The Newton-Gauss Algorithm

On a very different note, in Chapter 5, Model-Free Analyses, we introduce methods that attempt a model-free analysis of the data. Typically, a matrix Y is automatically decomposed into the product of the matrices C and A. These analyses usually are not as robust as the fitting discussed in this chapter, however, the results can guide the researcher in finding the correct model. 

Instead of developing a program that performs the task as just explained, we move to the 2-parameter case. Subsequently, we generalise to the np-parameter case and then we analyse the relationship with the Newton-Gauss algorithm for least-squares fitting. 

the generalisation of the nomenclature, using 2 instead of ssq. The parabolic surface approximates 2 at the point pi/p2 The quality of the approximation decreases with increasing distance from pi/p2. 

Having determined the first and second derivatives, either explicitly or numerically, the minimum of the parabolic surface has to be localised. 

In order to compute the minimum we need to know the coefficients a2 to as. 

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We are now in a position to devise a first, very crude program that should, starting from a set of initial guesses, move towards the best fit. Below, a flow diagram is given that represents the basic principle of the Newton-Gauss algorithm ... 

Figure 4-33. First version of the Newton-Gauss algorithm...

Figure 4-38. The Newton-Gauss algorithm after implementation of the Marquardt strategy...

We have already given the equations for the computation of the standard errors in the parameters optimised by linear regression, equation (4.32). The equations are very similar for parameters that are passed through the Newton-Gauss algorithm. In fact, at the end of the iterative fitting, the relevant information has already been calculated. 

The central part of the Newton-Gauss algorithm is the computation of the residuals, which are now collected in the matrix R. R is a function of the measurement Y, the model, and the parameters. For the example, the parameters include the two rate constants k and fe, which we collect in the vector p and all molar absorptivities, all elements of the matrix A. For a given model we can write... 

Note also that the Newton-Gauss algorithm for function optimisation is the standard option in Excel s solver. 

Resolving Factor Analysis, RFA, is an attempt to introduce the strengths of the Newton-Gauss algorithm into the model-free analysis methodology. As... 

For a three component system, the matrix T has nine elements and thus it appears that C and eventually the sum of squares are a function of nine parameters. As we will see in a moment there are actually fewer, only six, parameters to be fitted. The idea of RFA is to use the Newton-Gauss algorithm to fit this rather small number of parameters in T. 

The Newton-Gauss algorithm requires initial estimates for the parameters in T. These can be computed from the same estimated concentration profiles Cguess as before (Figure 5-44). It is determined by... 

The Newton-Gauss algorithm (ng Jm3. m), is called from Main RFA.m, and requires a Matlab function that computes the residuals as a function of the parameters T, as defined in equation (5.54). This calculation is performed in the Matlab function Rcalc RFA. m. 

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