On the Criterion of Proximity to the True Value: Information Approach

Ilya Feldman

doi:doi:10.11648/j.ajrs.20190701.11

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On the Criterion of Proximity to the True Value: Information Approach

Ilya Feldman

Published in American Journal of Remote Sensing (Volume 7, Issue 1)

Received: 25 March 2019 Accepted: 28 May 2019 Published: 9 July 2019

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Abstract

What is the criterion of proximity to the true value of the measured value: absolute or relative error? The least squares method traditionally operates with absolute values of corrections to measured values, and the equalization is carried out under the condition of the minimum of the sum of squares of absolute corrections. However, as shown in the article, the informational approach leads to the conclusion that the measure of proximity to the true value is a relative measurement error. Therefore, it is advisable to carry out an equalization under the condition of a minimum of the sum of squares of not absolute, but relative corrections. This is equivalent to equalization, in which the weight of the correction depends on the size of the object being measured: the larger the object being measured, the smaller the weight of the corresponding amendment, and its value can be increased during equalization. In this case, the described approach leads to a kind of “method of least relative squares” (MLRS). Another interesting consequence of the information approach is that the relative measurement error modulus has the meaning of the probability of a measurement result deviating from the true value. The article presents the required information approach formulas for the weights of the amendments when using the MLRS. In particular, it is shown that the angular discrepancy distribution in a triangle depends on the lengths of the sides.

Published in	American Journal of Remote Sensing (Volume 7, Issue 1)
DOI	10.11648/j.ajrs.20190701.11
Page(s)	1-4
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

The Criterion of Proximity, The True Value, Amount of Information, The Hartley-Shannon Formula, The Least Squares Method, The Weighting Factors to Amendments, The Distribution of Angular Discrepancy

References

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[2]	Legendre A. Nouvelles methodes pour la determination des orbites des kometes. (Legendre A. New methods for the determination of the orbits of kometes). – Paris, 1806.
[3]	Gauss C. F. Methode des moindres carres. – Traduits par J. Bertrand. (Gauss C. F. Method of the least squares. – Translated by J. Bertrand). – Paris, 1855.
[4]	Гаусс К. Ф. Избранные геодезические сочинения, т. 1. Способ наименьших квадратов. – Перевод с немецкого. Под редакцией Г. В.Багратуни. (Gauss K. F. Selected geodesic works, vol. 1. Method of least squares. – Translated from German. Edited by G. V. Bagratuni). – М: Геодезиздат, 1957.
[5]	Jordan W. Handbuch der Vermessungskunde. (Jordan W. Handbook of Surveying). – Stuttgart, 1920.
[6]	Иордан В. Руководство по геодезии. – Перевод с немецкого. (Jordan W. Surveying Guide. – Translation from German). – Редбюро ГУГК, 1939.
[7]	Helmert F. R.. Ausgleichungsrechnung nach der Methode der Kleinsten Quadrate. (Helmert F. R.. Equalization calculation according to the method of least squares). – Leipzig, 1872.
[8]	Шилов П. И. Способ наименьших квадратов. (Shilov P. I. Least squares method). – М: Геодезиздат, 1941.
[9]	Идельсон Н. И. Способ наименьших квадратов и теория математической обработки наблюдений. (Idelson N. I. Least squares method and theory of mathematical processing of observations). – М: Геодезиздат, 1947.
[10]	Ashby W. Ross. An Introduction to Cybernetics. – London, Chapman & Hall ltd, 1956.
[11]	Goldman St., Information Theory. – London, Constable and Company, 1953.
[12]	Маркузе Ю. И., Бойко Е. Г., Голубев В. В. Геодезия. Вычисление и уравнивание геодезических сетей. (Markuze Yu. I., Boyko E. G., Golubev V. V. Geodesy. Calculation and equalization of geodetic networks). – М.: Картгеоцентр – Геодезиздат, 1994.

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Ilya Feldman. (2019). On the Criterion of Proximity to the True Value: Information Approach. American Journal of Remote Sensing, 7(1), 1-4. https://doi.org/10.11648/j.ajrs.20190701.11

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Ilya Feldman. On the Criterion of Proximity to the True Value: Information Approach. Am. J. Remote Sens. 2019, 7(1), 1-4. doi: 10.11648/j.ajrs.20190701.11

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Ilya Feldman. On the Criterion of Proximity to the True Value: Information Approach. Am J Remote Sens. 2019;7(1):1-4. doi: 10.11648/j.ajrs.20190701.11

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@article{10.11648/j.ajrs.20190701.11,
  author = {Ilya Feldman},
  title = {On the Criterion of Proximity to the True Value: Information Approach},
  journal = {American Journal of Remote Sensing},
  volume = {7},
  number = {1},
  pages = {1-4},
  doi = {10.11648/j.ajrs.20190701.11},
  url = {https://doi.org/10.11648/j.ajrs.20190701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajrs.20190701.11},
  abstract = {What is the criterion of proximity to the true value of the measured value: absolute or relative error? The least squares method traditionally operates with absolute values of corrections to measured values, and the equalization is carried out under the condition of the minimum of the sum of squares of absolute corrections. However, as shown in the article, the informational approach leads to the conclusion that the measure of proximity to the true value is a relative measurement error. Therefore, it is advisable to carry out an equalization under the condition of a minimum of the sum of squares of not absolute, but relative corrections. This is equivalent to equalization, in which the weight of the correction depends on the size of the object being measured: the larger the object being measured, the smaller the weight of the corresponding amendment, and its value can be increased during equalization. In this case, the described approach leads to a kind of “method of least relative squares” (MLRS). Another interesting consequence of the information approach is that the relative measurement error modulus has the meaning of the probability of a measurement result deviating from the true value. The article presents the required information approach formulas for the weights of the amendments when using the MLRS. In particular, it is shown that the angular discrepancy distribution in a triangle depends on the lengths of the sides.},
 year = {2019}
}

