Application of methods of the non-classical error theory in absolute measurements of galilean acceleration

Lviv Polytechnic National University
International University of Economics and Humanities named after Academician S. Demianchuk

The aim of this research is to develop a method for a posteriori control over the stability of observation conditions in modern high-precise absolute measurements of the acceleration of the Earth's gravity on the basis of methods of the non-classical error theory (NET). These measurements are carried out in a complicated metrological situation, which is continuously broken under the influence of various causes: trends of the frequency spectrum and energy of microseisms, geological, atmospheric, tidal, and other space factors, including uncontrolled effects of the place of observation and possible malfunctions in the work of the gravimeter. Through such control, it is necessary to obtain such distributions of observation errors that provide effective, or, at least, admissible arithmetic mean estimates of the Galilean acceleration. The methodology of achieving this aim is provided by the NET algorithms, which are created to control the form of empirical distributions of errors of high-precise multiple large-scale observations based on the principles of the Neumann-Pearson hypothesis testing theory. The basic result of the study is the development of a method for diagnosing the metrological situation, which was in the course of observing, on the basis of the NET methods. These methods are based on the use of the posteriori estimates of statistical cumulates in the form of empirical distributions of errors with the further application of the χ2-criterion to control the significance of its deviations from the established norms. In accordance with the principles of the NET, such norms are the laws of: Gauss or Pearson-Jeffreys, since they provide the non-singularity of the weight function of observations, and therefore the possibility of estimates, in the mathematical processing of observations. The scientific novelty of this investigation: the NET procedures are used for the first time to improve the current absolute high-precise observations of Galilean acceleration, which are performed in complicated metrological conditions, simultaneously taking into account the number of non-stationary sources of systemic errors. Practical significance of the study: the development of an algorithm for controlling the form of the empirical distribution of errors in order to improve the realization of high-precise ballistic measurements of Galilean acceleration based on the axiomatics of the NET. The study of the reasons for the deviation of distribution of errors from the normal law has long been a necessary element of the theory of production accuracy and control over the stability of the operation of various aggregates. The introduction of such approaches, started by Kolmogorov and his school, has long been at the heart of the strategy, which ensures metrological literacy of the measurement process and ways of improving their accuracy.

