1. Introduction
Computer experiments are an effective method for exploring complex systems and scientific problems [
1,
2].The space-filling properties, which measure the uniformity of the design points in the experimental space, are critical for effectively exploring the experimental region of computer experiments [
2]. Latin hypercube designs (LHDs), proposed by [
3], are widely used space-filling designs for computer experiments. Such designs are often used in computer experiments with quantitative factors because they achieve optimal univariate uniformity. Computer experiments involving only quantitative factors have received considerable attention [
1,
2]. However, researchers usually encounter computer experiments involving both qualitative and quantitative factors; see [
1,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13].
Sliced Latin hypercube designs (SLHDs) proposed by [
14] are LHDs that can be partitioned into some LHD slices, which not only maintain the optimal univariate uniformity but for each slice as well. SLHDs are popular for computer experiments with both qualitative and quantitative factors; see [
9,
10,
15] and the references therein. Each slice of an SLHD can be used at one level combination of the qualitative factors. However, its number of runs increases dramatically with the number of level combinations of the qualitative factors. This is thus suitable for situations where there are few level combinations of the qualitative factors or where the cost of runs is low.
Inspired by the notion of SLHD, [
16] proposed the marginally coupled designs (MCDs). Their key feature is that for each level of any qualitative factor, the design points for the quantitative factors can form a small LHD, and they have fewer runs than SLHDs. In recent years, improvements for MCDs include, but are not limited to, its quantitative factors design with column orthogonality and multi-dimensional stratifications; for more details, refer to [
17,
18,
19,
20]. MCD, however, appears to be inapplicable when it is necessary to study the stratification between multiple qualitative factors and quantitative factors, whereas a design with such properties can be useful for studying the interaction between two qualitative factors and quantitative factors.
To this end, [
21] proposed the doubly coupled designs (DCDs). It not only maintains the properties of MCDs, but also ensures that the design points for the quantitative factors can form an LHD corresponding to any level combination of any two qualitative factors. In a DCD, the subdesign for qualitative factors is an orthogonal array (OA). Equal-level and mixed-level orthogonal arrays are called symmetric and asymmetric orthogonal arrays, respectively. In the DCDs constructed by [
21], the subdesign for qualitative factors is a symmetric orthogonal array. However, in real-world problems, there exist qualitative factors with mixed levels, and the design of the qualitative factors is usually an asymmetric OA. At present, there are no studies of DCDs with qualitative factors being asymmetric OAs. The latter construction cannot be a simple extension of the former. Moreover, the existing DCDs have an upper bound on the number of qualitative factors, namely, no more than the number of levels of qualitative factors. Therefore, existing DCDs are inapplicable when the qualitative factors are mixed-level or when the number of qualitative factors exceeds the number of their levels.
For a computer experiment with q s-level qualitative factors and p quantitative factors, an MCD is appropriate if there is no interaction effect between any two qualitative factors and all quantitative factors; if and there is the interaction effect between any two qualitative factors and all quantitative factors, a DCD is applicable. However, neither an MCD nor a DCD is suitable, when , some qualitative factors and all quantitative factors have such interaction effects, and some do not. Suppose that in an experiment there are four qualitative factors, the type of concentration of cell lysis reagent (A1, A2), the type of stain (Blue, Red, Pink), the shape of the cell slides (Thick, Moderate, Thin), and the cells’ activity (Dead, Alive) as well as other quantitative factors. We know that only the two qualitative factors, the type of concentration of cell lysis reagent and the shape of the cell slides, have the interaction effect with all quantitative factors. Obviously, both an MCD and a DCD are not suitable for such an experiment. Thus, we need to adopt a design that satisfies two properties: (i) the whole design is an MCD; and (ii) the columns of some qualitative factors and the columns of all quantitative factors form a DCD. In this paper, we focus on such designs and call them group DCD (GDCD).
In addition, not only can the GDCDs contain more qualitative factors, but the designs for the qualitative factors can be asymmetric OAs. Therefore, the level types of GDCDs are more flexible than those of DCDs. Our methods construct two forms of GDCDs, within-group DCDs and between-group DCDs. In a within-group DCD (WGDCD), the design of the qualitative factors can be divided into several groups, and the design of any two qualitative factors from the same group coupled with the design of the quantitative factors is a DCD. Thus, columns in the same group have excellent stratification properties between qualitative and quantitative factors. In a between-group DCD (BGDCD), the design of the qualitative factors can also be split into several groups, and the design of any two qualitative factors from different level groups combined with the design of quantitative factors is a DCD. The methods for constructing WGDCDs and BGDCDs are similar and easy to implement and are given in
Section 3.1 and
Section 3.2, respectively. Since the space-filling property of GDCDS is similar to that of DCDs, the space-filling property of GDCDs is better than that of MCDs.
