一般最小混杂的s水平正规设计理论
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摘要
部分因析设计在各个领域都有广泛的应用，为提高试验效率，如何选择设计非常关键。目前，主要有五种最优设计准则：最大分辨度(MaximumResolution，简记MR)准则、最小低阶混杂(Minimum Aberration，简记MA)准则、纯净效应(Clear Effects，简记CE)准则、最大估计容量(Maximum EstimationCapacity，简记MEC)准则和一般最小低阶混杂(General Minimum Lower OrderConfounding，简记GMC)准则。值得注意，MR和MA准则以字长型(Word-length Pattern，简记WLP)为基础。MEC准则看似与WLP没有直接的联系，然而，[31]证明如果只关心主效应和二阶交互效应并且MEC设计存在，那么这个设计等价于MA设计。在这种意义下，MEC准则也是建立在WLP上并且和MA等价。而CE准则不同，它是依据纯净的主效应和纯净的二因子交互效应个数来选择最优设计。
为了深入揭示部分因析设计中因子之间的混杂信息以及各准则之间的关系，[56]针对两水平正规设计提出效应别名个数的分类模式(Aliased Effect-NumberPattern，简记AENP)。基于AENP建立一个新的最优准则——GMC准则。一个设计的AENP包含各阶效应以不同程度与其他效应别名的基本信息，它充分、直接地反映设计中不同阶因子效应之间的混杂关系，所以WLP以及纯净因子的个数均是AENP的函数(证明过程见[56])。基于AENP的GMC准则比MR，MA及MEC准则更精确、客观地体现了效应等级原则，可以安排有先验信息的重要因子，因此GMC设计应用广泛。目前二水平的AENP和GMC准则逐渐发展并完善起来，应用于正规设计、区组设计、裂区设计、混水平设计和一般正交设计等，形成二水平GMC理论体系，已经有了大量成果。但是，三水平或一般水平的研究成果很少。
本博士论文旨在把GMC理论推广到三水平及素数或素数幂s水平的正规设计。然而，我们不能直接把二水平的AENP和GMC准则应用于三水平或s水平的设计，区别在于因子与因子成分之间的关系。对二水平设计，因子和因子成分是一一对应的关系，但三水平和s水平不同，如三水平的二阶因子交互效应对应于两个正交成分以及s水平的二阶因子交互效应对应于s1个成分，并且通过因子成分的别名关系反映设计中因子之间的混杂信息。由于低阶因子成分比高阶因子成分重要，同阶因子成分同等重要，建立成分效应等级原则(Component Hierarchy Principle,简记CHP)并引入成分别名个数的分类模式(Aliased Component-Number Pattern，简记ACNP)，特别的，二水平的AENP也称为ACNP。在ACNP的基础上提出三水平或s水平GMC准则。本文的主要贡献分成五部分：
第一部分由第二章组成。在三水平设计中，基于CHP引入ACNP以及三水平GMC准则，其中#iC(k)j表示与k个j阶因子成分混杂的i阶因子成分的个数，零阶因子成分表示均值及一阶因子成分为主效应。定理2.2.1给出ACNP与WLP的关系：其中A_i表示定义对照关系中字长为i字的个数。对于分辨度R≥III的三水平设计，定理2.2.2得到其纯净主效应的个数C1=#)1C(02和纯净二因子成分的个数定理2.3.3及定理2.3.4验证了三水平设计是GMC的必要条件。在这章最后一节，列出了27-run的全部GMC设计，因子数n=5,...,20时81-run GMC设计以及分辨度R≥IV的243-run GMC设计的表格，其中包括GMC设计和MA设计的比较。
第二部分是第三章。在二水平GMC设计中，和的计算存在着某种规律，为了揭示这种规律，分别对5N/16+1≤nFactorial designs have wide applications in many fields. For enhancing the effi-ciency of an experiment, it is important to employ optimal designs. Up to now, quite afew criteria for selecting optimal factorial designs are proposed: maximum resolution(MR), minimum aberration (MA), clear effects (CE), maximum estimation capacity(MEC) and general minimum lower order confounding (GMC) criteria. Note that MRand MA criteria are based on the word-length pattern (WLP). MEC criterion appearsto have no direct connection with the WLP. However,[31] proves that if attention isrestricted to main effects and two-factor interactions (2fi’s), then an MEC design hasMA. As for CE criterion, it selects designs by the numbers of clear main effects andclear2fi’s.
