Evolutionary Models of Color Categorization Based on Discrimination

INSTITUTE FOR MATHEMATICAL BEHAVIORAL SCIENCE UC IRVINE Evolutionary Models of Color Categorization Based on Discrimination Natalia L. Komarova Kimberly A. Jameson Louis Narens & Ragnar Steingrimsson Institute for Mathematical Behavioral Sciences, U.C. Irvine 1

Space of Perceived Colors

Space of Perceived Colors How does it become partitioned and named?

The Hering Color Model Universal Perceptual Features? 5

Color categories and Best-Exemplars are attributable to Hering primaries:... the six Hering primaries: white, black, red, yellow, green, and blue suggesting that these points in color space may constitute a universal foundation for color naming. Kay, Regier & Cook (2005). PNAS, 102.

Note: What remains unknown is the actual degree to which a universal perceptual basis, or shared cognitive dimensions, are large factors contributing to color categorization similarities reported across cultures. 8

The World Color Survey: Color Categorization in 110 nonindustrialized societies. Regier, Kay, & Cook (2005). PNAS, 102, 8386-8391. Etc.... 9

The World Color Survey stimulus: Image Credit: Kay, P. & Regier, T. (2003). Resolving the question of color naming universals. PNAS, 100, 9085-9089. 10

The Munsell Color Solid

An approximation of the Space of Perceived Colors

Individuals are asked to name each chip in isolation. This provides the experimenter with a list of color terms.

Individuals are asked: Where are the best-examples of... yellow green blue red

Where are the best-examples of... yellow green blue red at the individual level:

Where are the best-examples of... yellow green blue red at the group aggregate level:

Where are the best-examples of... yellow green blue red and across languages you find:

Different numbers of Color Terms: n=3 T. Regier et al, PNAS 104, 2007 18

Different numbers of Color Terms: n=3 n=4 T. Regier et al, PNAS 104, 2007 19

Different numbers of Color Terms: n=3 n=4 n=5 T. Regier et al, PNAS 104, 2007 20

Different numbers of Color Terms: n=3 n=4 n=5 n=6 T. Regier et al, PNAS 104, 2007 21

The Mainstream View: Different language groups use different numbers of color terms... But, best-example choices for color terms in different languages cluster near the prototypes for English white, black, red, green, yellow, and blue. Kay, Regier & Cook (2005). PNAS, 102.

English naming and choices of best exemplar focal points for each color category. Figure from Roberson, D., Davies, I. & Davidoff, J. (2000). J. Exp. Psych.: Gen., 129, 369-398.

Berinmo naming and choices of best exemplar focal points for each color category. Figure from Roberson, D., Davies, I. & Davidoff, J. (2000). J. Exp. Psych.: Gen., 129, 369-398.

Roberson (2005). Cog. & Culture.: There are culturally relative, pragmatic reasons why Nol and Wor are named partitions in Berinmo:... for Berinmo speakers,... tulip leaves, a favorite vegetable, are bright green when freshly picked and good to eat, but quickly yellow if kept. Agreement over the color term boundary coincides with agreement over when they are no longer good to eat and is highly salient in a community that talks little about color.

Human Empirical Results Summary: The mainstream view suggest that widely salient focal colors for the Hering primaries are the most likely universal foundation for color naming, and serve to explain a good deal of the cross-cultural similarities in color naming. When variation is recognized, there is acknowledgement that the Hering model does not help explain the wide range of differences seen in the categorization data.

Human Empirical Results Summary: And there is general awareness that other contributing factors exist that are linguistic, environmental, cultural, and so on. But so far these have been difficult to empirically measure and model.

