The Bootstrap Matrix

Equations 7.2.13 and 7.2.15 will often be needed in n — 1 dimensional matrix form. The required expressions are 

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It can be shown by inverting the bootstrap matrix [j8] using the Sherman-Morrison... 

The total flux is computed from Eq. 11.5.14 (cf. Eq. 7.2.13) with the term set to zero. For the record, the bootstrap matrix [/ q], although not used directly in the computation of the molar fluxes, is... 

Fig. 7. Bootstrap analysis among characters in a parsimony analysis. The tree to the right of each matrix is the most parsimonious tree for that matrix. The final results of the bootstrap analysis are shown in the tree at the bottom. The number of times each branch was supported in the bootstrap replication is shown as a percentage. Outgroup rooting carries the assumption of ingroup monophyly, so no confidence interval can be assigned to the branch that unites the ingroup.

This cumbersome expression (Eq. 7.3.12) is not particularly useful as it stands. Fortunately, it simplifies considerably in some special cases of interest. First, if the total flux is near zero (as it will often be in distillation), the bootstrap matrices [ ] reduce to the identity matrix and we have... 

Approximate (1 — a)100% confidence intervals can be developed using any of the methods presented in the bootstrapping section of the book appendix. Using the previous example, CL was simulated 1,000 times from a normal distribution with mean 50 L/h and variance 55 (L/h)2 while V was simulated 10,000 times with a mean of 150 L and variance 225 L2. The correlation between V and CL was fixed at 0.18 given the covariance matrix in Eq. (3.70). The simulated mean and variance of CL was 49.9 L/h and 55.5 (L/h)2, while the simulated mean and variance of V was 149.8 L with variance 227 L2. The simulated correlation between CL and V was 0.174. The mean estimated half life was 2.12 h with a variance of 0.137 h2, which was very close to the Taylor series approximation to the variance. The Sha-piro Wilk test for normality indicated that the distribution of half life was not normally distributed (p < 0.01). Hence, even though CL and V were normally distributed the resulting distribution for half life was not. Based on the 5 and 95% percentiles of the simulated half life... 

Bootstrapping is restrained to the use of the original data set to estimate confidence intervals. Part of the data (rows of the data matrix) is sorted out and used for later predictions, The missing rows in the matrix are replaced randomly by data vectors kept. The latter vectors are then used twice in a computational run (cf. Tree-Based Classification Section). 

In the case that the joint or marginal distribution of the test statistics is unknown, p-values can be estimated by resampling methods such as permutation and bootstrap. For example, consider a permutation algorithm to estimate p-values with large biological data in the following manner. First, permute the N sample columns of the data matrix and compute test statistics for each biomarker candidate. Let tij, be test statistics for the th permutation. When repeating this procedure many limes (e.g., B = 100 times), the permutation p-value for hypothesis Hj is... 

Chang and Lewis code the positions (x,y,z) of the heavy atoms within the unit cell in the chromosome. The solution proceeds in one of two ways. First is the bootstrap in which the position of one of the atoms is found and then the vicinity of that atom is excluded from further search. The problem with this. method is that errors in trial solutions of early positions can ruin later prediction. Their other approach, which seems preferable, is to code for all of the positions simultaneously. This seems to provide higher quality solutions. An interesting variant on the chromosome coding is to use (x, y, z) for the first atom and then reference the others from that with a rotation matrix and offset vector. This is useful when the relative positions of the heavy atoms are known at least approximately. Otherwise, a standard SGA is used. 

A) A distance matrix was constructed from the inferred amino acid sequences using a Poisson correction for multiple hits and the tree constructed using the minimum evolution approach. Five hundred bootstrap resamplings were carried out. Branches with bootstrap support values less than 50% are indicated with an asterisk. 

Fig. 2.3 Phylogeny of bacterial representatives of the peroxidase-cyclooxygenase superfamily. The reconstructed tree obtained from the NJ-method of the MEGA package [13] with JTT matrix and 1,000 bootstrap replications is presented. A similar tree was obtained also with ProML-method of the PHYLIP package [16] with 100 bootstraps. Numbers in the nodes indicate bootstrap values for NJ and ProML method, respectively. Abbreviations of protein names correspond to PeroxiBase...

In Section 8.3 we presented a derivation of an exact matrix solution of the Maxwell-Stefan equations for diffusion in ideal gas mixtures. Although the final expression for the composition profiles (Eq. 8.3.12), is valid whatever relationship exists between the fluxes (i.e., bootstrap condition), the derivation given in Section... 

More difficult is the estimation of errors for the nonlinear parameters, since no variance-covariance matrix exists. Frequently, the error estimations are restricted to a locally linear range. In the linearization range, the confidence bands for the parameters are then calculated as in the linear case (Eqs. (6.25)-(6.27)). An alternative consists in error estimations on the basis of Monte Carlo simulations or bootstrapping methods (cf. Section 8.2). 

The procedure implemented in MUSTADEPT to exploit the maintenance inspection records uses the MLE technique to estimate the stochastic transition parameters, and a bootstrap-based technique to estimate the Fisher Information Matrix, which allows evaluating the uncertainty in the MLE values. 

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