Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions

Kiplagat Nicholas Kipkosgei; Mutiso John Muindi; Rambaei Keny Silver Jeptoo

doi:doi:10.11648/j.ajtas.20200906.16

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Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions

Kiplagat Nicholas Kipkosgei, Mutiso John Muindi, Rambaei Keny Silver Jeptoo

Published in American Journal of Theoretical and Applied Statistics (Volume 9, Issue 6)

Received: 10 June 2020 Accepted: 7 July 2020 Published: 22 December 2020

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Abstract

This study uses the existing second order rotatable design to obtain optimum design based on the known classical optimality criteria that is the determinant criterion, the average-variance criterion, the smallest-Eigen value criterion and the trace criterion. These criteria measure the desirability of a design, D-optimum design minimizes the content of the ellipsoidal confidence region for the parameters of the linear model, A-optimum design minimizes the sum (or average) of the variances of the parameter estimates, E-criterion reduces the variance of each individual parameter estimate and T-criterion is one that has not enjoyed much use because of the linearity aspect of T-criterion. This study considers the already existing twenty four points three dimensional specific rotatable design of order two. The information matrices C₁, for this design is obtained from the moment matrix M, for the second order model for three factors using the relation C=(K¹M^-1K)^-1, where M=1/N(X¹X), is the moment matrix, K is the coefficient matrix of the parameter sub system of interest. Our parameter system of interest is that of the linear and pure quadratic factors only. The optimality criteria for the design with the corresponding information matrix C₁, is determined as A-optimal.

Published in	American Journal of Theoretical and Applied Statistics (Volume 9, Issue 6)
DOI	10.11648/j.ajtas.20200906.16
Page(s)	306-311
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), 2020. Published by Science Publishing Group

Keywords

Rotatable Design, Moment Matrix, Optimality Criteria, Order

References

[1]	Smith, K., On the standard deviations of adjusted and interpolated values of an observed polynomial function and its constants and the guidance they give towards a proper choice of the distribution of observations. Biometrika, 1918. 12 (1/2): p. 1-85.
[2]	Kiefer, J. and J. Wolfowitz, Optimum designs in regression problems. The Annals of Mathematical Statistics, 1959: p. 271-294.
[3]	Guest, P., The spacing of observations in polynomial regression. The Annals of Mathematical Statistics, 1958. 29 (1): p. 294-299.
[4]	Wald, A., On the efficient design of statistical investigations. The annals of mathematical statistics, 1943. 14 (2): p. 134-140.
[5]	Mood, A. M., On Hotelling's weighing problem. The Annals of Mathematical Statistics, 1946. 17 (4): p. 432-446.
[6]	Elfving, G., Optimum allocation in linear regression theory. The Annals of Mathematical Statistics, 1952. 23 (2): p. 255-262.
[7]	De la Garza, A., Spacing of information in polynomial regression. The Annals of Mathematical Statistics, 1954: p. 123-130.
[8]	Kiefer, J., Optimum experimental designs. Journal of the Royal Statistical Society: Series B (Methodological), 1959. 21 (2): p. 272-304.
[9]	Ehrenfeld, S., On the efficiency of experimental designs. The annals of mathematical statistics, 1955. 26 (2): p. 247-255.
[10]	Hoel, P. G., Efficiency problems in polynomial estimation. The Annals of Mathematical Statistics, 1958. 29 (4): p. 1134-1145.
[11]	John, R. S. and N. R. Draper, D-optimality for regression designs: a review. Technometrics, 1975. 17 (1): p. 15-23.
[12]	Fedorov, V. and M. Malyutov, Optimal design in regression experiments, Math. Operationsforsch. Stat, 1972. 14: p. 237-324.
[13]	Whittle, P., Some general points in the theory of optimal experimental design. Journal of the Royal Statistical Society: Series B (Methodological), 1973. 35 (1): p. 123-130.
[14]	Kiefer, J., General equivalence theory for optimum designs (approximate theory). The annals of Statistics, 1974: p. 849-879.
[15]	Draper, N. R., Introduction to Box and Wilson (1951) on the experimental attainment of optimum conditions, in Breakthroughs in Statistics. 1992, Springer. p. 267-269.
[16]	Box, G. E. and W. G. Hunter, Sequential design of experiments for nonlinear models. 1963, WISCONSIN UNIV MADISON.
[17]	Liang, W., S. Xu, and Y. Zhao, Experimental study of temperature effects on physical and mechanical characteristics of salt rock. Rock mechanics and rock engineering, 2006. 39 (5): p. 469-482.
[18]	Hill, W. J. and W. G. Hunter, Design of experiments for subsets of parameters. Technometrics, 1974. 16 (3): p. 425-434.
[19]	Hunter, W. G., W. J. Hill, and T. L. Henson, Designing experiments for precise estimation of all or some of the constants in a mechanistic model. The Canadian Journal of Chemical Engineering, 1969. 47 (1): p. 76-80.
[20]	van der Linden, W. J. and H. Ren, Optimal Bayesian adaptive design for test-item calibration. Psychometrika, 2015. 80 (2): p. 263-288.
[21]	Mutiso, J., Second and third order specific and sequential rotatabledesigns in K dimensions. 1998, D. Phil. Thesis, Moi university, Eldoret Kenya.
[22]	Draper, N. R., Third order rotatable designs in three dimensions. The Annals of Mathematical Statistics, 1960: p. 865-874.
[23]	Cornelious, N. O., A hypothetical sequential third order rotatable design in five dimensions. European International Journal of Science and Technology, 2019. 8 (4): p. 40-50.
[24]	Keny, S., I. Tum, and E. Chirchir, Construction of thirty two points specific optimum second order rotatable design in three dimensions with a practical example. International journal of current research, 2012. 4: p. 119-122.
[25]	Pukelsheim, F., Optimal Design of Experiments. © John Wiley and Sons. Inc., New York, New York, 1993.
[26]	Bose, R. and N. R. Draper, Second order rotatable designs in three dimensions. The Annals of Mathematical Statistics, 1959. 30 (4): p. 1097-1112.

