# Ordinal Regression

Hail should support ordinal regression. Ordinal regression deals with a dependent variable whose values are finite, discrete and ordered. This differs from a categorical variable where each value can be represented by a binary variable. As I understand it, ordinal regression of a dependent variable y is phrased in two steps: mapping from the independent variable x to a threshold-space, then mapping from threshold-space to the dependent variable y. For example:

\theta = x * w + \varepsilon \\ y = \begin{cases} y_1 & \theta < \theta_1 \\ y_2 & \theta_1 \leq \theta \lt \theta_2 \\ y_3 & \theta_2 \leq \theta \end{cases}

The choices of \theta_i and w are fit according to some loss function. I do not fully understand the power of the intermediate threshold space. At the very least, it involves projecting the x vector onto a line which is then partitioned.