4. Process Modeling 4.4. Data Analysis for Process Modeling 4.4.3. How are estimates of the unknown parameters obtained?
Parameter Estimation in General	After selecting the basic form of the functional part of the model, the next step in the model-building process is estimation of the unknown parameters in the function. In general this is accomplished by solving an optimization problem in which the objective function (the function being minimized or maximized) relates the response variable and the functional part of the model containing the unknown parameters in a way that will produce parameters estimates that will be close to the true, unknown parameter values. The unknown parameters are treated as variables to be solved for in the optimization and the data serve as known coefficients of the objective function in this stage of the modeling process.
	In theory there are as many different ways of estimating parameters as there are objective functions to be minimized or maximized. However, a few principles have dominated because they result in parameter estimates that have good statistical properties. The two major methods of parameter estimation for process models are maximum likelihood and least squares. Both of these methods have provide parameter estimates that have many good properties. Both maximum likelihood and least squares are sensitive to the presence of outliers, however. There are also many newer methods for parameter estimation, called robust methods, that try to balance the efficiency and desirable properties of least squares and maximum likelihood with a lower sensitivity to outliers.
Overview of Section 4.3	Although robust techniques are valuable, they are less well developed than the more traditional methods and often require specialized software that is not readily available. Maximum likelihood also requires specialized algorithms in general, although there are important special cases which do not. For example, for data with normally distributed random errors, the least squares and maximum likelihood parameter estimates are identical. As a result of these software and developmental issues, and the coincidence of maximum likelihood and least squares in many applications, this section currently focuses on parameter estimation only by least squares methods. The remainder of this section offers some intuition into how least squares methods work and illustrates the effectiveness of this method.
Contents of Section 4.3	Least Sum of Squares Weighted Least Sum of Squares