Modeling probability distributions

It is only natural to consider ways that would allow us to use our knowledge of the whole distribution P0(AU), rather than its lew-AU tail only. The simplest strategy is to represent the probability distribution as an analytical function or a power-series expansion. This would necessarily involve adjustable parameters that could be determined primarily from our knowledge of the function in the well-sampled region. Once these parameters are known, we can evaluate the function over the whole domain of interest. In a way, this approach to modeling P0(AU) constitutes an extrapolation strategy. 

In general, this type of extrapolation is not very successful, because its reliability deteriorates, often rapidly, as we move away from the region in which the function is known with a good accuracy. In the particular case of Pq AU), we might, however, be more successful, because this function is smooth and Gaussian-like. We shall exploit these features by considering three different representations of Pq(AU). 

In fact, we have already used a modeling strategy when Po(AU) was approximated as a Gaussian. This led to the second-order perturbation theory, which is only of limited accuracy. A simple extension of this approach is to represent Pq(AU) as a linear combination of n Gaussian functions, p, (AU), with different mean values and variances [40] 

A different expansion relies on using Gram-Charlier polynomials, which are the products of Hermite polynomials and a Gaussian function [41] These polynomials are particularly suitable for describing near-Gaussian functions. Even and odd terms of the expansion describe symmetric and asymmetric deformations of the Gaussian, respectively. To ensure that P0(AU) remains positive for all values of AU, we take 

In the last equation Hi(x) is the th Hermite polynomial. The reader may readily recognize that the functions / look familiar. Indeed, these functions are identical to the wave functions for the different excitation levels of the quantum harmonic oscillator. Using the expansion (2.56), it is possible to express AA as a series, as has been done before for the cumulant expansion. To do so, one takes advantage of the linearization theorem for Hermite polynomials [42] and the fact that exp(-t2 + 2tx) is the generating function for these polynomials. In practice, however, it is easier to carry out the integration in (2.12) numerically, using the representation of Po(AU) given by expressions (2.56) and (2.57). 

---

Dupont, P., Denis, F., Esposito, Y. Links between probabilistic automata and hidden Markov models probability distributions, learning models and induction algorithms. Patterns Recogn. 38(9), 1349-1371 (2005)... 

Contrary to t3q>ically deterministic mathematical models available in the literamre, real-world apphcations are usually surrounded with uncertainty. The two main approaches for dealing with uncertainty are stochastic programming and robust optimization. For developing stochastic programming models, probability distributions of uncertain parameters should be known in advance. However, in many practical situations, there is no information or enough information for obtaining probability distribution of uncertain parameters. Robust optimization models are viable answers to these situations via providing solutions that are always... 

Hi) Gaussian statistics. Chandler [39] has discussed a model for fluids in which the probability P(N,v) of observing Y particles within a molecular size volume v is a Gaussian fimction of N. The moments of the probability distribution fimction are related to the n-particle correlation functions and... 

Monte Carlo simulations generate a large number of confonnations of tire microscopic model under study that confonn to tire probability distribution dictated by macroscopic constrains imposed on tire systems. For example, a Monte Carlo simulation of a melt at a given temperature T produces an ensemble of confonnations in which confonnation with energy E. occurs witli a probability proportional to exp (- Ej / kT). An advantage of tire Monte Carlo metliod is tliat, by judicious choice of tire elementary moves, one can circumvent tire limitations of molecular dynamics techniques and effect rapid equilibration of multiple chain systems [65]. Flowever, Monte Carlo... 

We will refer to this model as to the semiclassical QCMD bundle. Eqs. (7) and (8) would suggest certain initial conditions for /,. However, those would not include any momentum uncertainty, resulting in a wrong disintegration of the probability distribution in g as compared to the full QD. Eor including an initial momentum uncertainty, a Gaussian distribution in position space is used... 

US model can be combined with the Monte Carlo simulation approach to calculate a r range of properties them is available from the simple matrix multiplication method. 2 RIS Monte Carlo method the statistical weight matrices are used to generate chain irmadons with a probability distribution that is implied in their statistical weights. 

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... 

Modeling the pore size in terms of a probability distribution function enables a mathematical description of the pore characteristics. The narrower the pore size distribution, the more likely the absoluteness of retention. The particle-size distribution represented by the rectangular block is the more securely retained, by sieve capture, the narrower the pore-size distribution. 

