Formulation of the Parameter 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. 

---

To complete the formulation of the parameter estimation problem and the differential equations, it is assumed that this set of data is described by the set of differential equations below (same as Eq, (11.65))... 

The formulation for the next three problems of the parameter estimation problem was given in Chapter 6. These examples were formulated with data from the literature and hence the reader is strongly recommended to read the original papers for a thorough understanding of the relevant physical and chemical phenomena. 

The previously mentioned causal inference part of the roadmap has now provided us with a formulation of the statistical estimation problem We observe n i.i.d. copies of O Pq EM and we want to estimate the target parameter value /q = P(Po) corresponding with a target parameter mapping P - M. In... 

With this book the reader can expect to learn how to formulate and solve parameter estimation problems, compute the statistical properties of the parameters, perform model adequacy tests, and design experiments for parameter estimation or model discrimination. 

In this chapter, the general problem of joint parameter estimation and data reconciliation will be discussed. 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, will be stated. Finally, joint parameter and state estimation in dynamic processes will be considered. 

Determination of T y. In the formulation of the phase equilibrium problem presented earlier, component chemical potentials were separated into three terms (1) 0, which expresses the primary temperature dependence, (2) solution mole fractions, which represent the primary composition dependence (ideal entropic contribution), and (3) 1, which accounts for relative mixture nonidealities. Because little data about the experimental properties of solutions exist, Tg is usually evaluated by imposing a model to describe the behavior of the liquid and solid mixtures and estimating model parameters by semiempirical methods or fitting limited segments of the phase diagram. Various solution models used to describe the liquid and solid mixtures are discussed in the following sections, and the behavior of T % is presented. 

Parameter estimation is one of the steps involved in the formulation and validation of a mathematical model that describes a process of interest. Parameter estimation refers to the process of obtaining values of the parameters from the matching of the model-based calculated values to the set of measurements (data). This is the classic parameter estimation or model fitting problem and it should be distinguished from the identification problem. The latter involves the development of a model from input/output data only. This case arises when there is no a priori information about the form of the model i.e. it is a black box. 

The above implicit formulation of maximum likelihood estimation is valid only under the assumption that the residuals are normally distributed and the model is adequate. From our own experience we have found that implicit estimation provides the easiest and computationally the most efficient solution to many parameter estimation problems. 

In this section the extension of the use of nonlinear programming techniques to solve the dynamic joint data reconciliation and parameter estimation problem is briefly discussed. As shown in Chapter 8, the general nonlinear dynamic data reconciliation (NDDR) formulation can be written as ... 

Tjoa and Biegler (1991) used this formulation within a simultaneous strategy for data reconciliation and gross error detection on nonlinear systems. Albuquerque and Biegler (1996) used the same approach within the context of solving an error-in-all-variable-parameter estimation problem constrained by differential and algebraic equations. 

As it has been mentioned, apart from point estimates there exist the parameter interval estimates. No matter how well the parameter estimate has been chosen, it is only logical to test the estimate deviation from its correct value, as obtained from the sample. For example, if in numerical analysis one obtains that the solution of an equation is approximately 3.24 and that 0.03 is the maximal possible deviation from the unknown correct solution of the equation, then we are absolutely sure that the range (3.24-0.03=3.21 3.24+0.03=3.27) contains the unknown correct solution of the equation. Therefore the problem of determining the interval estimate is formulated in the following way ... 

The classical approach to the fit parameter estimation problem in dielectric spectroscopy is generally formulated in terms of a minimization problem finding values of X which minimize some discrepancy measure S(, s) between the measured values, collected in the matrix s and the fitted values = [/(co,-, x(7 ))] of the complex dielectric permittivity. The choice of S(e,e) depends on noise statistics [132]. 

---