Suppose I have a set of $N$ observations for a circular variable $\theta_i$. Each trial for which an observation of $\theta_i$ is taken can lead to an outcome $x_i \in \{0,1\}$. Next, suppose that the set of trials for which $x_i = 1$ has a preferred direction, such that the mean resultant length $r$:
$r = \frac{1}{N} \sum_i^{N} e^{i\phi_i}$
is nonzero ($0.1 < r < 0.2$, to be concrete). The set of trials for which $x_i = 0$ is uniform.
Now, with the observations $\theta_i$, the outcomes $x_i$, and the mean resultant length $r$, intuitively it seems possible to predict the outcome for a new observation, either through a classifier, some closed-form expression in terms of $r$, or a parametric fitting approach. After searching, however, a reference clearly addressing this problem appears difficult to find. Could anyone suggest a potential start for thinking about this problem?