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Satoshi Kajimoto , Ryo Y amada

A rotation matrix , which is useful to interpret the restriction of marginal counts of multi-way tables. Satoshi Kajimoto , Ryo Y amada Unit of statistical genetics, Center for Genomic Medicine ,Graduate School of Medicine, Kyoto University , Kyoto, Japan. Objective.

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Satoshi Kajimoto , Ryo Y amada

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  1. 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

  2. Objective We introduce how to make the matrix X, that gives coordinates of df-dimensional space to multi-way tables.

  3. 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

  4. 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

  5. dimension (k=) 2 (k=) 3 shape 2×3×4 2×3 shape vector The number of the cells R= 24 R= 6 R= multi-way table (k-dimensional table)

  6. one-to-one 2×3 table The matrix X （square matrix） shape vector

  7. 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

  8. Now, We will explain 3 terms , which need to make the matrix X • Simplex • Simplex matrix • Kronecker product

  9. 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

  10. x n-simplex (1,0,0) z (0,0,1) 1-simplex 2-simplex 3-simplex y (0,1,0) A 2-simplex is a regular triangle. A 3-simplex is a regular tetrahedron. An (n-1)-simplex can be put in n-dimensional orthogonal coordinate system. 2-simplex in 3 dimensional orthogonal coordinate system

  11. Simplex matrix An n-simplex matrix is a matrix whose column vectors are the coordinates of vertices of the (n-1)- simplex. : coordinates of vertices of 3-simplex 4-simplex matrix =

  12. x x (1,0,0) z (0,0,1) y y 1 (0,1,0) Parallel to yz-plane rotation We can rotate an (n-1) simplex which is put in n-dimensional orthogonal coordinate system to plane whose x-axis is . 3-simplex matrix

  13. Our way to make n-simplex matrix An n-simplex matrix is the n×n matrix and defined as below.

  14. Examples of n-simplex matrix

  15. Kronecker product

  16. How to make the matrix X - (⊗ is the Kronecker product) X is matrix

  17. For example (the 2-simplex matrix) (the 3-simplex matrix) R

  18. 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

  19. 1. X is an R×R rotation matrix. We vectorize multi-way tables into vectors in R-dimensional space, and rotate them.

  20. z y 2×4×3 table 23 24 The contents of the table 24 22 21 20 19 22 17 18 16 20 18 14 17 18 12 8 10 6 9 10 4 2 x 1 2 vectorize

  21. 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’=Xa, b’ = Xb B vectorize vectorize A B

  22. The relationship between the matrix X and marginal counts of tables vectorized vector table A = 3 11 26 7 9 44 7 11 17 = 1012 53 table B vector vectorized

  23. vectorized vector table A = 3 11 26 7 9 44 7 11 17 = 1012 43 table B vector vectorized

  24. vectorized vector table A = 3 11 26 7 9 44 7 11 22 = 1012 38 table B vector vectorized

  25. vectorized vector table A = 3 11 26 7 9 44 7 11 17 = 3953 table B vector vectorized

  26. = 3 11 26 7 9 44 7 11 22 = 3 9 48 The degrees of freedom of this table is 2. So, 2 elements of these two vectors are different, and 4 elements are equal.

  27. Now, by using X, Tables are placed in df-dimensional space.

  28. 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

  29. 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

  30. df =2 Counter line of statistics Association test with df=2 χ2 →　pvalues χ2 pvalues But, by reducing the degrees of a vector and showing a diagram, We can calculate p values computaionally. Ryo Yamada, Yukinori Okada, 2009, An Optimal Dose-effect Mode Trend Test for SNP Genotype Tables, Genetic Epidemiology vol.33, p.114~127

  31. Thank you for listening.

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