Introduction and the General Method of Least Squares

It is assumed that x, the independent variable, is error free (this might be time or temperature, for example), and y, the dependent variable, contains experimental error. In chemical and biomolecular engineering applications, theoretical knowledge often exists of the function that should ht the data. If the deviations (errors) between the data and the fitting function are statistically distributed with a normal distribution with zero mean and constant variance, then it can be shown that a proper way to find the unknown coefficients of the function is to minimize the sum of squares of the errors. It is common nomenclature to call the errors residuals, which are defined as follows  

This is, therefore, called the method of least squares. If the htting function can be represented as 

FIGURE 7.1 Data with experimental error in the dependent variable. 

To minimize Equation 7.2, differentiate with respect to the unknown parameters and set the result to zero (seeking a stationary point)  

If 7 is expanded into a Taylor s series about a point c° (this is an initial guess of the c s), then 

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