Fitting data by the method of least squares

The dassical method for fitting data to theoretical curves is linear regression. This procedure allows the equation of the best straight line fitting the experimental data to be calculated directly  

Until relatively recently this was the only method that could be used conveniently to fit data by regression. This is the reason why so many classical approaches for evaluating biochemical data depended on linearising data, sometimes by quite complex transformations. The best known examples are the use of the Lineweaver-Burk transformation of the Michaelis-Menten model to derive enzyme kinetic data, and of the Scatchard plot to analyse ligand binding equilibria. These linearisation procedures are generally no longer recommended, or necessary. 

In contrast to the explicit analytical solution of least-squares fit used in linear regression, our present treatment of data analysis relies on an iterative optimization, which is a completely different approach as a result of the operations discussed in the previous section, theoretical data are calculated, dependent on the model and choice of parameters, which can be compared with the experimental results. The deviation between theoretical and experimental data is usually expressed as the sum of the errors squared for all the data points, alternatively called the sum of squared deviations (SSD)  

This deviation is now minimised by variation of the parameters. The combination of parameter values that best fit the experimental data using this deviation as the criterion of best fit is the desired solution of the analysis. This process of finding a solution is termed iteration because the solution is located by trying out many possible combinations of parameters since the equations being fitted are in general non-linear, the process is more specifically one of iterative non-linear least-squares fitting. 

Region A is the location of the global minimum, region B is a local minimum, and region C represents an area where the model is no longer valid and the slope of the error surface is directed away from the minimum. 

---