Abstract
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.
Highlights
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
When dealing with state space models that are subject to constraints, the Kalman filtering algorithm [12] can no longer be used, since it assumes linearity in the model
Model (2) and (3) adopts the use of non-negative definite quadratic forms to describe the dynamic evolution of the hidden two-sided components; that is, to ensure that the estimations of the components will be non-negative
Summary
State space modeling is used for estimating—revealing the dynamic evolution of hidden variables’ processes. The state vector is subject to non-negative constraints that have to be taken into account for its estimation in time Such a model could describe, for example, the evolution of a system where the positive component represents “gain”, while the negative one represents “loss”; the observation is derives from the difference between the two components, which stands for the “surplus”, under noise inclusion. An asset return can be defined as the difference between the two-sided nonnegative return jump components under noise inclusion, and the jump components are considered to be hidden variables Another example could be the one-dimensional random walk, where a positive jump could represent (the measure of) a move to the right and a negative jump (the measure of) a move to the left, while the observation could be a function of the two jump components given at discrete times.
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