A new biased estimation method based on Neumann series for solving ill-posed problems

Qiuwei Yang

doi:10.1177/1729881419872058

Abstract

The ill-posed least squares problems often arise in many engineering applications such as machine learning, intelligent navigation algorithms, surveying and mapping adjustment model, and linear regression model. A new biased estimation (BE) method based on Neumann series is proposed in this article to solve the ill-posed problems more effectively. Using Neumann series expansion, the unbiased estimate can be expressed as the sum of infinite items. When all the high-order items are omitted, the proposed method degenerates into the ridge estimation or generalized ridge estimation method, whereas a series of new biased estimates can be acquired by including some high-order items. Using the comparative analysis, the optimal biased estimate can be found out with less computation. The developed theory establishes the essential relationship between BE and unbiased estimation and can unify the existing unbiased and biased estimate formulas. Moreover, the proposed algorithm suits for not only ill-conditioned equations but also rank-defect equations. Numerical results show that the proposed BE method has improved accuracy over the existing robust estimation methods to a certain extent.

Highlights

Many engineering problems need to solve linear equations
The ill-posed least squares problems often arise in many engineering applications such as machine learning, intelligent navigation algorithms, surveying and mapping adjustment model, and linear regression model.[1,2,3,4,5]
Using Neumann series expansion, the unbiased estimate can be expressed as the sum of infinite items

Summary

Introduction

Least squares estimation (LSE) is the most commonly used method to solve linear equations It is called unbiased estimation since it satisfies the optimal linear unbiasedness. When the coefficient matrix of the equation system is ill-conditioned, the calculation results obtained by the LSE often have large errors or even complete distortion. This phenomenon is called the ill-posed least squares problem. A new BE based on Neumann series is proposed to solve the ill-posed least squares problems. 2. If we ignore all the high-order items in equation (8), we have x1BE 1⁄4 ðB þ KÞÀ1 Á z ð10Þ where x1BE denotes the first-order biased estimate derived from equation (8). Searching the optimal biased estimate is the second important issue in the proposed method, which will be discussed

Discussion on the special issues

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Conclusions