Network Reconstruction from Intrinsic Noise: Non-Minimum-Phase Systems

This paper considers the problem of inferring the structure and dynamics of an unknown network driven by unknown noise inputs. Equivalently we seek to identify direct causal dependencies among manifest variables only from observations of these variables. We consider
linear, time-invariant systems of minimal order and with one noise source per manifest state. It
is known that if the transfer matrix from the inputs to manifest states is minimum phase, then
this problem has a unique solution, irrespective of the network topology. Here we consider the
general case where the transfer matrix may be non-minimum phase and show that solutions are
characterized by an Algebraic Riccati Equation (ARE). Each solution to the ARE corresponds to at most one spectral factor of the output spectral density that satisfies the assumptions made.
Hence in general the problem may not have a unique solution, but all solutions can be computed
by solving an ARE and their number may be finite.