Naslov
Problem diskretne $L_p$ aproksimacije u nekim specijalnim matematičkim modelima
(Discrete $L_p$ Approximation Problem In Some Special Mathematical Models)

Autori
Marošević, Tomislav

Vrsta, podvrsta i kategorija rada
Ocjenski radovi, doktorska disertacija

Fakultet
PMF - Matematički odjel

Mjesto
Zagreb

Datum
14.07

Godina
1998

Stranica
81

Mentor
Scitovski, Rudolf

Ključne riječi
diskretna $L_p$ aproksimacija; matematički model; procjena parametara; problem egzistencije
(discrete $L_p$ approximation; mathematical model; parameter estimation; existence problem)

Sažetak
We consider the best discrete approximation problem in the sense of the least $L_p$-norm ($1leq p< infty$) in some special mathematical models. We look at the estimation problem of the unknown parameters of special model-functions on the basis of the given finite data points set, which can have weights. As a criterion for the optimality of parameters, one uses the least deviations in the sense of the ordinary $L_p$-norm ($1leq p< infty$), if only the measured values of the dependent variable contain errors. In the total approximation problem, when unknown additive errors occur in the measured values of both the dependent and the independent variable, the criterion of the least total $L_p$-norm ($1leq p < infty$) is used. In practice, the most usually used $L_p$-norm is for $p=2$ (least squares), but in many situations the use of some other $L_p$-norm is reasonable. For example, if the data can contain outlying points (``outliers''), then one should use the $L_1$-norm because of its robustness property. The main problem considered in this work is the existence of optimal parameters of some special model-functions. We prove theorems on the existence of optimal parameters in the sense of the least (weighted) ordinary $L_p$-norm ($1leq p < infty$) for an exponential model-function and for a generalized logistic model-function, under assumption that the data are either positive or negative. As corollaries, analogous results on the existence for both the exponential model-function and the generalized logistic model-function with in advancegiven asymptotes are stated. In addition, we prove the existence of optimal parameters in the sense of the least (weighted) total $L_p$-norm ($1leq p <infty$) for the affine model-function, for the exponential model-function and for the generalized logistic model-function, under the monotonicity condition on the data. Also, analogous corollaries on the existence for the exponential model-function and the generalized logistic model-function with in advance known asymptotes are stated. At the end, for the purpose of illustration of necessity and possibility for using the $L_p$-norm for various values $p$, we briefly describe some methods for the estimation of optimal parameters in the $L_p$-norm. We also consider a smoothing-the-data method as a moving $L_p$ method for various values of $p$. The usage of this method is illustrated with solving a parameter identification problem in a model described by a differential equation.

Izvorni jezik
Hrvatski

Znanstvena područja
Matematika

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