration sets and test sets. A calibration set is used as described above to create an optimized model and this is applied to the corresponding test set. All objects are principally used in training set, validation set, and test set however, an object is never simultaneously used for model creation and for test. This strategy (double CV or double bootstrap or a combination of CV and bootstrap) is applicable to a relatively small number of objects furthermore, the process can be repeated many times with different random splits resulting in a high number of test-set-predicted values (Section 4.2.5). 

In most cases, however, the protein molecules are larger or the resolution of the data is lower, and phasing becomes a two-stage process. If two or more intensity measurements are available for each reflection with differences arising only from some property of a small substructure, the positions of the substructure atoms can be found first, and then the substructure can serve as a bootstrap to initiate the phasing of the complete structure. Suitable substructures may consist of heavy atoms soaked into a crystal in an isomorphous replacement experiment, or they may consist of the set of atoms that exhibit... 

I will return to this diagram near the end of the chapter, particularly to amplify the meaning of error removal, which is indicated by dashed horizontal lines in Fig. 7.1. For now, I will illustrate the bootstrapping technique for improving phases, map, and model with an analogy the method of successive approximations for solving a complicated algebraic equation. Most mathematics education emphasizes equations that can be solved analytically for specific variables. Many realistic problems defy such analytic solutions but are amenable to numerical methods. The method of successive approximations has much in common with the iterative process that extracts a protein model from diffraction data. 

Figure 18.4 A Phylogenetic relationships of moth ABPXs and DmeIPBPRPs. The primary sequences of the proteins were aligned in Clustal X 1.8 and processed using PAUP 4.0d65 (Swofford, 1999). The tree represents equally most parsimonious trees of 909 steps and consistency index 0.52. The numbers above each branch indicate the percent bootstrap support above 50 percent for the supported node using maximum parsimony (Felsenstein,...

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