Gauss-Newton algorithm

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 4-36. Improved Newton-Gauss algorithm, including a termination criterion.

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... 

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. 

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

An additional observation for photon counting data there are no fractions of photons and thus the count can only include integer numbers. Thus the measurements in column B are rounded down to the nearest integer. It seems to be reasonable to do the same with the calculated values in column C. However, a test in Excel reveals that such an attempt does not work. The reason is, that the solver s Newton-Gauss algorithm requires the computation of the derivatives of the objective (x2 or ssq) with respect to the parameters. A rounding would destroy the continuity of the function and effectively wipe out the derivatives. 

The matrix C is defined by the non-linear parameters (rate constants). It is possible to minimise Ru, i.e. the corresponding ssq, as a function of these parameters in a normal Newton-Gauss algorithm. The chain of equations goes as follows... 

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. 

SCHEME 7.1 Flow diagram of a very basic Newton-Gauss algorithm. 

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