A probabilistic model

This model can be extended (/00) tQ take termination by combination into account, by assuming that a certain fraction, F, of the chain ends other than those occupied by initiator or transfer radicals are linked together in pairs at random. It is found that the weight-average DP Pw remains bounded if P Peru, where  

For F Fcrj cross-linked gel will be present, and there is no physical reason why this condition should not hold. When F Fcrjt, analytical expressions for the moments can be found, but when F Fcrit, these moments are only calculable numerically such moments refer to the sol fraction only. 

The results for this model suggest, though they do not conclusively demonstrate, that in the absence of termination by combination branching can only lead to finite values of MJMn, but that combination plus branching can lead to the production of gel. This is in agreement with the results of Kuchanov and Pismen for batch reactions (94). 

In a real free-radical polymerization, the probability that a given monomer unit in the polymer bears a branch is not constant, but increases with time. Mullikin and Mortimer (tOt) have extended their model to take into account the time-dependence of branching probability and the distribution of residence times in a continuous reactor. They assume that the branching probability b is given by  

Because of the difficulty of theoretical calculations of the properties of branched polymers, much experimental work has been done with polymers of more or less known structures, to test the results of theory. Several methods have been used to produce such polymers. 

The term r-i, is not a parameter of the model but is a single value sampled from the population of possible deviations [Natrella (1963)]. The magnitude of r-i, might be used to provide an estimate of a parameter associated with that population of residuals, the population variance of residuals, aj. The population standard deviation of residuals is a,. The estimates of these two parameters are designated s] and s respectively [Neter, Wasserman, and Kutner (1990)]. If DF, is the number of degrees of freedom associated with the residuals, then 

For this model and for only one experiment, n = I and DF, = 1 (we have not calculated a mean and thus have not taken away any degrees of freedom) so that s] = rj, and s = r,. In this example, s, = r, = 5. 

---

Before a probabilistic model can be developed, the variables involved must be determined. It is assumed that the variables all follow the Normal distribution and that they are statistically independent, i.e. not correlated in anyway. The scatter of the pre-load, F, using an air tool with a clutch is approximately 30% of the mean, which gives the coefficient of variation, = 0.1, assuming 3cr covers this range, therefore ... 

F. Rosenblatt, The perceptron a probabilistic model for information storage and organization in the brain. Psycholog. Rev., 65 (1958) 386-408. 

It does not contain a probabilistic modeling component that simulates variability therefore, it is not used to predict PbB probability distributions in exposed populations. Accordingly, the current version will not predict the probability that children exposed to lead in environmental media will have PbB concentrations exceeding a health-based level of concern (e.g., 10 pg/dL). Efforts are currently underway to explore applications of stochastic modeling methodologies to investigate variability in both exposure and biokinetic variables that will yield estimates of distributions of lead concentrations in blood, bone, and other tissues. 

To develop a probabilistic model, one has to assign probability distributions to model inputs such as degradation rates, partition coefficients, dose-response parameters (or dose-time-response parameters), exposure values, and so on, for a model relating impacts to exposure. This chapter is concerned with several kinds of technical decisions involved in the selection of distributions. 

A probabilistic model will typically require distributions for multiple inputs. Therefore, it is necessary to consider the joint distribution of multiple variables as well as the individual distributions, i.e., we must address possible dependencies among variables. At least, we want to avoid combinations of model inputs that are unreasonable on scientific grounds, such as the basal metabolic rate of a hummingbird combined with the body weight of a duck. 

A probabilistic model is available for predicting the average log-tissue residue as a function of the water concentration at the site, and a set of site-specific tissue residue measurements is available. The water concentration to which the fish were exposed is known, so the average log-tissue residue can be predicted with the model. A Monte Carlo simulation will provide a set of equally probable predictions of the average log-tissue residue. The BMC acceptance-rejection procedure then boils down to estimating, for each model prediction, the probability of getting the observed sample average log-tissue residue concentration if the model prediction is correct. 

When the EPA considered exposures to insecticide residues in the home they identified at least six possible sources and routes these are given in Table 2.6. Their original approach apportioned the acceptable daily intake (ADI) between the various routes but it soon became clear that this was unrealistic because an individual was unlikely to be exposed via all routes on any one day. The EPA s present strategy is to develop an approach called micro-exposure event modelling. Micro-exposure event modelling is based on statistical data on the frequencies and levels of contamination of food, water, etc. and on behavioural information about the frequency of use of lawn/pet/timber treatments, etc. The combined data are assembled in a probabilistic model called LIFELINE which is able to predict the frequency and level of exposure to a group of hypothetical individuals over their lifetime.12 The model is also able to take account of the relative proportions of different types of accommodation, the incidence of pet ownership or any other data that will affect real levels of exposure. The output from the LIFELINE model allows the exposures of individuals in a population to be modelled over any interval from a single occasion to a lifetime. 

N. Mehranbod, M. Soroush, M. Piovoso, and B.A. Ogunnaike. A probabilistic model for sensor fault detection and identification. AIChE Journal, 49(7) 1787, 2003. 

CATABOL is a knowledge-based expert system for the prediction of biotransformation pathways. It works in tandem with a probabilistic model that calculates the probabilities of the individual transformations and overall biochemical oxygen demand (BOD) and extent of C02 production (Jaworska et al., 2002). The model assesses biodegradation based on the entire pathway and not,... 

Modolo J., Garenne A., Henry J., Beuter A. Probabilistic model of the subthalamic nucleus with small-world networks properties (XXVIIIth Symposium of Computational Neuroscience, Montreal (Canada), 2006). 

Sampling The results of an experiment are determined by the small number of phage (typically 20-200) that are sampled from the eluted phage. Because this sample is such a minuscule fraction of the library, we include a probabilistic model of sampling in the model. 

A Markov chain is a probabilistic model applying to systems that exhibit a special type of dependence, that is, where the state of the system on the n+1 observation depends only on the state of the system on the nth observation. In other words, once this type of a system is in a given state, future changes in the system depend only on this state and not on the manner the system arrived at this particular state. This emphasizes the fact that the past history is immaterial and is completely ignored for predicting the future. 

Hughes and Waley (35) described a probabilistic model that was used to characterize dose-taking behavior of subjects (patients) in a Upid-lowering agent study. They used a random sampling of adherence patterns to drive the model that described the onset and offset of drug effects. Patients could either comply (with a probability P) or not comply (with a probability 1-P) when faced with their first dose. The probability of taking the next dose decreased as a function of time if the dose was missed as follows ... 

Rappaz M. and Gandin Ch.-A., Probabilistic Modelling of Microstructure Formation in Solidification Processes, Acta Metall. Mater 41, 345 (1993). 

Thomas A., Skolnick M. (1994) A probabilistic model for detecting coding regions in DNA sequences. IMA J. Math. Appl. Med. Biol, 11, 149-160. 

Hidden Markov model A probabilistic model that is often used as a prediction engine in bioinformatics and cheminformatics. The probability of transition between states is known although the states remain hidden. 

CLEA is a probabilistic model of human exposure to contaminants in soil. It is... 

A probabilistic model for the deactivation of a dual function catalyst by coke formation accounting for reaction and surface migration. 

---