Using of functional of the quasi-extent in the problems of approximation of distorted data

  • Serhii Vovk

Abstract

The quasi-extent functional is intended to process the data distorted by noise and anomalous values. To solve the approximation problem of distorted data described by a parametric model it is advisable to build this functional on the basis of a residual function. The goal of this paper is to formulate the recommendations to effective use of the quasi-extent functional for approximation of distorted data.
The effectiveness of using the various functionals in the data approximation problem is determined, first of all, by the presence of such local minima that correspond to the true values of unknown model parameters closely. In this paper, it is noted that the quasi-extent functional of solution residual contains such minima both for the case of a linear parameter of data model and for the case of a non-linear parameter of data model. In particular, by setting the corresponding values of free parameters, the quasi-extent functional can be tuned to a quasinorm of the space Lp; 0 <p <1. While the noise is absence, the local minima of this quasi-extent functional will exactly correspond to those values of unknown data model parameter, when using which the residual function strictly equals to zero on a certain argument interval, i.e. when at this interval the model coincides with the data exactly. However, in the presence of moderate noise, such minima are smoothed out, leading to a decrease in their depth and broadening. Moreover, in the presence of high noise, the neighboring local minima that correspond to the different values of the sought parameter can either merge into one minimum (for the case of linear parameter), or become indistinguishable against the general background of values (for the case of nonlinear parameter). Taking the features of the quasi-extent functional into account, the recommendations about its effective use to solve the data approximation problem are formulated. The effectiveness of using the quasi-extent functional to solve the data approximation problem is due to the possibility of tuning of approximation process to the current noise environment and the possibility of approximating of distorted data under condition when unknown model parameters take several values.

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Published
2020-05-04