A rotation matrix , which is useful to interpret the restriction of marginal counts of multi-way tables Satoshi Kajimoto , Ryo Yamada Unit of statistical genetics, Center for Genomic Medicine ,Graduate School of Medicine, Kyoto University , Kyoto, Japan Objective We introduce how to make the matrix X, that gives coordinates of df-dimensional space to multi-way tables. The chart of this presentation • Step1 Introduction • Step2 How to make the matrix X – Simplex – Simplex matrix – Kronecker product • These three terms are needed to make the matrix X. • Step3 The properties of the matrix X • Step4 Applications of the matrix X The chart of this presentation • Step1 Introduction • Step2 How to make the matrix X – Simplex – Simplex matrix – Kronecker product • These three terms are needed to make the matrix X. • Step3 The properties of the matrix X • Step4 Applications of the matrix X 10 17 13 11 14 12 dimension (k=) 2 (k=) 3 shape 2×3 2×3×4 shape vector 2 𝒓= 3 2 𝒓= 3 4 The number of the cells R= 6 R= 24 multi-way table (k-dimensional table) 𝒓= 𝑟1 𝑟2 ⋮ 𝑟𝑘 R= 𝑘 𝑖=1 𝑟𝑖 10 17 13 11 14 12 2×3 table one-to-one The matrix X shape vector (square matrix) 1 6 1 2 𝒓= 3 6 1 3 1 3 1 − 6 1 1 − 6 3 1 3 0 0 0 0 − − 1 1 1 6 1 6 1 6 1 6 1 6 1 6 1 2 3 1 2 3 1 2 1 2 − − 2 3 1 2 3 1 2 1 − 2 − − 2 3 1 1 − − 1 6 1 6 1 2 3 2 3 2 3 1 1 − − 2 2 1 1 − 2 2 The chart of this presentation • Step1 Introduction • Step2 How to make the matrix X – Simplex – Simplex matrix – Kronecker product • These three terms are needed to make the matrix X. • Step3 The property of the matrix X • Step4 Applications of the matrix X Now, We will explain 3 terms , which need to make the matrix X • Simplex • Simplex matrix • Kronecker product Simplex A simplex is a generalization of the notion of a regular triangle or regular tetrahedron to arbitrary dimension. 1-simplex 2-simplex 3-simplex n-simplex 1-simplex 2-simplex x 3-simplex A 2-simplex is a regular triangle. A 3-simplex is a regular tetrahedron. (1,0,0) z (0,0,1) (0,1,0) y An (n-1)-simplex can be put in ndimensional orthogonal coordinate system. 2-simplex in 3 dimensional orthogonal coordinate system Simplex matrix An n-simplex matrix is a matrix whose column vectors are the coordinates of vertices of the (n1)- simplex. 𝑣1 𝑣2 𝑣3 𝑣4 : coordinates of vertices of 3-simplex 𝑣1 𝑣4 𝑣2 𝑣3 4-simplex matrix = 𝑣1 𝑣2 𝑣3 𝑣4 x x 1 3 ,0 𝑥= (1,0,0) 1 3 z (0,0,1) 1 3 (0,1,0) y 1 y Parallel to yz-plane rotation We can rotate an (n-1) simplex which is put in n-dimensional orthogonal coordinate system 1 to plane whose x-axis is 𝑛. 3-simplex matrix Our way to make n-simplex matrix An n-simplex matrix is the n×n matrix and defined as below. 𝑗=1 𝑎𝑗𝑘 = 1 𝑛 𝑗>1 𝑎𝑗𝑘 = 0 𝑎𝑗𝑘 = 𝑎𝑗𝑘 = − (𝑤ℎ𝑒𝑛 𝑘 ≦ 𝑗 − 2) 𝑛−𝑗+1 𝑛−𝑗+2 (𝑤ℎ𝑒𝑛 𝑘 = 𝑗 − 1) 1 𝑛−𝑗+1 𝑛−𝑗+2 (𝑤ℎ𝑒𝑛 𝑘 ≧ 𝑗) Examples of n-simplex matrix 1 2 1 2 1 1 − 2 1 2 3 2 3 0 1 − 1 3 1 1 6 2 − − 3 1 1 1 2 2 3 1 − 2 2 3 6 1 0 2 0 2 3 0 − 1 2 1 − 1 2 1 2 3 2 3 1 1 − − 6 6 1 1 − 2 2 Kronecker product 𝐴= 𝑎11 𝑎21 ⋮ 𝑎𝑝1 𝑎12 𝑎22 ⋮ 𝑎𝑝2 ⋯ 𝑎1𝑝 ⋯ 𝑎2𝑝 ⋱ ⋮ ⋯ 𝑎𝑝𝑝 𝐵= 𝑏12 𝑏22 ⋮ 𝑏𝑞2 ⋯ 𝑏1𝑞 ⋯ 𝑏2𝑞 ⋱ ⋮ ⋯ 𝑏𝑞𝑞 𝑞 × 𝑞 𝑚𝑎𝑡𝑟𝑖𝑥 𝑝 × 𝑝 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴⊗𝐵 = 𝑏11 𝑏21 ⋮ 𝑏𝑞1 𝑎11 𝐵 𝑎12 𝐵 ⋯ 𝑎1𝑝 𝐵 𝑎21 𝐵 𝑎22 𝐵 ⋯ 𝑎2𝑝 𝐵 ⋮ ⋮ ⋱ ⋮ 𝑎𝑝1 𝐵 𝑎𝑝2 𝐵 ⋯ 𝑎𝑝𝑝 𝐵 𝑝𝑞 × 𝑝𝑞 𝑚𝑎𝑡𝑟𝑖𝑥 How to make the matrix X 𝒓= 𝑟1 𝑟2 ⋮ 𝑟𝑘 𝑋𝑙 ≝ 𝑟𝑙 -𝑠𝑖𝑚𝑝𝑙𝑒𝑥 𝑚𝑎𝑡𝑟𝑖𝑥 𝑋 = 𝑋𝑘 ⊗𝑋𝑘−1 ⊗ ⋯ ⋯ ⊗𝑋1 (⊗ is the Kronecker product) X is 𝑅 = 𝑟𝑙 × 𝑅 matrix 1 For example 1 𝒓= 2 3 2 1 𝑋1 = 𝑅 =2×3 =6 − 2 2 1 2 6 1 6 1 3 1 3 1 − 6 1 1 − 