Method parametric

Keywords deterministic methods, STOllP, GllP, reserves, ultimate recovery, net oil sands, area-depth and area-thickness methods, gross rock volume, expectation curves, probability of excedence curves, uncertainty, probability of success, annual reporting requirements, Monte-Carlo simulation, 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. 

Modified Negiect of Diatomic Overlap, Parametric Method Number 3 (MNDO-PM3)... 

Ideally, to characterize the spatial distribution of pollution, one would like to know at each location x within the site the probability distribution of the unknown concentration p(x). These distributions need to be conditional to the surrounding available information in terms of density, data configuration, and data values. Most traditional estimation techniques, including ordinary kriging, do not provide such probability distributions or "likelihood of the unknown values pC c). Utilization of these likelihood functions towards assessment of the spatial distribution of pollutants is presented first then a non-parametric method for deriving these likelihood functions is proposed. 

Non-linear models, such as described by the Michaelis-Menten equation, can sometimes be linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the use of non-linear regression can be obviated. As we have pointed out, the price for this convenience may have to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5). 

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

Again, the majority of these parameters are interrelated and highly dependent on the method used to determine them. Red blood cell count (RBC), platelet counts, and mean corpuscular volume (MCV) may be determined using a device such as a Coulter counter to take direct measurements, and the resulting data are usually stable for parametric methods. The hematocrit, however, may actually be a value calculated from the RBC and MCV values and, if so, is dependent on them. If the hematocrit is measured directly, instead of being calculated from the RBC and MCy it may be compared by parametric methods. 

It is fair to say that statisticians tend to disagree somewhat regarding the value of non-parametric methods. Some statisticians view them very favourably while others are reluctant to use them unless there is no other alternative. 

Clearly the main advantage of a non-parametric method is that it makes essentially no assumptions about the underlying distribution of the data. In contrast, the corresponding parametric method makes specific assumptions, for example, that the data are normally distributed. Does this matter Well, as mentioned earlier, the t-tests, even though in a strict sense they assume normality, are quite robust against departures from normality. In other words you have to be some way off normality for the p-values and associated confidence intervals to be become invalid, especially with the kinds of moderate to large sample sizes that we see in our trials. Most of the time in clinical studies, we are within those boundaries, particularly when we are also able to transform data to conform more closely to normality. 

Further, there are a number of disadvantages of non-parametric methods ... 

With parametric methods confidence intervals can be calculated which link directly with the p-values recall the discussion in Section 9.1. With non-parametric methods the p-values are based directly on the calculated ranks and it is not easy to obtain a confidence interval in relation to parameters that have a clinical meaning that link with this. This compromises our ability to provide an assessment of clinical benefit. 

Non-parametric methods reduce power. Therefore if the data are normally distributed, either on the original scale or following a transformation, the non-parametric test will be less able to detect differences should they exist. 

For these reasons non-parametric methods are used infrequently within the context of clinical trials and they tend only to be considered if it is clear that a corresponding parametric approach, either directly or following a data transformation, is unsuitable. 

The statistical methods discussed up to now have required certain assumptions about the populations from which the samples were obtained. Among these was that the population could be approximated by a normal distribution and that, when dealing with several populations, these have the same variance. There are many situations where these assumptions cannot be met, and methods have been developed that are not concerned with specific population parameters or the distribution of the population. These are referred to as non-parametric or distribution-free methods. They are the appropriate methods for ordinal data and for interval data where the requirements of normality cannot be assumed. A disadvantage of these methods is that they are less efficient than parametric methods. By less efficient is meant... 

Non-parametric methods statistical tests which make no assumptions about the distributions from which the data are obtained. These can be used to show iifferences, relationships, or association even when the characteristic observed can not be measured numerically. 

Quantitative methodology uses large or relatively large samples of subjects (as a rule students) and tests or questionnaires to which the subjects answer. Results are treated by statistical analysis, by means of a variety of parametric methods (when we have continuous data at the interval or at the ratio scale) or nonparametric methods (when we have categorical data at the nominal or at the ordinal scale) (30). Data are usually treated by standard commercial statistical packages. Tests and questionnaires have to satisfy the criteria for content and construct validity (this is analogous to lack of systematic errors in measurement), and for reliability (this controls for random errors) (31). 

