Supplementary Text 2. If the relationship between two random variables X and Y is approximately...

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Supplementary Text 2

Transcript of Supplementary Text 2. If the relationship between two random variables X and Y is approximately...

Page 1: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

Supplementary Text 2

Page 2: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

Supplementary Text 2 If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize the sample values of them?

μ1

μ2

N(μ1, σ1)

N(μ2, σ2)

t1,1 t1,2 t1,3 t1,4

t2,1 t2,2 t2,3 t2,4

Fig. A

Fig. B

Page 3: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

To uniformly describe the relation ship between X and Y, we can map them to normal distribution. And the corresponding relationship would be change into that illustrated in the figure below.

0

0

N(0, 1)

N(0, 1)

t1 t2 t3 t4

t1 t2 t3 t4

Fig. C

Fig. D

Page 4: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

The data transformation between Figure A and C is implemented with the Equation (14) in the paper. And the data transformation between Figure B and D is also implemented with Equation (14) in the paper.

Then our goal is to find the best common partitioning points t1, t2, …, tk.

The Purpose of Equation (15) in the paper is to determine the best common partitioning points for both variable X and Y, while simultaneously making both of them as evenly partitioned as possible.

The structures of the variables shown in Figure B and D reveal the essence of the relationship between the two random variables of the same kind. This is of more importance when the number of random variables becomes large. It makes the partition not “casual” just in order to adapt to the data, some of which are just noise.

Page 5: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

Example 3The gene expression data is partially listed below. Suppose that Gene #1 and #2 have been divided in the same group G1, while Gene #3, #4, and #5 are in the same group G2.

Time 1 Time 2 Time 16

Gene No.

Array1(Low)

Array2(High)

Array3(Low)

Array4(High)

…… Array31(Low)

Array32(High)

1 D1,1,L D1,1,H D1,2,L D1,2,H …… D1,16,L D1,16,H

2 D2,1,L D2,1,H D2,2,L D2,2,H …… D2,16,L D2,16,H

3 D3,1,L D3,1,H D3,2,L D3,2,H …… D3,16,L D3,16,H

4 D4,1,L D4,1,H D4,2,L D4,2,H …… D4,16,L D4,16,H

5 D5,1,L D5,1,H D5,2,L D5,2,H …… D5,16,L D5,16,H

Page 6: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

For the data in the table above, we define 15 vectors as follows.

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,16,5,16,5,2,5,2,5,1,5,1,55

,16,4,16,4,2,4,2,4,1,4,1,44

,16,3,16,3,2,3,2,3,1,3,1,33

,16,2,16,2,2,2,2,2,1,2,1,22

,16,1,16,1,2,1,2,1,1,1,1,11

,165,,25,,15,5,,165,,25,,15,5,

,164,,24,,14,4,,164,,24,,14,4,

,163,,23,,13,3,,163,,23,,13,3,

,162,,22,,12,2,,162,,22,,12,2,

,16,1,2,1,1,1,1,16,1,2,1,1,1,1

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

Time 1 Time 2 Time 16

Gene No.

Array1(Low)

Array2(High)

Array3(Low)

Array4(High)

…… Array31(Low)

Array32(High)

1 D1,1,L D1,1,H D1,2,L D1,2,H …… D1,16,L D1,16,H

2 D2,1,L D2,1,H D2,2,L D2,2,H …… D2,16,L D2,16,H

3 D3,1,L D3,1,H D3,2,L D3,2,H …… D3,16,L D3,16,H

4 D4,1,L D4,1,H D4,2,L D4,2,H …… D4,16,L D4,16,H

5 D5,1,L D5,1,H D5,2,L D5,2,H …… D5,16,L D5,16,H

Page 7: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

To compute the mutual information between the two groups G1 and G2 about low pressure treatment described in Equation (17) in the paper, we can use the following detailed equation in stead of (17):

