Branched-tree model

Figure 3.2. The branched-tree model (a) direct and (b) inverted to illustrate connected...

Frequency Phase 3 Use Branch Point Estimates to Develop a Ere-quency Estimate for the Accident Scenarios. The analysis team may choose to assign frequency values for initiating events and probability values for the branch points of the event trees without drawing fault tree models. These estimates are based on discussions with operating personnel, review of industrial equipment failure databases, and review of human reliability studies. This allows the team to provide initial estimates of scenario frequency and avoids the effort of the detailed analysis (Frequency Phase 4). In many cases, characterizing a few dominant accident scenarios in a layer of protection analysis will provide adequate frequency information. 

The event tree model is started from the initial occurrence and built upon by sequencing tlie possible events and safety systems tliat come into play. Tlie model displays at a glance, branches of events tliat relate tlie proper functioning or failure of a safety device or s) Stem and tlte ultimate consequence. 

Fig. 13 Hyperbranced graft architecture based on graft-on-graft technique, a Growing tree model (both stem and branch) b Growing hyperbranching model (controlled chain length) c Growing hyperbranching at stem end...

One of the earliest attempts to model chemical transformations in a living system was carried out in 1987. This system consists of a biotransformation database and one or more logic-based prediction tools.This system and other knowledge-based systems provide a branching tree of possible metabolites but provide no information on likelihood or quantitative rates of production. 

Figure 21. Mutant space high-value contour near local optimum. Diagram is multiply branched tree with different macromolecular sequences at vertices. Each line joins neighboring sequences whose values are within 0.5 of locally optimum sequence at lower center for linearized fitness function of type 2 [Eqn. (IV.7)] and reference fold that is cruciform, like tRNA, for sequence of length 72. Over 1300 branches shown extending up to 10 mutant shells away from central optimum. Better sequence (labeled optimum) was found in tenth mutant shell. Non-random sampling of mutant sequences demonstrated typical of population sampling in quasispecies model. Note small number of ridges that penetrate deeply into surrounding mutant space. (Additional connected paths due to hypercube topology of mutant space not shown.)...

In complex processes, defining tasks which the operator must perform should be optimizing a great deal by making a computerized census and description of these tasks. Tasks and their chaining can thus be represented on an tree model, on which the nodes are the tasks themselves, described by structured objects, and branches the successor or predecessor type relations linking them to one another, as weU as the conditions of progression from one task to another [12]. 

Note that Eqns. 33 to 35 of the Cayley tree model, and Eqns. 27 and 28 of the square-channel model are predictions for the same quantity the total current leaving the tree through its "reactive" side-walls. While the square-charmel model is restricted to describe branches of equal width and length, the Cayley the model carries no such limitation. This scaling is quantified by inclusion of the trees two fractal dimensions, Dtree and Damopy, implied by the length and width ratios p and q, respectively. 

Fig. 7. Scaled currents given by the Cayley tree model for trees of varying depth, n. Here, tq and Lq are the radius and length of the tree s entrance branch, respectively.

Rg.3 A branch model of primary chain crosslinking and compression into a tree model. 

A logic model that graphically portrays the range of outcomes from the combinations of events and circumstances in an accident sequence. For example, a flammable vapor release may result in a fire, an explosion, or in no consequence depending on meteorological conditions, the degree of confinement, the presence of ignition sources, etc. These trees are often shown with the probability of each outcome at each branch of the pathway... 

Accident progression scenarios are developed and modeled as event trees for each of these accident classes. System fault trees are developed to the component level for each branch point, and the plant response to the failure is identified. Generic subtrees are linked to the system fault trees. An example is "loss of clcciric power" which is analyzed in a Markov model that considers the frequencies of lo,sing normal power, the probabilities of failure of emergency power, and the mean times to repair parts of the electric power supply. 

Figure 1. An unrooted phylogenetic tree of the myosins based on the amino acid sequence comparison of their head domains demonstrating the division of the myosin superfamily into nine classes. The lengths of the branches are proportional to the percent of amino acid sequence divergence and a calibration bar for 5% sequence divergence is shovk n. The different classes of myosins have been numbered using Roman numerals in rough order of their discovery and hypothetical models of the different myosin structures are shown. Question marks indicate either hypothetical or unknown structural features, and only a fraction of the known myosins are shown. (Taken, in modified form, from Cheney et al., 1993).

In one of the more recent continuum models (69), an approach first suggested by Leonardo da Vinci in the fifteenth century, from his observations that the cross-sectional area (CSA) of the main stem of a tree equaled the total cross-sectional area of all the tree branches, was used. The more generalized rule was considered ... 

For acute releases, the fault tree analysis is a convenient tool for organizing the quantitative data needed for model selection and implementation. The fault tree represents a heirarchy of events that precede the release of concern. This heirarchy grows like the branches of a tree as we track back through one cause built upon another (hence the name, "fault tree"). Each level of the tree identifies each antecedent event, and the branches are characterized by probabilities attached to each causal link in the sequence. The model appiications are needed to describe the environmental consequences of each type of impulsive release of pollutants. Thus, combining the probability of each event with its quantitative consequences supplied by the model, one is led to the expected value of ambient concentrations in the environment. This distribution, in turn, can be used to generate a profile of exposure and risk. 

The formulation for this scenario entails 1411 constraints, 511 continuous and 120 binary variables. The reduction in continuous variables compared to scenario 1 is due to the absence of linearization variables, since no attempt was made to linearize the scenario 2 model as explained in Section 4.3. An average of 1100 nodes were explored in the branch and bound search tree during the three major iterations between the MILP master problem and the NLP subproblem. The problem was solved in 6.54 CPU seconds resulting in an optimal objective of 2052.31 kg, which corresponds to 13% reduction in freshwater requirement. The corresponding water recycle/reuse network is shown in Fig. 4.10. 

The overall model for this scenario involves 5614 constraints, 1132 continuous 280 binary variables. Three major iterations with an average of 1200 nodes in the branch and bound search tree were required in the solution. The objective value of 1560 kg, which corresponds to 33.89% reduction in freshwater requirement, was obtained in 60.24 CPU seconds. An equivalent of this scenario, without reusable water storage, i.e. scenario 2, resulted in 13% reduction in fresh water. Figure 4.12 shows the water recycle/reuse network corresponding to this solution. 

An initial suggestion made by Ford Doolittle shows a jumble of interconnections between the lines of development, rather than simple branches in the phylogenetic tree. These interconnections resemble a mycelium and have almost nothing in common with the original model, except for the termini of the three kingdoms. In a review article in Science, Elizabeth Pennisi (2001) chose the colourful metaphor of a tangled bramble bush to describe the new model. 

Model EFIMOD (Chertov, Komarov, 1997) has been applied to compute data on annual increase of wood stock in stems and large branches of main tree types widespread in the forests of the European part of Russia. We assume that these data after some improving and completing could be applied in the national database. As a cartographic layer a generalized version of the map of forest tree dominants is used (Figure 8). 

Statistical modeling by Gordon et al. [35, 36], Dusek [37], Burchard [38] and others reduced such branched species to graph theory designed to mimic the morphological branching of trees. These dendritic models were combined with... 

FIGURE 4.19 Tree of all possible models for three regressor variables with AIC values for a hypothetical example. If the AIC is used for model selection the right branch (models with x2 + x3 or x2 or x3) of the tree can he excluded. 

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