Mixed-Level Column Augmented Uniform Designs

08/17/2018 ∙ by Feng Yang, et al. ∙ Nankai University 0

In follow-up designs, some additional factors with two or three levels may be added in the follow-up stage since they are quite important but neglected in the first stage. Such follow-up designs are called mixed-level column augmented designs. In this paper, based on the initial designs, mixed-level column augmented uniform designs are proposed by using the uniformity criterion such as wrap-around L2-discrepancy (WD). The additional factors can be three-level or blocking, where the additional blocking factors can be regarded as additional common two-level factors. The multi-stage augmented situation is also investigated. We present the analytical expressions and the corresponding lower bounds of the WD of the column augmented designs. It is interesting to show that the column augmented uniform designs are also the optimal designs under the non-orthogonality criterion, E(f_NOD). Furthermore, a construction algorithm for the column augmented uniform design is provided. Some examples show that the lower bounds are tight and the construction algorithm is effective.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

In many industrial experiments such as semiconductor fabrication, the cost of each run is very expensive; therefore, small run sizes are preferred. Assume the number of runs in the initial design is much smaller than the number of parameters in the full second-order model, where the second-order model is commonly used to explore the nonlinear relationship between factors and responses. If analytical results based on the initial design satisfy the objective, experiment is terminated. Otherwise, a follow-up design may be needed to collect further information from the experimental system. Sometimes, a two-stage design may still not satisfy the needs, then more design points may be added stage by stage until meeting certain requirements. Such experimental strategy is called multi-stage augmented design.

Usually, in the field of the follow-up design, the researchers only considered adding some runs to the initial design but did not add any factor; see [17, 18, 4] and [21]. In many cases, however, the experimenters may only select some factors of interest, and ignore other possibly important factors because of the limited run size. For instance, Lai et al. [16] considered a real-world application of column augmented designs where there are seven factors in the initial design. Afterwards, in the second stage, a new glycerol factor was added to examine the possibility of enhancing the response lovastatin production. It turns out that the addition of is able to improve production significantly. Another example is from an industrial production, where the factor of reaction pressure is not considered to be significant initially. Thus, this factor was fixed as the standard atmosphere pressure (0.1 MPa), in order to limit the number of runs for saving cost. However, after analysis of the initial experiment, it shows that the reaction pressure may be an important factor and need further investigation in the follow-up stage. Usually, the initial design and the follow-up portions are usually conducted at different time, with different equipment or under different operators. The researchers may want to test whether there is a system difference or not for these nuisance variables between the different stages. Then a blocking factor should be added in the follow-up designs to analyze the significance of the nuisance variables. Therefore, in the follow-up stages, the experimenter may consider not only some additional runs but also some additional factors.

Yang et al. [27] considered augmenting not only some number of runs but also some two-level factors, in which the design structure for the two-level factors is very special such that they are completely correlated. Compared to [27], this paper discusses more complex situations with increased technical difficulty. Adding three-level factors is worth to be considered since they can be used to explore the nonlinear relationship based on second-order models. We consider the following types of follow-up designs:

  1. Augmenting several three-level factors to an initial design.

  2. Augmenting one or two blocking factors in the follow-up steps for assessing the system deviation.

  3. Constructing multi-stage augmented designs.

The blocking factors can be also treated as common two-level additional factors. Since the two-level, three-level and blocking factors may be added in the next stage, such follow-up designs are called mixed-level column augmented designs, which will have a wide range of applications. In particular, the resulting column augmented designs do not have any two completely correlated factors, which overcomes the shortcoming of that in [27].

As we know, for those expensive and time-consuming experiments, optimal designs are a widely used type of designs; see [23, 13, 11] and [20]. However, optimal designs need to specify a priori underlying models. When the relationship between the factors and the response is unknown, uniform design proposed by [7] is an effective experimental method. The main idea of uniform design is to scatter the design points uniformly on the experimental domain. Yue and Hickernel [28] showed that uniform design is robust against model specification. Xie and Fang [26] showed that uniform designs have the property of admissibility and minimaxity. Moreover, the number of runs in uniform designs is very flexible and can be chosen to be any integer. This appealing property often allows us to save experimental costs. Furthermore, the uniformity criterion has close relationship with the generalized minimum aberration criterion, which is widely used in orthogonal design theory; see [10] and [30]. Hence, the uniformity criterion is a reasonable consideration for assessing the goodness of the column augmented designs especially when the true model is unknown. In this paper, we discuss the best mixed-level column augmented designs under the uniformity criterion, wrap-around -discrepancy (WD, Hickernell, 1998), and call the resulting designs as mixed-level column augmented uniform designs.

