PROBLEM STATEMENTS OF DATA PROCESSING BASED ON CRITERION OF MINIMUM EXTENT
DOI:
https://doi.org/10.15588/1607-3274-2019-1-15Keywords:
processing, criterion, extent, approximation, inverse problem.Abstract
Context. In order to process the data containing anomalous values as well as to obtain the sparse solutions or solutions with small extent, the requirement to minimize the extent of the function used to find solution can be used. In this paper the object of the study is the process of setting the data processing problems on the basis of this requirement, which is further referred to as the criterion of minimum extent.
Objective. The goal of this work is the development of an approach to the formulation of the data processing problems based on criterion of minimum extent.
Method. On the basis of minimum extent criterion, a new approach is proposed. This approach allows to formulate the data approximation problem as well as the inverse problem with a direct linear operator and with a solution of small extent or with a sparse solution in conditions that the initial data contain noise and anomalous values. The statement of the approximation problem is obtained
by setting a parametric data model and applying the criterion of minimum extent to the solution residual. The statement of the inverse problem is obtained by applying the criterion of minimum extent to the solution of the problem and to the solution residual. The special cases of this statement are presented and it is noted that this statement generalizes the statement of the Tikhonov regularization
problem. The proposed problem statements are formulated as minimization problems for the corresponding functionals constructed on the basis of the “superset” of cost functions. In the general case, the indicated functionals are neither convex nor unimodal, and their minimization can be a laborious task.
Results. The proposed problem statements generalize those that are performed on the basis of the least squares criterion and/or least modules criterion. Numerical simulation of the problem of approximation by a linear function of noisy data in the presence of impulsive noise, as well as in the presence of an interfering fragment of exponential function, confirmed the feasibility of the proposed statement and its effectiveness. Numerical simulation of the inverse problem, corresponded to the overdetermined system of linear algebraic equations with gross errors in its right-hand side and sparse solution, also confirmed the feasibility of using the criterion
of minimum extent for its formulation.
Conclusions. The problem statement of data processing which is based on the criterion of minimum extent is expedient under conditions when the part of the data is roughly distorted and/or when the desired solution has a small extent. The statements based on the criterion of minimum extent allow us to expand the range of the problems to be solved.
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