Parametric identification of quasilinear difference equation

A. V. Panyukov ; Ya. A. Mezaal

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 11 (2019) no. 4, pp. 32-38

Résumé

Identification of the quasilinear difference equation is reduced to the problem of regression analysis with mutually dependent observable variables. This makes the classical solution schemes, based on the least squares method and its variations, ineffective. Finding estimates of the autoregressive equation coefficients is significantly complicated by poor conditionality of the system of equations, which represent necessary conditions for the minimum sum of squared deviations. In this case, estimates of the problem parameters are untenable. For solving such problems, it is possible to use the generalized least absolute deviations (GLAD) method, reduced to problems sequence of estimates of the autoregressive equation coefficients with the weighted least absolute deviations (WLAD) method. The article proposes an algorithm for solving the problem of WLAD-estimation, based on its conversion to the problem of linear programming (LP) of simple structure. The simplicity of the admissible set of the LP problem structure lies in the intersection of a linear subspace with a parallelepiped. It allows to propose an effective algorithm for its solution based on the gradient projection method. The algebraic computational complexity of the proposed algorithm does not exceed the value $O(N^2M^2)$, where $N$ is the number of coefficients in the equation under study, and $M$ is the number of the observed values. This WLAD computational complexity estimate is considered the best among the known ones.

Détail

Keywords: least absolute deviations, autoregressive model, linear programming, gradient projection method, computational complexity.