Regression Analysis Method for Estimating the Parameters of the Three-Parameter Size Distribution Equation

Since the original paper, two methods have been introduced to compute the parameters of the three parameter size distribution equation: [ ] Both of them are based on graphical means, which are time-consuming and of little statistical significance. A program has been developed, using the Gauss-Newton method to compute by nonlinear regression analysis the parameters of the equation. Although the method is quite general in curve-fitting applications, it does not appear to be used in the field of size analysis, and a brief description is given. As applied here, the Gauss-Newton method involves the minimization of the least squares: [ ] where n is the number of sieves used. The second member of Eq. 2 can be approximated by a multiple Taylor's series expansion of first order about a starting value [ ] The minimization of Q (r,s,Xo) means: [ ] Putting Eq. 3 into Eq. 2 and differentiating, we obtain a system of equations which can be represented in matrix symbolism (box) The system can be solved for: [ ] and the operations repeated with new values: [ ] The starting values for r and s are taken as 1, for Xo, a value higher than the maximum sieve opening value. Applied to the data given by Herbst 5,6 for batch grinding of 7 X 9 mesh dolomite, the parameters given in Table 1 were obtained. The discussion of the significance of the residual sum of squares is a difficult subject in multiple nonlinear