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Kiplagat Nicholas Kipkosgei, Mutiso John Muindi, Rambaei Keny Silver Jeptoo. (2020). Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions. American Journal of Theoretical and Applied Statistics, 9(6), 306-311. https://doi.org/10.11648/j.ajtas.20200906.16

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ACS Style

Kiplagat Nicholas Kipkosgei; Mutiso John Muindi; Rambaei Keny Silver Jeptoo. Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions. Am. J. Theor. Appl. Stat. 2020, 9(6), 306-311. doi: 10.11648/j.ajtas.20200906.16

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AMA Style

Kiplagat Nicholas Kipkosgei, Mutiso John Muindi, Rambaei Keny Silver Jeptoo. Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions. Am J Theor Appl Stat. 2020;9(6):306-311. doi: 10.11648/j.ajtas.20200906.16

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@article{10.11648/j.ajtas.20200906.16,
  author = {Kiplagat Nicholas Kipkosgei and Mutiso John Muindi and Rambaei Keny Silver Jeptoo},
  title = {Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {9},
  number = {6},
  pages = {306-311},
  doi = {10.11648/j.ajtas.20200906.16},
  url = {https://doi.org/10.11648/j.ajtas.20200906.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20200906.16},
  abstract = {This study uses the existing second order rotatable design to obtain optimum design based on the known classical optimality criteria that is the determinant criterion, the average-variance criterion, the smallest-Eigen value criterion and the trace criterion. These criteria measure the desirability of a design, D-optimum design minimizes the content of the ellipsoidal confidence region for the parameters of the linear model, A-optimum design minimizes the sum (or average) of the variances of the parameter estimates, E-criterion reduces the variance of each individual parameter estimate and T-criterion is one that has not enjoyed much use because of the linearity aspect of T-criterion. This study considers the already existing twenty four points three dimensional specific rotatable design of order two. The information matrices C1, for this design is obtained from the moment matrix M, for the second order model for three factors using the relation C=(K1M-1K)-1, where M=1/N(X1X), is the moment matrix, K is the coefficient matrix of the parameter sub system of interest. Our parameter system of interest is that of the linear and pure quadratic factors only. The optimality criteria for the design with the corresponding information matrix C1, is determined as A-optimal.},
 year = {2020}
}

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TY  - JOUR
T1  - Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions
AU  - Kiplagat Nicholas Kipkosgei
AU  - Mutiso John Muindi
AU  - Rambaei Keny Silver Jeptoo
Y1  - 2020/12/22
PY  - 2020
N1  - https://doi.org/10.11648/j.ajtas.20200906.16
DO  - 10.11648/j.ajtas.20200906.16
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 306
EP  - 311
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20200906.16
AB  - This study uses the existing second order rotatable design to obtain optimum design based on the known classical optimality criteria that is the determinant criterion, the average-variance criterion, the smallest-Eigen value criterion and the trace criterion. These criteria measure the desirability of a design, D-optimum design minimizes the content of the ellipsoidal confidence region for the parameters of the linear model, A-optimum design minimizes the sum (or average) of the variances of the parameter estimates, E-criterion reduces the variance of each individual parameter estimate and T-criterion is one that has not enjoyed much use because of the linearity aspect of T-criterion. This study considers the already existing twenty four points three dimensional specific rotatable design of order two. The information matrices C1, for this design is obtained from the moment matrix M, for the second order model for three factors using the relation C=(K1M-1K)-1, where M=1/N(X1X), is the moment matrix, K is the coefficient matrix of the parameter sub system of interest. Our parameter system of interest is that of the linear and pure quadratic factors only. The optimality criteria for the design with the corresponding information matrix C1, is determined as A-optimal.
VL  - 9
IS  - 6
ER  -