LLLLLL

LLLLLLL

DDIDDIDDI

DDIDDIDDIGGI

,5,2,4,2,3,2

,5,1,4,1,3,121

,,,

,,,32

1,

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,16,5,16,5,2,5,2,5,1,5,1,55

,16,4,16,4,2,4,2,4,1,4,1,44

,16,3,16,3,2,3,2,3,1,3,1,33

,16,2,16,2,2,2,2,2,1,2,1,22

,16,1,16,1,2,1,2,1,1,1,1,11

,165,,25,,15,5,,165,,25,,15,5,

,164,,24,,14,4,,164,,24,,14,4,

,163,,23,,13,3,,163,,23,,13,3,

,162,,22,,12,2,,162,,22,,12,2,

,16,1,2,1,1,1,1,16,1,2,1,1,1,1

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

Page 8: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

To compute the mutual information between the two groups G1 and G2 about high pressure treatment described in Equation (18) in the paper, we can use the following detailed equation in stead of (18):

HLHHHH

HHHHHHH

DDIDDIDDI

DDIDDIDDIGGI

,5,2,4,2,3,2

,5,1,4,1,3,121

,,,

,,,32

1,

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,16,5,16,5,2,5,2,5,1,5,1,55

,16,4,16,4,2,4,2,4,1,4,1,44

,16,3,16,3,2,3,2,3,1,3,1,33

,16,2,16,2,2,2,2,2,1,2,1,22

,16,1,16,1,2,1,2,1,1,1,1,11

,165,,25,,15,5,,165,,25,,15,5,

,164,,24,,14,4,,164,,24,,14,4,

,163,,23,,13,3,,163,,23,,13,3,

,162,,22,,12,2,,162,,22,,12,2,

,16,1,2,1,1,1,1,16,1,2,1,1,1,1

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

Page 9: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

To compute the mutual information between the two groups G1 and G2 about both high pressure and the high pressure treatment described in Equation (19) in the paper, we can use the following detailed equation in stead of (19):

524232

51413121

,,,

,,,32

1,

DDIDDIDDI

DDIDDIDDIGGI

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,16,5,16,5,2,5,2,5,1,5,1,55

,16,4,16,4,2,4,2,4,1,4,1,44

,16,3,16,3,2,3,2,3,1,3,1,33

,16,2,16,2,2,2,2,2,1,2,1,22

,16,1,16,1,2,1,2,1,1,1,1,11

,165,,25,,15,5,,165,,25,,15,5,

,164,,24,,14,4,,164,,24,,14,4,

,163,,23,,13,3,,163,,23,,13,3,

,162,,22,,12,2,,162,,22,,12,2,

,16,1,2,1,1,1,1,16,1,2,1,1,1,1

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

Page 10: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

To compute the mutual information between the treatment low pressure and the treatment high pressure within the group G1 described in Equation (20) in the paper, we can use the following detailed equation in stead of (20):

HLHLHL DDIDDIGI ,2,2,1,11, ,,2

1

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,16,5,16,5,2,5,2,5,1,5,1,55

,16,4,16,4,2,4,2,4,1,4,1,44

,16,3,16,3,2,3,2,3,1,3,1,33

,16,2,16,2,2,2,2,2,1,2,1,22

,16,1,16,1,2,1,2,1,1,1,1,11

,165,,25,,15,5,,165,,25,,15,5,

,164,,24,,14,4,,164,,24,,14,4,

,163,,23,,13,3,,163,,23,,13,3,

,162,,22,,12,2,,162,,22,,12,2,

,16,1,2,1,1,1,1,16,1,2,1,1,1,1

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

Page 11: Supplementary Text 2. If the relationship between two random variables X and Y is approximately illustrated in the following figure, then how can we discretize.

To compute the mutual information between the treatment low pressure and the treatment high pressure within the group G2 described in Equation (20) in the paper, we can use the following detailed equation in stead of (20):

HLHLHLHL DDIDDIDDIGI 5,5,4,4,3,3,2, ,,,3

1

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,,,......,,,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,......,,,......,

,16,5,16,5,2,5,2,5,1,5,1,55

,16,4,16,4,2,4,2,4,1,4,1,44

,16,3,16,3,2,3,2,3,1,3,1,33

,16,2,16,2,2,2,2,2,1,2,1,22

,16,1,16,1,2,1,2,1,1,1,1,11

,165,,25,,15,5,,165,,25,,15,5,

,164,,24,,14,4,,164,,24,,14,4,

,163,,23,,13,3,,163,,23,,13,3,

,162,,22,,12,2,,162,,22,,12,2,

,16,1,2,1,1,1,1,16,1,2,1,1,1,1

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HLHLHL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

HHHHLLLL

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD

DDDDDDDD


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