An algorithm for fitting the shefted multiplicative model

Abstract

The shifted multiplicative model may be fitted by least squares. The least squares solution can be obtained only by iterative methods. It has been shown that once the shift parameter p is known, the parametes of the multiplicative terms may be obtained by a singular value decomposition of the matrix of residuals . Thus, the residual sum of squares may be looked upon as a one–dimensional function depending only on the shift parameter. An occasional problem to be addressed by any algorithm for finding the least squares solution is movement of the shift parameter away from the value minimizing the residual sum of squares. It has been found in a number of real data sets that the residual sum of squares approaches an asymptote as the shift parameter moves to either + ∞, or - ∞ and usually has one maximum and one minimum. With this shape of the residual sum of squares, the shift parameter will move away from the minimizing value, provided the starting value is on the side of the maximum, which does not contain the minimum. This paper suggests an algorithm, which addresses this problem using standard procedures for minimizing a one dimensional function. An important aspect of the algorithm is that it searches for the minimum on both sides of the maximum. The algorithm has been found to locate the minimum even with very poor starting values.

Keywords:

• Least squares
• shift parameter
• singular value decomposition