An Information-theoretic Framework for Visualization - Google Sites

An Information-theoretic Framework for Visualization Min Chen and Heike Jänicke Swansea University and Universität Heidelberg

1 Motivation: A Theoretic Framework Quantitative measurement and reasoning Explanation of facts and phenomena Laws and guidelines for optimization Falsifiable predictions Looking for theories? Lookingfortheories?

1 Motivation 2 Communication n & Visualization 3 Quantifying Visual Info plot 4 Explanation n - logarithmic & redundancy 5 Laws – interaction & user study

6 Prediction n – DP inequality & ... 7 Conclusions & Future Work

2 Communication and Visualization Is a visualization

system a communication system?

Example Example: avisualcommunicationsystem imagefrom: http://chicagodesignintern.blogspot.com/

A General Visualization Pipeline (without interaction) raw data D

Source

N

information I

N

geometry & labels G

Visual Mapping

Filtering

image V

N

Rendering

optical signal S

N

Optical Transmission

Displaying

image V'

N Viewing

N

optical signal S'

N

information I'

Perception

N

knowledge K

Cognition

Noise message M

Source

signal S

Encoder (Transmitter)

signal S'

Channel

message M'

Decoder (Receiver)

A General Communication System

Destination

Destination

Three Visualization Subsystems raw data D

Source

information I

N Filtering

N

geometry & labels G

Visual Mapping

image V

N

Rendering

vis-encoder image V

N

optical signal S

N

optical signal S'

Optical O ti l Transmission

Displaying

image V'

N

Viewing

vis-channel image V'

N

information I'

Perception

N

knowledge K

Cognition

Destination

vis-decoder message M

Source

signal S


N Channel

signal S'

message M'

Decoder (Receiver)


Destination


image V

N visencoder

Source

message M

Source


N

image V'

vischannel

visdecoder

?

?

signal S

signal S'

N Channel

compactness

N

knowledge K

Destination

message M'

Decoder (Receiver)

error detection error correction


Destination

3 Quantifying Visual Information Random variable

X It takes values

x1 , x2 , , xm Probability mass function

p ( xi ) Entropy

H(X )

m

¦ p ( xi ) log 2 p ( xi ) i

ClaudeE.Shannon (19162001)

minimal 64 pixels 256

Entropy: Example

minimal 2 256 pixels

192

128

Ti Time Series S i z

64 independent samples

z

each sample is an integer in [0, 255]

z

probability mass function is i.i.d. (independent and identically identically-distributed) distributed)

Entropy

H (Z ) 64

0

8 16 24 32 40 48 56 64

1 1 ¦ ¦ log 2 256 t 0 i 0 256

512

Time Series Plot z

0

64 255

Minimal 256x64 p pixels ((214 p pixels))

minimal 64 pixels

76543210

256

byte 1

X X X

X X X X X X X X X X X X X X X X X X

The Most Compact, Compact Why Not?

X X

X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

byte 16


192

128

byte y 32

byte 48

64

byte 64

0 0

8 16 24 32 40 48 56 64

X X X X X

minimal 8 pixels

Th mostt compact: The t z

minimal 6 64 pixels

X X X X X X

One pixel per bit seems not enough for vision z

64 bytes (512 bits)

4x4 pixels per bit o 213 bits

Not a counter example: z

Sequential or parallel

z

Salient information o at o


image V

N visencoder

Source

perceptual efficiency cognitive load

vischannel

?

signal S

signal S'


N Channel

compactness

N

knowledge K

visdecoder

intuitiveness clarity

message M

Source

N

image V'

Destination

message M'

Decoder (Receiver)



Destination

Three Measures for Visualization

Entropy of Input Data Space: H(X)

Visualization Capacity: V(G)

Di l C Display Capacity: it D cf. (Yang-Peláez & Flowers, 2000)

Visual Mapping Ratio (VMR)

V (G ) H(X )

Information Loss Ratio (ILR)

max( H ( X ) V (G ),0) H(X )

V ((G G) Display Space Utilization (DSU) D

4 Explanation - Logarithmic Plot & Redundancy Logarithmic plot z

Why is it a useful in some situations?

z

In what situation?

z

What does it try to optimize?

Redundancy y in Visualization Design g z

Good and bad?

z

When is it useful?

z

Should this community pay more attention to “redundancy”?

minimal 64 pixels 256

Information Loss Ratio (ILR)

192

Di l S Display Space R Restriction t i ti


z

128

information loss:

25% 256

64

8 16 24 32 40 48 56 64

Evenly distributed probability mass function

Linear visual mapping

ILR is a p probabilistic measure about z

a data space X

128

z

not an instance xi

0 0

192

64

0

64x64 pixels

0

8 16 24 32 40 48 56 64

(a) evenly distributed p

Non uniform Distribution Non-uniform

Li Linear visual i l mapping i probability

D: information loss:

information loss:

25.0%

25.8% C:

256

256

192

192

B: 128

p

p

p

128

64

64

A A: 0

0 0

8 16 24 32 40 48 56 64


0

8 16 24 32 40 48 56 64

(b) unevenly distributed p

p

1 8 1 8 1 4 1 2

Z

linear

Z'