To deeply provide the confounding information among factors of designs and re-veal the relationships of the above criteria,[56] introduces an aliased effect-numberpattern (AENP) for two-level regular design. Based on the AENP, it provides a newoptimality criterion–GMC criterion. The AENP contains the basic information of thedifferent order effects aliased with the other order effects at various severity degrees.Thus, the WLP and the number of clear factors are respectively the functions of theAENP (see [56]). GMC criterion aims at finding optimal designs under effect hierarchyprinciple (EHP) in a more elaborate and explicit manner. GMC is flexible, accommo-dating to the prior information about the relative importance of factors. This kind ofprior information is often available in practice, so GMC designs are widely applicable.Now the AENP and GMC criterion have been developed well and formed GMC the-ory, which is applied in regular designs, block designs, split-plot designs, mixed-leveldesigns and orthogonal designs. The literature in two-level designs has grown copi-ously and included many results. However, the articles for three-or general-level GMCdesigns are few.
The doctoral dissertation aims at extending GMC theory to the cases of three-and prime or prime power s-level regular designs. However, we can not directly apply the AENP and GMC criterion of two-level case to three-or s-level designs since thedistinction lies in the correspondence between factorial effects and components. Inthe former, there is a one-to-one correspondence and use of components to representfactorial effects is unnecessary. In the latter, a2fi of three-level case corresponds totwo orthogonal components, and a2fi corresponds to s1components for s-level case.Moreover, we use the alias relation of components to reveal the confounding amongfactors. Since lower order components are more likely to be important than higher orderones and the same-order components are equally likely to be important, we establish thecomponent hierarchy principle (CHP). Based on the CHP, we introduce the concept ofthe alias component-number pattern (ACNP), especially, the AENP of two-level case isalso called the ACNP. Based on the ACNP we provide three-or s-level GMC criterion.The main contributions of this thesis are five parts:
The first part is formed by Chapter2. Based on the CHP, we introduce the conceptof ACNP as follows:and three-level GMC criterion. Here#)
iC(kjmeans the number of ith-order interactioncomponents that are aliased with jth-order interaction components at degree k. Espe-cially, the cases i=0,1respectively correspond to the grand mean and a main effect.Theorem2.2.1gets the relationship of the ACNP and the WLP:where Aiis the number of words with length i in the defining contrast identity relations.For any three-level designs with resolution R≥III, Theorem2.2.2gives the numberof clear main effects C1=#0)
2. Theorem2.3.3and Theorem2.3.4prove some necessary conditions for three-level designs to have GMC. In the last section, we list a catalogue of designs withall27-run GMC designs,81-run GMC designs with n=5,...,20and243-run GMCdesigns with resolution IV or higher, which includes the comparative case of GMCdesigns and MA designs.
Second part contains Chapter3. For GMC two-level designs, there are some prin-ciples for calculating the value of#2C2. To reveal the characters, we analyze the structure of the constructed2n mGMC designs with5N/16+1≤n 1C2and2C2, where N=2n m.Main results are Theorem3.2.6, Theorem3.2.7, Theorem3.3.1and Theorem3.3.2.
Third part consists of Chapter4. We study prime or prime power s-level GMCdesigns. Theorem4.1.1obtains the calculation formula of the ACNP for any sn mdesign D below:D, lt(=0)∈GF(s1),1≤t≤i} denotes the number of ith-order interaction compo-nents aliased with γ (γ∈Hq). Here Hqis an s-level saturated design.
