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D- and G- Optimal Axial Slope Designs for Four Ingredient Mixture

Njoroge Elizabeth Wambui*, Koske Joseph, Mutiso John

Dpartment of Physical Sciences, Chuka University, P.O. Box 109-60400, Chuka, Kenya

Abstract: This paper aims at investigating and comparing the D- and G-optimal criteria for non-pure blends slope designs. The study used a parameter subsystem of interest based on the second-degree Kronecker model to obtain the H-invariant information matrices for both Equally Weighted Simplex Centroid Axial Design and Un-equally Weighted Simplex Centroid Axial Design. The D- and G- optimal values worked out revealed that the centroid achieved the best D- and G-optimality values and that the best D-efficient and G-efficient design points were with 105.71% and 99.76% respectively. The latter design was more D-efficient while former design was more G-efficient.

Keywords: axial design, subsystem, H-invariant, centroid, D-optimal, G-optimal, efficiency

Received September 02, 2020; Revised October 04, 2020; Accepted October 13, 2020

Citation: Njoroge Elizabeth Wambui, Koske Joseph, Mutiso John, D- and G- Optimal Axial Slope Designs for Four Ingredient Mixture. Applied Mathematics and Physics, 8 (1). 20-25. January 2020. doi:10.12691/amp-8-1-4

1. Introduction

A mixture experiment with components according to [6] satisfy the condition where is the number of components in the experiments. The aim of any experimenter is to obtain levels of the ingredients that optimize the expected response. In so doing, some optimality criterions chosen either minimize the variance or maximize information of the information matrix of the model adopted. [10] studied the optimal slope designs for second order Kronecker model mixture experiments for

The researcher [8], introduced the q-component Simplex Centroid Design as designs where the points are located at the boundaries, the vertices and the Centre of the dimension simplex.

The q-component Simplex Centroid Design involves distinct design points in total. There are pure components the permutations of binary mixtures , the permutations of ternary mixture and the permutation of quaternary mixture and so on up to mixtures involving of equal proportion of all components or q-nary mixtures.

An Axial design is a design that consists mainly of complete or component blends where most of the points are located inside the simplex. Axial designs are recommended to be used when component effects are to be measured, and in screening experiments. The simplest form of the axial design is one whose points are equidistant from the Centroid , and has axes [1].

2. Methodology

The study adopted an inscribed tetrahedral Simplex Centroid design also referred to as an axial design, for the fact that all the corresponding points in each face vertex and line of the design are equidistant from the original Simplex Centroid. This study chose to let the vertices of the simplex to be a distance from the main vertices or from the base the axis of the original Simplex. This original simplex is a regular tetrahedron of vertices (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1). This value h also denoted by has a maximum value given by according to [1]. The design suggested by [3] was developed in the following way

The four-vertex design points generated as

has a corresponding moment matrix

The six edge mid points forming the design whose Moment Matrix is were given by

The four points on the faces of the inscribed tetrahedron forming the design whose moment matrix is were the mixture blends generated as;

The fifteenth point, which forms the design is the Centre of the simplex design referred to as the Centroid whose moment matrix is

The value arbitrarily chosen, developed the fifteen-point design given as

[picture] | (1)

The Kronecker models are better models because they have increased symmetry resulting from the repletion of cross product terms, which result to larger moment matrices. The models are less susceptible to ill conditioning, which results to highly correlated parameters and large standard errors according to [4].

The polynomial function related to the K-model is

[picture] | (2)

The moment matrices obtained at each design point by the summation of Kronecker product shown in (3) are matrices.

[picture] | (3)

The improved information matrices are the slope matrices which are obtained by utilizing the equation which is the improved information matrix of the slope obtained from the and H, the derivative of the elements of the design matrix that is where is given by (4)

[picture] | (4)

The general H derivative matrix of given as (5)

[picture] | (5)

For an arbitrary subset of matrices, we define a symmetric matrix C to be invariant if for all The set of all invariant symmetric matrices are expressed as as put by [5].

The optimality tests are usually done in order to locate the optimum values of a design according to each criterion, and are compared in order to obtain the design with the best characteristics. The D, and G-Optimal values for two designs namely, Equally Weighted Simplex Centroid Axial Design (EWSCAD) and Unequally Weighted Simplex Centroid Axial Design (UWSCAD) were calculated and compared in this study.

The designs obtaining the maximum information for the maximal parameter subsystem of interest are those that obtain maximum values through application of D and I optimality criterions, which satisfy the Kiefer-Wolfowitz equivalence theorem. The second-degree Kronecker full and subsystem of interest models for the four mixture components suggested by [2], are given in (6) and (7) respectively.

[picture] | (6)

[picture] | (7)

The experimenter uses the subsystem of interest with parameters, for three reasons; one, it is less expensive, two, there are no repeated terms that make the moment matrix rank deficient causing inefficiency in estimating the terms and three, it has the same characteristics as the full model.

The statistical package R-Gui version 4.0.0 was used to process the D- and G-optimal values.

2.1. Coefficient, Moment and Information Matrices

In this section coefficient, moment and information matrices K, M and C respectively were derived for the two designs.

