1.2 Accuracy (Rensink & Baldridge, 2010). For some properties (e.g., brightness), a logarithmic relation. - Fechner's law. Is this also true for correlation?
The Perception of Correlation in Datasets
Ronald A. Rensink Departments of Computer Science and Psychology University of British Columbia Vancouver, Canada
Visualizing information…
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What are the perceptual processes involved? European Conference on Visual Perception, 02 Sep 2012
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Domain: scatterplots - perception of correlation
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Previous Work Direct estimate of Pearson correlation: •
Pollack (1960)
•
Bobko & Kerren (1979)
•
Cleveland, Diaconis, & McGill (1982)
•
Lauer & Post (1989)
•
Meyer, Taieb, & Flascher (1997)
•
Knoblauch & Maloney (2008)
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Basic Behavior
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1.1 Discriminability (Rensink & Baldridge, 2010)
Within-subject design Scatterplots: 5.0° x 5.0° 100 points per plot -gaussian distribution
Which one has the higher correlation ? European Conference on Visual Perception, 02 Sep 2012
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Results (n=20) r = jnd (75% correct)
r = k(1/b - r) 0 .2
k: variability (= 0.22) b: offset (bias) (= 0.91) 0 .1 f rom above
(Let u = 1 - br)
ju s t n o ti c e a b le d if fe re n c e
u = ku u = k u
f rom below
0 0
Weber’s Law European Conference on Visual Perception, 02 Sep 2012
0 .5
1 .0
base correlat ion ( rAA) Average correlation
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How General is This? Four conditions: 1. gaussian distribution 2. uniform distribution 3. stretch (Y = 2x horizontal range) 4. 25 dots in scatterplot (cf. 100 dots)
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Greater variability when fewer points (n = 25)
r2 Values gaussian: 0.962 uniform: 0.936 doubleY: 0.957 N25: 0.955
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1.2 Accuracy (Rensink & Baldridge, 2010) Relate physical quantity r (Pearson correlation) to psychological quantity g (perceived correlation) For some properties (e.g., brightness), a logarithmic relation - Fechner’s law Is this also true for correlation?
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Measuring Accuracy: Bisection technique
Adjust test plot to be midway between reference plots Start with 0.0 & 1.0. Then 0.0 & 0.5 and 0.5 & 1.0. Then 0.0 & 0.25 and 0.25 & 0.5, etc, etc.
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Results (n=20)
1 .0
su b je c ti v e c o rr e la ti o n (g )
0 .5
0 0
0 .5
1 .0
object ive correlat ion ( r) European Conference on Visual Perception, 02 Sep 2012
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Results
1 .0
(n=20)
g(r) = log(1 - br) log(1 - b) (b = 0.9)
0 .5
Fechner’s law su b je c ti v e c o rr e la ti o n (g )
(with u = 1-br)
0 0
0 .5
1 .0
object ive correlat ion ( r) European Conference on Visual Perception, 02 Sep 2012
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How General is This? Four conditions: 1. gaussian distribution 2. uniform distribution 3. stretch (Y = 2x horizontal range) 4. 25 dots in scatterplot (cf. 100 dots)
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1.3 Timecourse (Rensink, 2011) Correlation perception in scatterplots follows linear/logarithmic laws ➞ based on a simple property How long does the underlying process take?
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Measuring Timecourse of Discrimination Scatterplot 2
Mask
Scatterplot 1
indefinite
200 ms 100 / 400 / 1600 ms European Conference on Visual Perception, 02 Sep 2012
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Results – uniform distribution (n=20)
0 .2
400 ms vs 1600 ms F(1,19) = 0.39; p > .5
0 .1 1 00 ms 4 00 ms ju s t n o ti c e a b le d if fe re n c e
1 60 0 ms
Later studies No difference for ≥ 150 ms
0 0
0 .5 base correlat ion ( r A
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1 .0 =
r + r/ 2) ) 19
Conclusions: Timecourse Correlation perception is entirely complete by 150 ms - mostly complete by 100 ms (already linear)
Correlation perception takes place rapidly - cf. time for perception of gist, summary averages
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Design Parameters
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Performance for Various Design Parameters Color, brightness, shape of dots… Measure k (precision) and b (accuracy) - within-subject design (5 conditions, or 4 conditions + 1 mixed) - 3 base correlations in each condition (0.3, 0.6, 0.9) - one bisection point (subjective 0.5) - 20 subjects per condition
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Number
n = 12
n = 24
n = 48
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n = 100
n = 200
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Results – precision .
