关于加权全最小一乘的探讨

ON WEIGHTED TOTAL LEAST ABSOLUTE DEVIATIONS APPROACH

设p（v;a;μ）为v∈Rk到超平面μTv+a=0的垂直距离,其中μ∈Rk,a∈R1.本文研讨的是寻找适当的a和μ使达到最小,这里{v1,v2,…,vn}(?)Rk而qi>0,i=1,2,…,n是给定的权.本文采用一条全新的途径来研讨上述加权全最小一乘问题的求解.文中导出了要使Q（u0,μ）达到最小,μ和u0应满足的若干本质性必要条件,而满足该条件的（μ,u0）只有有限多个;进而提出了一个求解加权全最小一乘问题的有限步终止算法.

Let p(v;a;μ) be the distance from v∈Rk to hyperplane μTv+a = 0, where μ∈Rk and a∈R1. In this paper, the problem of finding some suitable a and μ which minimizeis discussed, where {v1,v2,…,vn} (?) Rk and qi > 0,i = 1,2,…,n, are given weights. Some substantive necessary conditions that such a and μ must satisfy are derived. The range of finding such a and μ can be reduced substantively to a finite set. A finite step method for solving this problem is suggested then.