## 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.

• Each observaction $\in Z_n = \{(x_i, y_i), i = 1,...,n \} \subset \rm I\!R^2$ is tranformed to a line.
• The candidate fit is transformed to a point.

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.