And most recently Kay and colleagues pursue a new direction: Regier, Kay & Khetarpal. (2007). Color naming reflects optimal partitions of color space. PNAS, 104. They argue for the information processing approach of Jameson & D Andrade (1997) and Jameson (2005) which suggests that color categorization reflects roughly optimal divisions of the irregular perceptual color space. 28

Recent Color Category Simulations: Steels, L. & Belpaeme, T. (2005). Coordinating Perceptually Grounded Categories: A Case Study for Colour. Behavioral and Brain Sciences, 28, 469-529. Explored the potential for artifical agent communication with humans. They conclude:... the collective choice of a shared repertoire must integrate multiple constraints, including constraints coming from communication... 29

We Emphasize: Simulated Naming! Human Naming 30

We Emphasize: Simulated Naming! Human Naming The Suggestion is that... While simulated color naming does not tell us how humans categorize and name, it may actually help clarify many of the human issues just listed, and give further insight into some largely uninvestigated factors thought to play a role in human color categorization & naming behaviors. 31

Today we Present Results on Four Issues Relevant to the Empirical Literature: 32

(1) Individual color category learning: What features are needed for a simulated individual (or Agent) to learn a color category system? 33

(1) Individual color category learning: What features are needed for a simulated individual (or Agent) to learn a color category system? - a standard observer color model? - memory for observed colors? - what else?... 34

(2) Population color category learning: What does it take for a society of Agents to evolve and share a color category system? 35

(2) Population color category learning: What does it take for a society of Agents to evolve and share a color category system? - do all the Agents need to have the same identical individual model? - do they need exposure to one another s categorization systems? - etc.... 36

(3) Color category learning in Homogeneous versus Heterogeneous Populations: How does variation at the level of individual Agents influence a shared system? Jameson (2005). Cog. & Culture, 5. Jameson (2005). Beh. & Brain Sci., 28. 37

(3) Color category learning in Homogeneous versus Heterogeneous Populations: How does variation at the level of individual Agents influence a shared system? - does it make a difference in the development or robustness of the shared color categorization solution? 38

E.g.: Agent Heterogeneity resembling Normal Color Perception Variation: Normal Red weak Green weak

Agent Heterogeneity resembling Normal Color Perception Variation: Normal Red weak Green weak

Agent Heterogeneity resembling Normal Color Perception Variation: Normal Red weak Green weak Plus Color Deficient Dichromats

(4) Color category learning in Environments of nonuniform color utility: How does variation in environmental color-importance or -salience influence the development of shared system? 42

(4) Color category learning in Environments of nonuniform color utility: How does variation in environmental color-importance or -salience influence the development of shared system? - do pragmatic pressures influence color naming system solutions? - does it make a difference in the stabilization or robustness of a solution? 43

What about NonUniform Color Utility and NonUniform Environmental Color Distributions? Colors with High Nutrient Value... 44

Pragmatic Considerations... 45

Pragmatic Considerations... 46

Pragmatic Considerations... 47

Pragmatic Considerations... for Successful Communication. 48

Pragmatic Considerations... for Successful Communication. 49

Our Recent Investigations... Komarova, Jameson & Narens (under review). Evolutionary Models of Color Categorization Based on Discrimination. J. of Math. Psych. Emphasize: (1) What is the minimum one needs to evolve a color categorization systems? And (2) How do two or more factors which simultaneously influence color naming behaviors, trade off in the process of developing a stable color naming system? 50

We start with a Simplest-case Approach: Begin by assuming very little about the phenomenon at all levels of observer features, environmental stimulus, socio-pragmatic influences on color signaling systems... 51

We start by investigating... (1) Two topologically different Stimulus systems. 52

(1) Discretely Sampled Stimulus Gradients: light Lightness Continuum dark neutral Saturation Continuum intense 53

(2) Sampled Hue Circle Continuum: 54

Hue Circle Sampled: 1... i... j... n n = 40 sampled stimulus chips. 55

Color Space Dimensions Hue Circle and Saturation & Brightness Continua. Image Credit. http://www.handprint.com Bruce MacEvoy (2006) 56

Space of Perceived Colors

Part II...