Apart from the uniformity criterion, the orthogonality is also an important assessing criterion in experimental designs. criterion is popularly employed for comparing different designs from the viewpoint of non-orthogonality; see [9], [15] and [22]. The -value of any design is nonnegative, and less -value means the designs have better orthogonality. We will show that there exists some interesting relationship between the uniformity criterion WD and the criterion of the column augmented designs.

The rest of this paper is organized as follows. Section 2 gives the expressions of WD and the corresponding lower bounds of column augmented design with three-level additional factors and at most two additional blocking factors. Moreover, the multi-stage situation is also discussed. The criterion of the column augmented designs is discussed in Section 3, where the connection between WD and is also established. Section 4 presents the construction algorithm for the column augmented uniform designs, and shows some examples to demonstrate that the construction algorithm is powerful and the lower bounds in this paper are relatively tight. Some conclusions are summarized in Section 5. Some proofs are in Appendix A and the design matrices mentioned in Section 4 are listed in Appendix B.

2 WD criterion for column augmented design

According to the projection uniformity on one dimension, it is preferred to restrict the designs to be balanced, i.e., U-type designs. An asymmetric U-type design corresponds to a matrix , such that each column takes values from a set of integers, say , equally often. If some ’s are equal, we denote this asymmetrical U-type design by , where . Denote all of the and by and , respectively. For each , the runs of transform into points in by mapping . There are different criteria for measuring the uniformity. Among them the WD has many good properties; for example, it is invariant under reordering the runs, relabeling coordinates and coordinate shift. The squared WD-value of is where

2.1 Column augmented uniform designs with additional three-level factors

Let the initial design , . In the follow-up stage, one wants to add additional runs and additional three-level factors. It is reasonable to assume that each follow-up stage may not augment too many runs. Without loss of generality, let . Denote and be the matrix whose elements are zeros and ones, respectively. A design matrix means that each element of is chosen from the set .

Definition 1

A design is a column augmented design with additional three-level factors, if the initial design is augmented with , and . Denote all such column augmented designs by .

Remark 1

Each of the additional three-level factors can be a common quantitative or qualitative factor, or even a blocking factor. If , the column augmented designs become the row augmented designs defined by [27].

The explanation of Definition 1 is as follows. In the initial design, the level of initial ignored factors is usually fixed. For example, if the temperature is ignored in the initial stage, the researchers often set it to the room temperature in the initial runs. It can be easily shown that the WD-values of designs are not changed for mixed-level designs when permuting the levels of each column. Without loss of generality, assume all the levels of the additional factors be labeled as 0 in the initial design, then the initial design can be represented by with the factors. The design matrix in the follow-up stage is , where and are the design matrices in the second stage for the initial factors and the additional factors, respectively. When , to make the column augmented design as uniform as possible, the additional part should be limited to be U-type, and each column of the columns, , should occur the same number of 1 and 2 in the follow-up stage, because the design can be more uniform when the elements in each column are more balanced. Thereby, the number of adding runs meets the following requirements,

the cases of and the cases of
multiple of 2
multiple of 3
multiple of 6

Compared with other kinds of follow-up design, such as foldover and semifoldover designs which respectively limit and , our requirement for the number of additional runs is more flexible. Moreover, it should be mentioned that the restriction of to be a U-type design can be relaxed, i.e., one can augment any number of runs based on the initial design. In the rest of the paper, we will consider the cases when is U-type.

According to Definition 1, there are many alternative column augmented designs for a given initial design . Under the uniformity criterion, one tends to add the follow-up part such that the column augmented design with points is as uniform as possible.

