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Local linear approach: conditional density estimate for functional and censored data

Benkhaled Abdelkader, Madani Fethi

Demonstratio Mathematica

2022

Vol. 55, nr 1

315--327

Let Y be a random real response, which is subject to right censoring by another random variable C . In this paper, we study the nonparametric local linear estimation of the conditional density of a scalar response variable and when the covariable takes values in a semi-metric space. Our main aim is to prove under some regularity conditions both the pointwise and the uniform almost-sure consistencies with convergence rates of the conditional density estimator related by this estimation procedure.

State estimators and observers for continuous and discrete linear systems. Part 1. Differential asymptotic state estimators

Byrski J., Byrski W.

Science, Technology and Innovation

2018

Vol. 3, no. 2

62--68

In the paper an overview of state estimators and state observers used in linear systems, will be presented. The state estimators and observers can be used in many applications like the state reconstruction for the control purposes or for the diagnosis and fault detection in technical processes or for the virtual measurements of inaccessible variables of the system as well as for the best filtration of the differential equation solution. As the standard most commonly the Kalman filter and Luenberger type observers are used. Although the Kalman filter guarantees optimal filtering quality of the state, reconstructed from the noisy measurements, both Kalman filter and the Luenberger observer guarantee only asymptotic quality of the real state changes and tracking, basing on the current measurements of the system output and input signals. Unfortunately, the value of the estimation error at any moment of time cannot be calculated. The discussion on differences between continuous and two types of discrete Kalman Filter will be presented. This paper is planned to be the introduction to presentation of another type of the state observers which have the structure given by the integral operators. Based on measurements of the system output and input signals on some predefined finite time interval, they can reconstruct, after this interval, the observed state exactly.

Macierz precyzji jako narzędzie konstrukcji estymatorów liniowych

Karwatka J.

Prace Przemysłowego Instytutu Telekomunikacji

2007

Vol. 57, z. 140

44-51

W pracy przedstawiono pojęcie macierzy precyzji będącej odwrotnością macierzy kowariancji oraz zaprezentowano modyfikacje równań filtru Kalmana pozwalającą wykorzystać pomiary, których rozkład błędu jest opisany osobliwą macierzą precyzji. Taka sytuacja zachodzi, kiedy liczba stopni swobody wyniku pomiaru jest mniejsza niż wymiar przestrzeni, w której dokonywany jest pomiar. Macierz precyzji pozwala w łatwy sposób opisać tego typu pomiary, ponadto wzory estymatorów z wykorzystaniem macierzy precyzji są dużo prostsze.

In the work is presented the concept of a precision matrix being the reverse of a covariance matrix and presented modifications of Kalman futer equations enabling the use of measurements, for which the distribution of error is described by the singular precision matrix. Such a situation occurs, when the number of degrees of freedom of measuring result is less than dimensional space, in which the measurement is made. The precision matrix easily enables the description of this type of measurement, in addition estimators' formulae using the precision matrix are much more simple.