Likelihood

We should now consider what precisely is meant by severity and likelihood in the context of HIT. 

Severity represents the magnitude of the impact or potential impact on the patient -in other words, the extent to which a patient s wellbeing could be compromised. Note that the key word here is patient . One should only begin to determine the severity of a hazard when the potential impact on the patient can be and has been ascertained. If it cannot be established, further investigation is usually necessary (in association with a domain expert) or consideration should be given as to whether the scenario is indeed a hazard in the first place. 

Putting the patient at the centre of the analysis forces one to consider the impact on the patient rather than that on the system or its users. For example  

System A is used to schedule screening tests for adult patients. A hardware fault occurs which results in system downtime for a period of 6 h. 

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For all calculations reported here, binary parameters from VLE data were obtained using the principle of maximum likelihood as discussed in Chapter 6, Binary parameters for partially miscible pairs were obtained from mutual-solubility data alone. 

As indicated in Chapter 6, and discussed in detail by Anderson et al. (1978), optimum parameters, based on the maximum-likelihood principle, are those which minimize the objective function... 

The method used here is based on a general application of the maximum-likelihood principle. A rigorous discussion is given by Bard (1974) on nonlinear-parameter estimation based on the maximum-likelihood principle. The most important feature of this method is that it attempts properly to account for all measurement errors. A discussion of the background of this method and details of its implementation are given by Anderson et al. (1978). 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

When there is significant random error in all the variables, as in this example, the maximum-likelihood method can lead to better parameter estimates than those obtained by other methods. When Barker s method was used to estimate the van Laar parameters for the acetone-methanol system from these data, it was estimated that = 0.960 and A j = 0.633, compared with A 2 0.857 and A2- = 0.681 using the method of maximum likelihood. Barker s method uses only the P-T-x data and assumes that the T and x measurements are error free. 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

The maximum-likelihood method is not limited to phase equilibrium data. It is applicable to any type of data for which a model can be postulated and for which there are known random measurement errors in the variables. P-V-T data, enthalpy data, solid-liquid adsorption data, etc., can all be reduced by this method. The advantages indicated here for vapor-liquid equilibrium data apply also to other data. 

The maximum-likelihood method, like any statistical tool, is useful for correlating and critically examining experimental information. However, it can never be a substitute for that information. While a statistical tool is useful for minimizing the required experimental effort, reliable calculated phase equilibria can only be obtained if at least some pertinent and reliable experimental data are at hand. 

FORMAT (IH1,57HMAXlMUM LIKELIHOOD ESTIMATION OF PARAMETERS FROM VL IE DATA//1X,40HCONTROL PARAMETERS WERE SET AS FOLLOWS -/)... 

MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS FROM VLE DATA... 

In the first step, a screening process will be applied to separate the major potential hazards these will be addressed in more detail. QRA techniques are used to evaluate the extent of the risk arising from hazards with the potential to cause major accidents, based on the prediction of the likelihood and magnitude of the event. This assessment will be based on engineering judgement and statistics of previous performance. Where necessary, risk reduction measures will be applied until the level of risk is acceptable. This of course is an emotive subject, since it implies placing a value on human life. 

The producing gas oil ratio starts at the solution GOR, decreases until the critical gas saturation is reached, and then increases rapidly as the liberated gas is produced into the wells, either directly as it is liberated, or pulled into the producing wells from the secondary gas cap. The secondary gas cap expands with time, as more gas is liberated, and therefore moves closer to the producing wells, increasing the likelihood of gas being pulled In from the secondary gas cap. 

For clutter suppression, the test statistic used by the noncoherent detector has been proposed as an interesting output signal [1], This was motivated by the fact that, provided that transient and noise models are valid, the test statistic reflects the likelihood that a transient is present. 

The importance of distinct a priori knowledge account becomes more perceptible if noisy data are under restoration. The noise / ( shifts the solution of (1) from the Maximum Likelihood (ML) to the so called Default Model for which the function of the image constraint becomes more significant. 

Another question is the nature of the changes in the classical dynamics that occur with the breakdown of the polyad number. In all likelihood there are farther bifiircations. Apart from the identification of the individual polyad-breaking resonances, the bifiircation analysis itself presents new challenges. This is partly becanse with the breakdown... 

