病态问题解算精度的相对均方误差比较分析方法

Comparison and Analysis Method of Relative Mean Square Error for Solving Ill-Posed Problems

病态问题导致参数估值方差较大,在参数真值未知的情况下,难以对正则化与TSVD(truncated singular value decomposition)方法进行精度比较。针对此问题,提出一种均方误差相对比较分析方法。首先,基于正则化估值相对变化量与TSVD估值相对变化量确定二者相对于最小二乘估值的相对偏差,避免偏差计算对真值的依赖;然后,利用相对偏差以及相对标准差确定均方根误差相对下降量,通过比较相对下降量大小确定最优的解算方法;最后,通过两组实验验证均方误差相对比较分析方法的可行性与有效性。

The ill-posed problem results in a large variance in parameter estimation.It is difficult to conduct a precision comparison of regularization and truncated singular value decomposition(TSVD)methods when the true values of parameters are unknown.To address this problem,we propose a mean square error relative comparative method.Firstly,we determine the relative deviation of regularization estimation and TSVD estimation relative to least squares estimation,avoiding the dependency of deviation calculation on the true values.Secondly,we utilize the relative deviation and relative standard deviation to determine the relative decreases of root mean square errors,and the optimal solution is determined by comparing the magnitude of relative decreases.Finally,we verify the feasibility and effectiveness of root mean square error relative comparative method through two sets of experiments.