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Article  |   April 2014
Stimulus-specific variability in color working memory with delayed estimation
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Journal of Vision April 2014, Vol.14, 7. doi:10.1167/14.4.7
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      Gi-Yeul Bae, Maria Olkkonen, Sarah R. Allred, Colin Wilson, Jonathan I. Flombaum; Stimulus-specific variability in color working memory with delayed estimation. Journal of Vision 2014;14(4):7. doi: 10.1167/14.4.7.

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Abstract
Abstract
Abstract:

Abstract  Working memory for color has been the central focus in an ongoing debate concerning the structure and limits of visual working memory. Within this area, the delayed estimation task has played a key role. An implicit assumption in color working memory research generally, and delayed estimation in particular, is that the fidelity of memory does not depend on color value (and, relatedly, that experimental colors have been sampled homogeneously with respect to discriminability). This assumption is reflected in the common practice of collapsing across trials with different target colors when estimating memory precision and other model parameters. Here we investigated whether or not this assumption is secure. To do so, we conducted delayed estimation experiments following standard practice with a memory load of one. We discovered that different target colors evoked response distributions that differed widely in dispersion and that these stimulus-specific response properties were correlated across observers. Subsequent experiments demonstrated that stimulus-specific responses persist under higher memory loads and that at least part of the specificity arises in perception and is eventually propagated to working memory. Posthoc stimulus measurement revealed that rendered stimuli differed from nominal stimuli in both chromaticity and luminance. We discuss the implications of these deviations for both our results and those from other working memory studies.

Introduction
Color working memory has been the central focus in an ongoing and vigorous debate concerning the structure and limits of visual working memory. Initially, these limits were hypothesized to be discrete in nature, restricting the individual number of objects a person could store at once (Cowan, 2001; Luck & Vogel, 1997). More recently, the focus of much research has shifted to the quality of visual working memory—the precision with which an observer can store and report the specific value of an object feature. This shift in focus emerged to a large extent with the introduction of the delayed estimation paradigm (Wilken & Ma, 2004; Zhang & Luck, 2008). In this paradigm, an observer attempts to store the features of some number of objects and is then asked to identify the feature value of a probed item on a continuous scale. Because of the intuitive nature of continuous differences between colors, color working memory has enjoyed the lion's share of research with this paradigm (Figure 1; e.g., Anderson & Awh, 2012; Bays, Catalao, & Husain, 2009; Bays, Wu, & Husain, 2011; Emrich & Ferber, 2012; Fougnie & Alvarez, 2011; Fougnie, Asplund, & Marois, 2010; Fougnie, Suchow, & Alvarez, 2012; Gold et al., 2010; van den Berg, Shin, Chou, George, & Ma, 2012; Wilken & Ma, 2004; Zhang & Luck, 2008, 2009, 2011). While the debate concerning the limits of color working memory continues, there appears to be wide-ranging consensus that the working memory representation of color is noisy or probabilistic—that is, varying in fidelity—and that some of this variability is imposed by the structure of visual working memory. 
Figure 1
 
Schematic display of the delayed estimation procedure (with a memory load of one). Note that the display background, which is shown as white here, was a neutral gray in the experiments reported below, and its color has varied in the published literature.
Figure 1
 
