The parametric method

Figure 6.11 Schematic of Monte Carlo simulation 6.2.5 The parametric method...

The parametric method is an established statistical technique used for combining variables containing uncertainties, and has been advocated for use within the oil and gas industry as an alternative to Monte Carlo simulation. The main advantages of the method are its simplicity and its ability to identify the sensitivity of the result to the input variables. This allows a ranking of the variables in terms of their impact on the uncertainty of the result, and hence indicates where effort should be directed to better understand or manage the key variables in order to intervene to mitigate downside and/or take advantage of upside in the outcome. 

One significant feature of the Parametric Method is that it indicates, through the (1 + K 2) value, the relative contribution of each variable to the uncertainty in the result. Subscript i refers to any individual variable. (1 + K ) will be greater than 1.0 the higher the value, the more the variable contributes to the uncertainty in the result. In the following example, we can rank the variables in terms of their impact on the uncertainty in UR. We could also calculate the relative contribution to uncertainty. 

In order to maximize information obtained from raw spectroscopic data, analytical chemists and instrumental specialists depend an signal processing and apply a large number of specialized versions of the basic methods considered in this chapter, as well as the parametric methods discussed in the previous chapter, see, e.g. (ref. 22). Here we provide only an example of parametric methods. Table 4.4 shows 20 points of the electronic absorption spectrum of o-phenilenediamidine in ethanol (ref. 23). 

Expansion of this preprocessor-based evaluation procedure to include investigation of several crudes by the parametric method is the next appropriate application and is underway now. 

The ground-state dipole moments of BPHTs were calculated by evaluating the vector sum of the it moment (computed by the Pariser-Parr-Pople (PPP) method) and the a moment (from cr-bond moments) [17], a separate set of dipole moment values being obtained by the parametric method 3 (PM3) ([25] and references therein) (Table 5). By comparison, the calculated ground-state dipole moment values were considerably lower than the experimental ones. In some cases, the agreement between the experimental and calculated ground-state dipole moments was rather poor [17]. 

Figure 16-4 Central 95% reference interval.The 2,5 and 97,5 percentiles and their 0.90 confidence intervals of the 500 serum triglyceride concentrations (Figure i 6-3), as determined by the parametric method (see text). The curves are the estimated probability distributions.

The parametric method for the determination of percentiles and their confidence intervals assumes a certain type of distribution, and it is based on estimates of population parameters, such as the mean and the standard deviation. We are, for example, using a parametric method if we believe that the true distribution is Gaussian and determine the reference limits (percentiles) as the values located 2 standard... 

The parametric method is much more complicated than the simple nonparametric method and requires computer software. The method is presented here under separate headings for testing of type of distribution, transformation of data, and the estimation of percentiles and their confidence intervals. 

A critical phase in the parametric method is, therefore, to test the goodness-of-fit of the reference distribution to a hypothetical Gaussian distribution. If we must reject the Gaussian hypothesis at a specified significance level, we have... 

It is also possible to estimate the confidence limits of percentiles determined by the parametric method. The method is presented in a later section. 

As we have just seen, the disappearing-phase method of locating the boundary of the a field is based on a determination of the composition at which the P phase just disappears from a series of (a -f- P) alloys. The parametric method, on the other hand, is based on observations of the a solid solution itself. This method depends on the fact, previously mentioned, that the lattice parameter of a solid solution generally changes with composition up to the saturation limit, and then remains constant beyond that point. 

Fig. 12-9 Solvus curve determination in the copper-antimony system by the parametric method (a) parameter versus composition curve (b) solubility versus temperature curve. Mertz and Mathewson [12.2].

When rather extreme departures from required assumptions are noted, our choice of an appropriate statistical method should be one of first validity and second efficiency. The difference between an extreme departure from required assumptions and any departure from required assumptions is again a matter of judgment. It should be noted that many of the parametric methods in this book are robust to departures from distributional assumptions, meaning that the results are valid under a number of conditions. This is especially true with the larger sample sizes encountered in therapeutic exploratory and confirmatory trials. We should also note that all the methods described in this book require that observations in the analysis are independent. There are statistical methods to be used for dependent data, but they are not described in this book. 