CASRAM predicts discharge fractions, flash-entrainment quantities, and liquid pool evaporation rates used as input to the model s dispersion algorithm to estimate chemical hazard population exposure zones. The output of CASRAM is a deterministic estimate of the hazard zone (to estimate an associated population health risk value) or the probability distributions of hazard-zones (which is used to estimate an associated distribution population health risk). 

Uncertainly estimates are made for the total CDF by assigning probability distributions to basic events and propagating the distributions through a simplified model. Uncertainties are assumed to be either log-normal or "maximum entropy" distributions. Chi-squared confidence interval tests are used at 50% and 95% of these distributions. The simplified CDF model includes the dominant cutsets from all five contributing classes of accidents, and is within 97% of the CDF calculated with the full Level 1 model. 

In Section 20.2, equations for tlie reliability of series and parallel systems are established. Various reliability relations are developed in Section 20.3. Sections 20.4 and 20.5 introduce several probability distribution models lliat are extensively used in reliability calculations in hazard and risk analysis. Section 20.6 deals witli tlie Monte Carlo teclinique of mimicking observations on a random variable. Sections 20.7 and 20.8 are devoted to fault tree and event tree analyses, respectively. 

Bowron et al. [11] have performed neutron diffraction experiments on 1,3-dimethylimidazolium chloride ([MMIM]C1) in order to model the imidazolium room-temperature ionic liquids. The total structure factors, E(Q), for five 1,3-dimethylimidazolium chloride melts - fully probated, fully deuterated, a 1 1 fully deuterated/fully probated mixture, ring deuterated only, and side chain deuterated only - were measured. Figure 4.1-4 shows the probability distribution of chloride around a central imidazolium cation as determined by modeling of the neutron data. 

Various PIB architectures with aromatic finks are ideal model polymers for branching analysis, since they can be disassembled by selective link destmction (see Figure 7.7). For example, a monodisperse star would yield linear PIB arms of nearly equal MW, while polydisperse stars will yield linear arms with a polydispersity similar to the original star. Both a monodisperse and polydisperse randomly branched stmcture would yield linear PIB with the most-probable distribution of M jM = 2, provided the branches have the most-probable distribution. Indeed, this is what we found after selective link destruction of various DlBs with narrow and broad distribution. Recently we synthesized various PIB architectures for branching analysis. 

Many investigators have used the Ogston model and its fundamental idea as a basis for their models. Most recently, Johansson and Elvingson [182] obtained the probability distribution g(r) for spaces in a random suspension of fibers i.e., the probability that a randomly chosen point in a network of fibers is found at a radial distance r to the fiber of closest approach. For a cylindrical cell (CC) model, which consists of an infinite cylindrical cell, containing solvent and polymer, with the polymer represented as a rod centered in the cell, they obtained g(r) for one cylindrical cell as... 

We take a Bayesian approach to research process modeling, which encourages explicit statements about the prior degree of uncertainty, expressed as a probability distribution over possible outcomes. Simulation that builds in such uncertainty will be of a what-if nature, helping managers to explore different scenarios, to understand problem structure, and to see where the future is likely to be most sensitive to current choices, or indeed where outcomes are relatively indifferent to such choices. This determines where better information could best help improve decisions and how much to invest in internal research (research about process performance, and in particular, prediction reliability) that yields such information. 

The velocity gradient leads to an altered distribution of configuration. This distortion is in opposition to the thermal motions of the segments, which cause the configuration of the coil to drift towards the most probable distribution, i.e. the equilibrium s configurational distribution. Rouse derivations confirm that the motions of the macromolecule can be divided into (N-l) different modes, each associated with a characteristic relaxation time, iR p. In this case, a generalised Maxwell model is obtained with a discrete relaxation time distribution. 

According to the latter model, the crystal is described as formed of anumber of equal scatterers, all randomly, identically and independently distributed. This simplified picture and the interpretation of the electron density as a probability distribution to generate a statistical ensemble of structures lead to the selection of the map having maximum relative entropy with respect to some prior-prejudice distribution m(x) [27, 28],... 

The error-free likelihood gain, V,( /i Z2) gives the probability distribution for the structure factor amplitude as calculated from the random scatterer model (and from the model error estimates for any known substructure). To collect values of the likelihood gain from all values of R around Rohs, A, is weighted with P(R) ... 

---