Uniformity criterion for designs with both qualitative and quantitative factors
UUniformity criterion for designs with both qualitative and quantitativefactors
Mei Zhang , Feng Yang , Yong-Dao Zhou ∗ & KLMDASR, Nankai University, Tianjin 300071, China
Abstract
Experiments with both qualitative and quantitative factors occur frequently in practical applications.Many construction methods for this kind of designs, such as marginally coupled designs, were proposedto pursue some good space-filling structures. However, few criteria can be adapted to quantify thespace-filling property of designs involving both qualitative and quantitative factors. As the uniformityis an important space-filling property of a design, in this paper, a new uniformity criterion, qualitative-quantitative discrepancy (QQD), is proposed for assessing the uniformity of designs with both typesof factors. The closed form and lower bounds of the QQD are presented to calculate the exact QQDvalues of designs and recognize the uniform designs directly. In addition, a connection between theQQD and the balance pattern is derived, which not only helps to obtain a new lower bound but alsoprovides a statistical justification of the QQD. Several examples show that the proposed criterion isreasonable and useful since it can distinguish distinct designs very well.
Keywords:
Balance pattern, Lower bound, Marginally coupled design, Qualitative-quantitativediscrepancy
Primary 62K15 ∗ Corresponding author
Email address: [email protected] (Yong-Dao Zhou ) Preprint submitted to Statistics January 8, 2021 a r X i v : . [ m a t h . S T ] J a n . Introduction Experimental designs with qualitative and quantitative factors have received growing attention in re-cent years, see Han et al. (2009), Qian et al. (2008) and Zhou et al. (2011) for computer experiments;Wu and Ding (1998), Aggarwal et al. (2000), Chantarat et al. (2003) and Tunali and Batmaz (2003) forresponse surface designs. For computer experiments, there are two systematic approaches to handlethis issue. Qian (2012) suggested using sliced Latin hypercube designs to accommodate qualitativeand quantitative factors, where each slice of quantitative factors corresponds to a level combinationof qualitative factors. However, the run sizes of such designs can be very large even for a moderatenumber of qualitative factors. Inspired by this, Deng et al. (2015) proposed a new class of designs,marginally coupled designs (MCDs), of which the design points for quantitative factors form a Latinhypercube design (LHD), and for each level of any qualitative factor, the corresponding design pointsfor quantitative factors compose a smaller LHD. Intuitively, MCDs have some desirable space-fillingproperties. A series of construction methods for MCDs had been presented, see Deng et al. (2015), Heet al. (2017) and He et al. (2019). Except for designs for computer experiments with both qualitativeand quantitative factors, the response surface designs with qualitative and quantitative factors havebeen studied and applied extensively in science and engineering applications. Wu and Ding (1998)proposed a general approach for constructing response surface designs of economical size with quali-tative and quantitative factors. Aggarwal et al. (2000) applied a dual response surface optimizationtechnique for constructing robust response surface designs with quantitative and qualitative factors.Tunali and Batmaz (2003) suggested a methodology for developing a simulation metamodel involvingboth quantitative and qualitative factors to deal with various strategic issues, such as metamodel es-timation, analysis, comparison, and validation. Moreover, the response surface methodology was alsodeveloped in Choquette-Labbe et al. (2014), to investigate the photocatalytic degradation of phenoland phenol derivatives using a Nano-TiO catalyst. In their study, two quantitative factors (TiO particle size and temperature) and one qualitative factor (reactant type) were considered.Let D = ( D , D ) be a design with n runs for p qualitative and q quantitative factors, in which D and D are sub-designs for qualitative and quantitative factors, respectively. Given the parameters( n, p, q ), a great number of MCDs can be constructed. It arises a natural issue that: which one isthe best. In other words, it needs some proper criterion to distinguish these MCDs. For example,2 Figure 1: The scatter plots for the D (1)2 and D (2)2 in (1), where the points of D ( j )2 ( j = 1 ,
2) corresponding to level 0and 1 of D are represented by “ ◦ ” and “ × ”, respectively. consider two MCDs with 8 runs for one qualitative and two quantitative factors, D (1) = ( D , D (1)2 ) and D (2) = ( D , D (2)2 ) (1)where D = (0 , , , , , , , T , D (1)2 = (cid:0) (0 , , , , , , , T , (0 , , , , , , , T (cid:1) , and D (2)2 = (cid:0) (0 , , , , , , , T , (2 , , , , , , , T (cid:1) . Figure 1 shows the scatter plots of sub-designs D (1)2 and D (2)2 with respect to the levels of D . From the intuition, D (2) outperforms D (1) since the wholepoints of D (1)2 as well as the corresponding points of D (1)2 with respect to each level of D , locate onthe diagonal line, respectively.If the number of factors is relatively small, visually, the scatter plots can help us to make ajudgment, whereas for the high-dimensional cases (even for p > q > , m . For example, Hickernell (1998,1999) used the tool of reproducing kernel Hilbert spaces to define the centered L -discrepancy (CD)and the wrap-around L -discrepancy (WD), and Zhou et al. (2013) proposed the mixture discrepancy(MD). Nevertheless, these discrepancies are only applicable for designs with quantitative factors.For designs with a finite number of levels, similar to CD and WD, Hickernell and Liu (2002)and Liu and Hickernell (2002) proposed the discrete discrepancy (DD) that was directly defined on adiscrete domain. It seems that DD is the only discrepancy we can use when the inputs are qualitative.Certainly, if we recognize that the levels of quantitative inputs are taken from a finite set, the DDalso works for quantitative factors. Hence, the DD could be geared to the designs with both typesof factors. While Zhou et al. (2008) pointed out that the DD is not an effective uniformity criterionfor designs with multi-level quantitative factors as the DD is based on the Hamming distance. Forinstance, it can be easily checked that D (1) and D (2) in (1) have the same DD-value, which alsoreaches the lower bound of DD in Qin and Fang (2004). That is, D (1) and D (2) are the best designsunder the DD, simultaneously. This result is not in accord with the intuition since D (2) should bemore uniform than D (1) from Figure 1, which indicates that the DD is not suitable for assessingthe uniformity of the designs with both qualitative and multi-level quantitative factors. Therefore, itcalls for a new uniformity criterion for designs containing both quantitative and qualitative factors.To address this issue, in this paper, we propose a new uniformity criterion qualitative-quantitativediscrepancy , to assess the uniformity of designs with both types of factors.The remainder of this paper is organized as follows. Section 2 first uses the Hickernell’s approachto derive the proposed discrepancy, QQD, then gives the quadratic form of the QQD for finding theuniform designs. The lower bounds of the QQD that provides a simple way to recognize the uniformdesigns, are derived in Section 3. Section 4 shows the relationships between QQD and balance pattern,and induces another lower bound. Some illustrative examples are given in Section 5 to explain thereasonability and application of QQD, and demonstrate the tightness of the obtained lower bounds inSections 3 and 4. The last section gives some conclusions. All the proofs are listed in the Appendix.4 . The qualitative-quantitative discrepancy In the field of uniform designs, U-type designs are generally preferred because of their projectionuniformity on each dimension. A U-type design is an n × m matrix X = (d , . . . , d m ), of which the k th column takes values from the set of { , , . . . , s k − } equally often. Denote all of such designsby U ( n, s · · · s m ). If some s k ’s are equal, we rewrite them as U ( n, s m · · · s m l l ) with m = (cid:80) lk =1 m k .Denote U ( n, s · · · s p s p +1 · · · s p + q ) as all the U-type designs whose first p factors are qualitative andthe last q factors are quantitative, then D = ( D , D ) ∈ U ( n, s · · · s p s p +1 · · · s p + q ) implies that D ∈ U ( n, s · · · s p ) and D ∈ U ( n, s p +1 · · · s p + q ). Specially, when s p +1 = · · · = s p + q = n , D is anLHD( n, q ). According to the structure of MCDs, MCDs are also U-type designs.We first review the derivations of the discrepancies, and give the formula of the QQD. Then, thequadratic form for the QQD is induced to help to obtain the uniform design under the QQD. Let F be the uniform distribution function on the design region χ (= χ × · · · × χ m ), and F n be theempirical distribution of a set of points X = { x , . . . , x n } . For a given kernel function K ( t , z ), anydiscrepancy of design X with n runs and m factors can be defined by (see Fang et al. (2018)) D ( X , K ) = (cid:90) χ K ( t , z )d( F ( t ) − F n ( t ))d( F ( z ) − F n ( z ))= (cid:90) χ K ( t , z )d F ( t )d F ( z ) − n n (cid:88) i =1 (cid:90) χ K ( t , x i )d F ( t ) + 1 n n (cid:88) i,j =1 K ( x i , x j ) . (2)The discrepancy in (2) is devoted to measuring the difference between F n and F , hence, the smallerits value is, the more uniformly the design points spread on χ . A design is called a uniform design ifit has the minimal discrepancy over the design space. From (2), the discrepancy is determined by thekernel function K ( · , · ), that is, a kernel function will give rise to a type of discrepancy. Usually, K ( · , · )has a product form K ( t , z ) = m (cid:89) k =1 K k ( t k , z k ) , (3)where K k ( t k , z k ) represents the kernel of the k th factor. It should be noted that the D ( X , K )defined in (2) with the product formed kernel K ( · , · ) in (3) considers the space-filling properties onprojections to all subsets of factors. For any subset u ⊆ { m } , let χ u be the experimental region5f the corresponding factors whose indices are in u , and t u be the vector containing the componentsof t indexed by u . Denote (cid:101) K u ( t u , z u ) = (cid:81) k ∈ u [ K k ( t k , z k ) −
1] when u (cid:54) = ∅ and 1 otherwise. Then K ( t , z ) = (cid:80) u ⊆{ m } (cid:101) K u ( t u , z u ) by the induction. At this moment, the discrepancy in (2) could berewritten as D ( X , K ) = (cid:90) χ (cid:88) u ⊆{ m } (cid:101) K u ( t u , z u )d( F ( t ) − F n ( t ))d( F ( z ) − F n ( z ))= (cid:88) u ⊆{ m } (cid:90) χ u (cid:101) K u ( t u , z u )d( F ( t ) − F n ( t ))d( F ( z ) − F n ( z )) . It implies that the defined D ( X , K ) could measure not only the uniformity of X on χ , but alsoprojection uniformity of X on any χ u , where u is a non-empty subset of { , . . . , m } .Recall that the WD is valid for the design with continuous inputs. If one transforms its levels intothe unit cube [0 ,
1] by (2 x + 1) / (2 s k ) , x = 0 , . . . , s k −
1, the kernel function for the k th factor is definedby K k ( t k , z k ) = − | t k − z k | + | t k − z k | . In addition, the DD is intended for the designs with finitenumbers of levels, whose kernel function of the k th factor is K k ( t k , z k ) = a δ tkzk b − δ tkzk with a > b ,where δ t k z k equals 1 if t k = z k and 0 otherwise, and the design region is χ k = { , , . . . , s k − } . Fora design D = ( D , D ) ∈ U ( n, s · · · s p s p +1 · · · s p + q ). It is reasonable to allocate the kernel of DD tothe qualitative factors of D and the kernel of WD to the quantitative factors of D . By doing this,the kernel function for the k th factor of D is K k ( t k , z k ) = a δ tkzk b − δ tkzk , for k = 1 , . . . , p , − | t k − z k | + | t k − z k | , for k = p + 1 , . . . , p + q . (4)We call the discrepancy with respect to kernel function in (4) as the qualitative-quantitative discrep-ancy. Based on (2), (3) and (4), it can be seen that the value of QQD depends on the parameters a and b . Thus the choice of a and b is a crucial issue for the proposed QQD. Let us focus onthe kernel functions for the two types of factors. The kernel function for the qualitative factors, K k ( x ik , x jk ) , k = 1 , . . . , p, has the lower bound b and upper bound a , while the kernel function for thequantitative factors, K k ( x ik , x jk ) , k = p + 1 , . . . , p + q , has the lower bound 5 / /
2. In general, if we do not have any prior information about the importance of factors, just treatthem comparably. In this sense, let a = 3 / b = 5 / a = 3 / = 5 /
4. Under this, we can derive the closed form of the expression for QQD.
Theorem 1.
