  The inverse problem is studied in multi-body systems with nonlinear dynamics
representing, e.g., phase-locked wave systems, standard multimode and random
lasers. Using a general model for four-body interacting complex-valued
variables we test two methods based on pseudolikelihood, respectively with
regularization and with decimation, to determine the coupling constants from
sets of measured configurations. We test statistical inference predictions for
increasing number of sampled configurations and for an externally tunable {\em
temperature}-like parameter mimicing real data noise and helping minimization
procedures. Analyzed models with phasors and rotors are generalizations of
problems of real-valued spherical problems (e.g., density fluctuations),
discrete spins (Ising and vectorial Potts) or finite number of states (standard
Potts): inference methods presented here can, then, be straightforward applied
to a large class of inverse problems.
