stanford / cs448b

http://cs448b.stanford.edu2 February 2009

Using Space Effectively

Jeffrey Heerassistant: Jason Chuang

Topics

Displaying data in graphsBanking to 45 degreesZoomable viewsFocus+Context and spatial distortions

Displaying Data in Graphs

Effective use of spaceWhich graph is better?

Government payrolls in 1937 [How To Lie With Statistics. Huff 93]

Aspect ratio

Yearly CO2 concentrations [Cleveland 85]

Clearly mark scale breaks

Well marked scale break [Cleveland 85]

Poor scale break [Cleveland 85]

Scale break vs. Log scale

[Cleveland 85]

Scale break vs. Log scale

Both increase visual resolutionLog scale - easy comparisons of all dataScale break – more difficult to compare across break

Linear scale vs. Log scale

MSFT

MSFT0

10

2030

604050

0

10

20

30

60

40

50

Linear scale vs. Log scale Linear scaleAbsolute change

Log scaleSmall fluctuationsPercent changed(10,20) = d(30,60)

MSFT

MSFT0

10

2030

604050

0

10

20

30

60

40

50

Exponential functions (y = kamx) transform into lineslog(y) = log(k) + log(a)mxIntercept: log(k)Slope: log(a)m

Semilog graph: Exponential growth

y = 60.5x, slope in semilog space: log(6)*0.5 = 0.3891

Exponential functions (y = kamx) transform into lineslog(y) = log(k) + log(a)mxIntercept: log(k)Slope: log(a)m

Semilog graph: Exponential decay

y = 0.52x, slope in semilog space: log(0.7)*2 = -0.3098

Exponential functions (y = kamx) transform into lineslog(y) = log(k) + log(a)mxIntercept: log(k)Slope: log(a)m

Semilog graph: Lines

y = x, slope in semilog gives instantaneous : log(a)m

Power functions (y = kxa) transform into linesExample - Steven’s power laws:

S = kIp log S = log k + p log I

Log-Log graph

10

1

100

1

0

2

log(

Sens

atio

n)

Sens

atio

n

0 1 2

1 10 100Intensity

log(Intensity)

[The Elements of Graphing Data. Cleveland 94] [The Elements of Graphing Data. Cleveland 94]

[The Elements of Graphing Data. Cleveland 94] [The Elements of Graphing Data. Cleveland 94]

Transforming dataHow well does curve fit data?

[Cleveland 85]

Residual graphPlot vertical distance from best fit curveResidual graph shows accuracy of fit

[Cleveland 85]

Banking to 45 Degrees

William S. ClevelandThe Elements of Graphing Data

Banking to 45° [Cleveland]

Two line segments are maximally discriminable when their average absolute angle is 45°

Optimize the aspect ratio to bank to 45°

To facilitate perception of trends, maximize the discriminability of line segment orientations

Aspect-Ratio Banking Techniques

Median-Absolute-Slope

Average-Absolute-OrientationUnweighted

Weighted

Average-Absolute-Slope

Max-Orientation-ResolutionGlobal (over all i, j s.t. i≠j)

Local (over adjacent segments)

| ( ) | 45i

i nθ α

= °∑

| ( ) | ( )45

( )

i ii

ii

l

l

θ α α

α= °

∑

∑

2| ( ) ( ) |i ji j

θ α θ α−∑∑

21| ( ) ( ) |i i

iθ α θ α+−∑

mean | | /i x ys R Rα =median | | /i x ys R Rα =

Slopeless Line Culling

Exclude line segments with zero or infinite slope

Standard, Aspect Ratio = 1.97 Culled, Aspect Ratio = 4.00

Comparison (Data Sets)CO2 Measurements (co2) PRMTX Mutual Fund (prmtx)

Prefuse Downloads (prefuse) Sunspot Cycles (sunspot)

Comparison (Results)

ms msc as asc ao aoc awo awoc gor gorc lor lorc

5

10

15

20

25

Asp

ect

Ratio

co2 prefuse prmtx sunspot

ms msc as asc ao aoc awo awoc gor gorc lor lorcms msc as asc ao aoc awo awoc gor gorc lor lorc

5

10

15

20

25

Asp

ect

R atio

5

10

15

20

25

Asp

ect

R atio

co2 prefuse prmtx sunspot

Average-slope (as) and Average-weighted-orientation (awo) provide similar ratios

Comparison (Results)

ms msc as asc ao aoc awo awoc gor gorc lor lorc

5

10

15

20

25

Asp

ect

R atio

ms msc as asc ao aoc awo awoc gor gorc lor lorcms msc as asc ao aoc awo awoc gor gorc lor lorc

5

10

15

20

25

Asp

ect

R atio

5

10

15

20

25

Asp

ect

R atio

Average-orientation (ao) and Global-orientation-resolution (gor) provide similar ratios

co2 prefuse prmtx sunspot

Discussion

Due to computational complexity…Prefer avg-slope to avg-weighted-orientPrefer avg-orient to global-orient-resolution