1D subsets of the color space Hue continuum

Color categorization as a probabilistic strategy A B C D E F 0.1 0.2 0.5 0.3 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.1 0.2 0.5 0.3 0.0 0.0 0.0 0.0 0.0 0.1 0.5 0.4 0.1 0.2 0.5 0.3 0.0 0.0 0.1 0.2 0.5 0.3 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.7 0.0 0.0 0.1 0.2 0.5 0.3 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.9 0.1 0.2 0.5 0.3 0.0 0.0 0.0 0.2 0.8 0.0 0.0 0.0 0.1 0.1 0.1 0.7 0.0 0.0 0.1 0.2 0.5 0.3 0.0 0.0 0.0 0.2 0.8 0.0 0.0 0.0 0.1 0.2 0.5 0.3 0.0 0.0 0.1 0.1 0.1 0.7 0.0 0.0 0.1 0.1 0.1 0.7 0.0 0.0 0.0 0.0 0.0 0.1 0.5 0.4 0.1 0.2 0.5 0.3 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.1 0.5 0.1 0.1 0.3 0.0 0.1 0.5 0.1 0.1 0.3 0.1 0.2 0.5 0.3 0.0 0.0 0.1 0.2 0.5 0.3 0.0 0.0 0.1 0.2 0.5 0.3 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.1 0.5 0.1 0.1 0.3 0.1 0.1 0.1 0.7 0.0 0.0 0.0 0.1 0.5 0.1 0.1 0.3 0.0 1.0 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.7 0.0 0.0 0.0 0.1 0.5 0.1 0.1 0.3 0.1 0.1 0.1 0.7 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.1 0.5 0.1 0.1 0.3 0.1 0.2 0.5 0.3 0.0 0.0 0.1 0.2 0.5 0.3 0.0 0.0

The individual discrimination game Two colors chips are picked from a distribution The agent (probabilistically) assigns color categories Same category Different category Chips nearby Success Failure Chips far apart Failure Success

The individual discrimination game Two colors chips are picked from a distribution The agent (probabilistically) assigns color categories Same category Different category Chips closer than k-sim Success Failure Chips farther apart than k-sim Failure Success

The measure of similarity k-sim k-sim k-sim For success I need to put these in the same category

The measure of similarity k-sim k-sim k-sim For success I need to put these in different categories

The optimal categorization One theory is that Color categorization reflects roughly optimal divisions of a perceptual color space (Jameson 2005). Here the optimal categorization is the one that maximizes the probability of success.

Evolutionary dynamics A player starts from a random categorization Rounds of the discrimination game are played In case of success, the category is strengthened, in case of failure it is weakened

Reinforcement learners Only m=2 color categories: l = light d = dark

Rounds of individual discrimination game Start Finish

One agent s color categorization 4 categories after 44,000 runs

Optimal categorization We can prove that the optimal number of categories is: m opt = 2 k ( k + 1) sim n sim Categories have an equal size. Categorization is rotationally invariant n = the total number of distinguishable chips

Success rate can never be one k-sim

Communication game Two individuals play the discrimination game If one succeeded and the other failed, the failed individual learns from the successful one If both suceeded, the teacher is chosen at random If both failed, both update their categorization as in the individual game

Population Color Categorization: Agent 1 Agent 2 Agent 3 Three Agents solutions at an initial run.

Population Color Categorization: Agent 1 Agent 2 Agent 3 Column 2: Solutions at run 10,000.

Population Color Categorization: Column 3: Solutions at run 70,000.

In this simple model, The population of agents converges to a common, nearly-optimal color categorization The color space is split in a (predictable) number of equal color categories The solution is defined up to an arbitrary rotation, that is, the boundaries of the color categories can be anywhere along the circle

Introduce inhomogeneities 1. Inhomogeneity in the color space 2. Inhomogeneity in the population

1. Inhomogeneities in color space: hot spots The parameter k-sim is non-constant throughout the color space There is one region where k-sim is smaller than in the rest of the space k-sim small k-sim not small

The presence of a hot spot 5 categories after 10 million runs

Non-equal categories, and symmetry breaking Hot-spot: k-sim=small Otherwise: k-sim=not small No rotation. Size and number of categories are defined by the different k-sim values.

