Complex non-linear regression least squares

Complex Non-linear Regression Least-Squares (CNRLS) for the Analysis of impedance Data... 

Regression analysis is one of the main tools in generating mathematical models by fitting a model equation to experimental data. In general, the regression analysis is based on the application of the least squares method for the estimation of unknown coefficients in the model equation. This method minimizes the sum of squares of the differences between the experimental values of the dependent variable, and those estimated by the model, y . Polynomials of various degrees are often used to describe complex non-linear relationships between the dependent and independent variables because the model equation is linear with respect to the unknown coefficients, and therefore the procedure for the calculation of the coefficients reduces to the solution of a system of linear simultaneous equations. 

The solid curves in Fig. 3 were calculated by simultaneous non-linear least-squares regression of reaction-rate data at all alkali-metal cation concentrations (20 kobs values using three different cations) to the complex rate constant in eq 15. The calculation was performed subject to the stipulation that aU three curves converge to a single k value. (Convergence to a single kobs value, k, at infinite dilution is implicit to derivation ofeq 15.) Simultaneous analysis of the three curves (one for each of the cations, Li, Na and K ) provides unique values for rate and equilibrium constants, kMi and Kmi, i.e., for kui, knai, kxi, Kbii, KnbI and Kki. 

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