For a design D ∈ U ( n, s · · · s p s p +1 · · · s p + q ) , the expression of the squared QQD is asfollowsQQD ( D ) = C + 1 n n (cid:88) i,j =1 (cid:18) (cid:19) p (cid:18) (cid:19) δ ij ( D ) · p + q (cid:89) k = p +1 (cid:18) − | x ik − x jk | + | x ik − x jk | (cid:19) . (5) where C = − (cid:81) pk =1 (cid:16) s k +14 s k (cid:17) (cid:0) (cid:1) q , δ ij ( D ) represents the coincidence number between the i th and j throws of D . It is noted that, although the proposed QQD is motivated by comparing different MCDs, it is notlimited to MCDs. According to the expression in (5), QQD can be used for quantifying the uniformityof any design with both qualitative and quantitative factors.We now review the MCDs in (1) to illustrate that the proposed criterion, QQD, is more effectiveand appropriate than DD to measure the goodness of different designs with both types of factors.
Example 1.
Based on the expression of QQD in Theorem 1, for the two MCDs in (1), QQD ( D (1) ) =0 . ( D (2) ) = 0 . D (2) isbetter than design D (1) , which is coincident with the scatter plot in Figure 1. As discussed in Section1, the DD cannot distinguish the two different designs at all. Consequently, the QQD is more sensitivethan the DD to evaluate the uniformity of designs associated with qualitative and quantitative factors.Here, we also compute the extended MaxPro criterion value of D (1) and D (2) by the expression inJoseph et al. (2019) and obtain the corresponding values 8 . . D (2) isbetter than design D (1) under this criterion. The agreement with the result under the QQD criterionfurther certifies the reasonability of QQD.In Example 1, it is reflected that both the QQD and the extended MaxPro criterion performwell when they are used to measure the space-filling property of designs with both qualitative andquantitative factors, although the two criteria are defined from different points. In this subsection, the uniform designs among the design space under the QQD criterion are discussed.We first present the quadratic form of the QQD and then solve the optimization problem of a specialconvex quadratic programming to obtain the uniform designs.7or ease of presentation, given a design D = ( D , D ) ∈ U ( n, s · · · s p s p +1 · · · s p + q ), a new notationis introduced. Let y = y ( D ) be an N (= (cid:81) p + qk =1 s k )-dimensional column vector with components n ( i , . . . , i p + q ) arranged lexicographically, where n ( i , . . . , i p + q ) is the number of runs at the levelcombination ( i , . . . , i p + q ) in design D . Obviously, y is a non-negative vector and satisfies TN y = n .Moreover, if each element of y is an integer, the design is called an exact design, otherwise, a continuousdesign. Any U-type design is an exact design. Based on the concept of y ( D ), the expression ofQQD ( D ) in (5) can be rewritten as a quadratic form given in the following lemma. Lemma 1. If D ∈ U ( n, s · · · s p + q ) and y = y ( D ) , we haveQQD ( D ) = − p (cid:89) k =1 (cid:18) s k + 14 s k (cid:19) (cid:18) (cid:19) q + 1 n y T Ay , where A = A (cid:78) A (cid:78) · · · (cid:78) A p + q , A k = ( t kij ) , i, j = 1 , . . . , s k , k = 1 , . . . , p + q , and (cid:78) is theKronecker product, furthermore, t kij = (cid:0) (cid:1) δ ij (cid:0) (cid:1) − δ ij , for k = 1 , . . . , p, − | i − j | ( s k −| i − j | ) s k , for k = p + 1 , . . . , p + q . Lemma 1 indicates that, essentially, QQD ( D ) is a convex function of y with some constraints.Provided this quadratic form, the theory of convex optimization can be utilized to solve the optimiza-tion problem of minimizing QQD ( D ). The corresponding solution D ∗ is the uniform design underthe QQD criterion. The rest of this section aims at exploring the uniform designs under the QQD.The proof of Theorem 2 is analogous to that of Theorem 1 in Zhou et al. (2012), which relies onLemma 3 in Appendix. For simplicity, we omit it. Theorem 2.
The design D ∗ minimizes QQD ( D ) over U ( n, s · · · s p + q ) when y ( D ∗ ) = nN N . Theorem 2 provides a sufficient condition for a design D ∈ U ( n, s · · · s p + q ) being a uniform designunder QQD. It should be noted that, in the proof of Theorem 2, the constrains of y are relaxed to y ≥ N and TN y = n as in Zhou et al. (2012). Thus the obtained optimal design D ∗ under the QQDis not an exact design unless n (mod N ) = 0. The uniform design D ∗ with y ( D ∗ ) = nN N impliesthat if the points at each level combination have the same weight in a design, the design achievesthe best uniformity under the QQD criterion. For an exact uniform design, the squared QQD can becalculated by a concise expression, of parameters only, given by the following Corollary 1.8 orollary 1. When n = cN with c being a positive integer, the D ∗ in Theorem 2 is a repetition ofa full factorial design and its squared QQD isQQD ( D ∗ ) = − p (cid:89) k =1 (cid:18) s k + 14 s k (cid:19) (cid:18) (cid:19) q + p (cid:89) k =1 (cid:18) s k + 14 s k (cid:19) p + q (cid:89) k = p +1 (cid:18)
43 + 16 s k (cid:19) . Corollary 1 shows that all of the full factorial designs and their repetitions are uniform designsunder the QQD, since the formula of QQD in Corollary 1 is independent of the repetition times c .This result is natural since full factorial designs are the best designs under many existing criteria,such as WD and minimum aberration.
3. The lower bounds of QQD
The lower bounds of a discrepancy can be employed as a benchmark not only in searching for uniformdesigns but also in helping validate that some good designs are uniform. The issue of lower boundsfor all kinds of discrepancies have been considered and many researchers have made plenty of effortin finding the lower bounds for different discrepancies, such as CD, WD, MD and DD, see Fang et al.(2018) for a comprehensive review. In this section, some lower bounds of the QQD for U-type designswith qualitative and quantitative factors are explored.
Theorem 3.
Let D = ( D , D ) with D ∈ U ( n, s · · · s p ) and D ∈ U ( n, s p +1 · · · s p + q ) , and assume s p +1 , . . . , s t be odd and s t +1 , . . . , s p + q be even, where p ≤ t ≤ p + q , then QQD ( D ) ≥ LB , whereLB = C + 1 n (cid:18) (cid:19) p + q + ( n − n (cid:18) (cid:19) p (cid:18) (cid:19) (cid:80) pk =1 n − sksk ( n − (cid:18) (cid:19) (cid:80) p + qk = p +1 n − sksk ( n − (cid:18) (cid:19) (cid:80) p + qk = t +1 nsk ( n − × t (cid:89) k = p +1 ( s k − / (cid:89) i =1 (cid:18) − i (2 s k − i )4 s k (cid:19) nsk ( n − p + q (cid:89) k = t +1 ( s k / − (cid:89) i =1 (cid:18) − i (2 s k − i )4 s k (cid:19) nsk ( n − , where C is given in Theorem 1. Theorem 3 obtains the results for general cases, i.e., any U-type design D with both qualitativeand quantitative factors, where the number of quantitative factors with odd levels could be 0 , , . . . , q. Specially, t = p + q means the levels of quantitative factors are all odd and t = p means the levelsof quantitative factors are all even. This provides a benchmark for identifying the uniform designs.For instance, if the QQD of a design reaches the corresponding lower bound, such a design must be a9niform design. Under some special parameter settings, the associated results can be directly acquiredand described in Corollary 2 and Remark 1. Corollary 2.
