Asymmetry in Colour Similarity

Asymmetry in Colour Similarity

Student ID: 12616753

Date: 1/04/2021

Abstract

A significant scientific and philosophical question that aims to understand who we are is: what is consciousness? One way we can study conscious experience is through the relationships between experiences. Among the possible relationships between conscious experiences, similarity relations between experiences are one of the easiest to report. Reported similarity ratings can then be used to create geometric models of experience, a common model in psychology. However, whether this is appropriate is unclear, as the axioms of geometric models are often violated in psychological data. One of the most common violations is the assumption that similarity ratings are symmetric. For example, when we ask participants to rate how similar A is to B, it is often the case they differ from the rating on how similar B is to A. In response, the quantum model similarity model of cognition has been proposed to account for such an asymmetry in similarity judgements. The quantum similarity model predicts asymmetry in similarity judgments by applying quantum probability theory analysis to the similarity judgment data. Here, we tested if asymmetry in similarity judgment as described by the quantum similarly model can be observed empirically in a simple colour judgement task. We ran three similarity experiments (N=15 for each) using two colour stimuli presented in different temporal orders. In all experiments, we found asymmetry in colour similarity judgements with some particular combinations of colours. However the direction of asymmetry was opposite to the quantum similarity model. This asymmetry was highly reliable, as it was reproducible in our double pass paradigm, where the trials were repeated in the same order twice. Our results suggest colour stimuli do generate asymmetric similarity experiences based on the order they are presented. Lastly, we consider other models of the similarity relationships between colour experiences and discuss how they should represent asymmetry.

Contents

Introduction

2

Consciousness and similarity 2

Geometric model of colour experience 2

Symmetry in similarity challenged 4

Asymmetry in colour judgement 4

A quantum geometric model of similarity 5

Aims and Hypotheses 6

Methods

Results

Within Participant Variance 17

Double pass analysis 19

Discussion

Asymmetry in colour similarity 21

Potential cause of asymmetry 21

Quantum similarity and asymmetry 22

Reliability and asymmetry 23

Limitations 25

Conclusion

References

27

Introduction

Consciousness and similarity

Conscious experience is subjective and not directly communicable to others. When looking at a red chair, you cannot know if someone experiences the 'redness' of the chair as you do. If our conscious experience is entirely subjective, how can we study it with objective scientific methods? While we cannot directly know how you experience the red chair, we can determine how your experience of the red chair relates to other objects. The experience of the chair may be 'harder than' the couch. These relations can be used to define experiences as without them there would be no distinction between experiences. For instance, if we did not experience dark red as 'darker than' light red, there would be no separation of the two.

To study consciousness we can investigate the relationships between experiences. One relationship between experiences that is used in psychology research is how similar objects are, this is defined as their similarity (Goldstone & Son, 2012; Sloman & Rips, 1998). Similarity experiments obtain similarity ratings between experiences elicited by two or more stimuli or concepts. These relationships can then be represented using geometric models. According to such models, the psychological dissimilarity is a function of distance in a multidimensional space (Shepard, 1987). Geometric models provide a useful means of visualising and abstracting the data collected from similarity experiments that probe conscious experience.

Colour experience is fundamental to how we experience the world and a useful tool for providing insight into consciousness. A property we use to define and distinguish objects, colour is a common subject of phenomenological investigations. Given the importance of our experience of colour, it has been centered around research and discussions in psychology, philosophy, and science. Geometric models have been a focus of past experiments to represent colour experience (Palmer, 1999).

Geometric model of colour experience

Using a multidimensional spatial model to explain colour experience has been the standard within psychology since Newton created the colour circle in 1704. Although the geometric models used today are not as simplistic as Newton's, the same principles are applied (Palmer, 1999). Colours are positioned so that the location represents how dissimilar colours are to one another. Distance between the two colours corresponds to the psychological dissimilarity. For instance, red is positioned closer to orange than blue, as the experience of red is less dissimilar to orange than blue. In his work on a nonmetric method of multidimensional scaling, Shepard recreated the colour circle by obtaining similarity judgements of colour pairs (Shepard, 1962a, 1962b). Demonstrating the relationship between distance and dissimilarity,

Shepard mapped colour dissimilarity relationships with a geometric model. Figure 1 shows an example of a geometric model of colour. Colours are positioned using their hue (H), saturation (S), and value (V). The more similar the HSV of two colours, the closer together they are placed on the model.

Figure 1. Geometric model of colour using hue (H), saturation (S), and value (V). "Creative Commons HSV Cone" by (3ucky(3all is licensed under Creative Commons Attribution-Share Alike 3.0 Unported.

Geometric models of colour experience have three main assumptions (Shepard, 1987). The assumptions apply to all geometric models of experience, however this report will focus on the geometric model of colour. First is the minimality assumption: the distance between two colours is zero if and only if they are the same colour. The experience of primary red is the same as the experience of another primary red, therefore primary red exists at one location on the model. Second, the triangle inequality assumption. There is no shorter distance between two colours through a third colour. Third, the distance between two colours is symmetric. The distance from red to orange is the same as the distance from orange to red, implying the dissimilarity of red to orange is the same as the dissimilarity of orange to red. Researchers have challenged the assumptions of colour similarity proposed by the geometric model of colour experience (Johannesson, 2000; Nosofsky, 1991; Rosch, 1975).

