Inferring the Size of Stochastic Systems from Partial Measurements

Inferring the size of a complex system from partial measurements of some of its units is a common problem in engineering, with significant applications in the field of structural health monitoring (SHM), where one may attempt at relating system size (number of degrees of freedom) to the integrity of the structure. Here, we demonstrate the possibility of inferring the size of a stochastic system by assembling measurements of its response into a detection matrix. In deterministic systems, the rank of the detection matrix (number of non-zero singular values) equals the size of the largest observable system component. We extend this framework to reconstruct the number of states of an unknown Markov chain, where we cannot distinguish between two or more states. In this case, we only have access to an estimate of the detection matrix, but with a larger rank, since stochasticity generates a series of non-zero singular values. We establish conditions for the correct inference of system size, relating the number of realizations and the smallest true singular value. Our work highlights connections between SHM, system identification, and control theory, paving the way for new cross-disciplinary inquiries.