Statistical Measure of the Regression Quality

The minimum value of the statistic is commonly used to provide a figure of merit for the quality of regression. As shown in equation (19.9), the statistic accounts for the variance of the experimental data. In principle, for a model that well describes the data, the minimum value of the statistic has a mean value of the degree of freedom v with standard deviation /2v. Thus, for a good fit. 

The use of equation (19.35) to assess the quality of a regression is valid only for an accurate estimate of the variance of the stochastic errors in the impedance data. 

An independent method to identify the stochastic errors of impedance data is described in Chapter 21. An alternative approach has been to use the method of maximum likelihood, in which the regression procedure is used to obtain a joint estimate for the parameter vector P and the error structure of the data. The maximum likelihood method is recommended under conditions where the error structure is unknown, but the error structure obtained by simultaneous regression is severely constrained by the assumed form of the error-variance model. In addition, the assumption that the error variance model can be obtained by minimizing the objective function ignores the differences eimong the contributions to the residual errors shown in Chapter 21. Finedly, the use of the regression procedure to estimate the standard deviation of the data precludes use of the statistic 

2 Generate the synthetic data described in Problem 19.1, adding independent normally distributed random numbers N (0, cr) to both real and imaginary parts of the impedance where cr = a Z and the value of a is given below  

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