Symmetry in similarity challenged

Although geometric models of experience are common in psychology, their assumptions are often challenged. Most famously, Tversky, challenged the symmetry axiom of similarity (Tversky, 1977). His research involved various similarity experiments and found similarity was not symmetric. One experiment consisted of twenty one country pairs, for example China-Korea (Tversky's original paper used Red China and North Korea). Pairings were designed so one country was more prominent than the other, for instance, China is more prominent than Korea. Participants were asked which phrase they found more agreeable; 'China is similar to Korea' or 'Korea is similar to China'. Out of the 69 participants, 66 found 'Korea is similar to China' more agreeable. In all pairings, participants found the phrase more agreeable when the less

prominent country was the subject, and the more prominent country was being referred to. In the China-Korea example, Korea as the subject, referring to China was more agreeable than the reverse. The results confirmed there is asymmetry in similarity judgements. Asymmetry was due to an order effect, as the direction of asymmetry depended on which country was referring to the other country. Asymmetry in similarity was one example of Tversky's challenges to geometric models of similarity. However, asymmetry in similarity had been challenged prior to Tversky's study.

Asymmetry in colour judgement

Rosch reported asymmetry in colour similarity judgements in her research studying cognitive reference points (Rosch, 1975). Cognitive reference points are stimuli that may serve as reference for other stimuli within that category. For instance, focal colours may be the reference points for non-focal colours (Rosch, 1975). A focal colour is considered to be the best example of the colour, for example the primary colour red would be the best example of red. Non-focal red would be a non-primary colour red, such as a 'purple' red or a 'muddy' red. Rosch argues asymmetries in judgements would arise due to the relation of the reference point. To test this theory, Rosch developed both a linguistic and a spatial task. Each task tested colours, numbers, and line orientation. Here, we focus on their experiments with colour stimuli. The linguistic task required participants to place two colour cards between words that represented a metaphorical distance, such as 'almost' and 'virtually'. In the spatial task, participants physically placed a colour card on a board with another colour card positioned in the centre. The distance between the two colours represented how similar the participants found them. In both tasks, participants viewed both colours simultaneously for as long as it took to complete the task.

Both tasks resulted in asymmetries in colour judgements. Focal colours were judged to be less similar than non-focal colours. Within the linguistic task participants created the sentence frame 'non-focal red card is essentially focal red card', over the opposite placement. The results from the spatial task confirmed this finding as when participants were required to place a non-focal colour to a focal colour, the distance was smaller than if they were asked to do the reverse. Rosch notes these findings demonstrate judgments of distance can be asymmetrical due to order effects. Asymmetrical judgements provide complications for geometric models that use distance to represent the similarity of the stimuli. Rosch provides an explanation of asymmetries in judgements through reference points. Since Rosch's work there have been other attempts to understand the asymmetry in similarity judgments. Among them, a geometric model of quantum similarity has been recently proposed (Pothos & Busemeyer, 2013; Pothos et al., 2013; Trueblood et al., 2014).

A quantum geometric model of similarity

Quantum similarity models of cognition may provide a model that can address asymmetry in our similarity judgments (Pothos & Busemeyer, 2013; Pothos et al., 2013; Trueblood et al., 2014). While human reasoning does not always follow classical probability CP theory, however the axioms of many rational models of cognition are those of CP. In contrast, quantum cognition offers a model that does not apply CP (Pothos & Busemeyer, 2013). One motivation for quantum cognition is it can provide coherent predictions of human inferences in areas where CP fails to do so (Bruza et al., 2015). Many rational models of cognition resort to heuristics when their model fails to predict an outcome, whereas quantum cognition does not. Whilst still being based on a set of probabilistic axioms, quantum cognition offers a compelling alternative to traditional models. Additionally, the quantum model makes a priori predictions of order effects in judgements. Predictions made a priori further strengthen the model, than if it were only to provide a post hoc explanation.

By applying quantum probability (QP), asymmetries in similarity are predicted due to order effects. QP uses linear algebra along with certain assumptions to calculate probabilities. To further illustrate the quantum similarity model the example of China and Korea in Tversky's paper will be used. In the quantum similarity model, all of our knowledge is represented in a vector space. Different areas of knowledge correspond to different subspaces within the vector space. Sizes and dimensionalities of the subspaces correlate with the amount of knowledge we have of that concept. The subspace that relates to China contains all the information we have about China, including Chinese language, geography and so on. Given the extensive knowledge we have of China, the subspace is larger and has a higher dimensionality compared to the subspace of Korea.

