Semi-Latin squares, permutation groups, and statistical optimality - PowerPoint PPT Presentation

Semi-Latin squares, permutation groups, and statistical optimality Leonard Soicher Queen Mary, University of London Ottawa-Carleton Discrete Mathematics Days: 13-14 May 2011 Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 1 / 28

Introduction This talk is about an interaction between certain areas of combinatorial design theory, the statistical theory of optimal designs, computation, and permutation groups. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 2 / 28

Introduction This talk is about an interaction between certain areas of combinatorial design theory, the statistical theory of optimal designs, computation, and permutation groups. In particular, we shall be interested in semi-Latin squares, which are a generalisation of Latin squares, and are used in the design of comparative experiments. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 2 / 28

Introduction This talk is about an interaction between certain areas of combinatorial design theory, the statistical theory of optimal designs, computation, and permutation groups. In particular, we shall be interested in semi-Latin squares, which are a generalisation of Latin squares, and are used in the design of comparative experiments. Of particular interest will be “uniform” semi-Latin squares, which generalise complete sets of mutually orthogonal Latin squares. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 2 / 28

Block designs Definition A block design is an ordered pair ( V , B ), such that V is a finite non-empty set of points , and B is a (disjoint from V ) finite non-empty multiset (or collection) of non-empty subsets of V called blocks . Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 3 / 28

Block designs Definition A block design is an ordered pair ( V , B ), such that V is a finite non-empty set of points , and B is a (disjoint from V ) finite non-empty multiset (or collection) of non-empty subsets of V called blocks . Definition Let t be a non-negative integer. A t-design , or more specifically a t -( v , k , λ ) design , is a block design ( V , B ) such that v = | V | , each block has the same size k , and each t -subset of V is contained in the same positive number λ of blocks. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 3 / 28

Block designs Definition A block design is an ordered pair ( V , B ), such that V is a finite non-empty set of points , and B is a (disjoint from V ) finite non-empty multiset (or collection) of non-empty subsets of V called blocks . Definition Let t be a non-negative integer. A t-design , or more specifically a t -( v , k , λ ) design , is a block design ( V , B ) such that v = | V | , each block has the same size k , and each t -subset of V is contained in the same positive number λ of blocks. Example The block design ( V , B ), with V = { 1 , . . . , 7 } and B = [ { 1 , 2 , 4 } , { 2 , 3 , 5 } , { 3 , 4 , 6 } , { 4 , 5 , 7 } , { 1 , 5 , 6 } , { 2 , 6 , 7 } , { 1 , 3 , 7 } ] is a 2-(7 , 3 , 1) design (and also a 1-(7 , 3 , 3) design and a 0-(7 , 3 , 7) design). Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 3 / 28

Statistical design theory Block designs are of interest to both pure mathematicians and statisticians. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 4 / 28

Statistical design theory Block designs are of interest to both pure mathematicians and statisticians. They are used by statisticians in the design of comparative experiments: the points represent “treatments” to be compared, and blocks represent sets of homogenous “experimental units” in which treatments are placed. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 4 / 28

Statistical design theory Block designs are of interest to both pure mathematicians and statisticians. They are used by statisticians in the design of comparative experiments: the points represent “treatments” to be compared, and blocks represent sets of homogenous “experimental units” in which treatments are placed. Statisticians sometimes allow points to be repeated within a block, which here we do not. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 4 / 28

Statistical design theory Block designs are of interest to both pure mathematicians and statisticians. They are used by statisticians in the design of comparative experiments: the points represent “treatments” to be compared, and blocks represent sets of homogenous “experimental units” in which treatments are placed. Statisticians sometimes allow points to be repeated within a block, which here we do not. There are various measures of the “efficiency” of a block design for the design of an experiment, and we discuss these now in the context of 1-designs. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 4 / 28

The (scaled) information matrix Let ∆ be a 1-( v , k , r ) design, with v ≥ 2. Definition The concurrence matrix of ∆ is the v × v matrix whose rows and columns are indexed by the points, and whose ( α, β )-entry is the number of blocks containing both α and β . Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 5 / 28

The (scaled) information matrix Let ∆ be a 1-( v , k , r ) design, with v ≥ 2. Definition The concurrence matrix of ∆ is the v × v matrix whose rows and columns are indexed by the points, and whose ( α, β )-entry is the number of blocks containing both α and β . Definition The scaled information matrix of ∆ is F (∆) := I v − ( rk ) − 1 Λ , where Λ is the concurrence matrix of ∆. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 5 / 28

Canonical efficiency factors The eigenvalues of F (∆) are all real and lie in [0 , 1]. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 6 / 28

Canonical efficiency factors The eigenvalues of F (∆) are all real and lie in [0 , 1]. F (∆) has constant row-sum 0, so the all-1 vector is an eigenvector with corresponding eigenvalue 0. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 6 / 28

Canonical efficiency factors The eigenvalues of F (∆) are all real and lie in [0 , 1]. F (∆) has constant row-sum 0, so the all-1 vector is an eigenvector with corresponding eigenvalue 0. It can be shown that the remaining eigenvalues are all non-zero if and only if ∆ is connected (i.e. its point-block incidence graph is connected). Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 6 / 28

Canonical efficiency factors The eigenvalues of F (∆) are all real and lie in [0 , 1]. F (∆) has constant row-sum 0, so the all-1 vector is an eigenvector with corresponding eigenvalue 0. It can be shown that the remaining eigenvalues are all non-zero if and only if ∆ is connected (i.e. its point-block incidence graph is connected). Definition Omitting the zero eigenvalue corresponding to the all-1 vector, the eigenvalues δ 1 ≤ δ 2 ≤ · · · ≤ δ v − 1 of F (∆) are called the canonical efficiency factors of ∆. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 6 / 28

Efficiency measures for 1-designs Let ∆ be a 1-( v , k , r ) design with v ≥ 2 and canonical efficiency factors δ 1 ≤ · · · ≤ δ v − 1 . Definition If ∆ is not connected, then A ∆ = D ∆ = E ∆ := 0. Otherwise, we define these efficiency measures by v − 1 � A ∆ := ( v − 1) / 1 /δ i , i =1 � 1 / ( v − 1) � v − 1 � D ∆ := δ i , i =1 E ∆ := δ 1 = min { δ 1 , . . . , δ v − 1 } . Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 7 / 28

On efficiency measures If k = v (so ∆ has complete blocks) then each of its canonical efficiency factors is equal to 1 and so are each of the efficiency measures above. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 8 / 28

On efficiency measures If k = v (so ∆ has complete blocks) then each of its canonical efficiency factors is equal to 1 and so are each of the efficiency measures above. If k < v (so ∆ has incomplete blocks), we want to minimize the loss of “information” due to blocking, and want the above efficiency measures to be as close to 1 as possible. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 8 / 28

On efficiency measures If k = v (so ∆ has complete blocks) then each of its canonical efficiency factors is equal to 1 and so are each of the efficiency measures above. If k < v (so ∆ has incomplete blocks), we want to minimize the loss of “information” due to blocking, and want the above efficiency measures to be as close to 1 as possible. To test conjectures and rank designs we have classified, we need to be able to compare efficiency measures exactly . I can do this using algebraic computation in GAP, and this functionality should be included in the next release of the DESIGN package for GAP. Leonard Soicher (QMUL) Semi-Latin squares, groups, and optimality DMD 2011 8 / 28