256

64

224

56

192

48

160

40

128

32

96

24

64

16

32

8

0

0

Non uniform Distribution Non-uniform N li Nonlinear visual i l mapping i

information loss:

information loss:

information loss:

25.0%

25.8%

22.6%

256

256

256 192

192

192

128

128

128

64

64

64

0

0 0

8 16 24 32 40 48 56 64


0

8 16 24 32 40 48 56 64


0 0

8 16 24 32 40 48 56 64

(c) 4 regional mappings

D:

C:

B:

A:

p

1 8

p

1 8

p p

1 4

k of

1 2

p

1

p

1

p

1

2k

Non uniform Distribution Non-uniform

2 k 1 ......

p

1

p

1

p

2k

23

L Logarithmic ith i visual i l mapping i

22 1 2

information loss:

information loss:

information loss:

information loss:

25.0%

25.8%

22.6%

0%

256

256

28

256 192

192

192

128

26

128

128

64

24

64

64

0

0 0

8 16 24 32 40 48 56 64


22

0

8 16 24 32 40 48 56 64


20

0 0

8 16 24 32 40 48 56 64

(c) 4 regional mappings

0

8 16 24 32 40 48 56 64

(d) logarithmic plot


image V

N visencoder

Source

vischannel

?

signal S

signal S'


N Channel

compactness

N

knowledge K

visdecoder

intuitiveness clarity low info. loss

message M

Source

N

image V'

Destination

message M'

Decoder (Receiver)



Destination


image V

N visencoder

Source

vischannel

N


signal S

signal S'


N Channel

compactness

knowledge K

visdecoder

intuitiveness clarity low info. loss

message M

Source

N

image V'

Destination

message M'

Decoder (Receiver)



Destination

300

£M 271

262

How?

250 199

200

141

SMrt grp p

GreenG grp p

FoodXP grp p

2008

138

SMrt grp p

50

GreenG grp p

100

FoodXP grp p

150

200

Redundancy y z

see Rheingans & Landreth, 1995

Display Space Utilization (DSU) is generally very low

25-50 fps refresh rate can be utilized

2009

5 Laws - Interaction and User Studies Law 3: The information about an overview in one of its detailed view is the same as that about that detailed view in the overview.

Example 1 Mutual Information

p(x,y)

Hint

No Hint

Vortex

25%

25% negative

No V t Vortex

5%

45%

false positive

false

I ( X ,Y )

p ( x, y ) ¦ ¦ p( x, y ) log 2 p( x) p( y ) xX yY

Example 1: I = 0.147

Example 1 Mutual Information

p(x,y)

Hint

No Hint

Vortex

25%

25% negative

No V t Vortex

5%

45%

false

false positive

Example 2 p(x,y)

Hint

No Hint

Vortex

40%

10% negative

No V t Vortex

10%

40%

false positive

false

E Example l 1 1: I(X;Y) I(X Y) = 0.147 0 147

Example 2: I(X;Y) = 0.278

The Role of User Studies

Do quantitative D tit ti measurements make user studies less important?

Example 1 p(x,y)

Hint

No Hint

Vortex

25%

25%

No Vortex

5%

45%

Example 2 p(x,y)

Hint

No Hint

Vortex

40%

10%

No Vortex

10%

40%

If we can measure p(x,y), p(x y) p(x), p(x) p(y), p(y) ...

E Example l 1 1: I(X;Y) I(X Y) = 0.147 0 147

Example 2: I(X;Y) = 0.278

6 Prediction – DP Inequality & a Prediction Data processing inequality z “No clever manipulation of data can improve the inferences that can be made from the data” data [Cover and Thomas, 2006] Markov chain conditions z Closed coupling: (X, Y), (Y,Z) z X and Z are conditionally independent d d

p(x, y, z) = p(x) p(y|x) p(z|y) p(x)

p(y|x)

X

Process 1

p(z|y) Y

I (X; Y)

Process 2

Z

I (Y; Z)

I ( X ;Y ) t I ( X ; Z ) interaction U1 X

Process 1

interaction U2 Y

Process 2

Z

What if the condition is broken? domain knowledge about X

I ( X ;Y ) t I ( X ; Z )

X

Process 1

Y

Process 2

Z

IEEE CG&A CG&A, Jan Jan. 2009

I t Interactive ti visualization i li ti

Information-assisted visualization

Knowledge-assisted visualization

7 Conclusions and Future Work The process of gaining insight can be

i improved db by using i numbers. b Information theory can explain many

phenomena in visualization visualization, z

but be careful with naive use.

The current limitations: z

assumption of memoryless, z lack of real p(xi). This talk is an overview of the paper. Details of related work, z

in particular, Matt Ward’s suggestion (see Purchase et al. 2008)

RichardW.Hamming (19151998)

“The purpose of computing is insight, not numbers.”

The Role of a Theoretic Framework Facts W W W W W W W W W W W W W W W W W W W W

T

T

T

T

T

T

T

T

T

T

W T

W T

Wisdom

W T

T

8T T T

Theory Theoretic Framework

Th nkk You Thank u !

An Information-theoretic Framework for Visualization Min Chen and Heike Jänicke Swansea University and Universität Heidelberg