Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments

Wambua Alex Mwaniki; Njoroge Elizabeth; Koske Joseph; John Mutiso; Kuria Joseph Gikonyo; Muriungi Robert Gitunga; Cheruiyot Kipkoech

doi:doi:10.11648/j.ijamtp.20170304.12

Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments

Wambua Alex Mwaniki, Njoroge Elizabeth, Koske Joseph, John Mutiso, Kuria Joseph Gikonyo, Muriungi Robert Gitunga, Cheruiyot Kipkoech

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 3, Issue 4)

Received: 31 March 2017 Accepted: 14 April 2017 Published: 26 October 2017

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Abstract

The aim of this paper is to investigate some optimal slope mixture designs in the second degree Kronecker model for mixture experiments. The study is restricted to weighted centroid designs, with the second degree Kronecker model. For the selected maximal parameter subsystem in the model, a method is devised for identifying the ingredients ratio that leads to an optimal response. The study also seeks to establish equivalence relations for the existence of optimal designs for the various optimality criteria. To achieve this for the feasible weighted centroid designs the information matrix of the designs is obtained. Derivations of D-, A- and E-optimal weighted centroid designs are then obtained from the information matrix. Basically this would be limited to classical optimality criteria. Results on a quadratic subspace of H-invariant symmetric matrices containing the information matrices involved in the design problem was used to obtain optimal designs for mixture experiments analytically. The discussion is based on Kronecker product algebra which clearly reflects the symmetries of the simplex experimental region.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 3, Issue 4)
DOI	10.11648/j.ijamtp.20170304.12
Page(s)	86-91
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), 2017. Published by Science Publishing Group

Keywords

Slope Mixture designs Kronecker product, Optimal Designs, Weighted Centroid Designs, A-, D-, E-Optimality and H- invariant Symmetric Matrices

References

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[2]	Cornell, J. A. (1981). Experiments with Mixtures designs: Models and analysis of mixture data. John Willy & Sons, New York.
[3]	Cornell, J. A. (1990). Experiments with Mixtures. Wiley, New York.
[4]	Draper N. R., and Pukelsheim, F., (1998), Kiefer ordering of simplex designs for first-and second degree mixture models, Journal of statistical planning and inference, 79:325-348.
[5]	Draper, N. R. and Pukelsheim, F. (1998a). Polynomial representations for response surface modelling. In New Developments and Applications in Experimental Design (N. Flournoy, W. F. Rosenberger and W. K. Wong, eds.), 34 199–212. IMS, Hayward, CA.
[6]	Draper, N. R. and Pukelsheim, F. (1998b). Mixture models based on homogeneous polynomials. J. Statist. Plann. Inference 71 303–311.
[7]	Draper, N. R. and Pukelsheim, F. (1999). Kiefer ordering of simplex designs for first- and second degree mixture models. J. Statist. Plann. Inference 79 325–348.
[8]	Draper, N. R., Heiligers, B. and Pukelsheim, F. (2000). Kiefer ordering of second-degree mixture designs for four ingredients. In Proceedings of the American Statistical Association, Annual Meeting, Baltimore MD, August 1999. Amer. Statist. Assoc., Alexandria, VA.
[9]	Friedrich Pukelsheim, (1993) “Optimal designs of experiment”.
[10]	Hader, R. J and Park, S. H slope rotatable central composite designs, Technometrics, 1978; 20:413-417.
[11]	John A. Cornell, (1990) Experiments with Mixtures, second edition.
[12]	Kinyanjui J. K., (2007), Some Optimal Designs for Second-Degree Kronecker Model Mixture Experiments, PhD, Thesis, Moi University, Eldoret.
[13]	Koech Eliud, Koech Milton, Koske Joseph, Kerich Gregory, Argwings Otieno (2014) E-Optimal designs for maximal parameter subsystem second degree Kronecker model mixture experiments.
[14]	Norman R. Draper, Berthold Heiligers and Fredrich Pukelshiem (2000) Kiefer ordering of Simplex designs for second degree mixture models with four or more ingredients.
[15]	Park S. H and Kim H. J, A measure of slope rotatability for second order response surface experimental designs, Journal of Applied Statistics, 1992; 19: 391-404
[16]	Peter Goos, Poradley Jones and Utain Syafitri (2013) I-Optimal mixture designs.
[17]	Scheffé, H., 1958. Experiments with Mixtures, Journal of Royal Statistical Society. Series B, 20, 344-360.
[18]	Scheffé, H., 1963. Simplex-centroid designs for experiments with Mixtures, Journal of Royal Statistical Society. Series B, 25, 235-263.
[19]	Thomas Klein (2001) Invariant Symmetric block matrices for design of mixture of mixture experiments.