Similarities are calculated by projecting a vector corresponding to one’s current mental content from the subspace of one concept to the subspace of another concept. The length of projection from one subspace to the other determines the angle between the two subspaces. In turn, the angle determines how much of the first subspace corresponds to the second subspace; this corresponds to how similar the two concepts are. Projecting a vector from a small subspace onto a larger subspace would result in a longer projection length. Whereas, projecting a vector from a large subspace onto a smaller subspace would typically result in a smaller projection length. According to this framework, asymmetries in judgments are explained as a result of the order effects. For example, in Figure 2A, when asked 'How similar is Korea to China?', we project from the small subspace of Korea to the larger subspace of China, as we first consider Korea and then China. However, if we are asked 'How similar is China to Korea?', we project from the large subspace of China, to the smaller subspace of Korea, as we first consider China and then Korea. The former results in a projection that retains greater

amplitude of the original state vector, which corresponds to a higher similarity judgment. Whereas the latter retains less amplitude of the original state vector, therefore results in more dissimilar judgement.

Figure 2. Quantum similarity projections. China has a larger higher dimensionality subspace compared to Korea. The red line represents the resulting projection from each projection order. Figure 2.A. shows the projection from Korea to China which results in a larger projection and greater similarity contrasted to the projection from China to Korea in Figure 2.B

The quantum similarity model predicts asymmetry in similarity due to order effects. If we assume that the subspace size and dimensionality is attributed to the knowledge of the concept, we should predict that focal colours would have a larger subspace than non-focal colours. The direction of asymmetry seen in Rosch's experiment can then be explained by the quantum similarity model (Rosch, 1975). When a focal colour card was centered on the board and a non-focal colour card was placed by the subject, they would project from the larger focal colour subspace to the smaller non-focal colour subspace. This results in a smaller projection length than the reverse order, hence why the non-focal colour card was placed at a shorter distance. As the quantum similarity model provides a promising explanation of asymmetry, further research can aid the robustness and understanding of the model.

Aims and Hypotheses

This research project aims to investigate asymmetry in colour similarity due to the order presentation of the colour stimuli. Proposing the quantum similarity model will predict the asymmetry in colour similarity, the temporal order of stimuli will determine the direction asymmetry. Our assumption is that the quantum similarity model would presume a focal colour will have a larger subspace than a non-focal. This would imply that if a focal colour is presented first, followed by a non-focal colour, the colours will be rated more dissimilar than the reverse order. By investigating asymmetry in colour judgements, we aim to further understand the relationships of our colour experience.

Methods

Participants

Participants for all experiments were recruited with the online platform Prolific (www.prolific.co). Prolific users with a 100% past approval rate were selected to help ensure participants were trustworthy. Participants were incentivised with a monetary reward of £5.75/hr for their participation.

Design

The experiment used a similarity task where participants were asked to rate how similar two colours were. The design of the task is shown in Figure 3.

Figure 3. Time course of the stimuli presentation screens and response screens upon button press from the previous trial. First, a foveal colour stimulus is displayed for 250ms together with the fixation cross. Foveal stimuli were presented on the right of the fixation cross and peripheral stimuli were presented on the left, with a random polar angle. This was followed by the fixation cross for an additional 250ms (no masking). The second stimulus is shown in the centre of the response screen and is displayed for an unlimited amount of time until a response is selected. The final screen is shown for an unlimited amount of time until the grey rectangle is selected. Clicking the gray rectangle initiated the next trial. NB: Objects in this figure are exaggerated in size for illustrative purposes relative to what subjects were presented.

The experiment consisted of 324 trials. There were 9 colours used, resulting in 81 (9x9) colour comparisons. The two colours were presented separately. The first colour was displayed once in the periphery and once in the fovea for 250ms. The fovea stimulus had a radius of the inner circle to 1 degree of visual angle (DVA) and was positioned 1 DVA away from the fixation cross. The periphery stimulus had a radius of 4.5 DVA and was positioned 10 DVA away from the fixation cross. The second colour

was displayed in the centre of the screen 500ms after the first colour and remained on the screen until the participant selected a response. This resulted in 162 (81x2) unique trials. These 162 trials were completed twice with 20 catch trials. The catch trials required participants to select a specified number from 0 – 7.

Participants were asked to rate the similarity of the two colours. Participants were given no instructions or information on how to distribute the 0 – 7 ratings to the colour pairs. The experiment was created in PsychoPy (Peirce et al., 2019) and uploaded to Pavlovia (Pavlovia.org) to be run online. See Supplementary Table 1 for colour patches used with their name and hex code.

Procedure

Calibration

The experiment began with an online viewing distance calibration, which was adapted from the Virtual Chinrest (Li et al., 2020).

The first component of the calibration determined the screen resolution by requiring participants to place a physical credit card (or another wallet card) on the image. Participants had to change the size of the image of the card (by pressing ‘j’ or ‘k’) until it matched the size of the physical card.