But due to perceptual effectiveness… ?Cleveland recommends weighted-avg-orientBut, goal is to maximize discriminability

Perceptual experiments needed to clarifyCO2 Measurements

William S. ClevelandVisualizing Data

Aspect Ratio = 1.17

Aspect Ratio = 7.87

Multi-Scale Banking to 45°Goal

Optimized aspect ratios for varying scales

ApproachIdentify Scales of Interest Generate Scale-Specific Trend LinesBank Trend Lines to 45°Filter Resulting Aspect Ratios

Multi-Scale Banking to 45°

Use Spectral Analysis to identify trendsFind strong frequency componentsLowpass filter to create trend lines

Compute Power Spectrum

Take Discrete Fourier TransformCompute squared magnitudes

Original Data

Power Spectrum

Smooth the Spectrum

Convolve with Gaussian filterwindow size = 3, σ = 1

Smoothed Power Spectrum

Power Spectrum

Smoothed Power Spectrum

Thresholded Power Spectrum

Threshold the Spectrum

Threshold at µ + kσ (k=0 by default)Retain last values of contiguous runs

Generate Trend Lines

Generate trend lines with lowpass filterTrend Lines

Thresholded Power Spectrum

Trend Lines

Bank Trend Lines to 45°

Bank each trend line to 45°

Aspect Ratios Filtered Aspect Ratios

Aspect Ratios

Filter Aspect Ratios

Filter similar aspect ratiosKeep if αi+1 > cαi (c=1.25 by default)

Yearly values 1700-1987

Aspect Ratios

Power Spectrum

Aspect Ratio = 14.55

Aspect Ratio = 4.23

Sunspot CyclesMonthly concentrationsfrom the Mauna Loa Observatory, 1950-1990

Aspect Ratios

Power Spectrum

Aspect Ratio = 7.87

Aspect Ratio = 1.17

CO2

Small Multiples Displays

Sparklines

ApplicationsTrend Explorer

Banking can be applied to create sparklines, data‐intense, word‐sized graphics. A plot might be included inline    , supporting uninterrupted reading.

Zooming

Zooming

Eames’ Powers of Ten [http://www.powersof10.com/]

Overview + Detail

[Hornbaek et al. 2002]

Interactive zooming

Pad++ [Bederson and Hollan 94]

Semantic zooming

Change representations as zoom level changes

PAD [Perlin and Fox 93]

Speed-Dependent Zooming

Integrate Pan and Zoom into single interationAutomatically zoom to maintain optical flowSemantic zooming can simplify zoomed-out view

http://www-ui.is.s.u-tokyo.ac.jp/~takeo/java/autozoom/autozoom.htm [Igarashi 00]

Halo [Baudisch 03]

Focus + Context

Degree-of-Interest [Furnas 81, 06]

Estimate the saliency of information to displayCan affect what is shown and/or how to show it

DOI ~ f(Current Focus, A Priori Importance)

Example: Google SearchCurrent Focus = Query Hits (e.g., TF.IDF score)A Priori Importance = PageRankWhat: Top N results, How: List

TableLens [Rao & Card 94]

DateLens

[Bederson et al. 04]

Single view focus + contextFocus area – local detailsDe-magnified area – surrounding contextLike a rubber sheet with borders tacked down

Nonlinear Magnification Infocenter [http://www.cs.indiana.edu/~tkeahey/research/nlm/nlm.html]

Bifocal display [Leung and Apperley 94]

2D distortion

Multifocal display [Leung and Apperley 94]

Fisheye [Leung and Apperley 94]

1D 2D

Polar

Nonlinear magnification [Leung and Apperley 94]

1D 2D

Multifocal

Perspective wall Perspective allows more context

Perspective Wall [Mackinlay et al. 91]

Cartographic Distortions

Cartograms: Distort areas

Scale area by data[From Cartography, Dent]

Election 2004 map

http://www-personal.umich.edu/~mejn/election/

% voted democrat% voted republican

Election 2004 map

http://www-personal.umich.edu/~mejn/election/

% voted democrat% voted republican

Election 2004 map

http://www-personal.umich.edu/~mejn/election/

Statistical map with shading

[Cleveland and McGill 84]

Framed rectangle chart

[Cleveland and McGill 84]

Rectangular cartogram

American population [van Kreveld and Speckmann 04]

Rectangular cartogram

Native American population [van Kreveld and Speckmann 04]

Dorling cartogram

http://www.ncgia.ucsb.edu/projects/Cartogram_Central/types.html

States as nodes in a graph

Graphical fisheye views of graphs [Sarkar & Brown 92]

Distorting distances

Scale distance by data[From Cartography, Dent]

London underground

http://www.thetube.com/content/history/map.asp

Comparison to geographic map

Distorted Undistorted

Summary

Spatial layout is the most important visual encodingGeometric properties of spatial transforms support

geometric reasoningShow data with as much resolution as possible Emphasize important information

Important to consider what to show, not just how