Table 1 Description of M-estimation robust regression.
Methods | Objective function | Weight function |
|---|---|---|
Bi-Square | \(\rho_{BS} = \left\{ {\begin{array}{*{20}c} {\frac{{k^{2} }}{6}\left\{ {1 - \left[ {1 - \left( \frac{e}{k} \right)^{2} } \right]^{3} } \right\} for \left| e \right| \le k} \\ {\frac{{k^{2} }}{6} for \left| e \right| > k} \\ \end{array} } \right.\) | \(w_{BS} = \left\{ {\begin{array}{*{20}c} {\left[ {1 - \left( \frac{e}{k} \right)^{2} } \right]^{2} for \left| e \right| \le k} \\ { 0 for\left| e \right| > k} \\ \end{array} } \right.\) |
Huber | \(\rho_{Huber} = \left\{ {\begin{array}{*{20}c} { \frac{1}{2}e^{2} for \left| e \right| \le k} \\ {k\left| e \right| - \frac{1}{2}k^{2} for \left| e \right| > k} \\ \end{array} } \right.\) | \(w_{Hub} = \left\{ {\begin{array}{*{20}c} {1 for \left| e \right| \le k} \\ {\frac{k}{\left| e \right|} for \left| e \right| < k} \\ \end{array} } \right.\) |
Hampel | \(\rho_{Ham} = \left\{ {\begin{array}{*{20}c} {\frac{{e^{2} }}{2} , 0 < \left| e \right| < a} \\ {a\left| e \right| - \frac{{e^{2} }}{2} , b < \left| e \right| \le c } \\ {\frac{ - a}{{2\left( {c - b} \right)}}\left( {c - e} \right)^{2} + \frac{a}{2}\left( {b + c - a} \right) , b < \left| e \right| \le c} \\ \end{array} } \right.\) | \(w_{Ham} = \left\{ {\begin{array}{*{20}c} {1 for 0 < \left| e \right| < a } \\ {\frac{a}{\left| e \right|} for b < \left| e \right| \le c} \\ {a\frac{{\frac{c}{\left| e \right|} - 1}}{c - b} for b < \left| e \right| \le c } \\ \end{array} } \right.\) |