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Author Information

Kiplagat Nicholas Kipkosgei

Department of Mathematics, Physics and Computing, Moi University, Eldoret, Kenya
Mutiso John Muindi

Department of Mathematics, Physics and Computing, Moi University, Eldoret, Kenya
Rambaei Keny Silver Jeptoo

Department of Mathematics, Physics and Computing, Moi University, Eldoret, Kenya

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APA Style

Kiplagat Nicholas Kipkosgei, Mutiso John Muindi, Rambaei Keny Silver Jeptoo. (2020). Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions. American Journal of Theoretical and Applied Statistics, 9(6), 306-311. https://doi.org/10.11648/j.ajtas.20200906.16

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ACS Style

Kiplagat Nicholas Kipkosgei; Mutiso John Muindi; Rambaei Keny Silver Jeptoo. Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions. Am. J. Theor. Appl. Stat. 2020, 9(6), 306-311. doi: 10.11648/j.ajtas.20200906.16

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AMA Style

Kiplagat Nicholas Kipkosgei, Mutiso John Muindi, Rambaei Keny Silver Jeptoo. Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions. Am J Theor Appl Stat. 2020;9(6):306-311. doi: 10.11648/j.ajtas.20200906.16

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@article{10.11648/j.ajtas.20200906.16,
  author = {Kiplagat Nicholas Kipkosgei and Mutiso John Muindi and Rambaei Keny Silver Jeptoo},
  title = {Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {9},
  number = {6},
  pages = {306-311},
  doi = {10.11648/j.ajtas.20200906.16},
  url = {https://doi.org/10.11648/j.ajtas.20200906.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20200906.16},
  abstract = {This study uses the existing second order rotatable design to obtain optimum design based on the known classical optimality criteria that is the determinant criterion, the average-variance criterion, the smallest-Eigen value criterion and the trace criterion. These criteria measure the desirability of a design, D-optimum design minimizes the content of the ellipsoidal confidence region for the parameters of the linear model, A-optimum design minimizes the sum (or average) of the variances of the parameter estimates, E-criterion reduces the variance of each individual parameter estimate and T-criterion is one that has not enjoyed much use because of the linearity aspect of T-criterion. This study considers the already existing twenty four points three dimensional specific rotatable design of order two. The information matrices C1, for this design is obtained from the moment matrix M, for the second order model for three factors using the relation C=(K1M-1K)-1, where M=1/N(X1X), is the moment matrix, K is the coefficient matrix of the parameter sub system of interest. Our parameter system of interest is that of the linear and pure quadratic factors only. The optimality criteria for the design with the corresponding information matrix C1, is determined as A-optimal.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Calculus Optimum Values Optimality Criteria for Twenty Four Points Specific Second Order Rotatable Design in Three Dimensions
AU  - Kiplagat Nicholas Kipkosgei
AU  - Mutiso John Muindi
AU  - Rambaei Keny Silver Jeptoo
Y1  - 2020/12/22
PY  - 2020
N1  - https://doi.org/10.11648/j.ajtas.20200906.16
DO  - 10.11648/j.ajtas.20200906.16
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 306
EP  - 311
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20200906.16
AB  - This study uses the existing second order rotatable design to obtain optimum design based on the known classical optimality criteria that is the determinant criterion, the average-variance criterion, the smallest-Eigen value criterion and the trace criterion. These criteria measure the desirability of a design, D-optimum design minimizes the content of the ellipsoidal confidence region for the parameters of the linear model, A-optimum design minimizes the sum (or average) of the variances of the parameter estimates, E-criterion reduces the variance of each individual parameter estimate and T-criterion is one that has not enjoyed much use because of the linearity aspect of T-criterion. This study considers the already existing twenty four points three dimensional specific rotatable design of order two. The information matrices C1, for this design is obtained from the moment matrix M, for the second order model for three factors using the relation C=(K1M-1K)-1, where M=1/N(X1X), is the moment matrix, K is the coefficient matrix of the parameter sub system of interest. Our parameter system of interest is that of the linear and pure quadratic factors only. The optimality criteria for the design with the corresponding information matrix C1, is determined as A-optimal.
VL  - 9
IS  - 6
ER  -

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