From Theorem4.2.1to Theorem4.2.8, we analyze the relationships betweenGMC criterion and other criteria for s-level case. Theorem4.2.1and Theorem4.2.2an-alyze the connection among the ACNP and resolution R. Theorem4.2.3gives the factthat the WLP is a function of the ACNP. Theorem4.2.4reveals the inherent property ofMA designs, which has the average minimum lower order confounding property. Theconnections between GMC and CE criteria are obtained in Theorem4.2.5and Theorem4.2.6. Moreover, Theorem4.2.7and Theorem4.2.8obtain the relationship of GMC andMEC criteria. By the corresponding complementary set, we get some results for s-levelGMC designs, such as Theorem4.1.3and Theorem4.3.4. To develop the s-level GMCtheory, we study sn mGMC designs for two cases:(1)(N/s+1)/2 Forth part is formed by Chapter5. It first provides a useful theorem to constructthree-level GMC designs. By this theorem, we know that any design Sqr=Hq\Hr(r Chapter6is fifth part. An experimenter who has prior knowledge may suggestthat main effects and2fi’s are potentially important and should be estimated. To solve this problem,[47] introduces CE criterion. However, CE criterion can not distinguishdesigns with the same numbers of clear main effects and2fi’s or no clear effects. Ac-cording to the confounding degree of components, a new ordering of three-level ACNPsolves this problem:0. It is called the aliased component-number pattern based on degrees (ACNP-D).Based on the ACNP-D we introduce the general minimum lower order confoundingbased on degrees (GMC-D) criterion. Main results are obtained in Theorem6.2.4andTheorem6.2.5. In the last section, we provide a catalogue of all27-run GMC-D de-signs,81-run GMC-D designs with n=5,...,20and243-run GMC-D designs for res-olution IV or higher, which contains the comparison of GMC-D and GMC designs.
为了深入揭示部分因析设计中因子之间的混杂信息以及各准则之间的关系，[56]针对两水平正规设计提出效应别名个数的分类模式(Aliased Effect-NumberPattern，简记AENP)。基于AENP建立一个新的最优准则——GMC准则。一个设计的AENP包含各阶效应以不同程度与其他效应别名的基本信息，它充分、直接地反映设计中不同阶因子效应之间的混杂关系，所以WLP以及纯净因子的个数均是AENP的函数(证明过程见[56])。基于AENP的GMC准则比MR，MA及MEC准则更精确、客观地体现了效应等级原则，可以安排有先验信息的重要因子，因此GMC设计应用广泛。目前二水平的AENP和GMC准则逐渐发展并完善起来，应用于正规设计、区组设计、裂区设计、混水平设计和一般正交设计等，形成二水平GMC理论体系，已经有了大量成果。但是，三水平或一般水平的研究成果很少。
本博士论文旨在把GMC理论推广到三水平及素数或素数幂s水平的正规设计。然而，我们不能直接把二水平的AENP和GMC准则应用于三水平或s水平的设计，区别在于因子与因子成分之间的关系。对二水平设计，因子和因子成分是一一对应的关系，但三水平和s水平不同，如三水平的二阶因子交互效应对应于两个正交成分以及s水平的二阶因子交互效应对应于s1个成分，并且通过因子成分的别名关系反映设计中因子之间的混杂信息。由于低阶因子成分比高阶因子成分重要，同阶因子成分同等重要，建立成分效应等级原则(Component Hierarchy Principle,简记CHP)并引入成分别名个数的分类模式(Aliased Component-Number Pattern，简记ACNP)，特别的，二水平的AENP也称为ACNP。在ACNP的基础上提出三水平或s水平GMC准则。本文的主要贡献分成五部分：
第一部分由第二章组成。在三水平设计中，基于CHP引入ACNP以及三水平GMC准则，其中#iC(k)j表示与k个j阶因子成分混杂的i阶因子成分的个数，零阶因子成分表示均值及一阶因子成分为主效应。定理2.2.1给出ACNP与WLP的关系：其中A_i表示定义对照关系中字长为i字的个数。对于分辨度R≥III的三水平设计，定理2.2.2得到其纯净主效应的个数C1=#)1C(02和纯净二因子成分的个数定理2.3.3及定理2.3.4验证了三水平设计是GMC的必要条件。在这章最后一节，列出了27-run的全部GMC设计，因子数n=5,...,20时81-run GMC设计以及分辨度R≥IV的243-run GMC设计的表格，其中包括GMC设计和MA设计的比较。
第二部分是第三章。在二水平GMC设计中，和的计算存在着某种规律，为了揭示这种规律，分别对5N/16+1≤n
To deeply provide the confounding information among factors of designs and re-veal the relationships of the above criteria,[56] introduces an aliased effect-numberpattern (AENP) for two-level regular design. Based on the AENP, it provides a newoptimality criterion–GMC criterion. The AENP contains the basic information of thedifferent order effects aliased with the other order effects at various severity degrees.Thus, the WLP and the number of clear factors are respectively the functions of theAENP (see [56]). GMC criterion aims at finding optimal designs under effect hierarchyprinciple (EHP) in a more elaborate and explicit manner. GMC is flexible, accommo-dating to the prior information about the relative importance of factors. This kind ofprior information is often available in practice, so GMC designs are widely applicable.Now the AENP and GMC criterion have been developed well and formed GMC the-ory, which is applied in regular designs, block designs, split-plot designs, mixed-leveldesigns and orthogonal designs. The literature in two-level designs has grown copi-ously and included many results. However, the articles for three-or general-level GMCdesigns are few.