The article is organized as follows:
Section 2 introduces the basic notation and definitions. Methods for constructing GDCDs and the corresponding examples are given in
Section 3. Comparison was made in
Section 4.
Section 5 provides conclusions and discussion. All proofs are deferred to
Appendix A.
2. Definitions and Notation
Let
denote the Galois field of order
s, where
and
. An
matrix
D is called a difference scheme over
, denoted by
, if it has the property that every element of
in the vector difference between any two distinct columns in
D occurs
times equally. For details of the difference schemes, refer to Section 6.1 of [
22]. An
matrix is called an asymmetric orthogonal array of strength
t, denoted by
,
, if any of its
submatrix satisfies all possible t-tuples occur equally often, where the level of the first
columns is taken from
, the level of the next
columns is taken from
, and so on. When all the
’s are equal to
s, the orthogonal array is symmetric, denoted by
.
We now review the Rao-Hamming construction in Section 3.4 of [
22]. For a prime power
s, let
and
be two
s-level columns of length
with entries from
,
, where
and
. Suppose that
and
are independent. We apply the Rao-Hamming construction in [
22] to obtain an
Y, i.e.,
over
, where
.
An matrix that each column is a permutation of integers is called an LHD, denoted by . and are two n-dimensional column vectors with all entries being zeros and ones, respectively. Let represent the transposition of matrix A. For an matrix A and an matrix B, = and = represent the Kronecker sum and Kronecker product, respectively, where is the th entry of A.
Suppose there is an , if its rows can be divided into ’s, and for j = 1, 2, remains the same, then called the array as -resolvable orthogonal array, denoted by . Especially, when , then the array reduces to . If , the array is called a completely resolvable orthogonal array (CROA).
Let be an n-run design with q qualitative factors and p quantitative factors, where the subdesigns A and L are qualitative factors and quantitative factors, respectively. The design D is called a marginally coupled design, denoted by , , if it satisfies: (i) A is an ; (ii) L is an ; and (iii) the rows in L, corresponding to each level of any factor in A, form a small LHD. When , , the MCD is denoted as .
Let be an . The design is called a doubly coupled design, denoted by , if it satisfies that the rows in , corresponding to each level combination of any two factors in , form a small LHD. When , , the DCD is denoted as . Obviously, is an , is an , and the rows in , corresponding to each level combination of any t factors in , form a small LHD for .
Definition 1. Let be an , , where is an , L is an , is an , .
(i) The D is called a within-group DCD (WGDCD), denoted by , if is a DCD for . When and , then we denote such D by .
(ii) The D is called a between-group DCD (BGDCD), denoted by , if is a DCD, where and are the jth column in and the nth column in , respectively, for , j = , n = . When and , then we denote such D by .
From Definition 1, it is easy to see that
is a
in a
, for
. For any
, Corollary 1 of [
21] shows that
. Similarly, we have the following Corollary 1.
Corollary 1. If a WGDCD with A being an exists, then , .
Corollary 1 above tells us that the can accommodate up to qualitative factors.
Here we provide some results on the existence of GDCDs. Recall the definition of GDCDs, a GDCD of n runs has two subdesigns, A and L, which are for q qualitative factors and p quantitative factors, respectively. Theorems 1 and 2 below establish the necessary and sufficient conditions of the existence of WGDCDs and BGDCDs, respectively. For ease of expression, for an column vector d, define and based on d. Let be the vth entry of d, and , , where and are the vth entries in and , respectively, and represents the largest integer not exceeding a.
Theorem 1. Suppose is an , is an , where is the kth column of L, . Let be the jth column of for , j = . Then design is a if and only if:
(i) is an , for any , , ; and
(ii) is an , for any , , .
Theorem 2. Suppose is an , is an , where is the kth column of L, . Let be the jth column of for , j = . Then design is a if and only if:
(i) is an , for any , , ; and
(ii) is an , for any , , , .
Theorems 1 and 2 establish the existence of GDCDs in terms of the relations between the individual columns in A and , and between any pair of columns in A and or .