1. Bolshev L. N., Smirnov N. V. Tablitsy matematicheskoy statistiki [Tables of mathematical Statistics]. Moscow: Nauka, 1983, 416 p.
2. Borodachev N. A. Osovnyie voprosy teorii tochnosti proizvodstva. [The main questions of the theory of production accuracy]. Pod red. A. N. Kolmogorova. Moscow– Leningrad: Izd. AN SSSR. 1950, 360 p.
3. Dzhun I. V. Analiz parallelnyih shirotnyih nablyudeniy, vypolnennyh po obschey programme [Analysis of parallel latitudinal observations performed under the general program:]: avtoref. dis... na soiskanie uch. stepeni kand. fiz. – matem. nauk: spets. 01.03.01 «Astrometriya i nebesnaya mehanika» ["Astrometry and Celestial Mechanics"]. Kyiv: Institut matematiki AN USSR, 1974, 19 p.
4. Dzhun I. V., Arnautov G. P., Stus Yu. F., Scheglov S. N. Osobennost zakona raspredeleniya rezultatov ballisticheskih izmereniy uskoreniya silyi tyazhesti [A feature of the distribution law of the results of ballistic measurements of acceleration due to the gravity]. Povtornyie gravimetricheskie nablyudeniya [Repeated Gravimetric Observations]. Izd. MGK pri Prezidiume AN SSSR i NPO «Neftegeofizika». Moscow: 1984, pp. 87–100.
5. Dzhun I. V. Neklassichnaya teoriya pogreshnostey izmereniy [The nonclassical error theory of measurements]. Rovno: Estero, 2015, 168 p.
6. Kramer G. Matematichiskie metody statistiki [Mathematical methods of Statistics]. Moscow: Mir, 1975, 648 p.
7. Sheffe G. Dispersionnyi analiz. Per. s angl. [Dispersion analysis]. Moscow: Fizmatgiz, 1963, 628 p.
8. Bessel F. W. Fundamenta Astronomiae. Konigsberg: Nicolovius, 1818.
9. Bessel F. W., Baeyer J. J. Gradmessung in Ostpreussen und ihre Verbindung mit Preussischen und Russischen Dreiecksketten. Druckerei der Koniglichen Akademie der Weissenschaften. Berlin, 1838. Reprinted in part in: Abhandlungen von F. W. Bessel. Vol. 3. Ed. by R. Engelmann. Leipzig: W. Engelmann, 1876, pp. 62–138.
10. Dzhun I. V. Method for diagnostics of mathematical models in theoretical Astronomy and Astrometry. Kinematics and Physics of Celestial Bodies. New York: Allerton Press, Inc., 2011, Vol. 27, no. 5, pp. 260–264.
11. Dzhun I. V. What are the differences "Observation-Calculation" bound to be during modern experiments in Astrometry. Kinematics and Physics of Celestial Bodies. New York: Allerton Press, Inc., 2012, Vol. 28, no. 1. pp. 70–78.
12. Edgeworth F. Y. The Law of Errors. Proceeding of the Cambridge Philosophical Society. 1905, Vol. 20, no. 36.
13. Gauss C. F. Theoria motus corporum coelestium in sectionibus conicis Solem ambientium. Hamburgi: 1809.
14. Geary R. C. Distribution of Student's ratio for non-normal samples. Journal of the Royal Statistical Society, 1936. Suppl. 3.
15. Hampel F. R., Ronchetti E. M., Rousseeuw P. J., Stahel W. A. Robust Statistics. The approach based on influence functions. New York: John Wiley & Sons. 1986, 488 p.
16. Jeffreys H. The Law of Errors in the Greenwich Variation of Latitude Observations. Mon. Not. of the RAS, 1939, Vol. 99, no. 9, pp. 703– 709.
17. Jeffreys H. Theory of Probability. Sec. Eddition. Oxford: 1940, 468 p.
18. Lukacs E. A. A characterization of the normal distribution. Annals of Mathematical Statistics, 1942, Vol. 13, no. 91.
19. Newcomb S. Generalized theory of the combination of observations so as to obtain the best Result. Amer. J. Math. 1986, no.1/14, pp: 1–249.
20. Pearson K. On the mathematical theory of errors of judgment with special reference to the personal equation. Philosophical Transactions of the Royal Sosiety of London. Ser. A., 1902, Vol. 198, pp: 253–296.
21. Poincare H. Calcus des probabilities. Paris: 1912 (2 ed).
22. Romanowski M., Green E. Practical applications of the modified normal distribution. Bull. Geodesique, 1965, Vol. 76, pp. 1–20.
23. Romanowski M. The Theory of Random Errors based on the Concept of Modulated Normal Distributions. Ottawa: National Research Council of Canada (NRC-11432), Division Phys., 1970.
24. Stigler S. M. Contribution to the discussion of the meeting of Robust Statistics. In: Proceedings of the 40th Session of the ISI, Warsaw. Bull. Int. Statist. Inst., 1975, Vol. XLVI, book 1, pp. 383–384.
25. Student. Errors of Routine Analysis. Biometrika, 1927, Vol. 19, pp. 151–164.
26. Tukey J. W. A Survey of Sampling from Contaminated Distributions. In: Contributions to Probability and Statistics. Ed. by I. Olkin. Stanford: Stanford Univ. Press, 1960, pp. 448–485.
27. Tukey J. W. The future of data analysis. – Ann. Math. Statist., 1962, Vol. 33, pp. 1–67.
28. Tukey J. W. Data analysis and the frontiers of Geophysics. Science, 1965, Vol. 148. No. 3675, pp. 1283–1289.