Schematic display of the delayed estimation procedure (with a memory load of one). Note that the display background, which is shown as white here, was a neutral gray in the experiments reported below, and its color has varied in the published literature.
An implicit assumption underlying this research has been that perceptual and memory fidelity are independent of particular stimulus values. This assumption is most clear in the method by which delayed estimation responses have been modeled. Typically, a mixture model is used to characterize responses as arising either from a noisy representation of a probed item or from unbiased guessing (possibly with a third component reflecting misbinding or misremembering of color position, e.g., Bays et al., 2009; alternative models remove the guessing component altogether and sometimes include other sources of variability such as motor imprecision, e.g., van den Berg et al., 2012). Because a single set of model parameters is estimated from all responses for a given memory load, the implication is that all color values are represented with the same fidelity on average (e.g., Bays et al., 2009; van den Berg et al., 2012; Zhang & Luck, 2008). 
The assumption of value-independent fidelity is predicated on a more practical assumption: namely, that any value-dependent effects have been controlled for by sampling stimuli from a perceptually homogeneous set of colors. In fact, this assumption motivates the use of color and delayed estimation in many studies because, as Zhang and Luck (2011) put it, in these experiments “precision can be unambiguously operationalized” (p. 1434). In particular, researchers have most often used the CIELAB color space (Commission Internationale de l'éclairage L* a* b*) to sample values around a center point (defined by L*, a*, and b* values) with a fixed radius. This choice is presumably motivated by the assumption that the CIELAB space is perceptually uniform, with equal physical distances in the space corresponding to equal perceptual distances. However, there are at least three general reasons to doubt the validity of this assumption. 
First, the nature of color perception could introduce stimulus-specific effects. Although CIELAB—the most commonly used space in studies of delayed estimation—was developed to be a perceptually uniform color space, this is known to be only an approximation (e.g., Brainard, 2003; Fairchild, 1998; Wyszecki & Stiles, 1982). Furthermore, there are substantial individual differences in color perception between observers judged to be color normal by standard assessment techniques (Webster, Miyahara, Malkoc, & Raker, 2000a,b). Because two colors judged as equally discriminable by one color normal observer may not be judged so by another observer, no physically defined color space can be perceptually uniform across all observers. It is for this reason that studies seeking to link discrimination and color appearance measure discrimination thresholds directly rather than inferring them from standard color spaces (e.g., Bachy, Dias, Alleysson, & Bonnardel, 2012; Danilova & Mollon, 2012; Witzel & Gegenfurtner, 2013). Importantly, these studies have identified different just noticeable differences for different colors. 
Second, the technical difficulty of rendering colors accurately could introduce stimulus-specific effects. This is because stimuli on emissive displays are specified in units that are device specific (e.g., RGB [red, green, blue triplet] values) rather than in physical units (e.g., energy per wavelength). Because there is considerable variation in the hardware and software that govern stimulus display, two monitors requesting identical RGB values will likely produce different color signals and thus different CIELAB coordinates. Fortunately, there are standard calibration techniques that enable the reliable production of stimuli specified in a variety of color spaces. However, such calibration techniques have not been widely employed in the working memory literature. Some working memory studies report engaging in aspects of display calibration such as gamma correction (Zhang & Luck, 2008), but we were unable to find a single working memory paper that described utilizing the full calibration procedure that is standard in the literature on color perception (Allen, Beilock, & Shevell, 2012; Olkkonen & Allred, 2014; Witzel & Gegenfurtner, 2013; Xiao, Hurst, MacIntyre, & Brainard, 2012). 
Third, memory itself could introduce stimulus-specific variability in precision. There is a literature on color memory that is largely distinct from the working memory literature. In this literature, it has been reported that memory for colors is shifted in systematic yet complex ways relative to presented colors (e.g., Burnham & Clark, 1954, 1955; Collins, 1931; Jin & Shevell, 1996; Ling & Hurlbert, 2008; Nemes, Parry, Whitaker, & McKeefry, 2012; Nilsson & Nelson, 1981; Prinzmetal, Amiri, Allen, & Edwards, 1998). 
For these reasons, we investigated whether memory fidelity for color is independent of stimulus value in delayed estimation. To do so, we employed two typical versions of the delayed estimation task and two commonly used color spaces for stimulus sampling, using procedures standard in the working memory literature (van den Berg et al., 2012; Wilken & Ma, 2004; Zhang & Luck, 2008). In Experiment 1, contrary to the implicit assumption in color working memory research, we found that variability in memory was systematically dependent on color. In Experiments 2 through 4, we investigated the extent to which this dependence on stimulus generalized to perception and other memory loads. Posthoc measurements of color stimuli revealed significant deviations between intended and displayed stimuli. We explore the causes and likely effects of these deviations and discuss the reasons why such deviations are likely endemic in delayed estimation studies using color as the stimulus of interest. 
Experiment 1: Color-specific variability with a memory load of one
