Bootstrapping techniques

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. 

Cross validation and bootstrap techniques can be applied for a statistically based estimation of the optimum number of PCA components. The idea is to randomly split the data into training and test data. PCA is then applied to the training data and the observations from the test data are reconstmcted using 1 to m PCs. The prediction error to the real test data can be computed. Repeating this procedure many times indicates the distribution of the prediction errors when using 1 to m components, which then allows deciding on the optimal number of components. For more details see Section 3.7.1. 

If PCA is used for dimension reduction and creation of uncorrelated variables, the optimum number of components is crucial. This value can be estimated from a scree plot showing the accumulated variance of the scores as a function of the number of used components. More laborious but safer methods use cross validation or bootstrap techniques. 

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. 

Perform parsimony analysis with bootstrapping technique on gastrin precursors with nucleotide sequences given in Exercise 1. 

In the first multiple-blend study, the multiple-sample bootstrap technique correlated with the reference UV assay, correctly identifying that the appropriate potency was not reached until the 20-min time point, although standard deviations levelled off at 10 min. The 10- and 15-min samples had... 

Figure 14-9 DoD plot for comparison of two drug assays nonparametric analysis. A histogram shows the relative frequency of N = 65 differences with demarcated 2.5 and 97.5 percentiles determined nonparametrleally.The 90% CIs of the percentiles are shown. These were derived by the bootstrap technique.

The Cl is [-0.144, -0.108] and does not contain zero, supporting the notion that the two elimination rate constants do differ. An alternative approach to the above would be to replace the Wald based confidence intervals with those produced using the nonparametric bootstrap technique. With this technique the data set is sampled with replacement at the subject level many times, and the model is fit to each of these resampled data sets, generating an empirical distribution for each model parameter. Confidence intervals can then be constructed for the model parameters based on the percentiles of their empirical distributions. 

In a real HMM tagger system, we have to determine these probabilities from data. In general, this data has to be labelled with a POS tag for each word, but in cases where labelled data is scarce, we can use a pre-existing tagger to label more data, or use a bootstrapping technique where we use one HMM tagger to help label the data for another iteration of training. 

Sequence-based phylogenies are determined using an parsimony algorithm (Cedergren al.. 1988). Most parsimonious solutions are evaluated using a bootstrapping technique and occasionally branch points are validated by the invariants method of Cavender and Felsenstein (1987) and of Lake (1987). 

The second dimension, denoted Dim 2 in Fig. 9.3, explains 6.69% of the total variance. Although this value seems apparently low, this dimension will be kept in the analysis since it has been tested as statistically significant by using bootstrap technique (not presented in this chapter), and is obviously interesting in terms of interpretation. 

The simulation-on-simulation technique is computationally intensive. The information contained in Figures 2-4 required approximately 16 hours on an IBM-PC 386 clone operating at 20 MHz with an 80387 coprocessor running NDP-386 fortran code. Similar analyses have been carried out using the maximum likelihood estimator and using non-paranietric bootstrap techniques for both linear regression and maximum likelihood estimators. 

---