正则化的奇异值分解参数构造法

Tikhonov正则化法引入正则化参数和稳定泛函来改善矩阵的病态性。稳定泛函表示为参数的二范约束时,正则化矩阵为单位阵的正则化法即为岭估计法。通过对岭估计的方差与偏差进行分析可知,岭估计改善矩阵病态性的同时也过度地引入了偏差,降低了解的可靠性,对较大奇异值的修正不能有效地减小估计的方差,却引入了偏差,而对较小奇异值的修正可有效地减小估计的方差。因此,选择较小奇异值特征向量构造正则化矩阵,调节各奇异值的修正,可有效减小参数估计的方差,减少偏差的引入,得到更为可靠的参数估计。通过试验证明了该方法的有效性。

Construction Method of Regularization by Singular Value Decomposition of Design Matrix

Tikhonov regularization introduces regularization parameter and stable functional to improve the ill-condition.When the stable functional expressed as two-norm constraint,the regularization method is the same as ridge estimation.The analysis of the variance and bias of the ridge estimation shows that ridge estimation improved the ill-condition but introduced more bias.The estimation reliability is lowered.We get that correct the larger singular values cannot decrease the variance effectively but introduced more bias, correcting the smaller singular values can decrease the variance effectively.We choose the eigenvectors of the smaller singular values to construct the regularization matrix.It can adjust the correction of the singular values,decrease the variance and biases and finally get a more reliable estimation.