Bootstrap problem

The major part of the next few chapters are devoted to methods of estimating the low flux mass transfer coefficients k and [A ] and of calculating the high flux coefficients k and [/c ]. In practical applications we will need these coefficients to calculate the diffusion fluxes 7, and the all important molar fluxes N. The are needed because it is these fluxes that appear in the material balance equations for particular processes (Chapters 12-14). Thus, even if we know (or have an estimate of) the diffusion fluxes 7 we cannot immediately calculate the molar fluxes because all n of these fluxes are independent, whereas only n — 1 of the J I are independent. We need one other piece of information if we are to calculate the N. Usually, the form of this additional relationship is dictated by the context of the particular mass transfer process. The problem of determining the knowing the 7 has been called the bootstrap problem. Here, we consider its solution by considering some particular cases of practical importance. 

There is a close relationship between the bootstrap problem discussed in Chapter 7 and the interphase energy balance Eq. 11.5.7. To demonstrate this relationship we rewrite Eq. 11.5.7 as... 

The first part deals with our non equilibrium model focused on the diffusional layer near the interface, where complete multi component reactive mass and heat transfer is described. The numeric resolution, avoid the bootstrap problem, is discussed in the second part. Finally, our experimental pilot and the different experiments to validate the model are presented in the last part. 

This, then, became the task of a small core of UCLA faculty to create an actual academic program which, by contrast with chance or the slowly accumulating scars of experience, would efficiently prepare people for the transition from the idealized, contemplative world of the university to the harsh and often political realities of environmental problem-solving. Program ideas were tested and the concept took on a recognizable structure. A program existed, even if in embryo form and only as a "bootstrap" operation, largely fueled by the after-hours effort of a number of the principals. 

Within the scope of biopharmaceutics and IVIVC, bootstrap techniques have been applied to several specific problems related to the estimation of confidence intervals of, e.g., the similarity factor/ (21), the Chinchilli metric (27), parameters of an open two-compartment system (32), and the SD in general (33). From these few applications, it cannot be judged how much is actually gained from these new techniques. 

The most serious problem is the ultimate interpretation. Will a 90% or even 99% bootstrap support for a grouping be real or biologically... 

Cross-validation is an alternative to the split-sample method of estimating prediction accuracy (5). Molinaro et al. describe and evaluate many variants of cross-validation and bootstrap re-sampling for classification problems where the number of candidate predictors vastly exceeds the number of cases (13). The cross-validated prediction error is an estimate of the prediction error associated with application of the algorithm for model building to the entire dataset. 

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. 

A Monte Carlo study demonstrated the problem of estimating the number of clusters [DUBES, 1987]. One principal reason for this problem is that clustering algorithms tend to generate clusters even when applied to random data [DUBES and JAIN, 1979]. JAIN and MOREAU [1987] therefore used the bootstrap technique [EFRON and GONG, 1983] for cluster validation. 

DM can be applied to "small" structures (< 1000 atoms in the asymmetric unit). Since a crystal with, say, 10 C atoms requires finding only x, y, and z variables, but typically several thousand intensity data can be collected, then, statistically, this is a vastly overdetermined problem. There are relationships between the contributions to the scattering intensities of two diffraction peaks (with different Miller indices h, k, l, and h, k, / ), due to the same atom at (xm, ym, zm). DM solves the phase problem by a bootstrap algorithm, which guesses the phases of a few reflections and uses statistical tools to find all other phases and, thus, all atom positions xm, ym, zm. How to start ... 

It is common for patients initially to experience a change in self-identity. Our culture values the rugged individual who can pull himself up by his bootstraps. Patients may worry that it is a sign of weakness or disability to try medication for an invisible problem. Other patients may express concerns about the cost of medication and wonder how they can afford to pay for it. 

The classical method for solving the phase problem in macromolecular crystal structures, known as isomorphous replacement, dates back to the earliest days of protein crystallography.10,16 The concept is simple enough we introduce into the protein crystal an atom or atoms heavy enough to affect the diffraction pattern measurably. We aim to figure out first where those atoms are (the heavy atom substructure) by subtracting away the protein component, and then bootstrap — use the phases based on the heavy atom substructure to solve — the structure of the protein. 

The resampling approaches of cross-validation (CV) and bootstrapping do not have the drawback of data splitting in that all available data are used for model development so that the model provides an adequate description of the information contained in the gathered data. Cross-validation and bootstrapping are addressed in Chapter 15. One problem with CV deserves attention. Repeated CV has been demonstrated to be inconsistent if one validates a model by CV and then randomly shuffles the data, after shuffling, the model may not be validated. 

However, in parametric problems the bootstrap adds little or nothing to the theory or application and one cannot explain why the typical approach to estimating parameters via formulas should be replaced by bootstrap estimates. Consequently, it is uncommon to see the parametric bootstrap used in real problems. When applied to population pharmacometric (PPM) modeling, a weakness of the parametric bootstrap is that it assumes that the model is known with a high degree of certainty. This is seldom true. 

The Bayesian bootstrap was introduced by Rubin (26) in 1981 and subsequently used by Rubin and Schenker (29) for multiple imputation in missing-data problems. The Bayesian bootstrap is not covered because its application is for multiple imputation of missing data and this is addressed in Chapter 9. 

To execute this, an estimate of the sample distribution of the LED under the null hypothesis must be derived to perform a test. The bootstrap method for estimating sample distribution of the difference of the objective function given the observations is used to solve the problem. This allows one to reject the null hypothesis of equal noncentrality parameters, that is, of equality of fit if zero is not contained in the confidence interval so derived. One thousand bootstrap pseudosamples were constructed, the nonhierarchical models of interest were applied, and the percentile method for computing the bootstrap confidence intervals was used. 

Another problem that can occur is when the 95% confidence intervals cross 0 and values below 0 make no sense. For example, one may obtain estimates of a random effect, say, the coefficient of variation for a parameter, and along with this the standard error for the coefficient of variation. Sometimes when asymptotic normality is assumed, the 95% confidence interval for the coefficient of variation can be less than 0 at its lower bound. It does not make sense to have a coefficient of variation for a parameter that is less than 0. The bootstrap can be used to avoid this error and estimate confidence intervals that make sense. An example of applying bootstrapping to deal with the concerns described here can be found in a paper by Ette and Onyiah (8). 

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