Let D = ( D , D ) ∈ U ( n, s p s q ) , where D and D are symmetric designs with s levels and s levels, respectively, then the lower bound of QQD ( D ) is given by LB odd = − (cid:18) s + 14 s (cid:19) p (cid:18) (cid:19) q + 1 n (cid:18) (cid:19) p + q + ( n − n (cid:18) (cid:19) p (cid:18) (cid:19) p ( n − s s n − (cid:18) (cid:19) q ( n − s s n − × ( s − / (cid:89) i =1 (cid:18) − i (2 s − i )4( s ) (cid:19) nqs n − , for odd s ,and LB even = − (cid:18) s + 14 s (cid:19) p (cid:18) (cid:19) q + 1 n (cid:18) (cid:19) p + q + ( n − n (cid:18) (cid:19) p (cid:18) (cid:19) p ( n − s s n − (cid:18) (cid:19) q ( n − s s n − × (cid:18) (cid:19) nqs n − ( s / − (cid:89) i =1 (cid:18) − i (2 s − i )4( s ) (cid:19) nqs n − , for even s . Remark 1. If q = 0, the QQD reduces to DD in Hickernell and Liu (2002); Liu and Hickernell (2002)and the lower bound in Theorem 3 becomes that for DD of Theorem 2 in Fang et al. (2003a). When p = 0, the QQD becomes WD in Hickernell (1998) and the lower bound for QQD becomes that forWD of Theorem 1 in Zhou and Ning (2008).
4. The connection between QQD and balance pattern
The concept of balance pattern was introduced by Fang et al. (2003b), which characterizes the columnbalance of a design. For a two-level or three-level U-type design, Fang et al. (2003b) pointed out thatWD can be expressed as a function of the balance pattern. Qin and Li (2006) derived the relationshipbetween DD and the balance pattern for symmetric designs. Furthermore, they calculated a new lowerbound of discrepancies by using these connections. In this section, we give similar results on the QQDcriterion. To our knowledge, this is the first time that the balance pattern of an asymmetric design isstudied, and the extension from the symmetric situation to the asymmetric one is not trivial.Consider an asymmetric design D ∈ U ( n, s p s q ), where the first p factors are qualitative and thelast q factors are quantitative. For each k columns of D , d l , . . . , d l k such that 1 ≤ l < · · · < l k ≤ < l k +1 < · · · < l k ≤ p + q , modify the definition of B l ,...,l k in Fang et al. (2003b) by B l ,...,l k = (cid:88) ∆ (cid:32) n ( l ,...,l k ) a ,...,a k − ns k s k (cid:33) , (6)to make it applicable to asymmetric situation, where ∆ = { ( a , . . . , a k ) | ≤ a , . . . , a k ≤ s , ≤ a k +1 , . . . , a k ≤ s } , k = k + k , and n ( l ,...,l k ) a ,...,a k is the number of rows in which the column group { d l , . . . , d l k } takes the level combination { a , . . . , a k } . The summation is taken across all the possiblelevel combinations. If B l ,...,l k = 0, the sub-design { d l , . . . , d l k } is an orthogonal array with strength k . Obviously, B l ,...,l k assesses the closeness to the orthogonality of strength k of the sub-design formedby the columns d l , . . . , d l k . Now, we define the balance pattern by B ( D ) = ( B ( D ) , . . . , B p + q ( D )),where B k ( D ) = (cid:88) Ω B l ,...,l k (cid:44)(cid:18) p + qk (cid:19) , (7)with Ω = { ( l , . . . , l k ) | ≤ l · · · ≤ l k ≤ p < l k +1 · · · ≤ l k ≤ p + q } . The summation is taken overall the possible k columns. As a result, B k ( D ) evaluates the nearness between the orthogonality ofstrength k and the design D , and B k ( D ) = 0 implies that the design D is an orthogonal array ofstrength k .The B ( D ) is defined based on the columns of a design, interestingly, the following lemma illustratesthat it has a close relationship with the rows of a design. Lemma 2.
For a design D ∈ U ( n, s p s q ) , the entries of B ( D ) could be reshaped as B k ( D ) = n (cid:88) i,j =1 (cid:88) Ω δ ( l ,...,l k ) ij (cid:44)(cid:18) p + qk (cid:19) − (cid:88) Ω n s k s k (cid:44)(cid:18) p + qk (cid:19) , for k = 1 , . . . , p + q, (8) where Ω = { ( l , . . . , l k ) | ≤ l · · · ≤ l k ≤ p < l k +1 · · · ≤ l k ≤ p + q } , δ ( l ,...,l k ) ij equals if ( x il , . . . , x il k ) = ( x jl , . . . , x jl k ) , and otherwise. The proof of this lemma is very similar to that of Lemma 3.1 in Fang et al. (2003b). The maindifference is that D ∈ U ( n, s p s q ) is an asymmetric design when s (cid:54) = s . Thus we omit it. Note that,the QQD in (5) is defined by rows. Now we can obtain another form of expression of QQD, whichrelies on the balance pattern. Theorem 4. If D = ( D , D ) with D ∈ U ( n, s p ) and D ∈ U ( n, q ) , thenQQD ( D ) = − (cid:18) s + 14 s (cid:19) p (cid:18) (cid:19) q + (cid:18) s + 14 s (cid:19) p (cid:18) (cid:19) q + 1 n (cid:18) (cid:19) p + q p + q (cid:88) k =1 (cid:18) (cid:19) k (cid:18) p + qk (cid:19) B k ( D ) . (9)11heorem 4 reveals the QQD is a function of B k ( D ), based on which we can derive a new lowerbound for the QQD. Theorem 5. If D = ( D , D ) with D ∈ U ( n, s p ) and D ∈ U ( n, q ) , then QQD ( D ) ≥ LB , where LB = − (cid:18) s + 14 s (cid:19) p (cid:18) (cid:19) q + (cid:18) s + 14 s (cid:19) p (cid:18) (cid:19) q + 1 n (cid:18) (cid:19) p + q p + q (cid:88) k =1 (cid:18) (cid:19) k × (cid:88) k + k = k (cid:18) pk (cid:19)(cid:18) qk (cid:19) r n,k ,k ,s, (cid:16) − r n,k ,k ,s, s k k (cid:17) , with r n,k ,k ,s, being the residual of n ( mod s k k ) . For those designs D ∈ U ( n, s p q ), two lower bounds, LB and LB , are applicable, then define LB = max { LB , LB } . Remark 2.
