Regression equations parameter estimates

When estimates of k°, k, k", Ky, and K2 have been obtained, a calculated pH-rate curve is developed with Eq. (6-80). If the experimental points follow closely the calculated curve, it may be concluded that the data are consistent with the assumed rate equation. The constants may be considered adjustable parameters that are modified to achieve the best possible fit, and one approach is to use these initial parameter estimates in an iterative nonlinear regression program. The dissociation constants K and K2 derived from kinetic data should be in reasonable agreement with the dissociation constants obtained (under the same experimental conditions) by other means. 

Experimental polymer rheology data obtained in a capillary rheometer at different temperatures is used to determine the unknown coefficients in Equations 11 - 12. Multiple linear regression is used for parameter estimation. The values of these coefficients for three different polymers is shown in Table I. The polymer rheology is shown in Figures 2 - 4. 

It is well known that cubic equations of state may predict erroneous binary vapor liquid equilibria when using interaction parameter estimates from an unconstrained regression of binary VLE data (Schwartzentruber et al.. 1987 Englezos et al. 1989). In other words, the liquid phase stability criterion is violated. Modell and Reid (1983) discuss extensively the phase stability criteria. A general method to alleviate the problem is to perform the least squares estimation subject to satisfying the liquid phase stability criterion. In other... 

The implicit LS, ML and Constrained LS (CLS) estimation methods are now used to synthesize a systematic approach for the parameter estimation problem when no prior knowledge regarding the adequacy of the thermodynamic model is available. Given the availability of methods to estimate the interaction parameters in equations of state there is a need to follow a systematic and computationally efficient approach to deal with all possible cases that could be encountered during the regression of binary VLE data. The following step by step systematic approach is proposed (Englezos et al. 1993)... 

Kittrell et al. (1965a) also performed two types of estimation. First the data at each isotherm were used separately and subsequently all data were regressed simultaneously. The regression of the isothermal data was also done with linear least squares by linearizing the model equation. In Tables 16.7 and 16.8 the reported parameter estimates are given together with the reported standard error. Ayen and Peters (1962) have also reported values for the unknown parameters and they are given here in Table 16.9. 

The MO concentrations versus time profiles were fitted to second order polynomial equations and the parameters estimated by nonlinear regression analysis. The initial rates of reactions were obtained by taking the derivative at t=0. The reaction is first order with respect to hydrogen pressure changing to zero order dependence above about 3.45 MPa hydrogen pressure. This was attributed to saturation of the catalyst sites. Experiments were conducted in which HPLC grade MIBK was added to the initial reactant mixture, there was no evidence of product inhibition. 

The values calculated for various Group 14 substituents provide the only evidence we have for the validity of our parameter estimates. Table 10 presents the values of gobs, gcalc, A, the number of the regression equation used to obtain the calculated value and the parameter types used in the calculation. The agreement between observed and calculated values is described in terms of the number of standard deviations, Nsd, defined as ... 

The solution of problems in chemical reactor design and kinetics often requires the use of computer software. In chemical kinetics, a typical objective is to determine kinetics rate parameters from a set of experimental data. In such a case, software capable of parameter estimation by regression analysis is extremely usefiil. In chemical reactor design, or in the analysis of reactor performance, solution of sets of algebraic or differential equations may be required. In some cases, these equations can be solved an-... 

In equation 3.4-18, the right side is linear with respect to both the parameters and the variables, j/the variables are interpreted as 1/T, In cA, In cB,.. . . However, the transformation of the function from a nonlinear to a linear form may result in a poorer fit. For example, in the Arrhenius equation, it is usually better to estimate A and EA by nonlinear regression applied to k = A exp( —EJRT), equation 3.1-8, than by linear regression applied to Ini = In A — EJRT, equation 3.1-7. This is because the linearization is statistically valid only if the experimental data are subject to constant relative errors (i.e., measurements are subject to fixed percentage errors) if, as is more often the case, constant absolute errors are observed, linearization misrepresents the error distribution, and leads to incorrect parameter estimates. 

As introduced in sections 3.1.3 and 4.2.3, the Arrhenius equation is the normal means of representing the effect of T on rate of reaction, through the dependence of the rate constant k on T. This equation contains two parameters, A and EA, which are usually stipulated to be independent of T. Values of A and EA can be established from a minimum of two measurements of A at two temperatures. However, more than two results are required to establish the validity of the equation, and the values of A and EA are then obtained by parameter estimation from several results. The linear form of equation 3.1-7 may be used for this purpose, either graphically or (better) by linear regression. Alternatively, the exponential form of equation 3.1-8 may be used in conjunction with nonlinear regression (Section 3.5). Seme values are given in Table 4.2. 

