关于多元加权全最小一乘法的最优解

Optimal solution on weighted total least absolute deviations

对给定n+1维欧氏空间Rn+1中的m个点x1=(x11,x12,…,x1n+1),x2=(x21,x22,…,x2,n+1),…,xm=(xm1,xm2,…,xmn+1),证明了存在最优超平面β0+β1x1+…+βn+1xn+1=0,使这组点到此超平面的加权垂直距离和Q(β)=(∑n+1j=1βj2)-21∑mi=1wi|β0+∑n+1j=1βjxij|=min(wi>0,i=1,2,…,m);提出并证明了最优超平面β0+β1x1+…+βn+1xn+1=0应满足的3个必要条件,从而给出了求最优超平面的方法.

It was demonstrated the existence of the optimal hyperplanes β0＋β1x1＋…＋βn＋1xn＋1=0,which minimize Q（β）=（n＋1∑j=1βj^2）^-1/2m∑j=1wi｜β0＋n＋1∑j=1βjxij｜=min（wi〉0,i=1,2…,m） where（x1,x2,…,xm）∩→R^n＋1 are given;x1=（x11,x12,…,x1n＋1）,x2=（x21,x22,…x2,n＋1）,…xm=（xm1,xm2,…,xmn＋1）It suggested the three necessary conditions that the optimal hyperplanes should satisfy providing a method for the calculation of an accurate and optimal solution.