Let p = 0 or q = 0, then the results related to QQD in Theorems 4 and 5 could degenerateinto the corresponding results in terms of WD in Fang et al. (2003b) or DD in Qin and Li (2006),respectively.
5. Illustrative examples
In this section, we give some examples to demonstrate the reasonability of QQD for measuring theuniformity of designs associated with both qualitative and quantitative factors. Particularly, it turnsout that the proposed QQD could distinguish different MCDs generated by the same constructionprocedure. In addition, it is shown that the design generated by column juxtaposition of two separatelyuniform sub-designs may not be uniform over the design space.Many construction methods of MCDs were proposed. For example, He et al. (2017) gave severaleffective construction methods for MCDs with some appealing space-filling properties for the quanti-tative factors from the view of stratification on grids. The generated MCDs are not unique becauseof some structures and randomness. Even if the MCDs share the same stratifications on grids for thesub-designs of quantitative factors, they may have distinct uniformity. This declaration is interpretedby three MCDs visually.
Example 2.
Construct an MCD D M = ( D M , D M ) via Construction 3 in He et al. (2017) under theparameter settings: s = 2 , u = 4 , k = 2. Then D M is an OA(16 , , ,
2) and D M is an LHD(16 , D M , to form a small MCD, D =( D , D ) with one qualitative factor and two quantitative factors. The D with one qualitative factorshould be chosen from D M , such as its 1st column. For the quantitative part D , which should be asub-design of D M , there are (cid:18) (cid:19) = 15 different choices. For explanation, let D M = (d , . . . , d ) andcompare the following three situations, D (1)2 = (d , d ), D (2)2 = (d , d ), and D (3)2 = (d , d ), where D = (0 , , , , , , , , , , , , , , , T , d = (0 , , , , , , , , , , , , , , , T , d =(1 , , , , , , , , , , , , , , , T and d = (0 , , , , , , , , , , , , , , , T .Denote the resulting three small MCDs by D (1) = ( D , D (1)2 ), D (2) = ( D , D (2)2 ) and D (3) =( D , D (3)2 ). It can be easily calculated that QQD ( D (1) ) = 0 . ( D (2) ) = 0 . ( D (3) ) = 0 . D (3) is the most uniform design among thethree designs, followed by D (2) , and D (1) is the worst one.Figure 2 displays the scatter plots of the three sub-designs D (1)2 , D (2)2 and D (3)2 for the quantitativefactors. According to Theorem 5 in He et al. (2017), D (1)2 achieves stratification on 2 × D (2)2 and D (3)2 can achieve stratification on 8 × × D (1)2 isthe worst among the three sub-designs, and D (2)2 and D (3)2 possess the same performance in terms ofspace-filling property on grids. We further consider the points of D ( i )2 , i = 1 , ,
3, corresponding tolevel 0 and 1 of D , respectively, which form a small LHD since D ( i ) are all MCDs. Additionally, thepoints of D (1)2 and D (3)2 corresponding to any level of D can achieve stratification on 2 × D (1)2 . From Figure 2, we can conclude that D (3) is the most outstanding,followed by D (2) and D (1) is the worst, which is coincident with the results in comparison under theQQD criterion.Example 2 reflects that the QQD can distinguish the MCDs constructed by the same constructionmethod even though they have the same stratifications on grids for the quantitative factors, like the D (2) and D (3) . Originally, although the proposed QQD is motivated by comparing different MCDs,it is not limited to MCDs, and can be used for quantifying the uniformity of any design with bothtypes of factors.Since D consists of two sub-designs, D and D , one may consider searching uniform designs D ∗ and D ∗ for two sub-designs, separately, then juxtaposing them by column to obtain D ∗ = ( D ∗ , D ∗ ).13 -0.5 7.5 15.5-0.57.5 -0.5 7.5 15.5-0.57.5 -0.5 7.5 15.5-0.57.5 Figure 2: The scatter plots for the quantitative sub-designs, D (1)2 , D (2)2 and D (3)2 in Example 2. The points of D ( j )2 ( j = 1 , ,
3) corresponding to level 0 and 1 of D are represented by “ ◦ ” and “ × ”, respectively. The panels in thefirst row shows stratification on 8 × × However, we find some unsatisfactory results about this method.
