顾及截断偏差影响的TSVD截断参数确定方法
发布时间：2019-03-09 16:20
【摘要】：TSVD通过截断参数截掉较小的奇异值来改善病态性对估计的影响,其本质是通过引入少量偏差来降低方差,以提高估值的稳定性和可靠性。截断参数是影响TSVD解算效果的关键因素,常用的广义交叉核实法(GCV法)和L曲线法未从TSVD改善模型参数估值质量的角度确定截断参数,稳定性和可靠性不足,而最小MSE法理论依据充分但受限于MSE计算的准确性。通过分析TSVD由小到大截掉奇异值后,相应的估值方差与偏差变化,本文提出了引入偏差量小于降低方差量来确定截断参数的思想,并通过估计出较大奇异值截掉后的偏差引入量建立偏差估值可信区间,利用可信区间内偏差估值与方差下降量进行比较,避免较小奇异值截掉后的方差下降量与偏差引入量的直接比较,从而解决参数真值未知截掉较小奇异值引入偏差量难以准确计算的问题。最后通过试验验证了新方法的可行性和有效性,相比于GCV法和L曲线法,新方法确定的截断参数稳定性和可靠性更高,可有效提高TSVD的解算效果。
[Abstract]:TSVD improves the ill-conditioned effect on estimation by truncating small singular values. Its essence is to reduce the variance by introducing a small amount of deviation to improve the stability and reliability of the estimation. Truncation parameters are the key factors that affect the calculation of TSVD. The commonly used generalized cross-validation method (GCV) and L-curve method do not determine the truncation parameters from the point of view of improving the estimation quality of the model parameters by TSVD, so the stability and reliability of the truncation parameters are insufficient. However, the minimum MSE method has sufficient theoretical basis but is limited by the accuracy of MSE calculation. Based on the analysis of the variation of estimation variance and deviation after the singular value is truncated from small to large by TSVD, the idea of introducing the deviation less than the reduced variance to determine the truncation parameter is presented in this paper. The error estimation confidence interval is established by estimating the deviation input after the larger singular value is truncated, and the deviation estimation within the confidence interval is compared with the variance decline. In order to avoid the direct comparison between the decrease of variance after the truncation of the smaller singular value and the amount of deviation introduced, the problem that the parameter truth value is unknown truncated the smaller singular value and the introduced deviation is difficult to accurately calculate is solved. Finally, the feasibility and effectiveness of the new method are verified by experiments. Compared with the GCV method and the L-curve method, the stability and reliability of the truncation parameters determined by the new method are higher than those of the GCV method and the L-curve method, which can effectively improve the calculation effect of TSVD.
【作者单位】： 中南大学地球科学与信息物理学院;中南大学有色金属成矿预测与地质环境监测教育部重点实验室;
【基金】：国家自然科学基金(41531068;41474008;41574006;41674012)~~
【分类号】：P207
[Abstract]:TSVD improves the ill-conditioned effect on estimation by truncating small singular values. Its essence is to reduce the variance by introducing a small amount of deviation to improve the stability and reliability of the estimation. Truncation parameters are the key factors that affect the calculation of TSVD. The commonly used generalized cross-validation method (GCV) and L-curve method do not determine the truncation parameters from the point of view of improving the estimation quality of the model parameters by TSVD, so the stability and reliability of the truncation parameters are insufficient. However, the minimum MSE method has sufficient theoretical basis but is limited by the accuracy of MSE calculation. Based on the analysis of the variation of estimation variance and deviation after the singular value is truncated from small to large by TSVD, the idea of introducing the deviation less than the reduced variance to determine the truncation parameter is presented in this paper. The error estimation confidence interval is established by estimating the deviation input after the larger singular value is truncated, and the deviation estimation within the confidence interval is compared with the variance decline. In order to avoid the direct comparison between the decrease of variance after the truncation of the smaller singular value and the amount of deviation introduced, the problem that the parameter truth value is unknown truncated the smaller singular value and the introduced deviation is difficult to accurately calculate is solved. Finally, the feasibility and effectiveness of the new method are verified by experiments. Compared with the GCV method and the L-curve method, the stability and reliability of the truncation parameters determined by the new method are higher than those of the GCV method and the L-curve method, which can effectively improve the calculation effect of TSVD.
【作者单位】： 中南大学地球科学与信息物理学院;中南大学有色金属成矿预测与地质环境监测教育部重点实验室;
【基金】：国家自然科学基金(41531068;41474008;41574006;41674012)~~
【分类号】：P207
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