U-Type Designs via New Generalized Partially Balanced Incomplete Block Designs with <i>m</i> = 4, 5 and 7 Associated Classes

The traditional combinatorial designs can be used as basic designs for constructing designs of computer experiments which have been used successfully till now in various domains such as engineering, pharmaceutical industry, etc. In this paper, a new series of generalized partially balanced incomplete blocks PBIB designs with m associated classes ( m = 4, 5 and 7) based on new generalized association schemes with number of treatments v arranged in w arrays of n rows and l columns ( w ≥ 2, n ≥ 2, l ≥ 2) is defined. Some construction methods of these new PBIB are given and their parameters are specified using the Combinatory Method ( s). For n or l even and s divisor of n or l, the obtained PBIB designs are resolvable PBIB designs. So the Fang RBIBD method is applied to obtain a series of particular U-type designs U ( wnl; ) ( r is the repetition number of each treatment in our resolvable PBIB design).

• Association Scheme
• Combinatory Method (s)
• Resolvable Partially Balanced Incomplete Block Design
• <i>U</i>-Type Design

1. Introduction

Designs of computer experiments drew a wide attention in the previous two decades and were still being used successfully till now in various domains. Among the various construction methods of these designs, the traditional combinatorial designs can be used as basic designs (example: [1] [2] ) and particularly the PBIB designs. The association schemes of two or three associated classes have been widely studied, while it is not the case of those over three associated classes. However, some association schemes with five associated classes have been studied (example: [3] ). Besides, a method to obtain new association schemes by crossing or nesting other association schemes was given by Bailey [4] using the character tables and strata of the initial association schemes to find the parameters of the obtained association schemes.

In this paper, new association schemes with 4, 5 and 7 associated classes are described starting by a geometric representation. The parameter expressions of these association schemes are given. Moreover, some methods to construct the PBIB designs based on these association schemes are explained using an accessible construction method called the Combinatory Method (s) [5] , which allows obtaining a series of PBIB designs by only using these association schemes. The parameters expressions of these new designs are given. In addition, for n or l even

and s divisor of n or l, the obtained PBIB are resolvable PBIB. Then, a series of U-type designs U

(r is the repetition number of each treatment in the resolvable PBIB design) is obtained by applying the RBIBD method [6] on these designs.

The paper is organized as follows. In Section 2, we give new definitions of generalized association schemes with m (= 4, 5 and 7) associated classes, starting by geometric representation and we give their parameters as properties. Section 3 describes a series of construction method using the Combinatory Method (s) for obtaining the PBIB designs associated to our generalized association schemes. We give the series of the U-type designs associated to our constructed PBIB designs in Section 4. We achieve our paper with a Conclusion.

Recall some definitions:

Definition 1. An m-association scheme (m ≥ 2) of v treatments [7] is a relation satisfying the following conditions:

1) Any two treatments are either 1^st, 2^nd, ∙∙∙, or m^th associates. The relation of association is symmetric, i.e., if the treatment is an i^th associate of β, then β is an i^th associate of.

2) Each treatment has n_i i^th associates, the number n_i being independent of.

3) If any two treatments and β are i^th associates, then the number of treatments that are j^th associates of and k^th associates of β is and is independent of the pair of i^th associates and β.

The numbers v, n_i and are called the parameters of the association scheme.

Definition 2. A PBIB design [7] , based on an m-association scheme (m ≥ 2), with parameters v, b, r, k, λ_i, , is a block design with v treatments and b blocks of size k each such that every treatment occurs in blocks and any two distinct treatments being i^th associate occur together in exactly λ_i blocks. The number λ_i is independent of the pair of i^th associates.

A parallel class of PBIB is a collection of disjoint blocks from the b blocks whose union is V. A partition of the b blocks into parallel classes is called a resolution, and PBIB design is resolvable if it has at least one resolution and it denoted by RPBIB design.

The Combinatory Method (s) [5] :

Let an array of n rows and l columns as follows:

Consider s different elements of the same row i, and associate with them s other elements of a row , respecting the correspondence between the elements a_ij and. Bringing together the 2s elements in the same block and making all possible combinations, we obtain a partially balanced incomplete block design of size.