Example 3. If n = 16 , p = 2 , q = 2 , s = s = 2 , s = s = 4, consider column juxtaposition of twoseparately uniform designs.Let D and D (1)2 be the full factorial designs as below. Permute the rows in D (1)2 and obtainanother 4-level full factorial design D (2)2 , where D = T , D (1)2 = T , D (2)2 = T . Obviously, D , D (1)2 and D (2)2 are uniform designs, respectively. Let D (1) = ( D , D (1)2 ) and D (2) =( D , D (2)2 ), that is, both resulting designs D (1) and D (2) consist of two separately uniform designs.Figure 3 shows the scatter points. Scatter points for the quantitative factors of both D (1) and D (2) have the same uniformity. However, we can see that the points of D (2)2 corresponding to level 0 and1 of each factor of D have better uniformity than that of D (1)2 , namely, the points “ ◦ ” and “ × ” in D (2)2 is more uniform than the points “ ◦ ” and “ × ” in D (1)2 , respectively. For instance, with respectto the level 0 of the first factor of D , the corresponding points of D (1)2 , i.e., points “ ◦ ” in the lefttop panel, only explore half of the region of x , while the points of D (2)2 corresponding to the level0 of the first factor of D i.e., points “ ◦ ” in the right top panel, can probe the whole region of x .Therefore, we can assert that D (2) looks more uniformly than D (1) from the scatter points.On the other hand, from Theorem 1, we have QQD ( D (1) ) = 0 . ( D (2) ) = 0 . D (2) is more uniform than D (1) under the QQD criterion, agreeing with our intuition.This example not only shows that the proposed QQD can discriminate different designs, but alsoindicates that column juxtaposition of two separately uniform designs does not work for constructingthe uniform designs with both types of factors since the combination mode of the qualitative andquantitative factors could affect the uniformity of the joint designs.For comparison, a naive criterion for designs with both qualitative and quantitative factors, thesum of the space-filling criterion of the quantitative factors over each level of each qualitative factor,may be considered. Such a criterion only measures the space-filling properties of all the quantitativefactors and the space-filling properties between each qualitative factor and all the quantitative factors.Consider the sum of the WD values of the quantitative factors over each level of each qualitative factoras one naive criterion, denoted by SWD. Then SWD( D (2) ) sums the WD values of the four 8-runsub-designs, represented by “ ◦ ” in the right top and right bottom panel, and “ × ” in the right topand the right bottom panel, respectively. It can be easily calculated that SWD( D (2) ) = 1 . (cid:101) D (2) = ( (cid:101) D , D (2)2 ), where (cid:101) D = ( × , × ) T . Obviously, D (2) has better space-filling property than (cid:101) D (2) since the two columns for the qualitative factors in (cid:101) D (2) remain the same. By computation, SWD( (cid:101) D (2) ) = 1 . < SWD( D (2) ) implies that (cid:101) D (2) is15 -0.51.53.5-0.5 1.5 3.5-0.51.53.5 -0.5 1.5 3.5 -0.51.53.5-0.5 1.5 3.5-0.51.53.5 Figure 3: The scatter plots for the quantitative factors D (1)2 and D (2)2 in Example 3. The points of D ( j )2 , j = 1 , D are represented by “ ◦ ” and “ × ”, respectively. The top and bottom panels showplots with respect to the first and the second factor of D , respectively. better than D (2) under SWD. This result is contrary to the objective truth. In terms of QQD criterion,it could be calculated that QQD ( (cid:101) D (2) ) = 0 . > QQD ( D (2) ). The comparison result under QQDis still agrees with the facts. It confirms that the advantage of QQD over the naive criterion in theprojection space-filling properties on all subspaces are important.Next, we give two designs, whose QQD values achieve the lower bounds obtained in this paper.This further indicates the lower bounds we derived are tight. Example 4.
Consider the design D = ( D , D ) ∈ U (4 , × ) with D = (0 , , , T and D = (cid:0) (0 , , , T , (1 , , , T (cid:1) . It reaches the lower bound LB with QQD ( D ) = LB = 0 . Example 5.
Consider the design D listed in Table 1, which can reach the lower bound LB , and16QD ( D ) = LB = 17 . Table 1: D = ( D , D ) ∈ U (8 , × ) in Example 5 D D The designs presented in Examples 4 and 5 show that the lower bounds LB and LB obtainedin this paper are tight and achievable. Furthermore, we give the following example to illustrate theuse and effectiveness of the QQD in comparing response surface designs. Example 6.
Consider a response surface design with one qualitative ( z ) and two quantitative factors( x and x ). Table 2 lists the four designs, among which the quantitative factors are fixed. Thesub-design composed by x and x is a central composite design (CCD), which consists of a 2 factorial design, two center runs and four axial runs. The difference among the four designs existsin the assignment for the qualitative factor. The z (1) , . . . , z (4) in Table 2 are respectively representedas the qualitative factor z in the four designs. Denote the four designs by D (1) = ( x , x , z (1) ), D (2) = ( x , x , z (2) ), D (3) = ( x , x , z (3) ), and D (4) = ( x , x , z (4) ), respectively. Wu and Ding(1998) compared the efficiency of different designs by using the determinant criterion (D-criterion).They obtained that the first design D (1) is the optimal because of its maximum D-criterion value.It is noted that the D-criterion heavily relies on the pre-determined model. A small misspecificationmay seriously affect the performance of the chosen design. Then, the space-filling property may beconsidered. Next, we will choose a new design by the QQD criterion.According to the structure of the CCD, the sub-design composed by the two quantitative factorsconsists of three portions, factorial, center, and axial portion, see the three portions of ( x , x ) listed17 able 2: The values of x , x , and z in the four designs in Example 6 Run x x z (1) z (2) z (3) z (4) − − − −
12 1 − − − − − − − −
15 0 0 1 1 1 16 0 0 − − − − √ − − −√ − − − −
19 0 √ − − −
110 0 −√ − − in Table 2. Corresponding to the factorial portion of CCD, denote the sub-designs composed by thefirst four runs of the four designs by D (1)1 , D (2)1 , D (3)1 and D (4)1 , respectively. When the space-fillingproperty of the factorial portion is considered, we can use the QQD to assess the uniformity of the foursub-designs D (1)1 , . . . , D (4)1 . After transforming the level − / / x and x falling into [0,1], the QQD values could be calculated according to the expression of QQD.Table 3 lists the QQD values of the four sub-designs. It can be seen that D (4)1 owes the minimumQQD value. Moreover, QQD( D (4)1 ) reaches the lower bound of QQD for designs in U (4 , ). Theseresults imply that D (4)1 is the best among the four sub-designs under QQD, and the four sub-designsare ranked as D (4)1 , D (3)1 , D (1)1 , D (2)1 . Next, consider the space-filling property of the four full designs, D (1) , . . . , D (4) . At this time, all the four designs are not U-type. Transform each level of x and x into [0 ,
1] by (cid:0) x + √ (cid:1) / (cid:0) √ (cid:1) , x ∈ (cid:8) − √ , − , , , √ (cid:9) , then, we can compute the QQD valuesof D (1) , . . . , D (4) . Based on the QQD values listed in Table 3, D (4) is the best design. Combiningthe comparison result of the four sub-designs, the fourth design is suggested when the space-fillingproperty is considered. 18 able 3: The criterion values of designs under two space-filling criteria Criterion D (1)1 D (2)1 D (3)1 D (4)1 D (1) D (2) D (3) D (4) Squared QQD 0.2255 0.2255 0.1766 0.1571 0.0763 0.0795 0.0792 0.0653Extended MaxPro 1.8821 1.8821 1.8246 1.3606 3.5810 3.6441 3.7086 3.5587
Additionally, the extended MaxPro criterion proposed in Joseph et al. (2019) could be consideredhere to assess the space-filling property. The corresponding criterion values of the four sub-designs D (1)1 , . . . , D (4)1 , and the four designs D (1) , . . . , D (4) , are also listed in Table 3. It can be seen that,under the extended MaxPro criterion, the order of the four sub-designs is accordant with that underthe QQD, and the best design among the four designs D (1) , . . . , D (4) is the same with that under theQQD. It is noted that there is a small difference of the order of D (2) and D (3) under the two criteria.It means that the two criteria have a litter bit difference, and which does not affect the suggesteddesign.