Direct dissociation reactions are affected by surface temperature largely tlirough the motion of the substrate atoms [72]. Motion of the surface atom towards the incoming molecule mcreases the likelihood of (activated) dissociation, while motion away decreases the dissociation probability. For low dissociation probabilities, the net effect is an enliancement of the dissociation by increasing surface temperature, as observed in the system 02/Pt 100]-hex-R0.7° [73]. 

By examining the expression for Q ( equation (B1.16.4)). it should now be clear that the nuclear spin state influences the difference in precessional frequencies and, ultimately, the likelihood of intersystem crossing, tlnough the hyperfme tenn. It is this influence of nuclear spin states on electronic intersystem crossing which will eventually lead to non-equilibrium distributions of nuclear spin states, i.e. spin polarization, in the products of radical reactions, as we shall see below. 

Enderiein J, Goodwin P M, Van Orden A, Ambrose W P, Erdmann R and Keller R A 1997 A maximum likelihood estimator to distinguish single molecules by their fluorescence decays Chem. Phys. Lett. 270 464-70... 

MES)==10 These results suggest tliat C(MES) grows (in all likelihood) only as In N with N. Thus tlie restriction of compactness and low energy of tlie native states may impose an upper bound on tlie number of distinct protein folds. 

For example, for iron in aqueous electrolytes, tlie tliennodynamic warning of tlie likelihood of corrosion is given by comparing tlie standard electrode potential of tlie metal oxidation, witli tlie potential of possible reduction reactions. 

The simplest condensed phase VER system is a dilute solution of a diatomic in an atomic (e.g. Ar or Xe) liquid or crystal. Other simple systems include neat diatomic liquids or crystals, or a diatomic molecule bound to a surface. A major step up in complexity occurs with poly atomics, with several vibrations on the same molecule. This feature guarantees enonnous qualitative differences between diatomic and polyatomic VER, and casts doubt on the likelihood of understanding poly atomics by studying diatomics alone. 

For many reasons, including the Woodward-IIoffm an rules that describe the likelihood of reaction based on arguments about the shapes of orbitals, it is desirable to be able to visualize molecular orbitals. 

One limitation of clique detection is that it needs to be run repeatedly with differei reference conformations and the run-time scales with the number of conformations pt molecule. The maximum likelihood method [Bamum et al. 1996] eliminates the need for reference conformation, effectively enabling every conformation of every molecule to a< as the reference. Despite this, the algorithm scales linearly with the number of conformatior per molecule, so enabling a larger number of conformations (up to a few hundred) to b handled. In addition, the method scores each of the possible pharmacophores based upo the extent to which it fits the set of input molecules and an estimate of its rarity. It is nc required that every molecule has to be able to match every feature for the pharmacophor to be considered. 

With the more concentrated solution the results, as regards loss of intermolecular selectivity, were similar to those obtained with nitronium salts (table 4.1, column a), whilst with the more dilute solution a more usual situation was revealed. The significance of the former observations is again open to doubt because of the likelihood that mixing was relatively slow, and also because reaction upon encounter is here a serious probability. 

The carbon that bears the leaving group and a carbon ortho to it become equiva lent m the benzyne intermediate Thus when chlorobenzene 1 is the substrate the ammo group may be introduced with equal likelihood at either position... 

One of the most important uses of models is to show how electrons are distributed inside molecules The laws of quantum mechanics state that an electron s spatial location can not be precisely specified but the likelihood of detecting an electron at a particular loca tion can be calculated (and measured) This likelihood is called the electron density (see Chapter 1) and SpartanView can display three dimensional graphs that show regions of high and low electron density inside a molecule... 

When collecting a sample, for instance, only a small portion of the available material is taken, increasing the likelihood that small-scale inhomogeneities in the sample will affect the repeatability of the analysis. Individual pennies, for example, are expected to show variation from several sources, including the manufacturing process, and the loss of small amounts of metal or the addition of dirt during circulation. These variations are sources of indeterminate error associated with the sampling process. 

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