Signal processing may also involve parameter estimation methods (e.g., resolution of a spectral curve into the sum of Gaussian functions). Even in such cases, however, we may need non-parametric methods to approximate the position, height and half-width of the peaks, used as initial estimates in the parameter estimation procedure. 

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). 

Since the convolution integral is symmetrical in u(t) and hit), this problem is similar to the one of system identification considered in the previous section. Nevertheless, it is usually easier to find the weighting function h(t) since its form is more - or - less known (e.g., as a sum of polyexponentials), and hence parametric methods apply, whereas the input function u(t) is a priori arbitrary. Therefore, the non - parametric point -area method is a popular way of performing numerical deconvolution. It is really simple evaluating the integral means h, f, . .., fy, of the weighting function over the subinterval [t. ., t ] we can easily solve the set (6.68)... 

In Section 1.1, parametric methods for improving tlie quality of correlated electronic-structure calculations were discussed in detail. Similarly, in Section 8.4.3, the mild parameterization of density functional methods to give maximal accuracy was described. Given that background, and the substantial data presented in diose earlier chapters, this section will only recapitulate in a rough categorical fashion tlie various approaches whose development was motivated by a desire to compute more accurate thermochemical quantities. 

Optimization techniques may be classified as parametric statistical methods and nonparametric search methods. Parametric statistical methods, usually employed for optimization, are full factorial designs, half factorial designs, simplex designs, and Lagrangian multiple regression analysis [21]. Parametric methods are best suited for formula optimization in the early stages of product development. Constraint analysis, described previously, is used to simplify the testing protocol and the analysis of experimental results. 

The most commonly employed univariate statistical methods are analysis of variance (ANOVA) and Student s r-test [8]. These methods are parametric, that is, they require that the populations studied be approximately normally distributed. Some non-parametric methods are also popular, as, f r example, Kruskal-Wallis ANOVA and Mann-Whitney s U-test [9]. A key feature of univariate statistical methods is that data are analysed one variable at a rime (OVAT). This means that any information contained in the relation between the variables is not included in the OVAT analysis. Univariate methods are the most commonly used methods, irrespective of the nature of the data. Thus, in a recent issue of the European Journal of Pharmacology (Vol. 137), 20 out of 23 research reports used multivariate measurement. However, all of them were analysed by univariate methods. 

Two non-parametric methods for hypothesis testing with PCA and PLS are cross-validation and the jackknife estimate of variance. Both methods are described in some detail in the sections describing the PCA and PLS algorithms. Cross-validation is used to assess the predictive property of a PCA or a PLS model. The distribution function of the cross-validation test-statistic cvd-sd under the null-hypothesis is not well known. However, for PLS, the distribution of cvd-sd has been empirically determined by computer simulation technique [24] for some particular types of experimental designs. In particular, the discriminant analysis (or ANOVA-like) PLS analysis has been investigated in some detail as well as the situation with Y one-dimensional. This simulation study is referred to for detailed information. However, some tables of the critical values of cvd-sd at the 5 % level are given in Appendix C. 

Parametric core Very low bit-rate coder based on parametric methods... 

Various parameterizations of NDDO have been proposed. Among these are modified neglect of diatomic overlap (MNDO),152 Austin Model 1 (AMI),153 and parametric method number 3 (PM3),154 all of which often perform better than those based on INDO. The parameterizations in these methods are based on atomic and molecular data. All three methods include only valence s and p functions, which are taken as Slater-type orbitals. The difference in the methods is in how the core-core repulsions are treated. These methods involve at least 12 parameters per atom, of which some are obtained from experimental data and others by fitting to experimental data. The AMI, MNDO, and PM3 methods have been focused on ground state properties such as enthalpies of formation and geometries. One of the limitations of these methods is that they can be used only with molecules that have s and p valence electrons, although MNDO has been extended to d electrons, as mentioned below. 

Be fore using parametric methods, the data have to be transformed and tested to see if normal distribution occurs. Sometimes multimodal distributions exist, as illustrated in Fig. 1-8 for the occurrence of particulate matter in the atmosphere. A further difficulty in this case is that the type of distribution may change during the transmission process in the air. If normal... 

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