Approximation of nonnegative systems by moving averages of fixed order

TL;DR

This paper addresses the approximation of scalar nonnegative systems using finite order moving averages, formulating it as a structured nonnegative matrix factorization problem solved via an iterative algorithm, with proven convergence and statistical properties.

Contribution

It introduces a novel approach to approximate nonnegative systems through structured matrix factorization and develops an efficient alternating minimization algorithm with theoretical guarantees.

Findings

Algorithm converges quickly in numerical experiments.

Theoretical conditions ensure existence and uniqueness of solutions.

Asymptotic analysis confirms statistical properties under noisy data.

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.