  We present a way to visualize and quantify renormalization group flows in a
space of observables computed using Monte Carlo simulations. We apply the
method to classical three-dimensional clock models, i.e., the planar (XY) spin
model perturbed by a $Z_q$ symmetric anisotropy field. The method performs
significantly better than standard techniques for determining the scaling
dimension $y_q$ of the $Z_q$ field at the critical point if it is irrelevant
($q\ge4$). Furthermore, we analyze all stages of the complex renormalization
flow, including the cross-over from the U(1) Nambu-Goldstone fixed point to the
ultimate $Z_q$ symmetry-breaking fixed point due to the relevance of the $Z_q$
field inside the ordered phase. We expect our method to be particularly useful
in the context of quantum-critical points with inherent dangerously irrelevant
operators that cannot be tuned away microscopically but whose renormalization
flows can be analyzed exactly as we do here for the clock models.
