We study the problem of completing partially known state statistics of complex dynamical systems using low-complexity linearized models. State statistics of linear systems satisfy certain structural constraints that arise from the underlying dynamics and the directionality of input disturbances. The dynamical interaction between state variables is known while the directionality of input excitation is uncertain. Thus, the goal of the inverse problem that we formulate is to identify the dynamics and directionality of input excitation so as to explain the observed sample statistics. In particular, we seek to explain the data with the least number of possible input disturbance channels. This can be formulated as a rank minimization problem, and for its solution, we employ a convex relaxation based on the nuclear norm. The resulting optimization problem can be cast as a semidefinite program and solved efficiently using general-purpose solvers for small- and medium-size problems. We develop a customized alternating minimization algorithm (AMA) to solve the problem for large-scale systems. We demonstrate that AMA works as a proximal gradient for the dual problem and provide examples to illustrate that identified colored-in-time stochastic disturbances represent an effective means for explaining available second-order state statistics.