2. Inhomogeneities in the population: the Dichromats Minimal Intermediate The color space of a dichromat Extreme

What happens in a mixed population of normals and dichromats? If there are too many dichromats, a coherent categorization does not arise. There is a large degree of disagreement

What happens in a mixed population of normals and dichromats? If there are too many dichromats, a coherent categorization does not arise. There is a large degree of disagreement If the proportion of dichromats is not too large, then we have a coherent, close-tooptimal categorization

Normals+Dichromats red yellow lime green orange blue blue Solid lines a normal agent Crosses a dichromat agent

Normals+Dichromats red yellow yellow lime green orange blue blue Solid lines a normal agent Crosses a dichromat agent blue blue

Normals+Dichromats red red-green yellow yellow lime green red-green orange blue blue Solid lines a normal agent Crosses a dichromat agent blue blue

Normals+Dichromats red yellow yellow lime lime-orange red-green lime-orange orange green red-green blue blue Solid lines a normal agent Crosses a dichromat agent blue blue

Dichromats remove rotational invariance Normals: any rotation of the optimal solution is a solution

Dichromats remove rotational invariance Normals: any rotation of the optimal solution is a solution Normals+dichromats: Only one solution

Dichromats remove rotational invariance The non-ambiguous axis Normals: any rotation of the optimal solution is a solution Normals+dichromats: Only one solution

They can do more Suppose, the normals come up with an odd number of categories

Solution found by a population of normals 5 equal categories

Solution found by a population of normals+dichromats Rotation

Solution found by a population of normals+dichromats Loss of a category

Solution found by a population of normals+dichromats Solution #1: loss of a color category blue yellow Non-ambiguous axis

Solution found by a population of normals 5 equal categories

Solution found by a population of normals+dichromats Rotation

Solution found by a population of normals+dichromats Compressing

Solution found by a population of normals+dichromats Solution #2: Categories squished blue yellow Non-ambiguous axis

The role of dichromats Dichromats remove rotational invariance (fix the category boundaries) Their presence ensures that the two least ambiguous color chips are at the center of two polar categories This is achieved by: Rotating categories Squishing categories Removing categories

Hot spots and dichromats Inhomogeneity in the k-sim measure (a hot-spot): Inhomogeneity in the population (dichromats):

Hot spots and dichromats Inhomogeneity in the k-sim measure (a hot-spot): Removes rotational invariance Changes category number and sizes Inhomogeneity in the population (dichromats): Removes rotational invariance Changes category number and sizes

Hot spots and dichromats Inhomogeneity in the k-sim measure (a hot-spot): Removes rotational invariance by aligning category boundaries with the hot spot Changes category number and sizes (smaller categories inside the hot spot) Inhomogeneity in the population (dichromats): Removes rotational invariance by aligning category boundaries with the ambiguity axis Changes category number and sizes to place non-ambiguous chips at the center of categories

Interations between the two types of inhomogeneity Consider a population of normals and dichromats Include a hot spot in the color space

Solution found by a population of normals with a hot spot

Add dichromats

Add dichromats Shift Extend + shift

Solution for an inhomogeneous population blue yellow Non-ambiguous axis

The interplay between two kinds of inhomogeneity Consider different positions of the hot spot: red-yellow blue-green blue green

Position of the hot spot with respect to the non-ambiguous axis: The hot spot covers ambiguous and nonambiguous chips (red-yellow): rotation and squishing blue yellow No dichromats Normals+ dichromats

Position of the hot spot with respect to the non-ambiguous axis: The hot spot is centered around a nonambiguous chip (blue): more categories emerge inside the hot spot blue yellow No dichromats Normals+ dichromats

Position of the hot spot with respect to the non-ambiguous axis: The hot spot is centered around the most ambiguous chip (red): no change blue yellow No dichromats Normals+ dichromats

These are testable hypotheses

Summary For a uniform color space and a homogeneous population of viewers, the categorization is a rotationally invariant equipartitioning of the circle. In the presence of inhomogeneities in the color space (hot-spots), or in the population (dichromats), the color boundaries are fixed, and color categories may have different size.

