最小一乘线性回归模型研究

A Linear Regress Model Based on Least-Absolute Criteria

用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归纳法得到如下定理:设有n(n>P)个样本点(x1i, x2i, ? xP i, yi,),则由最小一乘准则确定的线性非奇次模型y=b1x1+b2x2+?bPxP+a经过其P+1个样本点,而相应的奇次模型必经过其P个样本点。通过大量工程实例证实了最小一乘具有较强的稳健性,同时也证实了定理的正确性。

Five theorems about the properties of the linear regress model based on least-absolute criteria (LRMLAC) are proposed. These theorems lay a foundation for LRMLAC抯 engineering application. At first, we prove the following the properties:搕he LRMLAC y=b1x1+ b2x2 must pass through two sample points?and 搕he LRMLAC y=b1x1+ b2x2+a must pass through three sample points? Then Applying the mathematic induction method the main properties of LALRM, which are “the LRMLAC y=b1x1+b2x2+?bPxP+a must pass through the P+1 sample points” and “the LRMLAC y=b1x1+b2x2+?bPxP must pass through the P sample points”, are proposed. From some engineering examples, the robustness of the LRMLAC and the correctness of these theorems are verified.