Relations among partitions. IV: Adjusting for more than one - PowerPoint PPT Presentation

Relations among partitions. IV: Adjusting for more than one partition R. A. Bailey University of St Andrews Combinatorics Seminar, Shanghai Jiao Tong University, November 2017 Bailey Relations among partitions 1/28

Abstract Adjusted orthogonality and adjusted uniformity both generalize to adjusting for more than one partition. This leads to the notion of universal balance, in which every partition has adjusted uniformity after adjusting for any subset of the others. Multi-stage Youden rectangles and multi-layered Youden rectangles combine this idea with orthogonality. Bailey Relations among partitions 2/28

Outline ◮ Questions and answers. ◮ More general versions of adjusted orthogonality and adjusted uniformity. ◮ Universal balance. ◮ Multi-stage Youden rectangles and multi-layered Youden rectangles. Bailey Relations among partitions 3/28

Outline ◮ Questions and answers. ◮ More general versions of adjusted orthogonality and adjusted uniformity. ◮ Universal balance. ◮ Multi-stage Youden rectangles and multi-layered Youden rectangles. Bailey Relations among partitions 3/28

Question 1 Question Can we have three partitions, with each pair having adjusted orthogonality with respect to the third one? Bailey Relations among partitions 4/28

Question 1 Question Can we have three partitions, with each pair having adjusted orthogonality with respect to the third one? Answer Yes. Here are two examples. Bailey Relations among partitions 4/28

Question 1 Question Can we have three partitions, with each pair having adjusted orthogonality with respect to the third one? Answer Yes. Here are two examples. 1. F ⊥ G but F �≺ G and G �≺ F , so that F ∨ G is not F or G . Bailey Relations among partitions 4/28

Question 1 Question Can we have three partitions, with each pair having adjusted orthogonality with respect to the third one? Answer Yes. Here are two examples. 1. F ⊥ G but F �≺ G and G �≺ F , so that F ∨ G is not F or G . Then X ⊤ F ( I − P F ∨ G ) X G = 0 (by definition of F ⊥ G ) Bailey Relations among partitions 4/28

Question 1 Question Can we have three partitions, with each pair having adjusted orthogonality with respect to the third one? Answer Yes. Here are two examples. 1. F ⊥ G but F �≺ G and G �≺ F , so that F ∨ G is not F or G . Then X ⊤ F ( I − P F ∨ G ) X G = 0 (by definition of F ⊥ G ) and X ⊤ F ∨ G ( I − P F ) = X ⊤ F ∨ G ( I − P G ) = 0 because V F ∨ G ≤ V F ∩ V G . Bailey Relations among partitions 4/28

Question 1 Question Can we have three partitions, with each pair having adjusted orthogonality with respect to the third one? Answer Yes. Here are two examples. 1. F ⊥ G but F �≺ G and G �≺ F , so that F ∨ G is not F or G . Then X ⊤ F ( I − P F ∨ G ) X G = 0 (by definition of F ⊥ G ) and X ⊤ F ∨ G ( I − P F ) = X ⊤ F ∨ G ( I − P G ) = 0 because V F ∨ G ≤ V F ∩ V G . 2. Suppose that R , C and L are pairwise strictly orthogonal (e.g. the three 2-dimensional slicings of a cube, or the rows, columns and letters of a Latin square). Bailey Relations among partitions 4/28

Question 1 Question Can we have three partitions, with each pair having adjusted orthogonality with respect to the third one? Answer Yes. Here are two examples. 1. F ⊥ G but F �≺ G and G �≺ F , so that F ∨ G is not F or G . Then X ⊤ F ( I − P F ∨ G ) X G = 0 (by definition of F ⊥ G ) and X ⊤ F ∨ G ( I − P F ) = X ⊤ F ∨ G ( I − P G ) = 0 because V F ∨ G ≤ V F ∩ V G . 2. Suppose that R , C and L are pairwise strictly orthogonal (e.g. the three 2-dimensional slicings of a cube, or the rows, columns and letters of a Latin square). L ⊥ C ⇒ P L ( V C ) = P 0 ( V C ) ⇒ X ⊤ R ( I − P L ) X C = X ⊤ R ( I − P 0 ) X C = 0. Bailey Relations among partitions 4/28

