引用本文: 林东方, 朱建军. 附有奇异值修正限制的改进的岭估计方法[J]. 武汉大学学报 ( 信息科学版), 2017, 42(12): 1834-1839.
LIN Dongfang, ZHU Jianjun. Improved Ridge Estimation with Singular Value Correction Constraints[J]. Geomatics and Information Science of Wuhan University, 2017, 42(12): 1834-1839.
 Citation: LIN Dongfang, ZHU Jianjun. Improved Ridge Estimation with Singular Value Correction Constraints[J]. Geomatics and Information Science of Wuhan University, 2017, 42(12): 1834-1839.

## Improved Ridge Estimation with Singular Value Correction Constraints

• 摘要: 最小二乘估计具有无偏性，而岭估计是一种有偏估计，它通过引入偏差降低方差来降低均方误差。在模型出现病态时，岭估计优于最小二乘估计。对岭估计的方差与偏差进行分析发现，岭估计通过修正病态矩阵的奇异值降低均方误差，但对部分较大奇异值的修正不能有效降低均方误差。通过比较修正奇异值的方差下降量与偏差引入量的大小关系确定需要修正的小奇异值，进而改进岭估计方法，实现选择性地修正小奇异值，提出附有奇异值修正限制的改进的岭估计方法，可有效改善岭估计的解算效果和可靠性，实验验证了新方法的可行性和有效性。

Abstract: It is well known that ridge estimation is better than least squares estimation for ill posed problems, however the least squares estimation is unbiased compare to ridge estimation which is biased. Actually, ridge estimation reduces the variance by introducing bias so as to improve the MSE (mean square error). Therefore, ridge estimation is better than least squares estimation in term of MSE. Since the MSE is composed of variance and bias, the performance of the ridge estimation can be shown clearly through computing the variance and bias. Through analyzing the changes of variance and bias of ridge estimation, we know that ridge estimation correct the singular values of the ill posed matrix to reduce the variance and introduce the bias. When the reduced variance is much more than the introduced bias, the MSE can be reduced. However, ridge estimation correct all the singular values in the ill-conditioned matrix. Correcting the big singular values cannot reduce the variance of the estimation effectively but introduce much bias into the estimation. In view of this, improved ridge estimation is proposed in this paper to constrain the correction of the singular values. The new ridge estimation only correct the small singular values which are determined by comparing the variance reduction with bias introduction of singular value correction. Theoretical analysis clearly shows the feasibility of the improved ridge estimation. The experiment on the basis of the Fredholm integral equation of the first kind is carried out to demonstrate the effectiveness of the new ridge estimation. The results show that the improved ridge estimation performs much better than ridge estimation in stability and accuracy.

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