6. Conclusion
For measuring the space-filling property of designs with both qualitative and quantitative factors, inthis paper, we propose a new uniformity criterion, the qualitative-quantitative discrepancy, and giveits explicit expression. From several intuitive examples, we show the reasonability and effectivenessof the proposed QQD. In fact, the newly proposed discrepancy can measure the uniformity of anydesign containing two types of factors apart from that of MCDs and response surface designs whenthe space-filling properties are considered. In addition, according to the closed form of the QQD andthe connections between QQD and the balance pattern, two tight lower bounds of QQD are obtainedby strict mathematic deductions, which can be regarded as a benchmark for identifying the uniformdesigns. When the QQD is applied to stochastic optimization algorithms, as similar as the CD, WDand MD, an iteration formula could be derived to greatly simplified the calculation of the QQD valuefor a new design in each iteration. In the future, we can use some stochastic optimization methods orsome systematic construction methods to search uniform designs under QQD criterion.19 cknowledgements
This work was supported by National Natural Science Foundation of China (11871288) and NaturalScience Foundation of Tianjin (19JCZDJC31100). The first two authors contributed equally to thiswork.
Appendix
Proof of Theorem 1 . Note that χ = χ ×· · ·× χ p + q , where χ k = { , , . . . , s k − } , for k = 1 , , . . . , p and χ k = [0 , k = p + 1 , . . . , p + q . For the first term of (2), we have (cid:90) χ k K k ( t k , z k )d F ( t k )d F ( z k ) = s k +14 s k , for k = 1 , . . . , p, , for k = p + 1 , . . . , p + q. (10)Hence, (cid:82) χ K ( t , z )d F ( t )d F ( z ) = (cid:81) pk =1 (cid:16) s k +14 s k (cid:17) (cid:0) (cid:1) q . For the second term, (cid:90) χ k K k ( t k , x ik )d F ( t k ) = s k +14 s k , for k = 1 , . . . , p, , for k = p + 1 , . . . , p + q. (11)Thus, n (cid:80) ni =1 (cid:82) χ K ( t , x i )d F ( t ) = 2 (cid:81) pk =1 (cid:16) s k +14 s k (cid:17) (cid:0) (cid:1) q . With regard to the last term, n (cid:88) i,j =1 K ( x i , x j ) = n (cid:88) i,j =1 (cid:18) (cid:19) δ ij ( D ) (cid:18) (cid:19) p − δ ij ( D ) · p + q (cid:89) k = p +1 (cid:18) − | x ik − x jk | + | x ik − x jk | (cid:19) . (12)Substituting the Equations (10) - (12) into (2), we complete the proof. (cid:4) For proving Corollary 1, we give a lemma, which is the extension of related results in Zhou et al.(2012) and its proof is straightforward.
Lemma 3.
The matrices A and A k in Lemma 1 have the following properties, A k s k = (cid:0) + ( s k − (cid:1) s k , A − k s k = (cid:0) + ( s k − (cid:1) − s k , for k = 1 , . . . , p, A k s k = (cid:16) s k + s k (cid:17) s k , A − k s k = (cid:16) s k + s k (cid:17) − s k , for k = p + 1 , . . . , p + q, nd A N = (cid:81) pk =1 (cid:0) + ( s k − (cid:1) (cid:81) p + qk = p +1 (cid:16) s k + s k (cid:17) N , A − N = (cid:81) pk =1 (cid:0) + ( s k − (cid:1) − (cid:81) p + qk = p +1 (cid:16) s k + s k (cid:17) − N . Proof of Corollary 1 . From Lemma 1 and Theorem 2, we haveQQD ( D ∗ ) = − p (cid:89) k =1 (cid:18) s k + 14 s k (cid:19) (cid:18) (cid:19) q + 1 n (cid:16) nN N (cid:17) T A (cid:16) nN N (cid:17) = − p (cid:89) k =1 (cid:18) s k + 14 s k (cid:19) (cid:18) (cid:19) q + 1 N TN A N (13)Note that N = (cid:81) p + qk =1 s k and by Lemma 3, TN A N = p (cid:89) k =1 (cid:18)
32 + 54 ( s k − (cid:19) p + q (cid:89) k = p +1 (cid:18) s k s k (cid:19) TN N = N p (cid:89) k =1 (cid:18)
32 + 54 ( s k − (cid:19) p + q (cid:89) k = p +1 (cid:18) s k s k (cid:19) = N p (cid:89) k =1 (cid:18) s k + 14 s k (cid:19) p + q (cid:89) k = p +1 (cid:18)
43 + 16 s k (cid:19) (14)Combining (13) and (14), this proof can be finished. (cid:4) Proof of Theorem 3 . Note that (cid:0) (cid:1) δ ij ( D ) = (cid:81) pk =1 (cid:0) (cid:1) δ kij , where δ kij = δ x ik x jk . Let α kij = | x ik − x jk | − | x ik − x jk | , for k = p + 1 , . . . , p + q . The formula (5) can be reshaped asQQD ( D ) = C + 1 n (cid:18) (cid:19) p + q + 1 n (cid:18) (cid:19) p (cid:88) ≤ i (cid:54) = j ≤ n p (cid:89) k =1 (cid:18) (cid:19) δ kij p + q (cid:89) k = p +1 (cid:18) − α kij (cid:19) . From the above equation, it’s clear that the value of QQD is a function of (cid:81) pk =1 (cid:0) (cid:1) δ kij (cid:81) p + qk = p +1 (cid:0) − α kij (cid:1) , i, j = 1 , . . . , n , i (cid:54) = j , as the first two terms are constant for the given parameters. For the U-type de-sign D ∈ U ( n, s · · · s p ), there are n ( s k − s k number of δ kij being 0, and n ( n − s k ) s k number of δ kij being 1, i, j = 1 , . . . , n, i (cid:54) = j , for any k = 1 , . . . , p . For D ∈ U ( n, s p +1 · · · s p + q ), Table 4 shows the distributionof α kij , i, j = 1 , . . . , n, i (cid:54) = j , for any k = p +1 , . . . , p + q . Under the given parameters ( n, s , . . . , s p + q ) ofa design, minimizing QQD ( D ) is equivalent to minimizing (cid:80) ≤ i (cid:54) = j ≤ n (cid:81) pk =1 (cid:0) (cid:1) δ kij (cid:81) p + qk = p +1 (cid:0) − α kij (cid:1) .From the above discussion, we claim that (cid:81) ≤ i (cid:54) = j ≤ n (cid:81) pk =1 (cid:0) (cid:1) δ kij (cid:81) p + qk = p +1 (cid:0) − α kij (cid:1) is a constant forthe given ( n, s , . . . , s p + q ), say H . According to the geometric and arithmetic mean inequality, when21ach (cid:81) pk =1 (cid:0) (cid:1) δ kij (cid:81) p + qk = p +1 (cid:0) − α kij (cid:1) takes the same values for 1 ≤ i (cid:54) = j ≤ n , i.e., H n ( n − , QQD( D ) can reach its minimum, which completes the proof. Table 4: Distribution of α kij , for k = p + 1 , . . . , p + q even s k odd s k the values of α kij number the values of α kij number0 n ( n − s k ) s k n ( n − s k ) s k s k − s k n s k s k − s k n s k ... ... ... ... ( s k − s k +2)4 s k n s k ( s k − s k +3)4 s k n s k s k s k n s k ( s k − s k +1)4 s k n s k (cid:4) Proof of Theorem 4 . For D ∈ U ( n, s p ) and D ∈ U ( n, q ), from Theorem 1, we haveQQD ( D ) = C + 1 n (cid:18) (cid:19) p n (cid:88) i,j =1 (cid:18) (cid:19) δ ij ( D ) · p + q (cid:89) k = p +1 (cid:18) − | x ik − x jk | + | x ik − x jk | (cid:19) = C + 1 n (cid:18) (cid:19) p n (cid:88) i,j =1 p (cid:89) k =1 (cid:18) (cid:19) δ kij · p + q (cid:89) k = p +1 (cid:18) − | x ik − x jk | + | x ik − x jk | (cid:19) , (15)where C = − (cid:0) s +14 s (cid:1) p (cid:0) (cid:1) q . Note that, (cid:0) (cid:1) δ kij = 1 + δ kij , for k = 1 , . . . p , and | x ik − x jk | = (1 − δ kij ),for k = p + 1 , . . . p + q . Thus, (15) can be rewritten asQQD ( D ) = − (cid:18) s + 14 s (cid:19) p (cid:18) (cid:19) q + 1 n (cid:18) (cid:19) p n (cid:88) i,j =1 p (cid:89) k =1 (cid:18) δ kij (cid:19) · p + q (cid:89) k = p +1 (cid:18)
54 + 14 δ kij (cid:19) . (16)The value of (16) is dominated by the last term for given parameters, because the first term is a22onstant. Part of the last term n (cid:88) i,j =1 p (cid:89) k =1 (cid:18) δ kij (cid:19) · p + q (cid:89) k = p +1 (cid:18)
54 + 14 δ kij (cid:19) = n (cid:88) i,j =1 (cid:32)(cid:18) (cid:19) q + p + q (cid:88) k =1 (cid:88) Ω (cid:18) (cid:19) k (cid:18) (cid:19) k (cid:18) (cid:19) q − k δ ( l ,...,l k ) ij (cid:33) = n (cid:18) (cid:19) q + (cid:18) (cid:19) q p + q (cid:88) k =1 (cid:18) (cid:19) k n (cid:88) i,j =1 (cid:88) Ω δ ( l ,...,l k ) ij = n (cid:18) (cid:19) q + (cid:18) (cid:19) q p + q (cid:88) k =1 (cid:18) (cid:19) k (cid:32) B ( k ) + (cid:88) Ω n s k k (cid:44)(cid:18) p + qk (cid:19)(cid:33) (cid:18) p + qk (cid:19) (17)= n (cid:18) (cid:19) q + (cid:18) (cid:19) q p + q (cid:88) k =1 (cid:18) (cid:19) k (cid:18) p + qk (cid:19) B ( k ) + n (cid:18) (cid:19) q p + q (cid:88) k =1 (cid:18) (cid:19) k (cid:88) Ω s k k . (18)The Equation (17) holds because of (8) in Lemma 2. Furthermore, p + q (cid:88) k =1 (cid:18) (cid:19) k (cid:88) Ω s k k = p + q (cid:88) k =1 (cid:88) Ω s ) k k = (cid:18) s (cid:19) p (cid:18) (cid:19) q − (cid:18) s + 15 s (cid:19) p (cid:18) (cid:19) q − . (19)Substituting (19) into (18), and substituting (18) into (16), we obtain (9). (cid:4) Proof of Theorem 5 . For any k (= k + k )-tuple of columns in D , there are s k k level combinationsin total. Let n = ts k k + r n,k ,k ,s, , ≤ r n,k ,k ,s, < s k k . According to (6), to minimize B l ,...,l k ,the frequencies of all the possible level combinations in the column group { d l , . . . , d l k } should beas equal as possible. Therefore, if there are r n,k ,k ,s, level combinations that occur t + 1 timesand s k k − r n,k ,k ,s, level combinations that appear t times, B l ,...,l k would reach its lower bound, r n,k ,k ,s, (cid:0) − r n,k ,k ,s, s k k (cid:1) . By (7), the lower bound of B k ( D ) is obtained immediately, B k ( D ) ≥ (cid:88) Ω r n,k ,k ,s, (cid:16) − r n,k ,k ,s, s k k (cid:17) (cid:44)(cid:18) p + qk (cid:19) . (20)By combining (9) and (20), we haveQQD ( D ) ≥ − (cid:18) s + 14 s (cid:19) p (cid:18) (cid:19) q + (cid:18) s + 14 s (cid:19) p (cid:18) (cid:19) q + 1 n (cid:18) (cid:19) p + q p + q (cid:88) k =1 (cid:18) (cid:19) k × (cid:88) Ω r n,k ,k ,s, (cid:16) − r n,k ,k ,s, s k k (cid:17) . (21)23t’s noted that (cid:88) Ω = (cid:88) k + k = k (cid:18) pk (cid:19)(cid:18) qk (cid:19) . (22)The proof can be finished by substituting (22) into (21). (cid:4) ReferencesReferences
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