Multiresponse regression

The adaptive parameters in the model were estimated by nonlinear and multiresponse regression, performed using the Fortran subroutine BURENL23 based... 

A nonlinear multiresponse regression program (9) was used to search for the parameters which yield statistically the best accordance (maximum likelihood (10)) between the twenty interpolated and calculated responses. 

Dumez and Froment [ref 33] also determined the kinetics of the main reaction, rA, and the influence of Cg on this rate. Using the parameter values determined as explained above to calculate vs Cq led to an excellent agreement, indicating that the parameter estimation based upon the coking data only was quite accurate Nevertheless, it is advisable to use the rA/rA° vs Cg and the rc/rc° vs Cg-data simultaneously to derive the parameters by multiresponse regression. Such an approach was outlined in the paper by Marin et al. [ref 31] and by Froment [ref, 35]. 

Previously, we have considered only the analysis of single-response data. Here, we discuss multiresponse regression, focusing primarily upon the extension of the least-squares method to the case of multiple, perhaps correlated, responses in each experiment. 

Let us consider first the most general case of the multiresponse linear regression model represented by Equation 3.2. Namely, we assume that we have N measurements of the m-dimensional output vector (response variables), y , M.N. 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

The kinetic parameters of ammonia oxidation were fitted by multiresponse nonlinear regression, while the parameter estimates for the ammonia adsorption-desorption kinetics were kept unchanged with respect to those obtained from the fit in the previous section. Notably, in this case both the NH3 and the N2 outlet concentrations were regarded as regression responses. 

A global multiresponse non-linear regression was performed to fit Eq. (57) to all the runs with both 2% and 6% v/v 02 feed content to obtain the estimates of the kinetic parameters (Nova et al., 2006a). Figure 37 (solid lines) illustrates the adequacy of the global fit of the TRM runs with 2 and 6% 02 the MR rate law can evidently capture the complex maxima-minima NO and N2 traces (symbols) at low T at both NH3 startup, that a simple Eley-Rideal (ER), approach based on the equation... 

Procedure for Using SigmaPlot for Windows Solution of Multiresponse Linear Regression Problems Problems on Linear Regression... 

Predicted and measured values of Xj,buii( allow to undertake the regression analysis for estimating the kinetic parameters in Table 2. This was performed by employing the pack of routines GREGPAK [8] in multiresponse mode. 

Equation (10.19) implies that tNo becomes essentially independent of the ammonia surface coverage above a critical NH3 coverage identified by 0 nh3-Like for the ER rate expression, a global multiresponse nonlinear regression of all the TRM and TPR runs performed with 2 % O2 provided the estimates of the three rate parameters in Eq. (10.19) (k°No, E o, 0 nh3)-... 

The parameters included in the rate equations were estimated by global multiresponse nonlinear regressions based on the least squares method. For this purpose the BURENL routine, developed by Prof. Guido Buzzi-Ferraris, has been used [23, 24]. 

The parameters of the rate expressions (Eq. 18.1-18.15) for reactions (R.1-R.15) were estimated by multiresponse nonlinear regression of the transient microreactor runs, as detailed below. Notice that the rate equations do not include dependences on O2 and H2O, since the feed concentrations of both such species were kept constant at 8 % v/v. 

The prevailing reactions identified in the analyzed reacting systems were used to define the kinetic scheme in Table 18.2. The rate parameters of (Eq. 18.18) were independently estimated from NO oxidation tests, while a global multiresponse nonlinear regression on the whole set of runs involving NH3 provided estimates of the remaining rate parameters in (Eqs. 18.16-18.23). 

The regression analysis of experimental data then is performed according to one of the objective functions given in Fig. 15 for non-linear regression in different multiresponse situations. The selection of a specific objective function depends on the information available a priori concerning the different variances as given in the right column of Fig. 15 /29/. Naturally, from the statistical point of view, the different criteria are more efficient if more information about the variances is available. 

There are several techniques for minimization of the sum of squared residuals described by Eq. (7.160). We review some of these methods in this section. The methods developed in this section will enable us to fit models consisting of multiple dependent variables, such as the one described earlier, to multiresponse experimental data, in order to obtain the values of the parameters of the model that minimize the overall weighted) sum of squared residuals. In addition, a thorough statistical analysis of the regression results will enable us to... 

Least squares multiple nonlinear regression using the Marquardt and Gauss-Newton methods. The program can fit simultaneous ordinary differential equations and/or algebraic equations to multiresponse data. 

The basic parameter estimation, or regression, problem involves fitting the parameters of a proposed model to agree with the observed behavior of a system (Figure 8.1). We assume that, in any particular measurement of the system behavior, there is some set of predictor variables jc e 91 that fiiUy determines the behavior of the system (in the absence of any random noise or error). For each experiment, we measure some set of response variables yf.r) g fp— single-response data mAilL >, multiresponse data. t NxiXe... 

Above, we have used only the initial slope of cc(t) to measnre the reaction rate at the initial concentrations ca(0) and cb(0). We could fit the rate law to the complete data set of concentrations vs. time, such as is found in Table 8.3 for the experiment with ca(0) = 0.1 M and cb(0) = 0.1 M. If we use only the data for the concentration of C, we have a single-response data set. If we include all concentration values, we have a multiresponse data set with L = 3. For this regression problem, we obtain the model predictions by solving the IVP (8.5) numerically. 

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