Joint 95% probability region

Applications of the method to the estimation of reactivity ratios from diad sequence data obtained by NMR indicates that sequence distribution is more informative than composition data. The analysis of the data reported by Yamashita et al. shows that the joint 95% probability region is dependent upon the error structure. Hence this information should be reported and integrated into the analysis of the data. Furthermore reporting only point estimates is generally insufficient and joint probability regions are required. 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

Therefore to make meaningful inferences from experiments such as those reported by Yamashita et al. either the error structure must be known or sufficient data must be provided, preferably in the form of optimally designed replicates. This analysis confirms that it is generally insufficient to evaluate only point estimates. In fact these are secondary to evaluating and reporting joint probability regions. 

The joint probability region can have different orientations and extensions in the parameter space. This will correspond to different "quality" aspects of the estimated values of the model parameters. These quality aspects will depend on the properties of the dispersion matrix (X X) , see Fig. 5.8. 

Figure 1. Joint 95% posterior probability region— diad fractions. Shimmer bands shown at 95% probability. X, true value , point estimate.

This implies that the diad fraction measurements n and n, are made independently with constant standard deviation 0.05. Figure 3 shows the resulting joint 95% posterior probability region with 95% shimmer bands and point estimates. A second estimate of used here is... 

Figure 5 shows the joint 95% posterior probability region for the two parameter functions. Shimmer bands are also indicated at the 95% probability level. This analysis confirms the results of Hill et al. that both styrene and acrlyonitrile exhibit a penultimate effect. 

For linear models the joint confidence region is an Alp-dimensional ellipsoid. All parameters encapsulated within this hyperellipsoid do not differ significantly from the optimal estimates at the probability level of 1-a. 

Over the last few decades, it has become clear that the microscopic structure of the liquid phase has a direct influence on the thermodynamics and kinetics of chemical reactions[33-35]. One expects that similar relationships will hold in the inhomogeneous region. The microscopic structure of bulk liquids can be characterized experimentally and theoretically using the pair (and higher order) particle distribution functions, p-jlr r ) is the joint probability density of... 

If one does not wish to bias the boundaries of the NO region of a system, kernel density estimation (KDE) can be used to find the contours underneath the joint probability density of the PC pair, starting from the one that captures most of the information. Below, a brief review of KDE is presented first that will be used as part of the robust monitoring technique discussed in Section 7.7. Then, the use of kernel-based methods for formulating nonlinear Fisher s discriminant analysis (FDA) is discussed. 

This paper presents the results of a study to investigate and establish the reliability of both the description and performance data estimates from numerical reservoir simulators. Using optimal control theory, an algorithm was developed to perform automated matching of field observed data and reservoir simulator calculated data, thereby estimating reservoir parameters such as permeability and porosity. Well known statistical and probability methods were then used to establish individual confidence limits as well as joint confidence regions for the parameter estimates and the simulator predicted performance data. The results indicated that some reservoir input data can be reliably estimated from numerical reservoir simulators. Reliability was found to be inversely related to the number of unknown parameters in the model and the level of measurement error in the matched field observed data. 

To define the NO region (NOR) of the plant, kernel density estimation (KDE) is used. The joint probability density of the first and second, and... 

The proofs of Eqs. (9.59) and (9.60) explicitly rely on the linear character of the model. The above relations are thus only correct under the same conditions. Therefore, one speaks of the joint confidence region under linear assumptions. If the model is not linear in its parameters, the surface in the parameter space defined by Eq. (9.63) no longer is a contour of constant residual sums of squares. Although the probability level is correct, the contour itself has only been approximated. This property provides a qualitative measure of the degree of nonlinearity of the model it is rather simple to determine the coordinates of... 

Assume that X = (Xi,X2) and x is similarly defined. Let the Joint probability density function, f x), be defined over the region The joint probability density function satisfies the following three properties ... 

Temperature and residence time were varied in an initial halffactorial DoE to estimate the fitted reaction parameters. After these four experiments, subsequent experimental conditions were determined by the Bayesian approach, terminating once the Bayesian probability of one of the rate models passed 95%. The results from the fifth and sixth experiment are shown in Figure 4.8, which illustrates that after six experiments, only reaction model I matched with the experimental results, in agreement with the known rate law for the Diels-Alder reaction. Once the best rate law has been found, further experiments are performed to reduce the size of the joint confidence region of the fitted parameters using the d-optimal approach [57]. 

We define a bond as the region of space between a pair of atoms where the joint probability of finding electron density owned by each atom is nonzero. In the bonding region, the weighting function of both nuclei (Wa and Wg) must be nonzero (and nonnegligible) since both the space and electron density are shared. Thus, the bond density is defined in Eq. 11. Similarly, the radial bond density can be defined as in Eq. 12. 

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