State Space Modeling with Non-Negativity Constraints Using Quadratic Forms

• Published in: Mathematics (Basel) Vol. 9; no. 16; p. 1908
• Main Authors: Theodosiadou, Ourania, Tsaklidis, George
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.08.2021
• Subjects: Algorithms; constrained optimization; Constraint modelling; Evolution; Kalman filter; Kalman filters; Kuhn-Tucker method; Mathematics; Noise; Optimization; Quadratic forms; Random variables; state space model; State space models; State vectors; Stochastic processes; two-sided components
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math9161908

State space model representation is widely used for the estimation of nonobservable (hidden) random variables when noisy observations of the associated stochastic process are available. In case the state vector is subject to constraints, the standard Kalman filtering algorithm can no longer be used in the estimation procedure, since it assumes the linearity of the model. This kind of issue is considered in what follows for the case of hidden variables that have to be non-negative. This restriction, which is common in many real applications, can be faced by describing the dynamic system of the hidden variables through non-negative definite quadratic forms. Such a model could describe any process where a positive component represents “gain”, while the negative one represents “loss”; the observation is derived from the difference between the two components, which stands for the “surplus”. Here, a thorough analysis of the conditions that have to be satisfied regarding the existence of non-negative estimations of the hidden variables is presented via the use of the Karush–Kuhn–Tucker conditions.