The second component determined the distance the participant was situated from the screen by the blindspot procedure (Li et al., 2020). Participants were instructed to hold their arm out straight and touch the centre of the screen with their thumb. Participants were then told to lower their arm but keep their head in the same position for the remainder of the experiment. Then, upon instructions for the calibration task, participants closed their right eye and stared at a black box on the right of the screen. A white ball moved from the right to the left of the screen. The participant responded by pressing ‘enter’ when the white ball disappeared, repeating this process 5 times. Screen resolution and viewing distance were used to control the size and location of the stimuli presented to participants. If the calculated screen size from the screen resolution was less than 230mm or if the calculated viewing distance was less than 300mm, we terminated the experiment at this point as these results would suggest they were on a mobile phone or tablet.

Experiment 1

Before starting the experiment trials, participants were given instructions of the task. Participants were instructed to focus on the centre cross on the screen whilst a circle

would flash out at the periphery. The 9 colours used in the experiment were displayed at once, so participants could see the range of colours and consider how they would distribute the similarity ratings. They were told to rate the similarity of the colour between the previous circle to the circle on the next screen, using the range from 0 to 7 by clicking one of the triangles in the response screen (Figure 3). 0 indicated the most similar colour (least different colour) and 7 indicated the most different colour (least similar colour). Participants were asked to ignore any size differences. The instructions included a description of the catch trials where they would be asked to click on a particular value. All the instructions were accompanied with a video of the task.

Participants then completed 7 practice trials with one additional catch trial. For these practice trials, they received feedback, for example ‘You selected 7, 7 indicates you found the colours very different.’.

The first colour was displayed for 250ms in the periphery or the fovea with a centre cross that was displayed for 500ms. The centre cross remained for 250ms after the first colour. The periphery and fovea positions were diametrically opposite and the polar angle of the stimuli was randomised. The polar angle was randomized between -30 degrees to 30 degrees from the horizontal axis for each trial. The foveal colour patch was always displayed on the right side of the screen, and the peripheral patch was always displayed on the left. The second colour was always displayed in the centre of the screen and remained on the screen until the participant selected their response. After the participant had selected their similarity rating they had to click a rectangle in the middle of the response screen to proceed to the next trial.

The 324 trials were presented as a first and second pass. The 162 trials were repeated in the same order. The location of the first stimuli in the periphery or fovea was repeated in the same order, however the polar angle changed. Participants were not informed the 162 trials would repeat. The order of the catch trials were also designed as a double pass, however the number the participant was asked to select was randomised for each trial.

After completing the 344 trials (including 20 catch trials which were randomly inserted), participants answered a questionnaire. The questionnaire asked any vision impairments the participant may have.

All analysis was run in R. The data presented in this report for each experiment was collected as pilot data for future publication (but we plan to use these data as a basis for a registered report) and therefore was insufficiently powered to correct for multiple comparisons. We also analysed the pooled data of the three experiments with corrections for multiple comparisons. In our analysis we will define focal colours as the best example of that colour, for example red is a focal colour. Non-focal colours will be non-primary reds and include non-primary colours such as orange.

Experiment 2

Experiment 2 examined the effect of the fixed trial order as a control experiment in the double pass paradigm. As such, we randomized the order of the second pass (from trial 163 to trial 324). All other elements in Experiment 2 were the same as Experiment 1.

Experiment 3

Experiment 3 examined the effect of unlimited viewing time for the second stimuli. To achieve this, we limited the duration of the second trial into 250ms. All other elements (including the double pass design) in Experiment 3 were the same as Experiment 1.

Results

Descriptive

Experiment 1

15 participants completed the online study (9 males, mean age 25). Participants who either had a catch score lower than 80% or a mean response time of 700ms or less were excluded from the study. After the exclusion 14 participants remained.

Experiment 2

15 participants completed the online study (11 males, mean age 23). Experiment 2 had the same exclusion criteria as Experiment 1. After the exclusion criteria 14 participants remained.

Experiment 3

15 participants completed the online study (8 females, mean age 25). Experiment 3 had the same exclusion criteria as Experiment 1. After the exclusion criteria 12 participants remained.

Calibration Results

To calculate the screen resolution we calculated the ratio between the card image width in pixels and the physical card width of 86.5mm (the width of a standard wallet card). The ratio determined how many pixels are needed per mm.

To find the screen width in mm from the screen resolution we used the following calculation: ScreenWidth = (1 / ScreenResolution) * WindowSizePixels

All participants had a successful ScreenWidth result of 230mm or greater.

To calculate the distance the participant was from the screen we took the median of the distances the blind spot was to the black square. The median distance in pixels was converted into millimeters to obtain PhysicalDistance. The viewing distance was then calculated by the following equation, where a is the blind spot angle: ViewingDistance = PhysicalDistance / tan(a)

Following Li et al (2020), we assumed that the blind spot angle used was 13.5 degrees for all participants, as Li et al. found an angle of 13.5 degrees gave the most accurate results. All participants across experiments had a successful ViewingDistance result of 300mm or greater.

