We study the Hamiltonian dynamics of a many-body quantum system subjected to periodic projective measurements, which leads to probabilistic cellular automata dynamics. Given a sequence of measured values, we characterize their dynamics by performing a principal component analysis (PCA). The number of principal components required for an almost complete description of the system, which is a measure of complexity we refer to as PCA complexity, is studied as a function of the Hamiltonian parameters and measurement intervals. We consider different Hamiltonians that describe interacting, noninteracting, integrable, and nonintegrable systems, including random local Hamiltonians and translational invariant random local Hamiltonians. In all these scenarios, we find that the PCA complexity grows rapidly in time before approaching a plateau. The dynamics of the PCA complexity can vary quantitatively and qualitatively as a function of the Hamiltonian parameters and measurement protocol. Importantly, the dynamics of PCA complexity present behavior that is considerably less sensitive to the specific system parameters for models which lack simple local dynamics, as is often the case in nonintegrable models. In particular, we point out a figure of merit that considers the local dynamics and the measurement direction to predict the sensitivity of the PCA complexity dynamics to the system parameters.