Definition 3. Let U denote a design of v runs and r factors with respective levels. This design corresponds to an v × r matrix such that the i^th column xⁱ takes values from a set of q_i elements, say, equally often. The set of all such designs, called U-type designs in the statistical literature, is denoted by U. Obviously, v must be a multiple of q_i. When all the q_i are equal to q, we denote it by U. Note that the rows and columns of X are identified with the runs and factors respectively [6] .

2. Generalized Rectangular Right Angular Association Schemes (m) (m = 4, 5 and 7 Associated Classes)

2.1. Generalized Rectangular Right Angular Association Scheme (4)

Let V be a set of v = wnl treatments, (w ≥ 2, n ≥ 2, l ≥ 2), to which we associate a geometrical representation in the following way:

Each treatment of V is associated with a unique triplet of the set where:

Let be a treatment of coordinates

・ corresponds to the treatments 1^st associated to

・ corresponds to the treatments 2^nd associated to.

・ corresponds to the treatments 3^rd associated to.

・ corresponds to the treatments 4^th associated to.

This geometric representation describes a new association scheme, we call it for convenience, generalized rectangular right angular association scheme (4) with four associated classes, to which we give the following equivalent definition:

Definition 4. A generalized rectangular right angular association scheme (4) is an arrangement of v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments in w arrays of n rows and l columns such that, with respect to each treatment:

1) The first associates of are the other treatments of the same row in the same array.

2) The second associates of are the other treatments of the same column in the same array.

3) The third associates of are the remaining treatments in the same array.

4) The fourth associates of are the other treatments of the other arrays.

Property 1. The parameters of generalized rectangular right angular association schemes (4) are:

, , , ,

Definition 5. A PBIB design based on a generalized rectangular right angular association scheme (4) is called generalized rectangular right angular GPBIB₄ design.

2.2. Generalized Rectangular Right Angular Association Scheme (5)

Let V be a set of v = wnl treatments, (w ≥ 2, n ≥ 2, l ≥ 2), to which we associate a geometrical representation in the following way:

Each treatment of V is associated with a unique triplet (of coordinates) of the set where:

Let be a treatment of coordinates

・ corresponds to the treatments 1^st associated to

・ corresponds to the treatments 2^nd associated to.

・ corresponds to the treatments 3^rd associated to.

・ corresponds to the treatments 4^th associated to.

・ corresponds to the treatments 5^th associated to.

This geometric representation describes a new association scheme, we call it for convenience, generalized rectangular right angular association scheme (5) with five associated classes, to which we give the following equivalent definition:

Definition 6. A generalized rectangular right angular association scheme (5) is an arrangement of v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments in w (n × l) rectangular arrays such that, with respect to each treatment:

1) The first associates of are the other treatments of the same row in the same array.

2) The second associates of are the other treatments of the same column in the same array.

3) The third associates of are the remaining treatments in the same array.

4) The fourth associates of are the treatments of same row as, of the other arrays.

5) The fifth associates of are the remaining treatments in the other arrays.

Property 2. The parameters of the generalized rectangular right angular association schemes (5) are:

, , , , ,

Definition 7. A PBIB design based on a generalized rectangular right angular association scheme (5) is called generalized rectangular right angular GPBIB₅ design.

2.3. Generalized Rectangular Right Angular Association Scheme (7)

Let V be a set of v = wnl treatments, (w ≥ 2, n ≥ 2, l ≥ 2), to which we associate a geometrical representation in the following way:

Each treatment of V is associated with a unique triplet of the set where:

Let be a treatment of coordinates

・ corresponds to the treatments 1^st associated to

・ corresponds to the treatments 2^nd associated to.

・ corresponds to the treatments 3^th associated to.

・ corresponds to the treatments 4^th associated to.

・ corresponds to the treatments 5^th associated to.

・ corresponds to the treatments 6^th associated to.

・ corresponds to the treatments 7^th associated to.