The goal of this experiment was to determine empirically whether response variability in a standard delayed estimation task is stimulus dependent. Specifically, we sought to estimate response variability within and across observers for each of 180 colors in two of the sets of color samples typically utilized in delayed estimation experiments. We did so in a straightforward fashion: Using a minimal memory load of one, we collected measurements for all target colors from each observer. One group of observers each performed the experiment with colors nominally defined in CIELAB space and HSV (hue saturation value) space. 
Method
Observers
In exchange for course credit, three Johns Hopkins University undergraduates participated in the experiment with stimuli constructed with the CIELAB color space. A different group of three observers participated with stimuli constructed with the HSV color space. All observers had normal or corrected-to-normal visual acuity and reported no known deficits in color perception. The Johns Hopkins University Institutional Review Board (IRB) approved the protocol for this experiment. 
Apparatus
The experiment took place in a dark, sound-attenuated room. There was no light source except for a computer monitor. All stimuli were presented on a cathode ray tube (CRT) monitor at a viewing distance of 60 cm such that the display subtended approximately 28.64° by 19.09° of visual angle. 
Stimuli and procedure
For the CIELAB stimuli we followed the methods of Zhang and Luck (2008) and others. We specified a nominal set of 180 evenly spaced colors from CIELAB color space centered on L* = 70, a* = 20, and b* = 38 and with a radius of 60 (Anderson & Awh, 2012; Gold et al., 2010; Zhang & Luck, 2009, 2011). These CIELAB coordinates were then converted into 180 RGB values via a color conversion algorithm (Image Processing Toolbox, MATLAB, The MathWorks, Inc., Natick, MA). To do so, we used the International Color Consortium (ICC) standard white point, with an xyY value of 0.35, 0.36, 1 (i.e., D50; Y value is normalized). For the HSV wheel, 180 equally spaced hues were sampled, with the intensities of saturation and value fixed at 80%. 
Each trial began with a white fixation cross (0.5° × 0.5°) displayed in the center of a gray screen. After 500 ms, a colored square (2° × 2°) appeared at one of eight possible positions (4.5° from fixation). The square's color was one of the 180 stimuli. The square remained present for 500 ms, and observers were instructed to commit the color of the square to memory. After a 900 ms blank delay, a test display appeared with a black frame occupying the location of the square. A color wheel was drawn (8.2° radius and 2° thick) surrounding the space in which memory objects could appear. The color wheel consisted of all 180 stimuli, arranged so that each stimulus occupied 2°. The wheel was randomly rotated on each trial to avoid spatial encoding of the stimulus value. Observers were asked to click the target color on the color wheel as precisely as possible. After the observer's response, a black line superimposed on the wheel indicated the clicked position. Each observer completed 5 blocks of 360 trials, totaling 1,800 trials. Within a block, each of the 180 color values appeared twice in an order that was randomized by observer, producing 10 total measurements per color for each observer. 
Stimulus measurement
Working memory studies generally use the same procedures described above: A stimulus set is specified in CIELAB, and device-specific RGB values are determined using an industry standard white point. However, because the hardware and software of laboratory displays vary, there may be substantial differences between intended and presented colors. Color perception research standardly employs monitor calibration to ensure that device-specific commands render stimuli that match the physical characteristics of the requested stimuli (for a discussion of display calibration, see Brainard, Pelli, & Robson, 2002). We did not perform monitor calibration in our study, nor are such calibration practices common in the working memory literature. 
To assess the effects of incomplete calibration and color rendering, we made posthoc measurements of our stimuli using a PR-655 spectroradiometer (PR-655 SpectraScan, Photo Research Inc., Chatsworth, CA). The measurements were converted to Commission Internationale de l'éclairage XYZ color space (CIE XYZ) and CIELAB spaces using colorimetry routines in Psychophysics Toolbox (Brainard, 1997). The measured xyY values of our 180 stimuli (hereafter referred to as rendered stimuli) are reported in Appendix A, as are the intended (hereafter nominal) CIELAB values. In addition, we report the CIELAB values of the 180 stimuli when the background of the monitor during the experiment is used as the white point in the conversion between xyY coordinates and CIELAB coordinates. We do so because it is standard practice in color perception studies to take the color of the monitor background to be the white point in color conversions (e.g., Brainard, 1998; Giesel & Gegenfurtner, 2010; Wyszecki & Stiles, 1982). 
As can be seen in Figure 2, there were large deviations between nominal and rendered stimuli. Several features are of note. First, rendered stimuli were not circularly arranged in terms of their a* and b* coordinates in CIELAB space (Figure 2a). This can occur for several reasons, but one common reason is that a particular device cannot physically display a requested value. In Figure 2c, for example, we plot the gamut of the monitor on the xy plane of the CIE xyY space as determined by posthoc measurements with the radiometer. Any point outside the solid triangle cannot be produced by the monitor; clearly, some of the nominal colors fell outside this gamut. These colors were automatically mapped to others that do lie within the gamut, resulting in a mismatch between nominal and rendered colors and luminances. The mismatch between requested and actual luminance is evident in Figure 2b, which shows the L* values of rendered colors. The sum of these effects can be seen by plotting rendered colors in xyY space (Figure 2d). The rendered values varied systematically in both luminance (Y value) and chromaticity (x and y values). 
Figure 2
 
