Approximation of Nonnegative Systems by Finite Impulse Response Convolutions

We pose the deterministic, nonparametric, approximation problem for scalar nonnegative input/output systems via finite impulse response convolutions, based on repeated observations of input/output signal pairs. The problem is converted into a nonnegative matrix factorization with special structure for which we use Csiszár's I-divergenceas the criterion of optimality. Conditions are given, on the input/output data, that guarantee the existence and uniqueness of the minimum. We propose an algorithm of the alternating minimization type for I-divergence minimization, and study its asymptotic behavior. For the case of noisy observations, we give the large sample properties of the statistical version of the minimization problem. Numerical experiments confirm the asymptotic results and exhibit the fast convergence of the proposed algorithm.