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The look-up algorithm of monitoring an object described by non-linear ordinary differential equations

Hawro Przemysław, Kwater Tadeusz, Bartman Jacek, Kwiatkowski Bogdan

Bulletin of the Polish Academy of Sciences. Technical Sciences

2023

Vol. 71, nr 2

art. no. e144603

The article proposes an adaptive algorithm that generates all object signals, including those for which measurements are not performed due to the difficulties associated with on-line measurements. The algorithm is modeled on the idea of the Kalman filter using its equation, however, the selection of gains is optimized in a different way, i.e. the constant values depend on the adopted ranges of adaptation errors. Moreover, the knowledge of the statistics of all noise signals is not imposed and there is no linearity constraint. This approach allowed to reduce the complexity of calculations. This algorithm can be used in real-time systems to generate signals of objects described by non-linear differential equations and it is universal, which allows it to be used for various objects. In the conducted research, on the example of a biochemically contaminated river, only easily measurable signals were used to generated the object signals, and in addition, in the case of absence some measurements, the functioning of the algorithm did not destabilize.

Rüschendorf, Adaptive estimation of hazard functions

Döhler S., Rüschendorf L.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

355--379

In this paper we obtain convergence rates for sieved maximum-likelihood estimators of the log-hazard function in a censoring model. We also establish convergence results for an adaptive version of the estimator based on the method of structural risk-minimization. Applications are discussed to tensor product spline estimators as well as to neural net and radial basis function sieves. We obtain simplified bounds in comparison to the known literature. This allows us to derive several new classes of estimators and to obtain improved estimation rates. Our results extend to a more general class of estimation problems and estimation methods (minimum contrast estimators).