Summary Dichromats align the color categories along the non-ambiguous (blue-yellow) axis. k-sim inhomogeneities decrease category size inside the hot spot. Different populations may have different hot spots and different types of color vision variability. This approach has the potential to explain observed variations in color categorization.

Implications While simulated color naming does not tell us how humans categorize and name color, such investigations can help to clarify: Important features in individual category learning. How color naming systems may be shared among humans. How color naming systems take shape and evolve. What factors substantially shape color naming systems. Similarities & differences in categorization across cultures.

Implications It is possible that perceptual processing may not be the basis for human color categorization. Categorization systems evolved by our agents were not based on any complex color vision processing model, but nonetheless they resemble hue categories seen in some human societies.

Implications These investigations provide a rich approach for further evaluating empirical and theoretical results on the patterns of color naming within and across human cultures. E.g.,: Regier, Kay & Khetarpal (2007). PNAS, 104(4). Lindsey & Brown (2006). PNAS, 103(44). Jameson (2005). Cog. & Culture, 5. Roberson, Davidoff, Davies & Shapiro (2005). Cog. Psych., 50. Davidoff, Davies & Roberson (1999). Nature, 398.

Thank You for your Attention

References Cook, R. S., Kay, P. & Regier, T. (2005) The World Color Survey Database: History and Use. In Cohen, Henri and Claire Lefebvre (eds.) Handbook of Categorisation in the Cognitive Sciences. Amsterdam and London: Elsevier. Hering, E. (1920). Outlines of a theory of the light sense. Springer. (Translated by L. M. Hurvich & D. Jameson. Harvard University Press, 1964). Jameson, K. & D Andrade, R. G. (1997). It s not really Red, Green, Yellow, Blue: An Inquiry into cognitive color space. In Color Categories in Thought and Language. C.L. Hardin and L. Maffi (Eds.). Cambridge University Press: England. 295 319. Jameson, K. A. (2005). Culture and Cognition: What is Universal about the Representation of Color Experience? The Journal of Cognition & Culture, 5, (3 4), 293 347. Jameson, K. A. (2005). Sharing Perceptually Grounded Categories in Uniform and Nonuniform Populations. Commentary on Steels, L. & Belpaeme. T. (Target Article). Coordinating Perceptually Grounded Categories through Language. A Case Study for Colour. Behavioral and Brain Sciences, 28 (4), 501 502. Kay, P. (2005). Color Categories are Not Arbitrary. Cross Cultural Research, 39, 39 55. Kay, P. & Regier, T. (2003). Resolving the question of color naming universals. Proceedings of the National Academy of Science, 100, 9085 9089. Komarova, N. L., Jameson, K. A. & Narens, L. (2006). Evolutionary Models of Color Categorization based on Discrimination. Under review Journal of Mathematical Psychology. Lindsey, D. T. & Brown, A. M. (2006). Universality of color names, PNAS, October, 103(44): 16608 16613. Regier, T., Kay, P. & Cook, R. S. (2005). Focal colors are universal after all. Proceedings of the National Academy of Science, 102, 8386 8391. Regier, T., Kay, P. & Khetarpal, N. (2007). Color naming reflects optimal partitions of color space. Proceedings of the National Academy of Science, 104, 1436 1441. Roberson, D., Davies, I. R. L., & Davidoff, J. (2000). Color categories are not universal: Replications and new evidence from a stone-age culture. Journal of Experimental Psychology: General, 129,369-398. Roberson, D. (2005). Color categories are Culturally Diverse in Cognition as well as in Language. Cross Cultural Research, 39, 56 71. Roberson, D., Davies, I., & Davidoff, J. (2000). Color Categories are Not Universal: Replications and New Evidence from a Stone Age Culture. Journal of Experimental Psychology: General, 129(3), 369 398. Steels, L. & Belpaeme, T. (2005). Coordinating Perceptually Grounded Categories: A Case Study for Colour. Behavioral and Brain Sciences, 28, 469 529. 1