Resonating Vector Strength: How to Find Periodicity in a Time Sequence

For a given periodic stimulus with angular frequency $\omega_{\circ} = 2\pi/T_{\circ}$ we find responses as events at times $\{t_{1}, t_{2},\ldots, t_{n} \}$ located on the real axis $R$. How periodic are they? And do they repeat in ``some'' sense in accordance with the stimulus period $T_{\circ}$? The question and the answer are at least as old as a classical paper of von Mises dating back to 1918. The key idea is simply this. We map the events $t_{j}$ onto the unit circle or torus through $t_{j} \mapsto \exp (i \omega t_{j})$ and consider their center of gravity, $\rho(\omega)$, a complex number in the unit disk. Its absolute value $|\rho(\omega_{\circ})|$ with $\omega := \omega_{\circ}$ is what von Mises studied and is now called the vector strength. We prove that the nearer $|\rho(\omega_{\circ})|$ is to $1$ the more periodic the events $t_{j}$ are w.r.t. $T_{\circ}$. Furthermore, we also show why it is useful to study $\rho(\omega)$ as a function of $\omega$ so as to obtain a `resonating' vector strength, an idea strongly deviating from the classical characteristic function.