Predictive distribution

A FEM analysis was carried out and the predicted distribution of stresses on the pressure vessel compared with the stress distribution calibration using the SPATE technique. 

Evaluating the model in tenns of how well the model fits the data, including the use of posterior predictive simulations to determine whether data predicted from the posterior distribution resemble the data that generated them and look physically reasonable. Overfitting the data will produce unrealistic posterior predictive distributions. 

A common use of statistics in structural biology is as a tool for deriving predictive distributions of strucmral parameters based on sequence. The simplest of these are predictions of secondary structure and side-chain surface accessibility. Various algorithms that can learn from data and then make predictions have been used to predict secondary structure and surface accessibility, including ordinary statistics [79], infonnation theory [80], neural networks [81-86], and Bayesian methods [87-89]. A disadvantage of some neural network methods is that the parameters of the network sometimes have no physical meaning and are difficult to interpret. 

Thompson and Goldstein [89] improve on the calculations of Stolorz et al. by including the secondary structure of the entire window rather than just a central position and then sum over all secondary strucmre segment types with a particular secondary structure at the central position to achieve a prediction for this position. They also use information from multiple sequence alignments of proteins to improve secondary structure prediction. They use Bayes rule to fonnulate expressions for the probability of secondary structures, given a multiple alignment. Their work describes what is essentially a sophisticated prior distribution for 6 i(X), where X is a matrix of residue counts in a multiple alignment in a window about a central position. The PDB data are used to form this prior, which is used as the predictive distribution. No posterior is calculated with posterior = prior X likelihood. 

Figure 2 Data distribution and draws from the posterior distribution (mu sim) and posterior predictive distributions (data sim) for methionine side chain dihedral angles. The results for three ro-tamers are shown, (ri = 3 r2 = 1 = 3), (3, 2, 3), and (3, 3, 3). Each simulation consisted of...

This distribution resembies the data cioseiy for rotamer (3, 3, 3) but also forms a very reasonable distribution when there are only seven data points (3, 3, 1). A good posterior predictive distribution for any protein structural feature can be used in simulations of protein folding or strucmre prediction. 

The Burchell model s prediction of the tensile failure probability distribution for grade H-451 graphite, from the "SIFTING" code, is shown in Fig. 23. The predicted distribution (elosed cireles in Fig. 23) is a good representation of the experimental distribution (open cireles in Fig. 23)[19], especially at the mean strength (50% failure probability). Moreover, the predicted standard deviation of 1.1 MPa con ares favorably with the experimental distribution standard deviation of 1.6 MPa, indicating the predicted normal distribution has approximately the correct shape. 

As described above, the code "SIFTING" requires several microstructural inputs in order to ealculate a failure probability distribution. We are thus able to assess the physieal soundness of the Burchell model by determining the change in the predicted distribution when microstructural input parameters, such as particle or pore size, are varied in the "SIFTING" code. Each microstructural input parameter... 

Fig. 1. (left panel) [Eu/Fe] as a function of [Fe/H]. Gray-scale indicates predicted distribution of stellar fraction. The r-process site is assumed to be SNe of 8 — IOMq. The average stellar distributions are indicated by thick-solid lines with the 50% (solid lines) and 90% confidence intervals (thin-solid lines). The current observational data are given by large circles, with other previous data (small circles). 

Fig. 1. Predicted distribution of stars as a function of metallicity for several models.

A very sensitive test of the model is the comparison of calculated and observed structure functions this is shown in Figs. F and G. Note that the central force model yields the characteristic double peak in sh(s) near 2.5 A-1. That the theoretical curve oscillates with greater amplitude than the experimental data indicates that the predicted distribution of near neighbor 00 separations is too narrow. The comparison with neutron diffraction data shows that the theoretical... 

The predicted distribution corresponds to an unskewed Gaussian curve. Distributions for the other programs and MM3 with the vacuum dielectric constant were similar, but their maxima were located at about 4.55 A. We judged that distance to be too long, in part because the mean observed value is about 0.14 A (7.4 standard deviations) smaller. Also, those predicted distributions based on vacuum dielectric constants called for a large fraction of... 

An example of an exposure model to predict distribution in environmental media and estimation of the proportion of total exposure by various routes from consumer products is the BUSES (Section 7.2.4.3). It is important to recognize that the proportions of total intake from various media may vary, based on circumstances. 

Figure 4. Predicted distribution of the 10,000 catalysts in the first screening.

Initial charge distribution Carbon no. % Isomer formed Predicted distribution Observed distri- bution Predicted distribution from MO theory... 

Note that the predicted distributions of component D do not agree closely. 

Figure 11. Measured molecular-beam distribution from equation 25. Open circles represent the measured values. The solid curve represents the predicted distribution. (Reproduced with permission from reference 1. Copyright 1985 American Institute of Physics.)...

Mathematical construction of physical/ chemical processes that predict the range and probability density distribution of an exposure model outcome (e.g. predicted distribution of personal exposures within a study population)... 

Fig. 1.7 (a) Observed and model fitted percent responders in Phase II trials. (b)Predicted dose-response of response (% responders) in a Phase III population based on a population simulation of the Phase II model. The dotted lines are the 5th and 95th percentiles of the prediction distribution and represent uncertainty in the model parameters. 

In principle, the outputs from variability and uncertainty analysis are used to quantify the nature of the variability in the predicted distribution of the exposures and the uncertainties associated with different percentiles of the predicted population exposure or risk estimates. The combined sensitivity and uncertainty are also used in a two-dimensional probabilistic exposure or dose assessment, in order to determine either the uncertainty about exposure (or... 

QET (or RRKM theory) does predict the relative translational energies of the incipient fragments at the transition state. The predicted distribution of relative translational energies is given by [cf. eqn. (1)]... 

Assessment of the environmental fate of each group. Fate models may be used to predict environmental fate of mixture components on different spatial scales. Such models may yield a predicted distribution over air, water, soil, and sediment. 

Table 4 shows the predicted intensities from fn configurations in the final column the expected distribution among spin-orbit components is presented. It has been assumed that only the lowest J component of the ground state is appreciably populated. Since the predicted distribution is very uneven, some allowed components having a normalised intensity less than 10-4, only those components for which the intensity is >0.1 have been listed. In repeated (LS) states, for which only summed intensities are listed, the relative strengths for different J values should be the same. 

Table III also includes on line (d) for each composition the interpolated experimental result obtained from a computer fit to the 29si NMR data (11). The predicted distributions (b) in Table III are compared with the experimental data in Figure 4.

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