论文标题
最小二乘的变化
Variations on least squares
论文作者
论文摘要
检查了最小二乘的三种方法，以将线拟合到平面中的点。两种众所周知的方法是最大程度地减少垂直或水平距离的正方形总和。鲜为人知的是最大程度地减少线之间距离的正方形总和。使用第一个导数测试的组合为两个变量的功能和完成正方形的函数提供了简洁的证明。比较了这三种方法，在大多数情况下，对线方法的距离似乎很有利。他们通常绘制不同的回归线。垂直位移的方法通常会产生太小的斜率，而水平位移方法通常会产生太大的斜率。证明了涉及三个斜坡的不平等。用这两种方法以相同的方式旋转所有数据点不会导致回归线以相同的方式旋转。但是，在旋转下，与线方法的距离是不变的。
Three methods of least squares are examined for fitting a line to points in the plane. Two well known methods are to minimize sums of squares of vertical or horizontal distances to the line. Less known is to minimize sums of squares of distances to the line. Concise proofs are given for each method using a combination of the first derivative test for functions of two variables and completing the square. The three methods are compared and the distances to the line method appears to be favourable in most circumstances. They generally draw different regression lines. The method of vertical displacements typically gives a slope of too small magnitude while the method of horizontal displacements typically gives a slope of too large magnitude. An inequality involving the three slopes is proved. Rotating all the data points in the same way with these two methods does not result in the regression line being rotated the same way. However, the distance to the line method is invariant under rotations.
