Approximation of nonnegative systems by moving averages of fixed order

Abstract

We pose the approximation problem for scalar nonnegative input-output systems via impulse response convolutions of finite order, i.e. finite order moving averages, 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 Csisz\'ar's I-divergence as 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 present its asymptotic behavior. For the case of noisy observations we give the large sample properties of the statistical version of the minimization problem for different observation regimes. Numerical experiments confirm the asymptotic results and exhibit fast convergence of the proposed algorithm.