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 YamadaUnit 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

shape 2×3×42×3𝒓=(23) 𝒓=(234 )shape vector

The number of the cells

R=

R= 24

𝒓=(𝑟 1𝑟 2⋮𝑟𝑘

)

dimension (k=) 2 (k=) 3

R= 6

multi-way table(k-dimensional table)

10 17 13

11 14 12

one-to-oneThe matrix X（ square matrix）

shape vector

𝒓=(23) (1

√61

√61

√61√6

−1√6

1√6

1√3

1√3

−12√3

−

1

√61

√61

√61√6

1√6

−1√6

−12√3

−12√3

−12√3

1

√3−1

√3−12√3

0 012

0 012

12√3

−12√3

12√3

12

−12

−12

−12

−12

12

)

10 17 13

11 14 12

2×3 table



• 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 3-simplex

(0,1,0)

(1,0,0)

(0,0,1)

y

z

x

2-simplex in 3 dimensional orthogonal coordinate system

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.

Simplex matrix

An n-simplex matrix is a matrix whose column vectors are the coordinates of vertices of the (n-1)- simplex.

𝑣1

𝑣2

𝑣1

𝑣3

𝑣4

𝑣4𝑣3𝑣2 : coordinates of vertices of 3-simplex

4-simplex matrix =

(0,1,0)

(1,0,0)

(0,0,1)

3-simplex matrix

Parallel to yz-plane

x

y

z

x

y

rotation

𝑥=1

√3

1

√3

( 1√3 ,0)

1

We can rotate an (n-1) simplex which is put in n-dimensional orthogonal coordinate system to plane whose x-axis is .

Our way to make n-simplex matrix

An n-simplex matrix is the n×n matrix and defined as below.

𝑎 𝑗𝑘=1

√𝑛

{𝑎 𝑗𝑘=0( h𝑤 𝑒𝑛𝑘≦ 𝑗−2)

𝑎 𝑗𝑘=√𝑛− 𝑗+1𝑛− 𝑗+2

( h𝑤 𝑒𝑛𝑘= 𝑗−1)

𝑎 𝑗𝑘=−1

√(𝑛− 𝑗+1 ) (𝑛− 𝑗+2 )( h𝑤 𝑒𝑛𝑘≧ 𝑗)

𝑗>1

𝑗=1

(1

√31

√31

√3√2√3

−1√6

−1√6

01

√2−1

√2)(

1√2

1√2

1

√2−1

√2) (

12

12

√32

−12√3

12

12

−12√3

−12√3

0 √2√3

0 0

−1√6

−1√6

1√2

−1√2

)Examples of n-simplex matrix

Kronecker product

𝐴=(𝑎11 𝑎12𝑎21 𝑎22

⋯ 𝑎1𝑝⋯ 𝑎2𝑝

⋮ ⋮𝑎𝑝 1 𝑎𝑝 2

⋱ ⋮⋯ 𝑎𝑝𝑝

) 𝐵=(𝑏11 𝑏12𝑏21 𝑏22

⋯ 𝑏1𝑞⋯ 𝑏2𝑞

⋮ ⋮𝑏𝑞1 𝑏𝑞2

⋱ ⋮⋯ 𝑏𝑞𝑞

)𝑝×𝑝𝑚𝑎𝑡𝑟𝑖𝑥 𝑞×𝑞𝑚𝑎𝑡𝑟𝑖𝑥

𝑝𝑞×𝑝𝑞𝑚𝑎𝑡𝑟𝑖𝑥

𝐴⊗ 𝐵=(𝑎11𝐵 𝑎12𝐵𝑎21𝐵 𝑎22𝐵

⋯ 𝑎1𝑝𝐵⋯ 𝑎2𝑝 𝐵

⋮ ⋮ ⋱ ⋮𝑎𝑝 1𝐵 𝑎𝑝 2𝐵 ⋯ 𝑎𝑝𝑝𝐵

)

How to make the matrix X

𝒓=(𝑟 1𝑟 2⋮𝑟𝑘

) -

𝑋=𝑋𝑘⊗ 𝑋𝑘− 1⊗⋯⋯⊗ 𝑋 1( is the Kronecker product)⊗

X is matrix

For example

𝒓=(23) (the 2-simplex matrix)

(the 3-simplex matrix)

𝑋 1=(1√2

1√2

1

√2−1

√2) 𝑋 2=(

1

√31

√31

√3√2√3

−1√6

−1√6

01

√2−1

√2)

𝑋=𝑋 2⊗ 𝑋1=(1

√61

√61

√61√6

−1√6

1√6

1√3

1√3

−12√3

−

1

√61

√61

√61√6

1√6

−1√6

−12√3

−12√3

−12√3

1

√3−1

√3−12√3

0 012

0 012

12√3

−12√3

12√3

12

−12

−12

−12

−12

12

)

R

1. X is an R×R rotation matrix.

We vectorize multi-way tables into vectors in R-dimensional space, and rotate them.

23 24

21 22

19 20

17 18

15 16

13 14

11 12

9 10

7 8

5 6

3 4

1 2

vectorize

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)❑𝑡

2×4×3 tableThe contents of the table

z

x

2321

2422

20191817

1817

109

21

24

108

6

24

2220

18 1416

12

y

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

B𝒂 𝒃vectorizevectorize BA

40

60

10 20 70 100

3 11 267 9 44

35

75

17 23 70 110

7 11 1710 12 53

=

=

The relationship between the matrix X and marginal counts of tables

table A

table B

vector

vector

vectorized

vectorized

40

60

10 20 70 100

3 11 267 9 44

35

65

17 23 60 100

7 11 1710 12 43

=

=

table Avector vectorized

table B

vector vectorized

40

60

10 20 70 100

3 11 267 9 44

40

60

17 23 60 100

7 11 2210 12 38

=

=


table B

vector vectorized

40

60

10 20 70 100

3 11 267 9 44

35

65

10 20 70 100

7 11 173 9 53

=

=


table B

vector vectorized

40

60

10 20 70 100

3 11 267 9 44

40

60

10 20 70 100

7 11 223 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.

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

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 complexLectures on Algebraic StaticsISBN-13: 978-3764389048

df = 2

χ2 →　 p values 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

Counter line of statistics

Association test with df=2

But, by reducing the degrees of a vectorand showing a diagram,We can calculate p values computaionally.

Thank you for listening.

Satoshi Kajimoto , Ryo Y amada

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