t he structure of lightness - matches sunčica zdravković petar milin department of psychology,...
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THE STRUCTURE OF
LIGHTNESS - MATCHES
Sunčica Zdravković
Petar Milin
Department of Psychology, University of Novi Sad, Serbia
Laboratory for Experimental Psychology, University of Beolgrade, Serbia
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DISTRIBUTION
Normal distribution assumption psychological measurements statistical techniques
Perception not necessarily normally distributed mean of repeated measures
Measuring real objects Experimental procedure, same object, naïve subject… Experimental aesthetics, faces Lightness Katz (1911) ->…
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DISTRIBUTION
Normal distribution?
Distribution of lightness matches Usage of statistical techniques
Normal distribution assumption
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STIMULUS
ILLUSION: Simultaneous lightness contrast Identical grey targets on black and white backgrounds Robust effect demonstrated in various conditions
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Paper/CRT
Gilchrist et al. 1999Colors IlluminationAgostini & Bruno 1996
Targets Backgrounds
Arend & Goldstein, 1987;Bressan & Actis-Grosso, 2001
DarkLight
Dynamic(Zdravković et al. 2005)Static
Single-color Multy-colorNon-uniformSchirillo & Shevell, 1996; Bruno, Bernardis & Schirillo, 1997
Gradient Bressan, 2003
ArticultedCannon & Fullenkamp, 1991; Adelson, 2000; Salmela & Laurinen 2005; Bressan & Actis-Grosso, 2006
DecrementsIncrements
Gilchrist et al. 1999
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STIMULUS
Target Dark background
Light background
C-A paper 4.5 black white Reflectance (%) 15.6 3.1 90 Luminance (cd/m2) 1.2 0.24 7
Measured by Igor Smolić
7 x 7 cm
20 x 29 cm
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METHOD
Controlled illumination
220V, 60W
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METHOD
Munsell scale
Controlled illumination
220V, 60W
2.67º visual angle
150 cm
Perceptual experiment with larger number of subjects
61 subject
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DISTRIBUTION - ANALYSIS Illusion: 1.3 Munsell steps 1-way ANOVA (2 levels) reflectances
Raw data: F(1, 120) = 85.14 p<0.001 Log data: F(1, 120) = 85.07 p<0.001
0
1
2
3
4
5
6
7
1 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 21
6
11
16
21
26
31
36
41
1 2
Munsell data Reflectances Log Reflectances
veridical
datadata
}1. 3
veridical
datadataveridical
datadata
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DISTRIBUTION - ANALYSISGaussian, normal distributionParameters: mean (central tendency) and standard deviation (variability)Standard: AS = 0, SD = 1 Fixed number of parameters
Log-normal distribution
Single-tailed probability distribution of any random variable whose logarithm is normally distributedParameters: mean (central tendency) and standard deviation of the variable’s natural logarithm (variability)
Laplace distribution Continuous, double exponentialParameters: median (central tendency) and absolute deviation (variability)Models the symmetrical data with long tales
Narrower confidence intervalsNon-fixed number of parameters
exp ln(x)
2
/2
/(x 2 )
f (x,b) 1
2bexp
x b
1
2b
exp x
b,ifx
expx
b,ifx
1
2exp
(x )2
2 2
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DISTRIBUTION - RESULTS Tests (one-sample tests)
power Cramer-von Mises (W2), Anderson-Darling (A2)
for symmetrical distributions Watson (U2)
well known Kolmogorov-Smirnov (D), Kuiper (V)
Test Distribution W2 U2 A2 D V normal 0.271 0.265 1.513 1.397 2.284 lognormal 19.959 5.050 250.20 7.750 7.773 Laplace 0.463 0.308 2.792 1.732 2.463
Critical values 0.05 0.141 0.083 0.974 0.910 1.311
Df = 60
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DISTRIBUTION - RESULTS Bayesian inference Likelihood ratio for paired distributions Prior: P(Hynormal), P(Hylognormal), P(HyLaplace)
