Maximum Likelihood Parameter and State Estimation

Over the years two ML estimation approaches have evolved (a) parameter estimation based an implicit formulation of the objective function and (b) parameter and state estimation or error in variables method based on an explicit formulation of the objective function. In the first approach only the parameters are estimated whereas in the second the true values of the state variables as well as the values of the parameters are estimated. In this section, we are concerned with the latter approach. 

Only two of the four state variables measured in a binary VLE experiment are independent. Hence, one can arbitrarily select two as the independent variables and use the EoS and the phase equilibrium criteria to calculate values for the other two (dependent variables). Let Q, (i=l,2.N and j=l,2) be the independent variables. Then the dependent ones, g-, can be obtained from the phase equilibrium relationships (Modell and Reid, 1983) using the EoS. The relationship between the independent and dependent variables is nonlinear and is written as follows 

In this case, the ML parameter estimates are obtained by minimizing the following quadratic optimality criterion (Anderson et al., 1978 Salazar-Sotelo et al. 1986) 

This is the so-called error in variables method. The formulation of the above optimality criterion was based on the following assumptions  

Unless very few experimental data are available, the dimensionality of the problem is extremely large and hence difficult to treat with standard nonlinear minimization methods. Schwetlick and Tiller (1985), Salazar-Sotelo et al. (1986) and Valko and Vajda (1987) have exploited the structure of the problem and proposed computationally efficient algorithms. 

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