Implicit Maximum Likelihood Parameter Estimation

Implicit estimation offers the opportunity to avoid the computationally demanding state estimation by formulating a suitable optimality criterion. The penalty one pays is that additional distributional assumptions must be made. Implicit formulation is based on residuals that are implicit functions of the state variables as opposed to the explicit estimation where the residuals are the errors in the state variables. The assumptions that are made are the following  

The formulation of the residuals to be used in the objective function is based on the phase equilibrium criterion 

The above equations hold at equilibrium. However, when the measurements of the temperature, pressure and mole fractions are introduced into these expressions the resulting values are not zero even if the EoS were perfect. The reason is the random experimental error associated with each measurement of the state variables, Thus, Equation 14.18 is written as follows 

The estimation of the parameters is now accomplished by minimizing the implicit ML objective function 

The above form of the residuals was selected based on the following considerations  

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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. 

A comparison of the various fitting techniques is given in Table 5. Most of these techniques depend either explicitly or implicitly on a least-squares minimization. This is appropriate, provided the noise present is normally distributed. In this case, least-squares estimation is equivalent to maximum-likelihood estimation.147 If the noise is not normally distributed, a least-squares estimation is inappropriate. Table 5 includes an indication of how each technique scales with N, the number of data points, for the case in which N is large. A detailed discussion on how different techniques scale with N and also with the number of parameters, is given in the PhD thesis of Vanhamme.148... 

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