Asymmetry Results

The core results about the asymmetry in similarity ratings remained highly consistent across three experiments, thus we present them together in the following analyses. To visualise the similarity ratings for each pair of colour patches, distinguishing their order we created a dissimilarity matrices. (See Supplementary Figure 1 for individual participant matrices.) For each experiment, we took the mean similarity rating of all participants and plotted a dissimilarity matrices, shown in Figure 4. The colour presented first was shown along the x-axis and the colour presented second was shown along the y-axis. Cell colours represented how similar the two colours were rated, with the lighter the colour of the cell the more similar the rating. The diagonal entries tend to be light as this represents when participants were presented with two of the same colours at different locations.

Figure 4. A-D Dissimilarity matrices of mean similarity rating for all participants for each experiment and the mean across all experiments. The colour presented first is shown along the x axis and the colour presented second is shown along the y axis. The more similar the two colours were rated on average, the lighter the colour of the cell. The red arrows in 4.A show the dissimilarity ratings for spring bud and green differ due to the order they were presented. A. Experiment 1, N=14 B. Experiment 2, N=14 C. Experiment 3, N=12. D. All experiments, N = 40

Figure 4 gives us an insight into the overall colour relationships for the participants. For example, the dissimilarity matrices contain rich information, for example, implying that most participants found greens more similar to blues. Red was rated more similar to electric purple, hollywood cerise, and orange than the blues and greens. Red and hollywood cerise appeared to be rated least similar overall to the other colours, as indicated by the dark cells in their corresponding rows and columns.

The dissimilarity matrices shown in Figure 4 do not appear symmetrical in many entries along the light diagonal axis. For example, in Experiment 1 (Figure 4.A), if participants were shown green and then spring bud they found the colours more similar (≅2), than if they were shown spring bud and followed by green (≅3). Arrows in red on the matrices point to the relevant cells.

To further visualise the asymmetry, we created asymmetry matrices. We calculated the asymmetry by subtracting the transpose of the matrix from the original matrix. The greater the difference of the mean similarity rating for each temporal order, the higher the asymmetry value. Figure 5 shows the asymmetry matrices. The colour of the cells represent the direction of asymmetry.

To test if the asymmetry is statistically significant across the population sample we performed a one sample two-tailed t-test. The t-scores for Experiments 1, 2 and 3 were not corrected for multiple comparisons. The t-scores for the combined data were corrected with the Holm's correction. Figure 5 indicates the significance on the t-score matrices.

Figure 5 also shows the between participant response variance for each experiment as well as the combined data. To investigate the extent to which the participants agreed on the asymmetry we calculated the mean of the 4 asymmetry values for each participant at each cell. The variance of the mean asymmetry was then calculated across the participants. Darker green indicates more variance and lighter green indicates less variance.

Figure 5. A-D Asymmetry matrices of mean asymmetry for all participants for each experiment and combined data. The asymmetry matrix was created by taking the mean similarity rating across all participants and generating a 9x9 matrix. Asymmetry was calculated by subtracting the inverse of the matrix from the original matrix. Between participant response variance matrices for each experiment and combined data. The variance was calculated by taking the mean asymmetry for each participant at each cell. The variance of the mean asymmetry was then calculated across participants. Darker green indicates more variance and lighter green indicates less variance. T-score matrix of asymmetry for each experiment and combined data. The t-score matrix indicates significant asymmetry. A. Experiment 1, N=14 B. Experiment 2, N=14 C. Experiment 3, N=12. D. All data, N = 40 A-C ( * p<0.05, ** p <0.01, ***

p<0.001, uncorrected for multiple comparisons).D ( * p<0.05, uncorrected for multiple comparisons, ˟ p<0.05, Holm's correction).

The direction of asymmetry for all experiments indicates that if a focal colour was presented first and a non-focal colour second, the colours were judged more similar and if the order was reversed as shown in Figure 5. Figure 5 also shows colour pairings that have significant asymmetry do not have the highest between participant response variance. For example, in Experiment 1, medium spring green and spring bud, which have significant asymmetry, do not have a high between participant response variance.

To summarise the consistency of asymmetry across experiments as well as the combined data, Table 1 lists the significant colour pairing for each experiment. The direction of asymmetry is listed and compared to the direction of asymmetry in the other experiments as well as the effect size of the asymmetry. When no asymmetry was observed the direction is labelled N/A. As shown in the asymmetry matrices, the results suggest, with a few exceptions, when a focal colour was presented first, followed by a non-focal colour, they were rated more similar than the reverse order.

A. Experiment 1

Colour Pair Direction of Asymmetry

(More similar order)

Exp 2 Exp 3 Effect Size

(Asymmetry) Same Same 0.22* N/A Same 0.80** Opposite Same 0.78** Same N/A 0.20* B. Experiment 2 Colour Pair Direction of Asymmetry

(More similar order)

Exp 1 Exp 3 Effect Size

(Asymmetry) Same Same 0.48* Same Same 1.05** Same Opposite 0.57* C. Experiment 3 Colour Pair Direction of Asymmetry

(More similar order)

Exp 1 Exp 2 Effect Size

(Asymmetry)

Same Same 0.56*

Opposite Same 0.20*

N/A Same 0.33*

D. All data

Colour Pair Direction of Asymmetry

(More similar order)

Effect Size (Asymmetry) 0.33* 0.28* 0.33* 0.41** 0.64*** 0.44** 0.25*

Table 1 A-D. Significant asymmetry colour pairings for each experiment. The tables list the direction of asymmetry for the colours. Tables A-C list the agreement of asymmetry across experiments. ( * p<0.05, ** p <0.01, *** p<0.001, uncorrected for multiple comparisons).

