Linear algebra Generalized Least Squares

To compensate for the errors involved in experimental data, the number of data sets should be greater than the number of coefficients p in the model. Least squares is just the application of optimization to obtain the best solution of the equations, meaning that the sum of the squares of the errors between the predicted and the experimental values of the dependent variable y for each data point x is minimized. Consider a general algebraic model that is linear in the coefficients. 

Given the rates of reactions, it Is a simple matter to compute the species production rates with Equation 2.60. One cannot solve uniquely the re-verse problem, in general Given observed production rates, computing the corresponding reaction rates requires additional information, such as rate expressions for the elementary reactions in a reaction mecha-nism. If the set of chemical reactions Is linearly independent, then one can uniquely solve the reverse problem. If the observed production rates contain experimental errors, there may not exist an exact solution of reaction rates, r, that satisfy Equation 2.60. In this situation, one is normally interested in finding the reaction rates that most closely satisfy Equation 2.60. The closest solution in a least-squares sense is easily computed with standard linear algebra software. 

In general, it may be messy to find the simultaneous solution of these equations algebraically. Nelder and Wedderburn (1972) showed that in the generalized linear model, these maximum likelihood estimators could also be found by iteratively reweighted least squares. Let the observation vector and parameter vector be... 

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