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APA Style

Wambua Alex Mwaniki, Njoroge Elizabeth, Koske Joseph, John Mutiso, Kuria Joseph Gikonyo, et al. (2017). Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments. International Journal of Applied Mathematics and Theoretical Physics, 3(4), 86-91. https://doi.org/10.11648/j.ijamtp.20170304.12

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ACS Style

Wambua Alex Mwaniki; Njoroge Elizabeth; Koske Joseph; John Mutiso; Kuria Joseph Gikonyo, et al. Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments. Int. J. Appl. Math. Theor. Phys. 2017, 3(4), 86-91. doi: 10.11648/j.ijamtp.20170304.12

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AMA Style

Wambua Alex Mwaniki, Njoroge Elizabeth, Koske Joseph, John Mutiso, Kuria Joseph Gikonyo, et al. Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments. Int J Appl Math Theor Phys. 2017;3(4):86-91. doi: 10.11648/j.ijamtp.20170304.12

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@article{10.11648/j.ijamtp.20170304.12,
  author = {Wambua Alex Mwaniki and Njoroge Elizabeth and Koske Joseph and John Mutiso and Kuria Joseph Gikonyo and Muriungi Robert Gitunga and Cheruiyot Kipkoech},
  title = {Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {3},
  number = {4},
  pages = {86-91},
  doi = {10.11648/j.ijamtp.20170304.12},
  url = {https://doi.org/10.11648/j.ijamtp.20170304.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20170304.12},
  abstract = {The aim of this paper is to investigate some optimal slope mixture designs in the second degree Kronecker model for mixture experiments. The study is restricted to weighted centroid designs, with the second degree Kronecker model. For the selected maximal parameter subsystem in the model, a method is devised for identifying the ingredients ratio that leads to an optimal response. The study also seeks to establish equivalence relations for the existence of optimal designs for the various optimality criteria. To achieve this for the feasible weighted centroid designs the information matrix of the designs is obtained. Derivations of D-, A- and E-optimal weighted centroid designs are then obtained from the information matrix. Basically this would be limited to classical optimality criteria. Results on a quadratic subspace of H-invariant symmetric matrices containing the information matrices involved in the design problem was used to obtain optimal designs for mixture experiments analytically. The discussion is based on Kronecker product algebra which clearly reflects the symmetries of the simplex experimental region.},
 year = {2017}
}

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TY  - JOUR
T1  - Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments
AU  - Wambua Alex Mwaniki
AU  - Njoroge Elizabeth
AU  - Koske Joseph
AU  - John Mutiso
AU  - Kuria Joseph Gikonyo
AU  - Muriungi Robert Gitunga
AU  - Cheruiyot Kipkoech
Y1  - 2017/10/26
PY  - 2017
N1  - https://doi.org/10.11648/j.ijamtp.20170304.12
DO  - 10.11648/j.ijamtp.20170304.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 86
EP  - 91
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20170304.12
AB  - The aim of this paper is to investigate some optimal slope mixture designs in the second degree Kronecker model for mixture experiments. The study is restricted to weighted centroid designs, with the second degree Kronecker model. For the selected maximal parameter subsystem in the model, a method is devised for identifying the ingredients ratio that leads to an optimal response. The study also seeks to establish equivalence relations for the existence of optimal designs for the various optimality criteria. To achieve this for the feasible weighted centroid designs the information matrix of the designs is obtained. Derivations of D-, A- and E-optimal weighted centroid designs are then obtained from the information matrix. Basically this would be limited to classical optimality criteria. Results on a quadratic subspace of H-invariant symmetric matrices containing the information matrices involved in the design problem was used to obtain optimal designs for mixture experiments analytically. The discussion is based on Kronecker product algebra which clearly reflects the symmetries of the simplex experimental region.
VL  - 3
IS  - 4
ER  -

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Author Information

Wambua Alex Mwaniki

Department of Planning and Statistics, Ministry of Agriculture, Livestock and Fisheries, Nairobi, Kenya
Njoroge Elizabeth