2.1.1. Coefficient Matrix (K)

Due to repletion of terms in (6), experimenters find it more convenient to work with a subsystem of interest (7) whose coefficient matrix (8) less expensive. Therefore, they wish to study components out of the Equation (7) was achieved by integrating and averaging similar outcomes in (6). The linear parameter subsystem of interest for some matrix K referred to as the coefficient matrix of the parameter subsystem develops the coefficients (8) as a result. It is estimable when there exists an unbiased linear estimator for L is the left inverse of K such that The squares and the cross products of the components of for the subsystem of interest presented in Lexicographic order was given by (7).

[picture] | (8)

2.1.2. Moment Matrices

The general equivalence theorem of [5] defines the qualities and conditions for a competing moment matrix.

The moment matrices at the different design points obtained by (9)

[picture] | (9)

The general moment matrix is the equation (10)

[picture] | (10)

The moment matrices shown by (16) and (17) are worked out by the equations (11) and (12) for the two designs respectively

[picture] | (11)

[picture] | (12)

The in (13) were worked out using the general equation and

[picture] | (13)

The matrix (13), is the general moment matrix for four components.

The values of the fourth order moments, which correspond with the moment matrix of the unequally weighted centroid design, are as represented in the set of equations (14).

[picture] | (14)

Where and are given in (15)

[picture] | (15)

[picture] | (16)

[picture] | (17)

2.1.3. Information Matrices

The respective information matrices, and for the two designs are given by (18) and (19).

[picture] | (18)

[picture] | (19)

3. Results and Discussion

3.1. D-slope Optimal Values

This paper aims at obtaining slope information matrices at different points of the design in order to obtain the D- and G-optimal values for the two designs. The H- invariant slope matrices obtained from (6), for each design point were (20).

[picture] | (20)

Equation (21) below was used to obtain the determinants of the information matrices and hence the D-optimal values.

[picture] | (21)

P is the number of parameters under estimation, while and are as given in (18), (19) and (20) respectively. The following D-optimal values for both designs were calculated.

The efficiencies of the two designs were expressed using equation (22).

[picture] | (22)

At each design point, the efficiencies were; 105.71%, 105.34%, 105.38%, 105.42%.

3.2. G-slope Optimal Values

The G optimality criterion also known as the global optimality criterion is a prediction criterion first introduced by Smith in 1918. It is a design that minimizes the worst case expected error in prediction. [6] provided the G optimality criterion definition as which is equivalent to;

[picture] | (23)

for a full parameter system and the subsystem of interest

[picture] | (24)

The inverse of the improved information matrix substituted for the inverse of the information matrix to obtain the G-slope optimal values. The exquation (25) gave the results of the G-slope optimal values.

[picture] | (25)

[9] did define G-Optimal design as the design which minimizes the maximum variance of the estimated response function over the given design region. The minimum values of the maximum variances per design points were worked out and given as;

and

The -efficiencies worked out using the formula

[picture] | (26)

Were, 83.14%, 94.01%, 93.99% and 99.76%.

4. Conclusion

This paper established the H-invariant matrices needed to derive the slope optimal values for the D- and G- optimal criteria. The study revealed that for the two designs, the centroid had the best D- and G- slope optimal values, that the best D-efficient design was with 105.71% and the most G-efficient was the centroid with 99.76%. UWSCAD was a more D-efficient design while EWSCAD was a more G-efficient design. To further this research, experimental data obtained using the adopted design would compare with these results, while other criteria would necessitate further findings using the same design.

References

[1]   | Cornell, J.A. (2000). Experiments with Mixture Designs, Models and Analysis of Mixtures Data. John Wiley & Sons Inc, New York.
[2]   | Draper, N.R. and Pukelsheim, F. (1998). Mixture Models Based on Homogeneous Polynomials. J. Statist. Plann. Inference, 71, 303-311.
[3]   | Njoroge, E.W., Koske, J., Mutiso J. (2020). Quad-Axial Weighted Simplex Centroid Design Using Second Order Kronecker Model To Optimize The Plinth Concrete Mix Components For Low Cost Houses. ‘Unpublished’.
[4]   | Prescott, P., Dean, A.M., Draper, N.R., Lewis, S.M., (2002). Mixture Experiments; III Conditioning and Quadratic Model Specification. Technometrices. Pg. 260-268.
[5]   | Pukelsheim F. (1993). Optimal Design of Experiments, John Wiley & sons, Inc., New York.
[6]   | Rady, E.A., Abd EL-Monsef, M.M.E., Seyam, M.M., (2009). Relationship among several optimality criteria. Interstat Journal volume 15(6). pp1-11.
[7]   | Scheffe, H., (1958). Experiments with Mixtures. Journal of the Royal Staistical Society, ser B20: 344-359.
[8]   | Scheffe, H. (1963). Simplex Centroid Designs for Experiments with Mixtures. J. Royal statist. Soc. Ser, B25, 35-263.
[9]   | Thomas, A.U., Stephen, S.A., (2013). On the Comparison of Boundary and Interior Support points of a Response Surface under Optimality Criteria. International Journal of Mathematics and Statistics Studies, pp. 48-58.
[10]   | Wambua, A.M., Njoroge, E., Koske, J., Mutiso J., Kuria, J.G., Muriungi, R.G., Kipkoech, C.,(2017). Optimal Slope Designs for Second Degree Kronecker Model Mixture Experiments. International Journal of Applied Mathematics and Theoretical Physics, pp.86-91.3.