The doctoral dissertation aims at extending GMC theory to the cases of three-and prime or prime power s-level regular designs. However, we can not directly apply the AENP and GMC criterion of two-level case to three-or s-level designs since thedistinction lies in the correspondence between factorial effects and components. Inthe former, there is a one-to-one correspondence and use of components to representfactorial effects is unnecessary. In the latter, a2fi of three-level case corresponds totwo orthogonal components, and a2fi corresponds to s1components for s-level case.Moreover, we use the alias relation of components to reveal the confounding amongfactors. Since lower order components are more likely to be important than higher orderones and the same-order components are equally likely to be important, we establish thecomponent hierarchy principle (CHP). Based on the CHP, we introduce the concept ofthe alias component-number pattern (ACNP), especially, the AENP of two-level case isalso called the ACNP. Based on the ACNP we provide three-or s-level GMC criterion.The main contributions of this thesis are five parts:
The first part is formed by Chapter2. Based on the CHP, we introduce the conceptof ACNP as follows:and three-level GMC criterion. Here#)
iC(kjmeans the number of ith-order interactioncomponents that are aliased with jth-order interaction components at degree k. Espe-cially, the cases i=0,1respectively correspond to the grand mean and a main effect.Theorem2.2.1gets the relationship of the ACNP and the WLP:where Aiis the number of words with length i in the defining contrast identity relations.For any three-level designs with resolution R≥III, Theorem2.2.2gives the numberof clear main effects C1=#0)
2. Theorem2.3.3and Theorem2.3.4prove some necessary conditions for three-level designs to have GMC. In the last section, we list a catalogue of designs withall27-run GMC designs,81-run GMC designs with n=5,...,20and243-run GMCdesigns with resolution IV or higher, which includes the comparative case of GMCdesigns and MA designs.
Second part contains Chapter3. For GMC two-level designs, there are some prin-ciples for calculating the value of#2C2. To reveal the characters, we analyze the structure of the constructed2n mGMC designs with5N/16+1≤n
Third part consists of Chapter4. We study prime or prime power s-level GMCdesigns. Theorem4.1.1obtains the calculation formula of the ACNP for any sn mdesign D below:D, lt(=0)∈GF(s1),1≤t≤i} denotes the number of ith-order interaction compo-nents aliased with γ (γ∈Hq). Here Hqis an s-level saturated design.
From Theorem4.2.1to Theorem4.2.8, we analyze the relationships betweenGMC criterion and other criteria for s-level case. Theorem4.2.1and Theorem4.2.2an-alyze the connection among the ACNP and resolution R. Theorem4.2.3gives the factthat the WLP is a function of the ACNP. Theorem4.2.4reveals the inherent property ofMA designs, which has the average minimum lower order confounding property. Theconnections between GMC and CE criteria are obtained in Theorem4.2.5and Theorem4.2.6. Moreover, Theorem4.2.7and Theorem4.2.8obtain the relationship of GMC andMEC criteria. By the corresponding complementary set, we get some results for s-levelGMC designs, such as Theorem4.1.3and Theorem4.3.4. To develop the s-level GMCtheory, we study sn mGMC designs for two cases:(1)(N/s+1)/2
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