Alternative to mean and least squares methods used in processing the results of scientific and technical experiments


  • Ignatkin Valery
  • Dudnikov Volodymyr
  • Luchyshyn Taras
  • Alekseenko Serhii
  • Yushkevich Oleh
  • Karpova Tetyana
  • Khokhlova Tetyana
  • Khomosh Yuriy
  • Tikhonov Vasyl



method of mean and least squares; empirical formulas; pseudo reversal; experimental data; approximating polynomials.


Increasing the complexity and size of systems of various nature requires constant improvement of modeling and verification of the obtained results by experiment. It is possible to clearly conduct each experiment, objectively evaluate the summaries of the researched process, and spread the material obtained in one study to a series of other studies only if they are correctly set up and processed. On the basis of experimental data, algebraic expressions are selected, which are called empirical formulas, which are used if the analytical expression of some function is complex or does not exist at this stage of the description of the object, system or phenomenon. When selecting empirical formulas, polynomials of the form: у = А0 + А1х+ А2х2+ А3х3+…+ Аnхn are widely used, which can be used to approximate any measurement results if they are expressed as continuous functions. It is especially valuable that even if the exact expression of the solution (polynomial) is unknown, it is possible to determine the value of the coefficients An using the methods of mean and least squares. But in the method of least squares, there is a shift in estimates when the noise in the data is increased, as it is affected by the noise of the previous stages of information processing. Therefore, for real-time information processing procedures, a pseudo-reverse operation is proposed, which is performed using recurrent formulas. This procedure is a procedure of successive updating (with a shift) along the columns of the matrix of given sizes and pseudo-reversal at each step of information change. This approach is straightforward and takes advantage of the bounding method. With pseudo-inversion, it is possible to control the correctness of calculations at each step, using Penrose conditions. The need for pseudo-inversion may arise during optimization, forecasting of certain parameters and characteristics of systems of various purposes, in various problems of linear algebra, statistics, presentation of the structure of the obtained solutions, to understand the content of the incorrectness of the resulting solution, in the sense of Adomar-Tikhonov, and to see the ways of regularization of such solutions.


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