Statistical/probabilistic models

A more general statistical (probabilistic) model of the system takes into account both an offset (Pq) and uncertainty (r,). 

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 ... 

Probabilistic CA. Probabilistic CA are cellular automata in which the deterministic state-transitions are replaced with specifications of the probabilities of the cell-value assignments. Since such systems have much in common with certain statistical mechanical models, analysis tools from physics are often borrowed for their study. Probabilistic CA are introduced in chapter 8. 

This leads us to the other hand, which, it should be obvious, is that we feel that Chemometrics should be considered a subfield of Statistics, for the reasons given above. Questions currently plaguing us, such as How many MLR/PCA/PLS factors should I use in my model , Can I transfer my calibration model (or more importantly and fundamentally How can I tell if I can transfer my calibration model ), may never be answered in a completely rigorous and satisfactory fashion, but certainly improvements in the current state of knowledge should be attainable, with attendant improvements in the answers to such questions. New questions may arise which only fundamental statistical/probabilistic considerations may answer one that has recently come to our attention is, What is the best way to create a qualitative (i.e., identification) model, if there may be errors in the classifications of the samples used for training the algorithm ... 

Ho CK (2004) Probabilistic modeling of peracutaneous absorption for risk-based exposure assessments and transdermal drug delivery. Statistical Methodology 1 47-69... 

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. 

Figure 18 presents the results of the statistical analysis for the sequences having approximately 1 1 AB composition. Also, Fig. 18 demonstrates the fluctuation function Fd(A.) predicted by the probabilistic model [88] for N = 1024 and designed sequences do not correspond... 

For food allergens, validated animal models for dose-response assessment are not available and human studies (double-blind placebo-controlled food challenges [DBPCFCs]) are the standard way to establish thresholds. It is practically impossible to establish the real population thresholds this way. Such population threshold can be estimated, but this is associated with major statistical and other uncertainties of low dose-extrapolation and patient recruitment and selection. As a matter of fact, uncertainties are of such order of magnitude that a reliable estimate of population thresholds is currently not possible. The result of the dose-response assessment can also be described as a threshold distribution rather than a single population threshold. Such distribution can effectively be used in probabilistic modeling as a tool in quantitative risk assessment (see Section 15.2.5)... 

The probabilistic model of macromolecular association introduced in the previous section, for the case of large a and n B, may be recast into the formal language in terms of statistical thermodynamics. Recall from Chapter 1 that the chemical potential of a species has two terms, a structural energy (enthalpy) term and a concentration/entropy term ... 

It is believed that the potential offered by probabilistic modelling to overcome the inadequacies in the data will start to be fulfilled in the coming years. The issue could be explaining the use of statistics to resolve a very complex problem. 

Price ND, Shmulevich I. Biochemical and statistical network models for systems biology. Curr. Opin. Biotechnol. 2007. Shmulevich I, et al. Probabilistic Boolean Networks a rule-based uncertainty model for gene regulatory networks. Bioinformatics. 2002 18 261-274... 

Stochastic (Probabilistic) Models. One of the most significant advances in exposure estimation in the past 15 to 20 years has been the application of probabilistic statistical methods to many types of data analyses (Duan and Mage 1997 Finley and Paustenbach 1994 Morgan and Henrion 1990 US ERA 1995, 1997, 2000a). Stochastic or probabilistic techniques can help quantify variability and uncertainty in model inputs and outputs, can be used to better characterize the possible range of exposures for a particular scenario when measured data are minimal, and can be employed to better understand the uncertainty inherent in estimates developed from many different types of sources, whether quantitative or qualitative. 

To overcome the shortcoming of dependency on the initial number of clusters and cluster centroid selection, several researchers developed new algorithms to automatically detect the optimal number of clusters. Yeung and co-workers (2001a) applied probabilistic models to automatically detect the optimal number of clusters. Hastie et al. (2000) proposed the gene shaving method, a statistical method utilized to identify distinct clusters of genes. 

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