Definition 2

A column augmented design is a column augmented uniform design if has the smallest WD-value among the design set.

According to the expression of WD, we can get the squared WD-value of the column augmented design ,

(1)

where is the total number of runs and is the total number of factors for the column augmented design.

In practice, we often choose a uniform design for the initial design when the relationship between factors and response is unknown, and we may terminate the experiments based on the data analysis of the first stage. If more runs and factors should be added after , one searches the follow-up part such that the column augmented design is as uniform as possible. Moreover, the initial design is assumed to be known in the follow-up stage. Therefore, it is necessary to derive the expression of the WD of the column augmented design based on . According to the coincident numbers between any two rows, which represents the number of places where two rows of the design take the same value, we can rewrite the expression of the WD-value in (1).

Proposition 1

Given an initial design , its column augmented designs have

(2)

where

(3)

, is the number of elements in the set .

The WD-value of column augmented design  is a function of . The equation (2) can be used for obtaining the lower bound of WD-values of the column augmented designs, which can be served as a benchmark to judge that whether a design is uniform or not. If the WD-value of a design reaches the lower bound, then this design must have the smallest WD-value among the design space, i.e., it is a uniform design.

Theorem 1

For a given initial design , its column augmented designs have where

(4)

and refers to (3), , The lower bound can be achieved if and only if , , and , .

The column augmented design for initial mixed-level design can be reduced to that for initial symmetrical two-level or three-level designs, i.e., becomes when , or when , according to actual demands. For these cases, we can derive more accurate lower bounds for these cases.

Theorem 2

Given an initial design , the WD-value of column augmented designs  has the lower bound

(5)

and is in (3), The lower bound can be reached if and only if , , and there are number of take , number of take , .

If the number of initial factors is even, according to the discussion after Lemma 2 in the Appendix A, the lower bound in Theorem 2 is equivalent to that in Theorem 1. However, if

is odd, the lower bound in Theorem

2 is more tight. For instance, choose the design in Example 2 of [6] as the initial design , and assume ; then both of the lower bounds in Theorem 1 and Theorem 2 are 53.5134, because is even. However, consider the design in Example 1 of [6] as the initial design , and let , then the lower bounds in Theorem 1 and Theorem 2 are 3.4094 and 3.4307, respectively, since is odd.

Theorem 3

Given an initial design , the lower bound of WD-value of column augmented designs  is

(6)

and refers to (3), The lower bound can be achieved if and only if number of take , number of take , , and number of take , number of take ,.

If and are integers simultaneously, the lower bound in Theorem 3 is equivalent to that in Theorem 1. Otherwise, the lower bound in Theorem 3 is more tight. The proof of Theorem 3 is analogous to that of Theorem 2 and we omit it.

2.2 Column augmented uniform designs with one additional blocking factor

This subsection discusses the column augmented designs with one additional blocking factor. Similarly, the levels of the blocking factor can be fixed as 0 in the first stage. For the level of the blocking factor in the second stage, we can often take 1. This is because the level of the blocking factor in the follow-up stage is often different from that in the first stage. A design is called a column augmented design with one additional blocking factor. Denote all such column augmented designs by . Furthermore, to judge the uniformity of this type of column augmented designs, the lower bound is presented.

Theorem 4

Given an initial design , or , the lower bound of the column augmented design  with one additional blocking factor is

where when , when , and when , respectively. Here, all the parameters are defined in Theorems 1-3.

For the column augmented design with a blocking factor , denote . One wants to search the best additional part and such that is uniform under WD. There is only a different positive coefficient between the lower bounds in Theorems 1- 3 and that in Theorem 4 for each term. Then, we have the following result directly, and omit its proof.

Proposition 2

Given an initial design , or , the column augmented design  achieves its lower bounds in Theorems 1-3 if and only if the corresponding column augmented design achieves its lower bound in Theorem 4.

Proposition 2 means that one can construct through , i.e., for constructing , one only needs to construct the corresponding , then add into to obtain the design .