This geometric representation describes a new association scheme, we call it for convenience, generalized rectangular right angular association scheme (7) with seven associated classes, to which we give the following equivalent definition:

Definition 8. A generalized rectangular right angular association scheme (7) is an arrangement of v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments in w arrays of n rows and l columns such that, with respect to each treatment:

1) The first associates of are the other treatments of the same row in the same array.

2) The second associates of are the other treatments of the same column in the same array.

3) The third associates of are the remaining treatments in the same array.

4) The fourth associates of are the treatments in the same row and the same column as, of the other arrays.

5) The fifth associates of are the treatments of the same row as in the other arrays, that are different from the fourth associates of.

6) The sixth associates of are the treatments of the same column as in the other arrays, that are different from the fourth associates of.

7) The seventh associates of are the remaining treatments in the other arrays.

Property 3. the parameters of generalized rectangular right angular association schemes (7) are:

, , , , ,

, ,

Definition 9. A PBIB design based on a generalized rectangular right angular association scheme (7) is called generalized rectangular right angular GPBIB₇ design.

3. Construction Method of GPBIB m Designs (m = 4, 5 and 7 Associated Classes)

Let v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments be arranged in w arrays of n rows and l columns, ..., and written as follows:

3.1. Construction Method of GPBIB 4 Designs

3.1.1. First Construction Method of GPBIB 4 Designs

Applying the Combinatory Method (s) on each of the w arrays, with chosen, then we obtain w rectangular PBIB designs. The set of all blocks gives a PBIB design with 4 associated classes.

Theorem 1. The partially balanced incomplete block designs with the parameters:

, , , , , , ,

are generalized rectangular right angular GPBIB₄ designs.

Proof. For each array of the w arrays, we obtain a rectangular design with parameters:, , , , , ,. (see [5] ).

For the w arrays we obtain a generalized rectangular right angular GPBIB₄ design with parameters:, , , , , ,.

: Two treatments and from the arrays and respectively they never appear together in the same block thus.

Lemma 1. For the special case s = l, the previous method can also be used for the construction of nested group divisible designs, with parameters:

, , , , , ,

Remark 1.

・ For w = 1, the GPBIB₄ design of Theorem 1 is a rectangular design with parameters as in the Theorem 1 of [5] .

・ For w = 2, the GPBIB₄ design of Theorem 1 is a rectangular right angular PBIB₄ design with parameters as in Proposition 1 of [8] .

Proposition 2. Let GPBIB₄ be a design with parameters:

, , , , , , ,.

For n or l even and s divisor of l or s, the GPBIB₄ design is a resolvable PBIB designs (RGPBIB₄) with r parallel classes where each parallel classes contain blocks.

Proof.

n or l is even and s is divisor of n or l, then.

Example 1. Let v = 3 × 4 × 4 treatments be arranged in the three following arrays:

The construction method for (s = 2), give the following resolvable generalized rectangular right angular GPBIB₄ design, with the parameters:

3.1.2. Second Construction Method of GPBIB 4 Designs with

Let, and,.

Applying the Combinatory Method with chosen on each array of the form for, and, by only considering the combinations of s treatments that always contain a component of the vector, the set of all the obtained blocks provides a PBIPwith 4 associated classes.

Theorem 3. The partially balanced incomplete block designs with the parameters:

, , , , ,

, ,

are generalized rectangular right angular GPBIB₄ designs.

Proof.

・ The v and k values are obvious.

・ r: For each treatment of the array , we have:

○ On an array , applying the procedure with the l other elements of the same row. There is possibilities, each one being repeated times, with n permutations, then we have repetitions. Therefore, we have arrays of the form ; so we have repetitions of the treatment.

○ On an array , and, applying the procedure with the

other elements of the same row. There is possibilities each one being repeated times, with n permutations, then we have appearances repeated themselves n times. Therefore, for an array we have repetitions of. Considering all the arrays for and; so we have repetitions of.

Thus:.

