相关数据集的最小二乘处理方法

数据的最小二乘处理可以归结为求解线性方程组Ａｘ＝ｂ，不论在何种情形下（常定，超定或欠定），它都有最小二乘意义下的最优解．这要求数据矩阵Ａ的相关矩阵的逆矩阵存在，即欠定增况下的ＡＡＴ或超定情况下的ＡＴＡ是满秩的．对于降秩的ＡＡＴ或ＡＴＡ的情况，文中提出用奇异值分解的方法求其矩阵伪逆，使数据的最小二乘处理适应于相关数据集的处理．同直接对数据矩阵Ａ进行奇异值分解求ＡＸ＝ｂ的最小二乘解相比，本文提出的方法只需对阶数较低的对称方阵进行分解，可在微机上实现高维数据的处理．

A Generalized Method to Solve the Least-Square-Error Problem for a Linearly-Dependent Data Set

he least-square-error problem associated with a data set can be boiled down to find the optimal sulution, in view of least square error, of simultaneous equations Ax=b under the various conditions of determinic, overdeterminic or underdeterminic. Where the inverse of the matrix, AAT for underdeterminic case or ATA for overdeterminic case, is needed. It will be failed to process a linearly-dependent data set. To deal with such problem we present a generalized method in this paper, which is to find a matrix pseudoinverse, instead of matrix inverse, by means of singular value decomposition (SVD). Compared with another method of solving Ax=b by direct finding the pseudoinverse of matrix A, the present method needs performing SVD for a symmetry matrix with a small order, it is suitable for processing highdimensional data set with a microcomputer.