For collected data X = {x1, ..., xN}, m and s are estimates for the central tendency and variability measures
Assumption: equal probability for two distributions
L1(m, s) = log [ P(Hynormal | X) / P(HyLaplace | X)]
L2(m, s) = log [ P(Hynormal | X) / P(Hylognormal | X)]
L3(m, s) = log [ P(Hylognormal | X) / P(HyLaplace | X)]
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DISTRIBUTION - RESULTS Bayesian inference Likelihood ratio for paired distributions Prior: P(Hynormal), P(Hylognormal), P(HyLaplace)
For collected data X = {x1, ..., xN}, m and s are estimates for the central tendency and variability measures
Assumption: equal probability for two distributions
L1 = 5.1164 => favour normal over Laplace
L2 = 332.795 => favour normal over lognormal
L3 = 327.6847 => favour Laplace over lognormal
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DISTRIBUTION - RESULTS Bootstrapping 1000 replications
Kolmogorov-Smirnov test D-statistic between empirical and randomly redrawn data
p-value for empirical and theoretical distributions
(expected: normal, lognormal or Laplace)
Empirical values of location and scale
(central tendency and variability)
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DISTRIBUTION - RESULTS Bootstrapping
Adequate theoretical data distribution normal (p = 0.213)
Laplace (p = 0.158)
Lognormal not adequate (p = 0.000)
In accordance with Bayesian inference
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CONCLUSION Distribution of lightness matches Simultaneous lightness contrast Probability density distribution:
Normal - most likely Laplace - good approximation
Data transformed into log reflectances violate the normality assumption.
Standard repertoire of statistical techniques and additional
statistical techniques for Laplace distribution should be
applied only on raw reflectance data.
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This research was financed by Ministry of Science of Republic of Serbia (grant number: 149039D)
THANK YOU
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DISTRIBUTION - ANALYSIS
One-sample tests
Cramer-von Mises (W2)
Watson (U2)
Anderson-Darling (A2)
Kolmogorov-Smirnov (D)
Kuiper (V)
W 2 n {Fn
(x) F(x)}2 dF(x)
U 2 n {Fn (x)
F(x) [Fn (x
) F(x)]dF(x)}2 dF(x)
A2 n {Fn (x) F(x)
}2(x)dF(x)
Dmax1iN
F(Yi) i 1
N,
i
N F(Yi)
D,D
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DISTRIBUTION - RESULTS 1) White back, raw reflectances, expected Laplace's overall p-value: 0.096 Two-sample Kolmogorov-Smirnov test D = 0.2131, p-value = 0.1252 2) White back, raw reflectances, expected normal overall p-value: 0.038 Two-sample Kolmogorov-Smirnov test D = 0.2459, p-value = 0.05002 3) White back, log-reflectances, expected Laplace's overall p-value: 0.152 Two-sample Kolmogorov-Smirnov test D = 0.1967, p-value = 0.1886 4) White back, log-reflectances, expected normal overal p-value: 0.296 Two-sample Kolmogorov-Smirnov test D = 0.1639, p-value = 0.3854
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DISTRIBUTION - RESULTS 5) Black back, raw reflectances, expected Laplace's overall p-value: 0.097 Two-sample Kolmogorov-Smirnov test D = 0.2131, p-value = 0.1252 6) Black back, raw reflectances, expected normal overall p-value: 0.004 Two-sample Kolmogorov-Smirnov test D = 0.2951, p-value = 0.00987 7) Black back, log-reflectances, expected Laplace's overall p-value: 0.195 Two-sample Kolmogorov-Smirnov test D = 0.1803, p-value = 0.2745 8) Black back, log-reflectances, expected normal overall p-value: 0.089 Two-sample Kolmogorov-Smirnov test D = 0.2131, p-value = 0.1252
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RESULTS - NORMAL
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RESULTS - LAPLACE
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