Joint Estimation Method

An independent method to identify the stochastic errors of impedance data is described in Chapter 21. An alternative approach has been to use the method of maximum likelihood, in which the regression procedure is used to obtain a joint estimate for the parameter vector P and the error structure of the data. The maximum likelihood method is recommended under conditions where the error structure is unknown, but the error structure obtained by simultaneous regression is severely constrained by the assumed form of the error-variance model. In addition, the assumption that the error variance model can be obtained by minimizing the objective function ignores the differences eimong the contributions to the residual errors shown in Chapter 21. Finedly, the use of the regression procedure to estimate the standard deviation of the data precludes use of the statistic... 

Estimation methods that are based on simulation platforms, such as Markov chain Monte Carlo (MCMC), also allow for model discrimination to be based on predictive or posterior distributions. When using MCMC, competing models can be fitted simultaneously as a joint model with an added mixing parameter to indicate which model is preferred (42, 43). The posterior distribution of the mixing parameter will provide both the weight of evidence and the posterior probability in favor of one model. The expectation of the prediction from m models and a the mixing parameter can then be evaluated ... 

Naturally, one could think of using nonlinear regression to jointly estimate the parameters and F, without taking the means of the responses. This consideration leads to the following method. 

The training problem determines the set of model parameters given above for an observed set of wavelet coefficients. In other words, one first obtains the wavelet coefficients for the time series data that we are interested in and then, the model parameters that best explain the observed data are found by using the maximum likelihood principle. The expectation maximization (EM) approach that jointly estimates the model parameters and the hidden state probabilities is used. This is essentially an upward and downward EM method, which is extended from the Baum-Welch method developed for the chain structure HMM [43, 286]. 

The error-in-variables method was used to estimate the reactivity ratios. This method was developed by Reilly et al. (57, 58), and it was first applied for the determination of reactivity ratios by O Driscoll, Reilly, and co-workers (59, 60). In this work, a modified version by MacGregor and Sutton (61) adapted by Gloor (62) for a continuous stirred tank reactor was used. The error-in-variables method shows two important advantages compared to the other common methods for the determination of copolymer reactivity ratios, which are statistically incorrect, as for example, Fineman-Ross (63) or Kelen-Tiidos (64). First, it accounts for the errors in both dependent and independent variables the other estimation methods assume the measured values of monomer concentration and copolymer composition have no variance. Second, it computes the joint confidence region for the reactivity ratios, the area of which is proportional to the total estimation error. 

In this section we revise the imcertainty propagation theory of methods commented in the introduction. Since the objective of this paper is, in addition to this revision, the application in a normal bivariate case, methods to study will be, essentially those multivariate in their parametric version. The application case in next section concerns to the implementation of these methodologies to the Pareto front, solution of a multiobjective optimization problem. So, applying maxi-mmn likelihood estimation method, parameters of life time distributions and maintenance effectiveness can be jointly estimated obtaining, in addition to information about their means, estimations of their variability and correlations between them. This information is... 

C. Wright, Effectiveness of joint estimation outlier detection method for short time series with quality control applications , Ph.D. Dissertation, Kent State University, Kent, OH, 1997. 

Thus, carrying out tests of the samples shows that the acoustic emission method is quite effective at the quality estimation of carbon plastic and its adhesive joints. Depending on the chosen diagnostic diagram of the construction material loading, the criteria parameters are K, S or AS (a C). 

Estimating by cost per joint depends on the accumnlation of past data, analyzed and conveniently correlated for use. The main advantage of the method lies in the fact that good engineering flow sheets can be used for the estimation. 

The researchers of ECOHYNTOX jointly with the colleagues of the Research Industrial Enterprise Burevestnik (St. Petersburg) have developed, valided, and introduced into practice the methods of inversuve voltampermetric estimation of iodine and selenium in drinking water using AVA-2 apparatus. 

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. 

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. 

Based on the patch method to assess worker or re-entry exposure, researchers have developed a database, which may be used to estimate exposure. Each patch from an individual in a study can be entered into the database separately, the residue data from patches from various body areas can be summed to yield a whole-body exposure number, and the data may be sorted as to worker tasks, equipment used, protective clothing worn, formulation types and other parameters. This is the basis for the currently used Pesticide Handlers Data Base (PHED), which was developed through a joint effort in the 1980s of CropLife America [formerly known as American Crop Protection Association (ACPA) and National Agricultural Chemicals Association (NACA)], the Environmental Protection Agency (ERA) and Health Canada. " The PHED is discussed in detail in another article in this book. 

Chapter 9 deals with the general problem of joint parameter estimation data reconciliation. Starting from the typical parameter estimation problem, the more general formulation in terms of the error-in-variable methods is described, where measurement errors in all variables are considered. Some solution techniques are also described here. 

In this chapter, the general problem of joint parameter estimation and data reconciliation was discussed. First, the typical parameter estimation problem was analyzed, in which the independent variables are error-free, and aspects related to the sequential processing of the information were considered. Later, the more general formulation in terms of the error-in-variable method (EVM), where measurement errors in all variables are considered in the parameter estimation problem, was stated. Alternative solution techniques were briefly discussed. Finally, joint parameter-state estimation in dynamic processes was considered and two different approaches, based on filtering techniques and nonlinear programming techniques, were discussed. 

Since the mean velocity and Reynolds-stress fields are known given the joint velocity PDF /u(V x, t), the right-hand side of this expression is closed. Thus, in theory, a standard Poisson solver could be employed to find (p)(x, t). However, in practice, (U)(x, t) and (u,Uj)(x, t) must be estimated from a finite-sample Lagrangian particle simulation (Pope 2000), and therefore are subject to considerable statistical noise. The spatial derivatives on the right-hand side of (6.61) are consequently even noisier, and therefore are of no practical use when solving for the mean pressure field. The development of numerical methods to overcome this difficulty has been one of the key areas of research in the development of stand-alone transported PDF codes.38... 

Step 1 Business value assessment Identify and quantify the business value for the proposed new method or analyzer. This is done jointly by the customer and the project manager. Ideally the value of the technology can be expressed concretely in monetary terms (e.g. dollars) as, for example, a net present value (NPV) or an internal rate of return (IRR). It is critical to include a realistic estimate of the costs of implementing and maintaining the analyzer, as weU as the benefits to be realized from it. This assessment is used to prioritize effort and expenses and to justify any capifal purchases needed. The various ways that process analyzers can connibute to business value are discussed in Section 15.2.2. 

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