Greater variability when number of points n < 48
.5 .4 .3
(v a ri a b il it y )
k
.2 .1
12
24
48
100
200
number of dot s in display European Conference on Visual Perception, 02 Sep 2012
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Results – accuracy No change in accuracy (bias)
.
1 .0 .9 .8
b (b ia s )
.7 .6
12
24
48
100
200
number of dot s in display European Conference on Visual Perception, 02 Sep 2012
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Color No differences
Brightness No differences
Size No differences European Conference on Visual Perception, 02 Sep 2012
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Conclusions: Design Parameters Performance is invariant over all design parameters tested Only depends on the number of dots in the display - discrimination deteriorates when n < 48 (sampling?) - accuracy remains unaffected
Performance is not based on the outline of the dot cloud - correlation perception unlikely the result of blurring - possibility: centroids of proto-objects(?)
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Other Dimensions
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3.1 Carrier = Size (horizontal) (vertical) (horizontal) (carrier = size) Augmented stripplot European Conference on Visual Perception, 02 Sep 2012
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r=1
r=0
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Technique: Measure jnds (same as scatterplots)
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Results (precision): Value → diameter (n=18) .30 0.3
Same as for scatterplots!
.250.25
k = .24
.20 0.2
.
.150.15 0 .2
k = .22
Above Ab ove
.10 0.1
Below Bel ow
.050.05
0 .1
f rom below
0
ju s t n o ti c e a b le d if fe re n c e
.0
f rom above
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Correlation (rA)
0 0
0 .5 base correlat ion ( r A)
1 .0
Correlation
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Results (accuracy): Value → diameter (n=18)
subjective correlation (g)
Same as for scatterplots! .
1 .0
b = .91
b = .90
g(r) = log(1 - br) log(1 – b)
su b je c ti v e c o rr e la ti o n (g )
0 .5
0 0
0 .5
1 .0
object ive correlat ion ( r)
objective correlation (r)
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3.2 Carrier = Brightness (horizontal) (size) (horizontal) (brightness) European Conference on Visual Perception, 02 Sep 2012
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Results (precision): Value → intensity (n=9) .300.35 Just Noticeable Distance
.25
Similar to scatterplots
0.3
0.25
k = .30
.20
0.2
.150.15 .10
Above Above Below Below
0.1
0.05
.05
0
.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Correlation (rA)
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Results (accuracy): Value → intensity (n=9) Similar to scatterplots subjective correlation (g)
b = .79
g(r) = log(1 - br) log(1 – b)
objective correlation (r)
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Conclusions
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1. Perceptual Abilities Correlation perception is an interesting perceptual ability
- obeys simple laws (Weber/Fechner) - carried out rapidly (within c. 150 ms) - not due to blurring, but to a more sophisticated process - laws are general - invariant to a wide range of design parameters - largely invariant to carrier - true for all basic features?
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2. Methodology Techniques of vision science can be successfully applied to perception of correlation in scatterplots Extend to other kinds of statistical measure? - averages, standard deviations, outlier detection, etc - effects of secondary groups, etc.
Extend to other types of visualization? - line graphs, bar charts, parallel co-ordinates, etc - 3, 4, 5… dimensions
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Visualizations are an interesting class of stimuli
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A Huge Amount of Thanks to… • The members of the UBC Visual Cognition Lab who helped on various aspects of this project: Gideon Baldridge Adelena Leon Natália Lopes Praveena Manogaran
Kyle Melnick Theo Rosenfeld Benjamin Shear Ramyar Sigarchy
• The Boeing Company and NSERC Canada, for financial support
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