Within Participant Variance

The analysis thus far has shown there is significant asymmetry in colour similarity ratings based on temporal order. Participants agreed on the direction of asymmetry; if a focal colour was presented first, followed by a non-focal colour, they were rated more similar than the reverse order. We then wanted to see if the asymmetry was related to variability in responses. If asymmetry is associated with response variability it may imply asymmetry arises from uncertainty from the participant.

To quantify if participants are uncertain where we see asymmetry effects, we calculated the within participant response variance. The variance was calculated by taking the variance of the 4 similarity values for each participant at each cell. We then calculated the mean of the variance across the participants. Figure 6 shows the within participant variance in response for each experiment, darker orange indicates a higher variance in response and lighter orange indicates less variance in response.

Figure 6. A-D Within participant response variance matrix for each experiment. The variance was calculated by taking the variance of similarity rating for each participant at each cell. The mean of the similarity variance was then calculated across participants. Darker orange indicates more variance and lighter indicates less variance. A. Experiment 1, N=14 B. Experiment 2, N=14 C. Experiment 3, N=12. D. All experiments, N = 40

Figure 6 shows the highest within participant response variance was in Experiment 2. We also see that when participants were shown the same colour they did not always rate it 0. There is a variance in response in each experiment across the diagonal that corresponds to the same colour pair.

To visualise if high participant response variances were associated with asymmetry we plotted the within participant variance versus the asymmetry for each experiment as well as the combined data for all experiments. The scatter plot is shown in Figure 7.

Figure 7. Scatter plot of within participant response variance and the absolute asymmetry for each experiment. Experiment 1, N=14. Experiment 2, N=14, Experiment 3, N=12, All Experiments, N = 40.

Figure 7 shows that we do not see the highest within participant response variance where we see the highest asymmetry. We again see Experiment 2 has the highest response variance. The highest within participant response variance in Experiment 2 has an asymmetry close to 0. The analysis suggests participant uncertainty is not associated with asymmetry in colour similarity.

Double pass analysis

Experiments 1 and 3 were designed as double pass experiments, with each participant completing the 162 trial sequence twice in exactly the same order. To see how reliable subjects are in their similarity ratings, we can calculate the correlation between the first and second pass. As Experiment 2 did not have a first and second pass, with all 324 trials being randomised, these results provide a control comparison for Experiment 1. We will use Experiment 2 as a control for Experiment 3 but it is important to note they are not directly comparable given the difference in the second stimulus display time. We calculated the Spearman's rank order correlation coefficient (rho) for the similarity ratings for the first 162 (1-162) trials and the second 162 (163-324) trials for each participant in each experiment. Figure 8 shows a box plot of the rho values grouped by experiment, the individual points correspond to participant rho values. As shown in Figure 8, the first and second pass in Experiment 1 (median rho=0.891) and

Experiment 3 (median rho=0.911) were highly correlated. The first and second pass in Experiment 2 appear less correlated, the median rho was lower (0.833).

To test if this difference is statistically significant, we performed a one way ANOVA on the calculated Fisher-transformed correlations of the rho values for each experiment. There was a statistically significant difference between groups as determined by one-way ANOVA (F(2,37)=4.152, p=0.024). To determine which groups were significantly different we performed Tukey multiple pairwise-comparison. The Tukey analysis confirmed there was a significant difference in the correlation of the first and second pass in Experiments 2 and 3 (p=0.026) indicated in Figure 8. The experiments designed with a first and second pass had no difference in correlation between passes (p=0.786). Experiments 1 and 2 were close to being significantly different with p=0.095.

Figure 8. 1st and 2nd pass Spearman's rank order correlation for each experiment. The correlation was calculated for the similarity ratings for the first 162 (1-162) trials and the second 162 (163-324) trials for each participant in each experiment. Plotted is box plots of the corresponding correlation values. Individual points correspond to participant correlation values. The bold black line corresponds to the median correlation value for that experiment group. Experiment 1; median rho=0.891 Experiment 2; median rho=0.833, Experiment 3; median rho=0.911. ( * p<0.05, uncorrected for multiple comparisons).

Discussion

Asymmetry in colour similarity

The current study investigated asymmetry in similarity ratings between experiences induced by colour stimuli. By running a similarity task where subjects were presented with two colour stimuli one after the other, we examined if the degree of dissimilarity between two colours differed depending on the temporal order that the two colours were presented. We found there was asymmetry in colour similarity judgments due to order effects. With a few exceptions, the direction of asymmetry was the same for all three experiments. If participants were presented with a non-focal colour followed by a focal colour, they rated the pair more dissimilar than the reverse order. As the direction of the asymmetry was consistent across all experiments, the results provide support for asymmetry in colour similarity. It is necessary to note however, that all of the following discussion should be considered preliminary as the data collected was pilot data and the results were insufficiently powered to correct for multiple comparisons. Only the combined data was able to be corrected for multiple comparisons. To further support our findings, more participant data should be collected.

