TARU PUBLICATIONS
Journal of Statistics and Management Systems cover

Journal of Statistics and Management Systems

(Since 1998)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0014·ISSN (Print): 0972-0510

Indexed in ABDC Australia

Powered by:Powered by

Monthly Journal: Publishes peer-reviewed aticles on theoretical and applied statistics and management systems, expoloring industrial statistics, actuarial and decision sciences.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Taru Online First Current IssueSpecial IssuesBrowse all Issues
Journal of Statistics and Management Systems
Open Access Research Article

An instance of negative variance-like ‘components’ in optics by way of applying multivariable calculus to a statistical problem

*

* Corresponding author · click or hover a name for details

pp. 579–587Vol. 27Issue 3March 2024DOI: 10.47974/JSMS-976XML
Received:
09 Feb 2022
Accepted:
12 Jul 2022
Published Online:
30 Mar 2024
Article type:
Research Article
Language:
EN
Article no.:
JSMS-976
Pages:
579–587

Abstract

In the field of optics, the design of a lens often involves analyses of the variability in an optical parameter (such as lens focal length) under manufacturing conditions, which is done in order to identify sources of high variability that may cause the optical parameter to be out of its specification. In this interdisciplinary paper, the author shares his experience in lens design, where analyses of variability are often performed through examining the partial derivatives in a multivariable Taylor series expansion of an optical parameter to second order when interactions are expected. It is shown algebraically that when the second-order expansion is applied to express a sum of squares, negative terms in the sum can occur. This instance of negative-valued “variance-like components” is interesting pedagogically in applied mathematics and statistics, as well as from an engineering standpoint, as it suggests that, based on theory, the variance of a function of random variables may not necessarily be comprised of a simple sum of positive-valued squared deviations comprising its total variance.

Keywords

Analysis of variance and covarianceMultivariable systemsFunctions of several variablesReal-analytic functionsSeries expansionsGeometric optics

Subject Classifications

(2010) 62J10 (Analysis of variance and covariance)93C35 (Multivariable systems)26B99 (Functions of several variables)26E05 (Real-analytic functions)41A58 (Series expansions)78A051 (Geometric optics)

References

[1] C.M. Creveling, Tolerance Design: A Handbook for Developing Optimal Specifications, Addison Wesley Longman, 1997.
[2] R. Siew, Monte Carlo Simulation and Analysis in Modern Optical Tolerancing, SPIE Press, 2019.
[3] J. Sasian, Introduction to Lens Design, Cambridge University Press, 2019, pp. 110-125.
[4] R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed., Elsevier, 2010, pp. 6-7.
[5] W. J. Smith, Modern Lens Design, 2nd ed., New York, NY: McGraw-Hill (2005), pp. 573-583.
[6] E. Hecht, Optics, 4th ed., Addison Wesley, 2002, p. 246.
[7] H. Sahai and M. I. Ageel, The Analysis of Variance: Fixed, Random and Mixed Models, Birkhauser Boston, 2000, pp. 177-190.
[8] D. G. Koch, A statistical approach to lens tolerancing, Proceedings of SPIE 0147 (1978) 71-82.
[9] R. N. Youngworth and B. D. Stone, Cost-based tolerancing of optical systems, Applied Optics 29 (25) (2000) 4501-4512.
[10] R. N. Youngworth, Statistical truths of tolerance assignment in optical design, Proceedings of SPIE 8131 (2011) 81310E.
[11] J. Stewart, Multivariable Calculus, 5th ed., Thomson Learning, 2003, p. 1000.
[12] R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed., Elsevier, 2010, p. 70.
[13] See Ref. 12, pp. 388-397.
[14] W. J. Smith, (2005), Modern Lens Design, 2nd ed., McGraw-Hill, 2005, pp. 355-393.
[15] D. C. Montgomery and G. C. Runger, Applied Statistics and Probability for Engineers, 4th ed., John Wiley & Sons, 2007, p. 189.

Views: 394Downloads: 65Citations: 0