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TY  - JOUR
T1  - On the Criterion of Proximity to the True Value: Information Approach
AU  - Ilya Feldman
Y1  - 2019/07/09
PY  - 2019
N1  - https://doi.org/10.11648/j.ajrs.20190701.11
DO  - 10.11648/j.ajrs.20190701.11
T2  - American Journal of Remote Sensing
JF  - American Journal of Remote Sensing
JO  - American Journal of Remote Sensing
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PB  - Science Publishing Group
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UR  - https://doi.org/10.11648/j.ajrs.20190701.11
AB  - What is the criterion of proximity to the true value of the measured value: absolute or relative error? The least squares method traditionally operates with absolute values of corrections to measured values, and the equalization is carried out under the condition of the minimum of the sum of squares of absolute corrections. However, as shown in the article, the informational approach leads to the conclusion that the measure of proximity to the true value is a relative measurement error. Therefore, it is advisable to carry out an equalization under the condition of a minimum of the sum of squares of not absolute, but relative corrections. This is equivalent to equalization, in which the weight of the correction depends on the size of the object being measured: the larger the object being measured, the smaller the weight of the corresponding amendment, and its value can be increased during equalization. In this case, the described approach leads to a kind of “method of least relative squares” (MLRS). Another interesting consequence of the information approach is that the relative measurement error modulus has the meaning of the probability of a measurement result deviating from the true value. The article presents the required information approach formulas for the weights of the amendments when using the MLRS. In particular, it is shown that the angular discrepancy distribution in a triangle depends on the lengths of the sides.
VL  - 7
IS  - 1
ER  -

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Author Information

Ilya Feldman

Moscow State University of Geodesy and Cartography, Moscow, Russia

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Plain Text BibTeX RIS

APA Style

Ilya Feldman. (2019). On the Criterion of Proximity to the True Value: Information Approach. American Journal of Remote Sensing, 7(1), 1-4. https://doi.org/10.11648/j.ajrs.20190701.11

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ACS Style

Ilya Feldman. On the Criterion of Proximity to the True Value: Information Approach. Am. J. Remote Sens. 2019, 7(1), 1-4. doi: 10.11648/j.ajrs.20190701.11

Copy | Download

AMA Style

Ilya Feldman. On the Criterion of Proximity to the True Value: Information Approach. Am J Remote Sens. 2019;7(1):1-4. doi: 10.11648/j.ajrs.20190701.11

Copy | Download

@article{10.11648/j.ajrs.20190701.11,
  author = {Ilya Feldman},
  title = {On the Criterion of Proximity to the True Value: Information Approach},
  journal = {American Journal of Remote Sensing},
  volume = {7},
  number = {1},
  pages = {1-4},
  doi = {10.11648/j.ajrs.20190701.11},
  url = {https://doi.org/10.11648/j.ajrs.20190701.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajrs.20190701.11},
  abstract = {What is the criterion of proximity to the true value of the measured value: absolute or relative error? The least squares method traditionally operates with absolute values of corrections to measured values, and the equalization is carried out under the condition of the minimum of the sum of squares of absolute corrections. However, as shown in the article, the informational approach leads to the conclusion that the measure of proximity to the true value is a relative measurement error. Therefore, it is advisable to carry out an equalization under the condition of a minimum of the sum of squares of not absolute, but relative corrections. This is equivalent to equalization, in which the weight of the correction depends on the size of the object being measured: the larger the object being measured, the smaller the weight of the corresponding amendment, and its value can be increased during equalization. In this case, the described approach leads to a kind of “method of least relative squares” (MLRS). Another interesting consequence of the information approach is that the relative measurement error modulus has the meaning of the probability of a measurement result deviating from the true value. The article presents the required information approach formulas for the weights of the amendments when using the MLRS. In particular, it is shown that the angular discrepancy distribution in a triangle depends on the lengths of the sides.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - On the Criterion of Proximity to the True Value: Information Approach
AU  - Ilya Feldman
Y1  - 2019/07/09
PY  - 2019
N1  - https://doi.org/10.11648/j.ajrs.20190701.11
DO  - 10.11648/j.ajrs.20190701.11
T2  - American Journal of Remote Sensing
JF  - American Journal of Remote Sensing
JO  - American Journal of Remote Sensing
SP  - 1
EP  - 4
PB  - Science Publishing Group
SN  - 2328-580X
UR  - https://doi.org/10.11648/j.ajrs.20190701.11
AB  - What is the criterion of proximity to the true value of the measured value: absolute or relative error? The least squares method traditionally operates with absolute values of corrections to measured values, and the equalization is carried out under the condition of the minimum of the sum of squares of absolute corrections. However, as shown in the article, the informational approach leads to the conclusion that the measure of proximity to the true value is a relative measurement error. Therefore, it is advisable to carry out an equalization under the condition of a minimum of the sum of squares of not absolute, but relative corrections. This is equivalent to equalization, in which the weight of the correction depends on the size of the object being measured: the larger the object being measured, the smaller the weight of the corresponding amendment, and its value can be increased during equalization. In this case, the described approach leads to a kind of “method of least relative squares” (MLRS). Another interesting consequence of the information approach is that the relative measurement error modulus has the meaning of the probability of a measurement result deviating from the true value. The article presents the required information approach formulas for the weights of the amendments when using the MLRS. In particular, it is shown that the angular discrepancy distribution in a triangle depends on the lengths of the sides.
VL  - 7
IS  - 1
ER  -

Copy | Download