关键词: TSVD, 截断参数, 有偏估计, 方差, 偏差

Abstract: TSVD truncate small singular values by truncation parameter to improve the parameter estimation of ill-posed model. From the perspective of MSE (mean squared error), TSVD introduce biases to reduce variances, therefore the stability and reliability of the solution can be improved. Truncation parameter is key factor of TSVD, but it is difficult to determine in case of the gently declined singular values. The parameter determined by GCV (generalized cross-validation) and L-curve often unstable and unreliable. And the minimum MSE method is limited by the accuracy of the estimated MSE. This paper compares the changes of variance and bias produced by truncating the singular values in turn and determines the truncation parameter when the reduced variance is less than the introduced bias. In order to avoid the comparison between reduced variance and introduced bias of truncating small singular values, the confidence domain of bias is established through estimating the introduced bias of truncating big singular values that are proved to be reliable. The comparisons are replaced by comparing the reduced variance with the bias in the confidence domain. Therefore, the issue of introduced bias of truncating small singular values cannot be calculated without true values of unknown parameters is solved. Numerical examples proves the feasibility and effectiveness of the new method. Truncation parameters determined by new method are more stable and reliable than GCV and L-curve and improve the TSVD solution effectively.

Key words: TSVD, truncation parameter, biased estimation, variance, bias