Maximum Likelihood and Least Squares Criteria

This comparison never shows a perfect correspondence between models and experiments because of modeling and measurement errors. In fact, even if the presence of systematic experimental errors can be excluded, systematic errors generated by the inadequacy of the model must be added to random experimental errors for each measured variable (m = 1. Am) and each experimental time (j = 1. Ad), the errors generated by the model are defined as 

On the other hand, since the model must be considered as a flexible tool that can be adapted to the experimental data by changing the values of the adjustable parameters, the method consists in computing the optimal values of the parameters, 0, on the basis of a suitable optimality criterion and submitting to a statistical analysis the residual errors %n,j, i.e., the differences between the measured data and the corresponding optimal computed values, em, = dmj - ymj(0) = dmj —ym,j- 

If the model perfectly describes the experiments, the sample of residual errors does not contain systematic errors thus, it must be compatible with the statistical distribution of the random experimental errors. All the systematic discrepancies eventually observed are attributed to the mathematical model, thus allowing a comparison between alternative models, since systematic errors can be decreased if a better model becomes available. 

The optimality criterion used to compute the best form of the available model is based on the concept of likelihood, defined in Sect. 3.1 as the confidence p( ix Th) in obtaining the experimental result Ex if the theory Th is true. In the light of the above discussion, the likelihood can be intended as the probability of obtaining the residual errors, which depend on the experimental data and the model, through y. Since for any component and any time instant the following is true  

A few additional simplifying assumptions must be introduced, in order to obtain a workable expression for an objective function based on the concept of likelihood. First, it is assumed that the comparison between the experimental data and the optimized computed data produces random residual errors Tmj following a normal 

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