Alternatives to Least Squares Regression

Consider the following data set, with the least squares regression line:

x
 [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
y
 [1] 2.06 2.12 2.32 2.02 2.76 3.04 2.83 3.15 3.36 3.68 3.96

Problem 1: Orthogonal Regression

In a typical least squares regression we define the distance between a data point and the line as the horizontal line (red), and then we minimize the sum of the squared distances. However, in geometry one defines the distance of a point and a line via the perpendicular line (blue). For the data set below find the best “least squares perpendicular line”.

Problem 2: Least Absolute Values Regression

Here one uses the criterion

\[\text{RSS}=\sum_{i=1}^n \vert y_i-a-bx_i\vert\]

Again find the corresponding line for the data set above. Make an argument in favour of this solution.