Department of Business Administration, Chuka University, Chuka, Kenya
Koske Joseph

Department of Mathematics and Computer Science, Moi University, Eldoret, Kenya
John Mutiso

Department of Mathematics and Computer Science, Moi University, Eldoret, Kenya
Kuria Joseph Gikonyo

Department of Mathematics, Statistics and Actuarial Sciences, Karatina University, Karatina, Kenya
Muriungi Robert Gitunga

Department of Mathematics, Meru University of Science and Technology, Meru, Kenya
Cheruiyot Kipkoech

Department of Mathematics and Physical Sciences, Maasai Mara University, Narok, Kenya

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APA Style

Wambua Alex Mwaniki, Njoroge Elizabeth, Koske Joseph, John Mutiso, Kuria Joseph Gikonyo, et al. (2017). Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments. International Journal of Applied Mathematics and Theoretical Physics, 3(4), 86-91. https://doi.org/10.11648/j.ijamtp.20170304.12

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ACS Style

Wambua Alex Mwaniki; Njoroge Elizabeth; Koske Joseph; John Mutiso; Kuria Joseph Gikonyo, et al. Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments. Int. J. Appl. Math. Theor. Phys. 2017, 3(4), 86-91. doi: 10.11648/j.ijamtp.20170304.12

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AMA Style

Wambua Alex Mwaniki, Njoroge Elizabeth, Koske Joseph, John Mutiso, Kuria Joseph Gikonyo, et al. Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments. Int J Appl Math Theor Phys. 2017;3(4):86-91. doi: 10.11648/j.ijamtp.20170304.12

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@article{10.11648/j.ijamtp.20170304.12,
  author = {Wambua Alex Mwaniki and Njoroge Elizabeth and Koske Joseph and John Mutiso and Kuria Joseph Gikonyo and Muriungi Robert Gitunga and Cheruiyot Kipkoech},
  title = {Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {3},
  number = {4},
  pages = {86-91},
  doi = {10.11648/j.ijamtp.20170304.12},
  url = {https://doi.org/10.11648/j.ijamtp.20170304.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20170304.12},
  abstract = {The aim of this paper is to investigate some optimal slope mixture designs in the second degree Kronecker model for mixture experiments. The study is restricted to weighted centroid designs, with the second degree Kronecker model. For the selected maximal parameter subsystem in the model, a method is devised for identifying the ingredients ratio that leads to an optimal response. The study also seeks to establish equivalence relations for the existence of optimal designs for the various optimality criteria. To achieve this for the feasible weighted centroid designs the information matrix of the designs is obtained. Derivations of D-, A- and E-optimal weighted centroid designs are then obtained from the information matrix. Basically this would be limited to classical optimality criteria. Results on a quadratic subspace of H-invariant symmetric matrices containing the information matrices involved in the design problem was used to obtain optimal designs for mixture experiments analytically. The discussion is based on Kronecker product algebra which clearly reflects the symmetries of the simplex experimental region.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments
AU  - Wambua Alex Mwaniki
AU  - Njoroge Elizabeth
AU  - Koske Joseph
AU  - John Mutiso
AU  - Kuria Joseph Gikonyo
AU  - Muriungi Robert Gitunga
AU  - Cheruiyot Kipkoech
Y1  - 2017/10/26
PY  - 2017
N1  - https://doi.org/10.11648/j.ijamtp.20170304.12
DO  - 10.11648/j.ijamtp.20170304.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 86
EP  - 91
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20170304.12
AB  - The aim of this paper is to investigate some optimal slope mixture designs in the second degree Kronecker model for mixture experiments. The study is restricted to weighted centroid designs, with the second degree Kronecker model. For the selected maximal parameter subsystem in the model, a method is devised for identifying the ingredients ratio that leads to an optimal response. The study also seeks to establish equivalence relations for the existence of optimal designs for the various optimality criteria. To achieve this for the feasible weighted centroid designs the information matrix of the designs is obtained. Derivations of D-, A- and E-optimal weighted centroid designs are then obtained from the information matrix. Basically this would be limited to classical optimality criteria. Results on a quadratic subspace of H-invariant symmetric matrices containing the information matrices involved in the design problem was used to obtain optimal designs for mixture experiments analytically. The discussion is based on Kronecker product algebra which clearly reflects the symmetries of the simplex experimental region.
VL  - 3
IS  - 4
ER  -

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