・ : Consider two treatments and from the array (and), they appear together times with the other elements of the same row i of the arrays , with n permutations, we obtain times in which the two treatments appear together. Considering all the arrays for and; then we have times where the two treatments and appear together.

・ : Consider two treatments and from the array, (and), we have:

○ In an array : the two treatments appear together times, with the n permutations they appear together times. Considering all the arrays of the form for and, they appear together times.

○ In the array: both treatments appear together times, with the n permutations

they appear times. Considering all the arrays of the form for; they appear together times.

In total.

・ : Consider two treatments and from the array (,and), they appear together times for each array of the form , with the n permutations they appear together times. Considering all the arrays for and, then the two treatments and appear together times.

・ : Consider two treatments and from the arrays and respectively (g,):

○ If, for the array and h = 1 the two treatments appear together

times. For the array and h = 1 they also appear together time, so we

have times. On the other hand, for, the two treatments appear together times for the array and times for the array, so we have for one value of h. Taking all the values of, we obtain times where the two treatments appear together.

In total.

○ If, then the two treatments appear together times for the array of the form and appear together times for the array of the form for, so we have times. For, among the permutations of the vector, and for a given value of, the treatment takes the same row as then the two treatments appear together times, for the remaining values the two treatments appear together times. For, among the permutations of the vector and for a given value of, the treatment takes the same row as then the two treatments appear together times, for the remaining values the two treatments appear together times.

In total.

・ : Using the above construction method on each array of the form:, we obtain blocks. So for the l arrays of the form we have blocks, with the n permutations we obtain blocks. Considering all the arrays for, then in total we have blocks. Considering all the arrays for, then in total we have blocks.

Remark 2. For w = 2, the GPBIB₄ design of Theorem 3 is a rectangular right angular PBIB₄ design with parameters as in Proposition 2 of [8] .

Proposition 4 Let GPBIB₄ be a design with parameters:

, , , , ,

, ,

For n or l even and s divisor of l or s, the GPBIB₄ design is a resolvable PBIB designs (RGPBIB₄) with r parallel classes where each parallel classes contain blocks.

3.2. Construction Method of GPBIB 5 Designs

Let be the j^th column of the g^th array. Applying the Combinatory Me-

thod (s) with s chosen, on each array in the form for and by only considering the combinations of s treatments that contain a component of the vector, then the set of all the blocks obtained, gives a PBIB design with 5 associated classes.

Theorem 5. The incomplete block designs with parameters:

, , , , ,

, , ,

are generalized rectangular right angular GPBIB₅ designs.

Remark 3. For w = 2, the GPBIB₅ design of Theorem 5 is a rectangular right angular PBIB₅ design with parameters as in Proposition 3 of [8] .

Proposition 6. Let GPBIB₅ be a design with parameters:

, , , , ,

, , ,

For n or l even and s divisor of l or s, the GPBIB₅ design is a resolvable PBIB designs (RGPBIB₅) with r parallel classes where each parallel classes contain blocks.

Example 2. Let v = 3 × 2 × 3 treatments be arranged in the two following arrays:

The construction method for (s = 3), give the following generalized rectangular right angular GPBIB₅ design, with parameters:

3.3. Construction Method of GPBIB 7 Designs

3.3.1. First Construction Method of GPBIB 7 Designs

Applying the Combinatory Method (s) on each of the w arrays, with chosen and fixed, then we obtain w rectangular PBIB designs. The juxtaposition of the blocks of the w rectangular PBIB designs, such that the blocks containing treatment and are put side by side, gives a PBIB design with 7 associated classes.

Theorem 7. The incomplete block designs with parameters:

, , , ,

, ,

are generalized rectangular right angular GPBIB₇ designs.

Proof. The design parameters are deduced from the construction method.

Remark 4.

・ For w = 1, the GPBIB₇ design of Theorem 7 is a rectangular design with parameters as in Theorem 1 of [5] .

・ For w = 2, the GPBIB₇ design of Theorem 7 is a rectangular right angular PBIB₇ design with parameters as in Proposition 4 of [8] .

