Mixture Experiments in the Interior: Yantram Designs

Abstract

We introduce a very rich class of mixture designs in the interior of the simplex, which are useful when all experiments must constitute a complete mixture, that is, when none of the components is absent from the mixture. These designs are called yantram designs, as they are directly derivable from the Hindu yantrams, which are certain numerical configurations of positive integers having certain properties. Not only do these designs work well within the interior of the simplex, they can also be easily refined to satisfy the constraints, if there are any, on the mixture components. Leverage values for such designs are more evenly distributed among interior points compared to simplex-lattice or simplex-centroid designs, which tend to place higher leverages on the vertices or edge design points where the experiments may not be feasible.

AMS Subject Classification:

• 62K20

Keywords:

• Complete mixtures
• Extreme vertices
• Magic squares
• Mixture designs
• Mixtures in interior
• Procrustation
• Yantram designs

Notes

¹ In Sanskrit, the word yantram roughly means numerical configuration (or sometimes configuration of letters or symbols). Hosts of such configurations are used in Hindu religious practices and spiritual endeavors. A magic square of order q is an arrangement of integers up to q² in a q × q grid with certain property that all rows, all columns, and both diagonals individually add to the same fixed number (= q(q² + 1)/2). Yantrams in general do not have to be the arrangements of the first q² integers. Arrangements of other positive integers have been used that satisfy the constant sum requirements indicated. In that sense, yantrams are more general than magic squares and a q × q magic square is just the base q × q yantram. The term magic square for any base yantram is somewhat a misnomer, as it has nothing to do with magic. Instead, they represent spirituality, as Hindus in Vedic times noticed the attractive properties of such configurations and, assuming their beauty to be God’s unique creation/representation, associated them with divinity.

² Mangala, Budh, Sukra, and Sani are the Sanskrit names for the planets Mars, Mercury, Venus, and Saturn, respectively. Hindu astrology revolves around the various configurations of the planetary positions of nine (not seven) planets, including two planets that do not have counterparts in Western astrology. In such assignments Earth is not considered but Surya, representing Sun, although not a planet, is included. Surya yantram and Mangala yantram have been used to generate Tables 2 and 3, respectively.

³ We call it a Parshvanath design since the corresponding yantram has been inscribed on the wall of the at least 1500 years old Parshvanath Jain Temple in Khajuraho, India. This yantram is also commonly known in India as Chautisa, since in many North Indian languages, the word Chautis is used for number 34.

⁴ Table 11 presents various other configurations, apart from rows, columns, and diagonals listed in Table 10 in the Parshvanath yantram, which result in a sum of 34. To illustrate, all nine 2 × 2 subsquares consisting of four elements yield the sum of 34. Similarly all six cases of three-element diagonals and the opposite one-element diagonal will result in a sum of 34. Two leftmost (row number 1 and 2) and two rightmost (row number 3 and 4) rows and corresponding outer columns (column numbers 1 and 4) do the same. The same is true when in the previous statement, roles of rows and columns are interchanged. Similarly, corner elements add to 34. If one takes the extreme elements of top two rows, they add to 34. The same is true for the bottom two rows. Statements similar to the previous two statements can be made for the columns also. The middle two elements in the outer (number 1 and 4) rows or columns add to 34. Finally, elements obtained by skipping successive rows as well as columns add to 34.

⁵ According to Greek mythology, Procrustes was an innkeeper with unique knack for standardization and customer service. His inn had fixed size cots for the guests. If a guest was too short for the cot, his or her legs would be stretched to fit the cot. On the other hand, if a guest was too tall, his or her legs were chopped off. The approach here does the same in that it tries to size the design points to be contained within the predetermined experimental region specified by the confines of the constraints. Hence we coin the word Procrustation to define this phenomenon. We can only surmise that Procrustes did not charge his guests extra for this extra privilege.

⁶ In fact, without loss of generality d can be taken as 1. This is because Y* = dY + cE = d(Y + d⁻¹cE) and the scaling constant d outside the parentheses cancels when the division by corresponding denominator takes place while constructing a design point.