Formulation of the Functional Estimation Problem

The formulation of the parameter estimation problem is equally important to the actual solution of the problem (i.e., the determination of the unknown parameters). In the formulation of the parameter estimation problem we must answer two questions (a) what type of mathematical model do we have and (b) what type of objective function should we minimize In this chapter we address both these questions. Although the primary focus of this book is the treatment of mathematical models that are nonlinear with respect to the parameters nonlinear regression) consideration to linear models linear regression) will also be given. 

The remainder of this chapter is structured as follows. In Section II the problem of deriving an estimate of an unknown function from empirical data is posed and studied in a theoretical level. Then, following Vapnik s original work (Vapnik, 1982), the problem is formulated in mathematical terms and the sources of the error related to any proposed solution to the estimation problem are identified. Considerations on how to reduce these errors show the inadequacy of the NN solutions and lead in Section III to the formulation of the basic algorithm whose new element is the pointwise presentation of the data and the dynamic evolution of the solution itself. The algorithm is subsequently refined by incorporating the novel idea of structural adaptation guided by the use of the L" error measure. The need... 

In Chapter 3 we discussed the formulation of objective functions without going into much detail about how the terms in an objective function are obtained in practice. The purpose of this appendix is to provide some brief information that can be used to obtain the coefficients in objective functions in economic optimization problems. Various methods and sources of information are outlined that help establish values for the revenues and costs involved in practical problems in design and operations. After we describe ways of estimating capital costs, operating costs, and revenues, we look at the matter of project evaluation and discuss the many contributions that make up the net income from a project, including interest, depreciation, and taxes. Cash flow is distinguished from income. Finally, some examples illustrate the application of the basic principles. 

The molecular structure of the unknown chemical could be found by inverting these three relationships. However, an explicit inversion is not analytic (the molecular structure is described by integer variables denoting the presence or absence of specific atoms and bonds), and it accepts multiple solutions (there may be several molecules satisfying the constraints). Implicit inversion of Eqs. (1) is possible through the formulation of appropriate optimization problems. However, in such cases the complexity and nonlinear character of the functional relationships used to estimate the values of physical properties in conjunction with the integer variables description of molecular structures, yield very complex mixed-integer optimization formulations. 

We now formulate the associated identification problem by arranging all the velocity measurements in a vector Y and, in the vector F, the corresponding values are calculated from the solution to Eqs. (4.1.24 and 4.1.25) using an estimate of the permeability function. The performance index is ... 

Part I comprises three chapters that motivate the study of optimization by giving examples of different types of problems that may be encountered in chemical engineering. After discussing the three components in the previous list, we describe six steps that must be used in solving an optimization problem. A potential user of optimization must be able to translate a verbal description of the problem into the appropriate mathematical description. He or she should also understand how the problem formulation influences its solvability. We show how problem simplification, sensitivity analysis, and estimating the unknown parameters in models are important steps in model building. Chapter 3 discusses how the objective function should be developed. We focus on economic factors in this chapter and present several alternative methods of evaluating profitability. 

Rawlings and co-workers proposed to carry out parameter estimation using Newton s method, where the gradient can be cast in terms of the sensitivity of the mean (Haseltine, 2005). Estimation of one parameter in kinetic, well-mixed models showed that convergence was attained within a few iterations. As expected, the parameter values fluctuate around some average values once convergence has been reached. Finally, since control problems can also be formulated as minimization of a cost function over a control horizon, it was also suggested to use Newton s method with relatively smooth sensitivities to accomplish this task. The proposed method results in short computational times, and if local optimization is desired, it could be very useful. 

A simple estimate of the computational difficulties involved with the customary quantum mechanical approach to the many-electron problem illustrates vividly the point [255]. Consider a real-space representation of ( ii 2, , at) on a mesh in which each coordinate is discretized by using 20 mesh points (which is not very much). For N electrons, becomes a variable of 3N coordinates (ignoring spin), and 20 values are required to describe on the mesh. The density n(r) is a function of three coordinates and requires only 20 values on the same mesh. Cl and the Kohn-Sham formulation of DFT (see below) additionally employ sets of single-particle orbitals. N such orbitals, used to build the density, require 20 values on the same mesh. (A Cl calculation employs in addition unoccupied orbitals and requires more values.) For = 10 electrons, the many-body wave function thus requires 20 °/20 10 times more storage space than the density and sets of single-particle orbitals 20 °/10x 20 10 times more. Clever use of symmetries can reduce these ratios, but the full many-body wave function remains inaccessible for real systems with more than a few electrons. 

The focus of this section is on the uncertainty estimates of equilibria in solution, where the key problem is analytical, i.e. the determination of the stoichiometric composition and equilibrium constants of complexes that are in rapid equilibrium with one another. We can formulate analysis of the experimental data in the following way From N measurements, of the variable y we would like to determine a set of n equilibrium constants k r= n, assuming that we know the functional relationship ... 

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