Statistical bounding techniques

In Chapter 3 of this book we discussed the problem of multisite refinery integration under deterministic conditions. In this chapter, we extend the analysis to account for different parameter uncertainty. Robustness is quantified based on both model robustness and solution robustness, where each measure is assigned a scaling factor to analyze the sensitivity of the refinery plan and integration network due to variations. We make use of the sample average approximation (SAA) method with statistical bounding techniques to generate different scenarios. 

Ferson (1996) points out that there is a class of problems in which information may be known regarding upper and lower bounds for a particular variable, but not regarding a probability distribution. The model output in such cases is an interval, rather than a distribution. Such problems are more appropriately dealt with using interval methods rather than imposing probabilistic assumptions upon each input. Interval methods can be extended to situations in which marginal probability distributions are specified for each model input but for which the dependence between the distributions is not known. Thus, rather than assume statistical independence, or any particular correlation or more complex form of dependency, a bounding technique can be used to specify the range within which the model output distribution must be bounded. 

InfraAlyzer 400 data, having a maximum of 19 data points/spectrum, are restricted in the variety of mathematical manipulations available for the purpose of generating a predictive equation. Even so, many methods were tried [23]. Eventually, the branch and bound technique of Furnival and Wilson [24] was found both satisfactory in performance and easy to apply, being part of the BMDP statistical software package [25]. With careful choice of the limiting parameters stable equations were consistently produced. 

The technique known as time correlated single photon counting is based on the probability that one (single) photon emitted by a luminescent sample could be detected by a highly sensitive photomultiplier. This probability is statistically bound with the change in the emitting excited state concentration with time by a specihc operative procedure. 

One of the major questions that should always be asked of any estimation technique is whether is is optimal. This problem can be addressed using Cramer-Rao bounds and we illustrate this in Section 24.4. We conclude this chapter in Section 24.5 with the application of these statistical techniques to the Shack-Hartmann and curvature sensors. 

Microarrays are thousands of 150- to 250-micron spots of DNA bound to microscope slides in a precise and known pattern. Each DNA spot quantitatively hybridizes to a specific mRNA so that expression of thousands of individual genes can be measured simultaneously. Importantly, microarray techniques are sensitive typically, twofold differences in mRNA concentration are determined and each gene/ DNA spot has a sensitivity of 15 attomoles, or approximately one out of 300,000 transcripts can be measured. Application of statistical analyses with appropriate replication has improved these analyses such that less than 1.5-fold differences are readily discerned. The most common use of microarrays is to profile transcript levels on a genome-wide basis. The idea is that knowledge of where (what tissue) and when and how much a gene is expressed will inform its function. Comparisons of gene... 

This section is organized as follows in subsection A the approaches based on the assumption of heat bath statistical equilibrium and those which use the generalized Langevin equation are reviewed for the case of a bounded one-dimensional Brownian particle. A detailed analysis of the activation dynamics in both schemes is carried out by adopting AEP and CFP techniques. In subsection B we shall consider a case where the non-Markovian eharacter of the variable velocity stems from the finite duration of the coherence time of the light used to activate the chemical reaction process itself. 

IR microscopy analysis allows the spatially resolved and destruction-free characterization and quality control of polymer-bound compound libraries. Furthermore, by superpositioning of different IR maps, compounds immobilized on individual resin beads can be identified. Through improved embedding techniques, even larger areas with many resin beads can be mapped very quickly, thus improving the statistical evaluation by image analysis. 

One can calculate the acceptance criteria for the particle size distribution median and standard deviation by use of a technique described by Hahn and Meeker (11). Their work describes three types of statistical interval confidence, prediction, and tolerance. The authors maintain that the choice of the appropriate statistical interval to use depends on the nature of the parameters to be estimated. The confidence interval is used when hying to find bounds on a population parameter—for example, the population mean or standard deviation. The confidence interval is the most commonly appearing of the three intervals and is the interval... 

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