Duality of regression depth

As explained in the first chapter, each plot composed of lines and points can be transformed via a duality transformation. If we apply the $D$ tranformation previously defined to our regression depth problem we emerge with a dual problem.

An example is illustrated in the two following pictures.


Before
After

The regression depth in the dual plot

In the dual space, the regression depth is defined as follow.


Definition: regression depth in the dual space
The regression depth $rdepth(\eta, L_n)$ of any candidate fit $\eta$ relative to a set of $n$ lines $L_n$ is the minimal number of lines from $L_n$ that should be removed so that $\eta$ becomes a non-fit.
Definition: non-fit the dual space
In the dual space, a candidate fit is a non-fit if there exists a direction $u$ (such that $\mid\mid u \mid\mid = 1$) and such that the halfline $[\eta, \eta + u >$ does not intersect any line from the set $L_n$.

We can say in a more intuitive way that a point $\eta$ is a non-fit if it can escape to infinity without crossing any line of the arrangement.


Example

We present a application similar to the one from the previous chapter but one can check the dual plot and the regression depth of any fit in the dual space.