Projection-based Bayesian recursive estimation of ARX model with uniform innovations

Recursive Bayesian estimation of autoregressive model with uniform noise using approximation by parallelotopes

“…The noise half-width r, Figure 3, with regression coefficients either constant (left part, no forgetting) or varying (right, forgetting), is systematically overestimated. The reason may be in the special structure of the matrix V last row, see (21), and nonuniform marginal pdf of the half-width r (or Θ Ψ , see, eg, (11)). For other systems, it is possible that r will be underestimated, but this property was not methodically tested.…”

“…In other words, the scaling factor W Ψ Ψ is maximised to minimise the range of Θ Ψ . For a general matrix and the circumscription algorithm as described in the previous study by Vicino and Zappa, 9 minima of Kullback-Leibler divergence and the parallelotope volume would differ because of the powers in the approximate posterior pdf (11). Hence, minimisation of the Kullback-Leibler divergence represented by the term ln J(V t , L t , U t , t ) is equivalent to minimisation of the volume of a circumscribing parallelotope, with respect to the class of pdfs defined in (11), for the special form of the matrix V (21) and the data update algorithm.…”

“…The way to express the given admissible set of the parameter Θ in (11) in the [L, U] form is not unique. The way to express the given admissible set of the parameter Θ in (11) in the [L, U] form is not unique.…”

“…Therefore, when new data come (13), the last row of W is only tightened around the new polytope and then kept untouched. Hence, minimisation of the Kullback-Leibler divergence represented by the term ln J(V t , L t , U t , t ) is equivalent to minimisation of the volume of a circumscribing parallelotope, with respect to the class of pdfs defined in (11), for the special form of the matrix V (21) and the data update algorithm. The bounds and their powers U +1 Ψ − L +1 Ψ are given by the data history.…”

“…Second, it is shown that a minimum volume parallelotope circumscribing the polytope can be obtained by selecting such rows from + 1 candidates into the matrix W t that minimise | det W −1 t |. We modified the algorithm 9 to keep the matrix V t (equivalently W t ) in a special form (see (11)). After the tightening, the last row is always selected for the new parallelotope, together with those from remaining rows to minimise the volume.…”