Question 1 Question Can we have three partitions, with each pair having adjusted orthogonality with respect to the third one? Answer Yes. Here are two examples. 1. F ⊥ G but F �≺ G and G �≺ F , so that F ∨ G is not F or G . Then X ⊤ F ( I − P F ∨ G ) X G = 0 (by definition of F ⊥ G ) and X ⊤ F ∨ G ( I − P F ) = X ⊤ F ∨ G ( I − P G ) = 0 because V F ∨ G ≤ V F ∩ V G . 2. Suppose that R , C and L are pairwise strictly orthogonal (e.g. the three 2-dimensional slicings of a cube, or the rows, columns and letters of a Latin square). L ⊥ C ⇒ P L ( V C ) = P 0 ( V C ) ⇒ X ⊤ R ( I − P L ) X C = X ⊤ R ( I − P 0 ) X C = 0. In fact, so long as F i ⊥ F j and F k � F i ∨ F j for { i , j , k } = { 1, 2, 3 } , this is true. Bailey Relations among partitions 4/28

Question 1 continued Question Are there any other ways of achieving three-way adjusted orthgonality? Bailey Relations among partitions 5/28

Question 1 continued Question Are there any other ways of achieving three-way adjusted orthgonality? Answer I do not know the answer today, but it may be “obvious”. Bailey Relations among partitions 5/28

Question 2 Question Any Hadamard matrix of order 4 n defines a SBIBD with n B = n L = 4 n − 1 and k B = k L = 2 n − 1. Can we construct a triple array from every such SBIBD? Bailey Relations among partitions 6/28

Question 2 Question Any Hadamard matrix of order 4 n defines a SBIBD with n B = n L = 4 n − 1 and k B = k L = 2 n − 1. Can we construct a triple array from every such SBIBD? Answer 1. No if n = 2. 2. Yes if n ≥ 3 and 2 n − 1 is a prime power. 3. Otherwise, I do not think that the answer is known. Bailey Relations among partitions 6/28

Question 3 Question What do you mean by “describe the system” given by a collection of partitions of a set? Bailey Relations among partitions 7/28

Question 3 Question What do you mean by “describe the system” given by a collection of partitions of a set? Answer (First part) I do not mean “we know the structure up to isomorphism”. Example 1 If n B = n L = 16 and k B = k L = 6 and B ⊲ ⊳ L then there are three isomorphism classes of SBIBDS. Bailey Relations among partitions 7/28

Question 3 Question What do you mean by “describe the system” given by a collection of partitions of a set? Answer (First part) I do not mean “we know the structure up to isomorphism”. Example 1 If n B = n L = 16 and k B = k L = 6 and B ⊲ ⊳ L then there are three isomorphism classes of SBIBDS. Example 2 If n R = n C = n L = n and R ⊥ C , L ⊥ R , L ⊥ C and R ∧ C = R ∧ L = C ∧ L = E then we have a Latin square of order n . If n ≥ 4 then there is more than one isomorphism class of Latin squares of order n . Bailey Relations among partitions 7/28

Question 3 Question What do you mean by “describe the system” given by a collection of partitions of a set? Answer (First part) I do not mean “we know the structure up to isomorphism”. Example 1 If n B = n L = 16 and k B = k L = 6 and B ⊲ ⊳ L then there are three isomorphism classes of SBIBDS. Example 2 If n R = n C = n L = n and R ⊥ C , L ⊥ R , L ⊥ C and R ∧ C = R ∧ L = C ∧ L = E then we have a Latin square of order n . If n ≥ 4 then there is more than one isomorphism class of Latin squares of order n . Aside Fisher and Yates always said that a Latin square for an experiment should be chosen at random from among all Latin squares of that size. (I happen to disagree.) They never said that a BIBD should be chosen at random from among all BIBDS with those parameters. Bailey Relations among partitions 7/28

Question 3: my motivation Xu Guangqi experimented with sweet potatoes to see if they could be grown successfully here. Responses in agronomy are more variable than those in physics, so experiments need to be designed carefully. Bailey Relations among partitions 8/28

Question 3 continued Example Suppose that F = { R , C , L } = a family of partitions of Ω . I measure Y ω on each element ω of Ω . I assume that if ω is in row i , column j and letter k then E ( Y ω ) = α i + β j + γ k . In order to estimate the γ -parameters, I have to project the data vector Y onto the orthogonal complement of V R + V C . Let P { R , C } be the matrix of orthogonal projection onto V R + V C . Bailey Relations among partitions 9/28

Question 3 continued Example Suppose that F = { R , C , L } = a family of partitions of Ω . I measure Y ω on each element ω of Ω . I assume that if ω is in row i , column j and letter k then E ( Y ω ) = α i + β j + γ k . In order to estimate the γ -parameters, I have to project the data vector Y onto the orthogonal complement of V R + V C . Let P { R , C } be the matrix of orthogonal projection onto V R + V C . The information matrix for L is X ⊤ L ( I − P { R , C } ) X L , which is a generalized inverse of the variance-covariance matrix for the estimators of γ 1 , γ 2 , . . . , γ n L . Bailey Relations among partitions 9/28