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

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) ->…

DISTRIBUTION

Normal distribution?

Distribution of lightness matches Usage of statistical techniques

Normal distribution assumption

STIMULUS

ILLUSION: Simultaneous lightness contrast Identical grey targets on black and white backgrounds Robust effect demonstrated in various conditions

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

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

METHOD

Controlled illumination

220V, 60W

METHOD

Munsell scale

Controlled illumination

220V, 60W

2.67º visual angle

150 cm

Perceptual experiment with larger number of subjects

61 subject

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

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

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

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)]

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

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)

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

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.

This research was financed by Ministry of Science of Republic of Serbia (grant number: 149039D)

THANK YOU

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

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

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

RESULTS - NORMAL

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RESULTS - LAPLACE

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