2.3 Column augmented uniform designs with two additional blocking factors

If one wants to add two blocking factors in the follow-up design due to the practical requirements, as the same idea in Subsection 2.2, the level of the two blocking factors in the initial design should be 0 and that in the second stage should be 1, i.e., the design matrix of the two additional blocking factors is . However, the two columns in are fully correlated such that the effects of them cannot be distinguished. If we wish to assess these blocking effects accurately, we have to adjust the structure of the two additional blocking factors to reduce the correlation. Usually, if an initial design has been done, the design matrix is fixed and cannot be altered, we have to consider changing the structure in the second- stage. Naturally, replace by , denote . This solution sacrifices the level balance of the second additional blocking factor to reduce the non-orthogonality between the two additional blocking factors. For assessing the uniformity of , we have the following lower bound.

Theorem 5

Given an initial design , any has the following lower bound of WD-value,

and the function is defined in (3),  

The proof is similar to Theorem 1 and we omit it here. If the initial design is a symmetrical two-level or three-level , a lower bound can be derived in the same way.

Another method to solve the high correlation between the two additional blocking factors is replacing by where , which sacrifices the level balance of two blocking factors. Denote the corresponding column augmented design by . It can be easily seen that column augmented design  has smaller a WD-value than , because the former is more balanced than the latter.

Remark 2

The additional blocking factors in Subsections 2.2 and 2.3 can also serve as additional common two-level factors. Therefore, the column augmented design  and are column augmented designs with additional mixed-level factors.

In fact, whether the additional two-level factors are used for blocking or not does not affect the structure of the design matrix; it only influences the modeling aspect.

2.4 Multi-stage augmented designs

If there is no additional factor in the multi-stage design and only some rows are added in each stage, then the -stage row augmented design can defined as and the additional number of runs of the -th stage portion is . Specially, . For constructing three-stage row augmented design, one can take the first and the second stage design as the initial design, then add the third-stage portion as the follow-up portion. Next, take the first three stages portion as the initial design, and add the fourth-stage portion as the follow-up portion, and so on.

If there exists an additional blocking factor in the multi-stage column augmented design, as similar as the discussion before, let the levels of the blocking factor in stages take , respectively. For the initial design , the -stage column augmented design with one blocking factor, with three-level additional factors, and with three-level additional factors and one blocking factor can be respectively defined as follows,

It is assumed that the additional three-level factors may be considered in the second stage, after which no factor is added.

Proposition 3

Given an initial design , its -stage column augmented design  achieves its lower bound if and only if the corresponding -stage row augmented design achieves its lower bound, and we have where

(7)

and is the -stage row augmented design, ; , representing the total run number of the first stages, , ,, , .

The proof is in the Appendix B. The result in Proposition 3 is a generalization from two-stage to -stage situation. Thus, one can arrange -stage row augmented design stage by stage, and then construct -stage column augmented design with one additional blocking factor through adding to .

Similarly, the multi-stage column augmented design  reaches the lower bound if and only if the reaches the lower bound. However, its recursive lower bounds of multi-stage column augmented design  like (3) are too complicated, since the lower bounds closely depend on the total additional number of runs, . When , each column for each of the follow-up stages should take the same number of levels 1 and 2, for keeping the balance of the levels of the additional factors. When , let . Each column of the follow-up stages should take the same number of levels 1 and 2. While, each column of the follow-up stage should take values from . Hence, it has some technical difficulty for deriving the lower bounds of the -stage column augmented design  when . The Proposition 3 is based on the initial mixed-level design . Specially, under symmetrical two-level or three-level initial designs, similar results can be obtained easily.

3 criterion for column augmented design

In this section, we study the column augmented designs from the view of non-orthogonality, by using the criterion. For a design , define the non-orthogonality between the -th and -th columns of as where is the number of -pairs in the -th and -th columns in , and represents for the average frequency of all level-combinations in each pair of the -th and -th columns of . Here, the subscript stands for non-orthogonality of the design. The -th and -th columns of are orthogonal if and only if . Define , which measures the average non-orthogonality among the columns of design . Especially, is an orthogonal design if and only if . The smaller value implies that the design has better orthogonality. Therefore, one prefers a design with small . A design is -optimal if it has the smallest value of among the design space.

The lower bound plays a key role in detecting the