Potential cause of asymmetry

The adaptation hypothesis may be one explanation of the asymmetry in colour similarity we see in our results. Our visual system can change its sensory processing when responding to changes in the environment (Webster, 2015). Studies have found stimuli have aftereffects on perception after being removed (Clifford et al., 2007; Webster & Mollon, 1995; Webster 2015). For example, the tilt after effect (TAE) shows that if participants fixate on tilted lines for a period of time, horizontal lines presented subsequently appear to be tilted in the opposite direction (Clifford et al., 2000, Gibson & Radner, 1987). The aftereffect of the first stimulus causes changes in the orientation perception of the participant. Other studies have found aftereffects can cause change in colour and motion perception (Webster & Mollon, 1995; Winawer et al., 2008). In our experiment the location of the first and second colour stimuli differed in order to prevent aftereffects. However, as the colour stimuli in our experiment are displayed subsequently, there still may be a possibility the first stimulus has an aftereffect on the second stimulus. We will consider how the adaptation hypothesis may explain our results, even though the location of the stimuli differed. For example, a trial consists of a participant first viewing green, a focal colour, followed by a non-focal green, spring bud. The participant may first adapt to seeing green, their response to green is

fatigued. Therefore when the non-focal green, spring bud, is shown they will find the spring bud less green. This is due to the aftereffect of being fatigued by the first green stimuli. If participants are first shown spring bud and then green, the green will be perceived as slightly less green as their response to green will have fatigued via the spring bud stimulus. Both trials would result in participants rating the two greens as dissimilar, however the dissimilarity will be larger for green followed by spring bud than the reverse order. A focal green would induce more green fatigue than a non-focal green, therefore colours shown after appear more dissimilar to green. As our results show the opposite direction of asymmetry, the adaptation hypothesis fails to explain the asymmetry in colour similarly we observe.

Quantum similarity and asymmetry

Our experiment tested the predictions of the quantum similarity model, where the direction of asymmetry is a result of projecting from subspaces of different sizes. We assumed the quantum similarity model would attribute focal colours a larger subspace than non-focal colours. Given our assumption the theory would infer; if a focal colour is presented first, followed by a non-focal colour, the colours would be rated more dissimilar than the reverse order. Our results found the opposite direction of asymmetry to the quantum similarity model's predictions with our assumption.

As the subspace size for focal and non-focal colours was only assumed rather than formally defined, a different interpretation of the quantum similarity model may still explain our results. The size of subspace for a concept is related to how much knowledge we have associated with that concept. When considering the China-Korea example for western participants, we can understand why the subspace China would be larger due to more knowledge of China. However, when applying this idea to colours it is not as clear that non-focal colours should have a larger subspace than focal colours. Further studies investigating which colours have more associations would help to understand the size of colour subspaces. If non-focal colours were found to have more associations than focal colours we could assume they have a larger subspace size. Our results would then fit a quantum geometric model of similarity. Asymmetric similarity models

Our results were opposite to the direction of asymmetry reported by Rosch (Rosch, 1975). However, Rosch's experimental design did not incorporate a temporal order, participants were viewing both stimuli simultaneously. Rosch theorised that asymmetry in colour similarity was due to cognitive reference points, with focal colours being reference points for non-focal colours. Regardless of the difference in the presentation of the stimuli, this theory does not predict our results. Other asymmetric models of similarity have aimed to provide an account of asymmetry in colour similarity which may offer an insight into our findings.

The additive similarity and bias model proposed by Holman and demonstrated by Nosofsky, argues asymmetry is a result of stimulus bias (Nosofsky, 1991; Holman, 1979). The model states we have a bias towards certain stimuli. A bias towards a stimulus arises from a characteristic of that stimulus. Nosofsky claims some stimuli may be more salient in perception or memory than the other. Objects we have bias towards are experienced as being less similar to objects we do not have a bias towards than the other way around. To fit our data we would need an appropriate explanation to why we are biased towards non-focal colours.

The additive similarity and bias model is related to Tversky's theory that argues asymmetry occurs as a result of one stimulus being more prominent than the other (Tversky, 1977). According to Tversky, prominence is determined by intensity, familiarity, salience, informational content, and goodness in form. The China-Korea example was developed as Tversky argued China is more prominent than Korea. Inspired by Tversky's prominence notion, Johannesson has since developed the relative prominence model (Johannesson, 2000). The model predicts that if a stimulus with greater prominence is compared to a stimulus with less prominence they will be judged more dissimilar than the reverse order. Testing his model by conducting a colour similarity experiment, Johannesson found the same direction of asymmetry as we have reported. He claimed the result demonstrated that the prominence of colours is not due to their focality. If prominence was determined by focality in colour, non-focal colours would be rated more similar to non-focal colours. Johannesson says there must be another property of colour that establishes prominence. Although Johannenson demonstrates the same direction of asymmetry as our study, by claiming we must find another property of colour for prominence does not aid us in our understanding of colour asymmetry. The models presented offer insights into asymmetry, however they fail to give a thorough explanation to why we experience asymmetry in colour similarity.

