Implicit Least Squares Estimation

It is well known that cubic equations of state have inherent limitations in describing accurately the fluid phase behavior. Thus our objective is often restricted to the determination of a set of interaction parameters that will yield an acceptable fit of the binary VLE data. The following implicit least squares objective function is suitable for this purpose 

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Furthermore, as a first approximation one can use implicit least squares estimation to obtain very good estimates of the parameters (Englezos et al., 1990). Namely, the parameters are obtained by minimizing the following Implicit Least Squares (ILS) objective function,... 

The same two interaction parameters (ka, kd) were found to be adequate to correlate the VLE. LLE and VLLE data of the H2S-H20 system. Each data set was used separately to estimate the parameters by implicit least squares (LS). 

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... 

If we decide to only estimate a finite number of basis modes we implicitly assume the coefficients of all the other modes are zero and that the covariance of the modes estimated is very large. Thus QN Q becomes large relative to C and in this case Eq. 16 simplifies to a weighted least squares formula... 

Figure 14.3 Vapor-liquid equilibrium data and calculated values for the carbon dioxide-n-hexane system. Calculations were done using interaction parameters from implicit and constrained least squares (LS) estimation, x and y are the mote fractions in the liquid and vapor phase respectively [reprinted from the Canadian Journal of Chemical Engineering with permission]...

This system illustrates the use of simplified constrained least squares (CLS) estimation. In Figure 14.3, the experimental data by Li et al. (1981) together with the calculated phase diagram for the system carbon dioxide-n-hexane are shown. The calculations were done by using the best set of interaction parameter values obtained by implicit LS estimation. These parameter values together with standard deviations are given in Table 14.3. The values of the other parameters (k, kj) were equal to zero. As seen from Figure... 

ARR residuals may be obtained from a DBG. If ARRs cannot be deduced from a DBG in closed symbolic form because of nonlinear implicit equations indicated by causal paths, ARR residuals are given implicitly. Their numerical values can be obtained by solving the entire DBG model in each step of the parameter estimation iteration. Derivatives of measured variables with respect to time that are needed in the evaluation of the DBG model are to be performed in discrete time. Once residuals are available for the time points of an observation window, the cost function can be built. If a gradient based parameter estimation method is used, the gradient of the least squares cost function can be obtained by using discrete derivatives. 

The probability density functions of the observations are generally unknown, but the Gauss-Markov theorem ensures that least-squares is always an acceptable estimator. However, the results of least squares are strongly influenced by discordant observations, so-called outliers. The robust-resistant techniques use weight-modification functions of O—Cy which progressively down-weight outliers. Tnese functions implicitly define probability functions p. They may alternatively be interpreted as an appreciation of the reliability of certain measurements. This approaches the frequently used option to simply omit discordant observations because they are judged to be unreliable. 

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