Heuristic scaling method for efficient parameter estimation

In parameter estimation, an order-of-magnitude difference in parameter values makes it difficult to find the optimum because parameter estimation problem can be ill-conditioned by the difference. To avoid the difficulty, a proper scaling factor of parameter should be introduced into parameter estimation. Unfortunately there has not been an appropriate method for determining a set of scaling factor. In this work, we propose a new heuristic method for determining the scaling factor which converts parameter estimation problem well-conditioned. The proposed method provides a set of scaling factor to reduce the condition number of Hessian matrix since the small value of condition number is directly related to the conditions of the problem. Numerical implementation demonstrated that the proposed method drastically enhanced the solvability of original ill-conditioned problem.