Reliability and asymmetry

The experimental design of Experiments 1 and 3 consisted of a double pass in order to test how reliable subjects were in their similarity ratings. We found a strong correlation between the similarity ratings in the first and second pass, which suggested the similarity ratings had a high reliability. Further supporting this finding; the correlation between similarity ratings for Experiment 2, which served as a control with no double pass, was less than Experiments 1 and 3. One factor that may have contributed to the strong correlation was participants selecting extreme similarity ratings of 0s and 7s. If there is not a range of responses, high correlation would not necessarily indicate high reliability as only two values are being compared. Although some participants rated the colours on the ends of the similarity scale, many others

used the full range of ratings when selecting similarity (See Supplementary Figure 1). Strong correlation was observed in all participants, regardless of their rating system. Reliability in similarity ratings supports our finding of asymmetry in similarity judgements. Additionally, it further validates our claim that the asymmetry we have found is a result of asymmetry in colour experience as the ratings were consistent. Limitations

One limitation the research project was met with was running the colour experiment online. Given the importance of the stimuli in our experiment, it can be questioned if the correct experimental set up can be achieved at home. We implemented the calibration to help ensure the correct stimulus size was used for participant's screen size and viewing distance. It may be argued that even with the calibration we cannot guarantee participants maintained the correct distance throughout the experiment, nor did they complete the calibration correctly. However, as the results were highly consistent across participants, it suggests the experiment was performed effectively online.

Another limitation of the online study is participants may have had different colour settings on their monitors. Therefore they may not have seen the exact same colours as each other. However, as our experiment was asking for similarity ratings for the temporal order of colour stimuli, it should not affect our conclusions if participants were viewing slightly different colours to each other.

Another limitation was comparing our results to Rosch's study given the experimental design differences. Rosch's study did not incorporate temporal order as we did. Similarity was not determined by a rating, rather by the distance or linguistic hedging. These factors may lead to the opposite results we found. To overcome this limitation we plan to replicate Rosch's study to rule out any errors in coding or analysis that may be responsible for the opposite results.

Our research would also benefit from investigating the size of subspaces for colours to apply the quantum geometric model of similarity. One possible experiment is asking participants to list associations they have with certain colours. If non-focal colours have more associations than focal colours it may suggest that non-focal colours have a larger subspace. The current experiment results may then be explained by a quantum similarity model.

When considering the adaptation hypothesis as a potential explanation of asymmetry, our experimental design had constraints. We claimed that when viewing the stimulus, the colour response is fatigued thereby affecting the experience of the second colour. The experiment would be improved to test the adaptation theory if the two colours were presented in the same location. Another experiment to test the adaptation hypothesis

and colour similarity would be to display the first colour at different durations. If the adaptation hypothesis is correct, the longer the duration the first colour is displayed the more fatigued the response to the colour. Therefore the second colour would be rated more dissimilar the longer the first colour is displayed.

As the experiment collected pilot data there were limitations to the statistical analyses we could perform. To improve our analysis we can increase the number of participants. This will then allow us to perform multiple corrections without combining the data from the three experiments. As the asymmetry matrix shows the colours that produce significant asymmetry appear to be clustered in groups, we can perform a cluster analysis in the future. This will allow us to see any spatial property of the results and if groups of colours are more prone to asymmetry than others.

Conclusion

Our research has found there is asymmetry in colour similarity due to presentation order effects. When focal colours were displayed first followed by a non-focal colour, the pair were rated less dissimilar than the reverse order. Given our assumptions of the quantum geometric model of similarity, our results did not support the direction of asymmetry predicted by the model. In reviewing other models of asymmetric similarity, we found further insight into asymmetry in colour similarity. However, no model captured why we experience colour asymmetry in the direction we do. Additionally, we found high reliability in colour similarity ratings which added support to the asymmetry in our results.

As the current study collected pilot data, our findings would benefit from additional data being collected. The research would gain insights from new experiments run with more colour stimuli. Continuing to use similarity tasks as a way of understanding the relationships between our experiences will help us understand our experience not only colour but other phenomena. We have shown asymmetry in colour similarity is reliable and consistent, therefore a model of the similarity relationships between colour experiences should represent asymmetry.

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Supplementary figures

Colour Name Hex Code Colour Name Hex Code

Red #FF0000 Deep Sky Blue #00A9FF

Orange #FFAA00 Blue #0000FF

Spring Bud #AAFF00 Electric Purple #AA00FF

Green #00FF00 Hollywood Cerise #FF00AA Medium Spring Green #00FFA9 White Grey #808080

Supplementary Figure 2 A-C. Histogram of similarity ratings of all participants for each experiment. The dissimilarity rating is shown along the x-axis and the frequency of selection is shown on the y-axis.