【摘要】：TSVD通過截斷參數(shù)截掉較小的奇異值來改善病態(tài)性對估計的影響,其本質(zhì)是通過引入少量偏差來降低方差,以提高估值的穩(wěn)定性和可靠性。截斷參數(shù)是影響TSVD解算效果的關(guān)鍵因素,常用的廣義交叉核實法(GCV法)和L曲線法未從TSVD改善模型參數(shù)估值質(zhì)量的角度確定截斷參數(shù),穩(wěn)定性和可靠性不足,而最小MSE法理論依據(jù)充分但受限于MSE計算的準(zhǔn)確性。通過分析TSVD由小到大截掉奇異值后,相應(yīng)的估值方差與偏差變化,本文提出了引入偏差量小于降低方差量來確定截斷參數(shù)的思想,并通過估計出較大奇異值截掉后的偏差引入量建立偏差估值可信區(qū)間,利用可信區(qū)間內(nèi)偏差估值與方差下降量進(jìn)行比較,避免較小奇異值截掉后的方差下降量與偏差引入量的直接比較,從而解決參數(shù)真值未知截掉較小奇異值引入偏差量難以準(zhǔn)確計算的問題。最后通過試驗驗證了新方法的可行性和有效性,相比于GCV法和L曲線法,新方法確定的截斷參數(shù)穩(wěn)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.
【作者單位】： 中南大學(xué)地球科學(xué)與信息物理學(xué)院;中南大學(xué)有色金屬成礦預(yù)測與地質(zhì)環(huán)境監(jiān)測教育部重點實驗室;
【基金】：國家自然科學(xué)基金(41531068;41474008;41574006;41674012)~~
【分類號】：P207

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