带统计约束的一般加权最小二乘问题

Weighted Least Square Problem with Statistics Constraints

曲线拟合问题是一个经典且十分重要的研究热点.曲线拟合的数学模型为最小二乘问题.本文针对给定的数据,建立了新的最小二乘模型,该模型的目标函数是最小化加权的残量的平方和,约束条件为每个残量服从一定的概率分布.这个模型和以往的模型的不同之处在于考虑了残量的概率分布特点并且把这种性质作为一个约束条件.同时,我们也建立了求解此模型的算法,在这个方法的求解过程中,曲线拟合函数被逐次改进.

The curve fitting problem is a classical and very important research hotspot. The mathematical model of curve fitting is the least square problem. In this paper, a new model to find the best regression function for observed data is design. For the given data, a new least squares model is established. The objective function of this model is to minimize the square sum of the weighted residuals, and the constraint conditions is that each residue follows a certain probability distribution. The difference between this model and the previous model is that the probability distribution characteristic of the residue is considered and this nature is used as a constraint. In particular, when all the residual quantities are independent of the same normal distribution, the least square model can be transformed into a general nonlinear constrained optimization problem. At the same time, an algo- rithm for solving this model is established. In the process of solving this method, the curve fitting function is improved gradually.