On statistical principles in reduction of thermodynamic data

Abstract

Estimation of the parameters in mathematical models describing physical phenomena is often performed using a fairly simple empirical criterion. Alternatively, the parameter estimation may be performed using a more sophisticated criterion based on statistical considerations. The principle of maximum likelihood, M.L., offers the most appealing possibility for optimum parameter estimation. However, the application of this principle requires that the statistical distribution of the observations is known and that the observations are subject to random errors only. Furthermore, no systematic errors should be introduced by the model for which the parameters are estimated. In cases where the requirements are fulfilled the M.L. estimator gives the best estimates of the parameters in question. Most of the models used (e.g. models for physical properties of pure or mixed fluids) do not represent the connection between the observed variables within experimental accuracy. For models of this type the use of a criterion based on the M.L. principle reduces to an empirical procedure for weighting the available information. The possibility of detecting systematic errors by means of data reduction using the M.L. principle has been somewhat exaggerated. It is not possible to detect the source of systematic errors in the observed variables by means of the M.L. principle. This is due to the fact that the principle is based on the assumption of random errors. The M.L. principle offers a valuable tool in the limiting case of parameter estimation in a perfect model based on observations which are subject to random errors only. It does not, however, give any hope for estimation of reliable model parameters based on systematically erroneous data. Furthermore, it is not possible to give any general guidelines based on statistical considerations for reliable parameter estimation in models which are not capable of representing the connection between the true values of the observed variables. This paper gives a short survey of the principle of M.L. It is discussed how the principle may be applied to data evaluation and to the estimation of model parameters. The M.L. principle is exemplified by parameter estimation for an equation of state using binary VLE data. It is furthermore shown how the computation time may be reduced by economic use of the available information.