A major limitation of the linearized forms of the Michaelis-Menten equation is that none provides accurate estimates of both Km and Vmax. Furthermore, it is impossible to obtain meaningful error estimates for the parameters, since linear regression is not strictly appropriate. With the advent of more sophisticated computer tools, there is an increasing trend toward using the integrated rate equation and nonlinear regression analysis to estimate Km and While this type of analysis is more complex than the linear approaches, it has several benefits. First, accurate nonbiased estimates of Km and Vmax can be obtained. Second, nonlinear regression may allow the errors (or confidence intervals) of the parameter estimates to be determined. 

In any regression problem, there are more equations than unknown parameters. In this instance, there are four equations, representing the four data pairs. The software then returns the optimum values of parameters in the governing equation(s). The Solu-tions/Statistics menu can be consulted to determine the squared residual between each experimental data point and the corresponding predicted value obtained from the parameter estimates. 

When linear regression does not yield a good correlation, application of a non-linear function may be feasible (see Chapter 10). The parameter estimates for higher-order or polynomial equations may prove to be more difficult to interpret than for a linear relationship. Nevertheless, this approach may be preferable to using lower-order levels of correlation (B or C) for evaluating the relationship between dissolution and absorption data. 

After all of the data were tabulated on a given gas, the one atmosphere partial pressure values estimated from the moderate to high pressure measurements up to 600 K were combined with the selected data of Battino in the 273 to 350 K temperature range in a linear regression to obtain the constants of equation (2). The equation parameters are listed in Table V. 

An elaborate and novel system was devised by Richard and Coursin (168,169,170,171) whereby 19 constituents (minerals, sugars, acids, amino acids) were determined and evaluated by a heirarchial classification approach. By means of a series of inequalities, based on deviations from the mean, a region of authentic juice is defined in a multidimensional space. A series of regression equations between parameters (with R > 0.9) are considered next to verify that the relationships between constituents are normal. Finally, the above information may be combined in a matrix approach to give an estimate of juice content. 

Landrum et al. (1992) developed a kinetic bioaccumulation model for PAHs in the amphipod Diporeia, employing first-order kinetic rate constants for uptake of dissolved chemical from the overlying water, uptake by ingestion of sediment, and elimination of chemical via the gills and feces. In this model, diet is restricted to sediment, and chemical metabolism is considered negligable. The model and its parameters, as Table 9.3 summarizes, treat steady-state and time-variable conditions. Empirically derived regression equations (Landrum and Poore, 1988 and Landrum, 1989) are used to estimate the uptake and elimination rate constants. A field study in Lake Michigan revealed substantial differences between predicted and observed concentrations of PAHs in the amphipod Diporeia. Until more robust kinetic rate constant data are available for a variety of benthic invertebrates and chemicals, this model is unlikely to provide accurate estimates of chemical concentrations in benthic invertebrates under field conditions. 

Given a set of experimental data, we look for the time profile of A (t) and b(t) parameters in (C.l). To perform this key operation in the procedure, it is necessary to estimate the model on-line at the same time as the input-output data are received [600]. Identification techniques that comply with this context are called recursive identification methods, since the measured input-output data are processed recursively (sequentially) as they become available. Other commonly used terms for such techniques are on-line or real-time identification, or sequential parameter estimation [352]. Using these techniques, it may be possible to investigate time variations in the process in a real-time context. However, tools for recursive estimation are available for discrete-time models. If the input r (t) is piecewise constant over time intervals (this condition is fulfilled in our context), then the conversion of (C.l) to a discrete-time model is possible without any approximation or additional hypothesis. Most common discrete-time models are difference equation descriptions, such as the Auto-.Regression with eXtra inputs (ARX) model. The basic relationship is the linear difference equation ... 

After the formulation stage, we have all the equations of the model, but they are not useful yet, because parameters in the equations do not have a particular value. Consequently, the model cannot be used to reproduce the behavior of a physical entity. The parameter estimation procedure consists of obtaining a set of parameters that allows simulation with the model. In many cases, parameters can be found in literature, but in other cases it is required to fit the model to the experimental behavior by using mathematical procedures. The easier and more used types of procedures are those based on the use of optimization algorithms to make minimum the differences between the experimental observations and the model outputs. The more frequently used criterion to optimize the values of the parameters is the least square regression coefficient. In this procedure, a set of values is proposed for all model parameters (one for every parameter) and the model is run. After that, the error criterion is calculated as the sum of the squares of the residues (differences between the values of every experimental and modeled value). Then, an optimization procedure is used to change the values of the model parameters in order to get the minimum value of this criterion. 

Statistical parameters, when available, indicating the significance of each of the descriptor s contribution to the final regression equation are listed under its corresponding term in the equation. These include the standard errors written as values, the Student t test values, and the VIF. The significance of the equation will be indicated by the sample size, n the variance explained, r the standard error of the estimate, s the Fisher index, F and the cross-validated correlation coefficient, q. When known, outliers will be mentioned. The equations are followed by a discussion of the physical significance of the descriptor terms. 

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