Proposition 8. Let GPBIB₇ be a design with parameters:

, , , ,

, ,

For n or l even and s divisor of l or s, the GPBIB₇ design is a resolvable PBIB designs (RGPBIB₇) with r parallel classes where each parallel classes contain blocks.

Example 3. Let v = 3 × 4 × 4 treatments be arranged in the three following arrays:

The construction method for (s = 2), give the following resolvable generalized rectangular right angular GPBIB₇ design, with the parameters:

3.3.2. Second Construction Method of GPBIB 7 Designs with

Let be the j^th column of the g^th array and let be the i^th row

of the g^th array. Then applying the Combinatory Method (s) with chosen on each

array of the form for and g, (respectively

for and g,), by only considering the combinations of s

treatments that always contain a component of the column (respectively the row), the set of all the obtained blocks provides a PBIP with 7 associated classes.

Theorem 9. The partially balanced incomplete block designs with the parameters:

, , , ,

, , , ,

, ,

are generalized rectangular right angular GPBIB₇ designs.

Proof. The design parameters are deduced from the construction method.

Remark 5. For w = 2, the GPBIB₇ design of Theorem 9 is a rectangular right angular PBIB₇ design with parameters as in Proposition 5 of [8] .

Proposition 10. Let GPBIB₇ be a design with parameters:

, , , ,

, , , ,

, ,

For n or l even and s divisor of l or s, the GPBIB₇ design is a resolvable PBIB designs (RGPBIB₇) with r parallel classes where each parallel classes contain blocks.

Example 4. Let v = 3 × 3 × 3 treatments be arranged in the three following arrays:

The construction method for (s = 3), give a generalized rectangular right angular GPBIB₇ design with parameters:

To illustrate the method, we applying the construction method for the columns and rows of the first array, where each column represents a block:

4. Construction of the U-Type Designs Based on Resolvable GPBIB m Designs (m = 4, 5 and 7)

In this section we apply the RBIBDmethod (see [6] ) on our resolvable rectangular right angular RGPBIB_m designs (m = 4, 5 and 7) to obtain a series of U-type designs.

Let a resolvable GPBIB_m designs (m = 4, 5 and 7) with r parallel classes where each j-th class contains blocks (1 ≤ j ≤ r). Then we can construct a U-type design from resolvable GPBIB_m designs (m = 4, 5 and 7) as follows:

Algorithm RGPBIB_m − UD

・ Step 1. Give a natural order to the q blocks in each parallel class PC_j,.

・ Step 2. For each PC_j, construct a q-level column as follows: Set, if treatment is contained in the u-th block of PC_j,

・ Step 3. The r q-level columns constructed from PC_j, form a.

Proposition 11. For v = wnl runs (w ≥ 2, n ≥ 2, l ≥ 2), a series of U-type designs exist:

・ U, n or l even and s divisor of n or l.

・ U, n or l even and s divisor of n or l.

・ U, n or l even and s divisor of n or l.

・ U, n or l even and s divisor of n or l.

・ U, n or l even and s divisor of n or l.

Proof. applying the RGPBIB_m − UD Algorithm on each resolvable rectangular right angular GPBIB_m (m = 4, 5 and 7) of the Proposition 1, 5.

Example 5. Applying the RGPBIB_m − UD Algorithm on the resolvable rectangular right angular GPBIB₇ of Example 1, we obtain the following U-type U (48, 4⁹) with 48 runs and nine 4-level factors.

5. Conclusions

New association schemes with m = 4, 5 and 7 associated classes called generalized rectangular right angular association schemes for v = wnl treatments arranged in w ≥ 2 (n × l) arrays were described and their parameters expressions were given exactly and directly. Some construction methods of PBIB designs based on these association schemes accommodated by accessible method called the Combinatory Method (s) which facilitates the

construction application were explained. Moreover, a series of U-type designs U, by applying the

Fang RBIBD method on resolvable generalized rectangular right angular GPBIB_m designs (m = 4, 5 and 7) was constructed.

We note that all the construction methods described in this article were programmed with the R-package “CombinS” [9] (the ameliorated version).

References

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