递推最小二乘算法的补充性证明

The Complemental Proof of Recursive Least Square Algorithm

在使用递推最小二乘算法时,通常考虑的情况是训练样本所构成的方程组为矛盾方程组时该算法的收敛情况。本研究对递推最小二乘算法进行了理论证明及分析,指出了在任意第k步,未知参数估计值收敛于前k组数据的极小范数解(如果前k组数据所组成方程组为相容方程组)或者极小范数最小二乘解(如果前k组数据所组成方程组为矛盾方程组),并且此解是唯一的;仿真结果同样也验证了该结论的正确性。

When using Recursive Least Square Algorithm, it is merely considered the case that the equations composed by the preceding k sample data are contradictory equations. By further theoretical proof and analysis, it is concluded that at any k th step, the estimated values of the unknowns converge to the minimum norm solution if the equations composed by the preceding k sample data are compatible equations or minimum norm least square solution in case that they are contradictory equations,Moreover, the solution is unique. The simulations also testify the validity of this conclusion.