Since the distribution of the log-transformed data after removing these outliers satisfied the Shapiro Wilk normality test (Fig. 8. A2b), the cut point determination for these validation data was made using the parametric method (i.e., mean + 1.645 x SD). The threshold 1.645 for normally distributed data ensures 5% false-positive rate. This cut point of the log-transformed values of the sample replicate means was determined for each analyst, each assay run, and for all the data combined (Table 8.A2). 

We have shown [46] that if a multipole expansion of the interaction "u is adopted and the mass renormalization procedure carried out at the partial wave level, a convergent representation is obtained for Es which is in precise agreement with the parametric methods which were introduced by Feynman [47] and implemented in the numerical calculations of Mohr [48]. For both the self energy and vacuum polarization, the positive- and negative-energy states are equally important in the calculation of the physical effect, with respect to both the numerical value of the energy shift and the finiteness of the theory. 

Heats of formation (unpublished data) were calculated using G. DePaz and Ciller empirical equations where AHf in gas phase was considered from the parametrization method 3 (PM3) using GAUSSIAN-94 program. 

Two main methods, parametric and nonparametric, have been developed for this deconvolution. The parametric method involves development of a multi-parameter model for obtaining the value of reactivity distribution... 

The parametric methods assume that is multivariate normal. 

Now, consider the parametric method. As was already mentioned, the underlying assumption is that the distribution belongs to a certain class. The choice of the distribution class can be made on the basis of general considerations about the kind of distribution that is likely to be formed in a specific physical process. For example, for a turbulent flow of emulsion in a pipe, the distribution can be described with sufficient accuracy by a logarithmic normal distribution or a gamma-distribution. Consider these two cases successively. 

We now obtain an approximate solution by the method of moments (see Section 11.1), using the parametric method and assuming that the initial distribution is a gamma - distribution,... 

To close the system of equations represented by Eq. (15.41) it is necessary to express the right-hand part in terms of moments. To this end, the coagulation kernel should have a special form (for example, the form of a homogeneous polynomial of degrees V and co), or it is necessary to accept that the distribution conforms to a certain class (for example, a logarithmic normal distribution or a gamma distribution). The first method is called the method of fractional moments, and the second one the parametric method. 

In the parametric method, it is assumed that the distribution with time remains as a gamma distribution and that the parameter Vb changes with time. If only drop coagulation is taken into account, there is constancy of the volume concentration W, i.e. mi. When limited to a two-moment approximation, one obtains Eqs. (11.30) for n( V, t), Vb, and the numerical concentration of drops N. Substitution of Eqs. (15.40) and (15.30) into these equations and elimination of Vb leads to an ordinary differential equation for N(t) ... 

The fitted curve exhibits the constant long yield that we observed in the hypothetical yield curve in Figure 5.1, while the strip curve trades expensive at the long end, which as we noted is a common observation. Nevertheless, for the purposes of accurate fitting, the parametric method exhibits a significant difference to the... 

Certain progress in this line of research has been made as a result of development of the general quantum theory of molecular transformations [17], which formed the basis for formulation of the parametric method for the evaluation of photochemical processes and a prognostic computer simulation procedure [18-22],... 

Later, two modifications of MNDO were introduced [102,235]. The most prominent of these are the Austin model 1 (AMI) by Dewar et al. [246] and the parametric method 3 (PM3) by Stewart [248]. In principle MNDO, AMI and PM3 methods differ only in the parametrization and in the empirical function /mat in the core-core repulsion... 

After 60 years, Fisher s LDA method is still the workhorse of the parametric methods. Of all the statistical classifiers, we understand LDA s theoretical properties best. We can derive it from ordinary (linear) least-squares (OLS) regression. LDA creates those linear combinations of the M features that best separate the classes. For each class k there is one of these discriminant functions The decision surfaces = dy - = 0 between any pair j,k) of... 

The Aa(192 —188) was measured in 10 lines with isotopically highly enriched osmium by Aufmuth and Wobker [32] and analyzed by means of the parametric method. Due to the influence of crossed-second-order effects they have obtained for the difference between the field shifts of the 56 6s f and terms AT(5F)-AT(3F) = (-h53.4 + 9.6) x 10- cm- in contrast to the value —35x10 cm obtained for the same terms by taking into account configuration mixing and intermediate coupling. 

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