6 3 1 3 0 0 0 0 − − 3 2 𝑋2 = − 3 1 6 1 6 1 6 1 6 1 6 1 2 3 1 2 1 2 6 3 1 − 6 1 2 (the 3-simplex matrix) 1 2 3 1 − 2 1 6 1 1 3 1 1 0 (the 2-simplex matrix) 1 𝑋 = 𝑋2 ⊗𝑋1 = 1 1 − − 2 3 1 2 3 1 2 1 − 2 − − 2 3 1 1 − − 1 6 1 6 1 2 3 2 3 2 3 1 1 − − 2 2 1 1 − 2 2 R The chart of this presentation • Step1 Introduction • Step2 How to make the matrix X – Simplex – Simplex matrix – Kronecker product • These three terms are needed to make the matrix X. • Step3 The properties of the matrix X • Step4 Applications of the matrix X 1. X is an R×R rotation matrix. We vectorize multi-way tables into vectors in R-dimensional space, and rotate them. z 23 24 24 22 21 20 19 22 17 18 20 16 18 14 17 18 12 8 9 10 10 4 6 2 1 2 vectorize 𝑡 y 2×4×3 table The contents of the table x 23 24 21 22 19 20 17 18 15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2 (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24) What does the rotation by X do ? • When two table vectors a and b share marginal counts, • only df elements of rotated vectors a’ and b’ differ. A vectorize • a’=Xa, b’ = Xb • 𝒂′ − 𝒃′ = 𝒂 B B vectorize 𝒃 The relationship between the matrix X and marginal counts of tables vector 𝒂 vectorized table A 3 7 11 9 26 44 40 10 20 70 100 11 12 17 53 35 23 70 110 7 10 17 table B vectorized 60 75 40.82 3 −8.16 7 𝑋 11 = −72.17 9 11.55 26 −25.00 44 10.00 7 44.91 10 −16.33 𝑋 11 = −70.73 12 30.31 17 −23.50 53 17.50 vector 𝒃 vector 𝒂 vectorized table A 3 7 11 9 26 44 40 10 20 70 100 11 12 17 43 35 23 60 100 7 10 17 table B vectorized 60 65 40.82 3 −8.16 7 𝑋 11 = −72.17 9 11.55 26 −25.00 44 10.00 40.82 7 −12.25 10 𝑋 11 = −62.07 12 21.65 17 −18.50 43 12.50 vector 𝒃 vector 𝒂 vectorized table A 3 7 11 9 26 44 40 10 20 70 100 11 12 22 38 40 23 60 100 7 10 17 table B vectorized 60 60 40.82 3 −8.16 7 𝑋 11 = −72.17 9 11.55 26 −25.00 44 10.00 7 40.82 10 −8.16 𝑋 11 = −62.07 12 12.99 22 −18.50 38 7.50 vector 𝒃 vector 𝒂 vectorized table A 3 7 11 9 26 44 40 10 20 70 100 7 3 11 9 17 53 35 10 20 70 100 table B vectorized 60 65 40.82 3 −8.16 7 𝑋 11 = −72.17 9 11.55 26 −25.00 44 10.00 40.82 7 −12.25 3 𝑋 11 = −72.17 9 31.75 17 −25.00 53 19.00 vector 𝒃 3 7 11 9 26 44 40 10 20 70 100 7 3 11 9 22 48 40 10 20 70 100 60 60 40.82 3 −8.16 7 𝑋 11 = −72.17 9 11.55 26 −25.00 44 10.00 7 40.82 3 −8.16 𝑋 11 = −72.17 9 23.09 22 −25.00 48 14.00 The degrees of freedom of this table is 2. So, 2 elements of these two vectors are different, and 4 elements are equal. Now, by using X, Tables are placed in df-dimensional space. The chart of this presentation • Step1 Introduction • Step2 How to make the matrix X – Simplex – Simplex matrix – Kronecker product • These three terms are needed to make the matrix X. • Step3 The properties of the matrix X • Step4 Applications of the matrix X Variations of df patterns in multi-way table “Lectures on Algebraic Statistics” express a restriction of marginal counts in simplicial complex. The matrix X is useful to grasp such a complex restriction. Simplicial complex Lectures on Algebraic Statics ISBN-13: 978-3764389048 df = 2 Counter line of statistics Association test with df=2 χ2 → p values Ryo Yamada, Yukinori Okada, 2009, An Optimal Dose-effect Mode Trend Test for SNP Genotype Tables, Genetic Epidemiology vol.33, p.114~127 χ2 p values But, by reducing the degrees of a vector and showing a diagram, We can calculate p values computaionally. Thank you for listening.
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