Comparison between intended and measured color values. (a) Distortions in a* and b* dimensions. Black circle represents a* and b* coordinates for intended color values. Colored outline represents measured color values. Gray dot in the middle represents the background color. Note that rendered colors were not circularly arranged and were not equally distant from the background. (b) Distortions in L* value. Plotted here is a* versus L* for measured (colored shape) colors. Rendered colors had different L* values, violating the isoluminance assumption. (c) Measured xy values (colored shape) and nominal xy values (black shape). Nominal xy values were calculated from nominal CIELAB values using the measured background xyY of the monitor as the white point in the conversion. The triangle in the figure represents the gamut of the monitor used in the study. Colors outside the triangle cannot be rendered correctly on the monitor, and some of the nominal colors lay outside this gamut. (d) Measured color values in xyY dimensions. Again, rendered colors were not isoluminant. Note: Although rendered colors deviated from nominal colors, the posthoc radiometer measurements demonstrate that the xy coordinates of all rendered stimuli were measurably different from each other.
Figure 2
 
Comparison between intended and measured color values. (a) Distortions in a* and b* dimensions. Black circle represents a* and b* coordinates for intended color values. Colored outline represents measured color values. Gray dot in the middle represents the background color. Note that rendered colors were not circularly arranged and were not equally distant from the background. (b) Distortions in L* value. Plotted here is a* versus L* for measured (colored shape) colors. Rendered colors had different L* values, violating the isoluminance assumption. (c) Measured xy values (colored shape) and nominal xy values (black shape). Nominal xy values were calculated from nominal CIELAB values using the measured background xyY of the monitor as the white point in the conversion. The triangle in the figure represents the gamut of the monitor used in the study. Colors outside the triangle cannot be rendered correctly on the monitor, and some of the nominal colors lay outside this gamut. (d) Measured color values in xyY dimensions. Again, rendered colors were not isoluminant. Note: Although rendered colors deviated from nominal colors, the posthoc radiometer measurements demonstrate that the xy coordinates of all rendered stimuli were measurably different from each other.
Because the rendered values, measured posthoc, deviated so substantially from nominal values and because standard analysis techniques depend heavily on the assumption that stimuli vary only in hue (i.e., that they are on a circle of constant L* in CIELAB space), we were faced with a difficult decision about how to report and interpret our data. On the one hand, our stimuli clearly did not differ from each other only in hue (and thus, angular distance). However, the entire analysis process as detailed below, and in the color working memory literature generally, rests on the assumption that stimuli can be described fully in terms of their angular distance from one other. Because of this, we proceed by analyzing data as though we had rendered stimuli correctly. We then carefully note the likely implications of rendering inaccuracies, which may be present in color working memory research quite generally, for the interpretation of our results. 
Despite this “worst case” scenario of large deviations between rendered and nominal stimuli, we believe our data to be useful for at least two reasons. Most importantly, for reasons we outline later, we believe that the conclusions we draw survive these deviations and make an important contribution to the literature. In addition, since we followed standard stimulus practices in the delayed estimation literature, it is likely that many other working memory studies fall prey to similar inaccuracies. Indeed, the authors of several influential reports were generous enough to discuss their methods with us in detail, confirming the absence of monitor calibration, the absence of nominal and rendered stimulus comparison, and the use of a default white point instead of a measured white point. Moreover, since we used the nominal CIELAB samples employed by several related reports, and because many of the stimuli fell outside our CRT monitor gamut, it is likely that at least some stimuli were out of gamut in those studies as well. 
Analysis
The raw data generated in these experiments were distributions of response positions elicited by each target stimulus. We characterized the spread of these response distributions in terms of angular precision (inverse variance), with interest in whether some stimuli elicited more concentrated response distributions than others. In keeping with the literature in this area, we also quantified response precision with a model-based measure. With only a single memory item, these two methods are very similar (see Figures 3 and 4b). But the working memory literature is typically interested in memory for several items (see Experiment 3), and the analysis of multiple-item data is thought to require a model in which representational precision combines with other sources of variability (e.g., lapse trials). 
Figure 3
 
Reciprocal of circular variance of target-response differences for each color value in CIELAB (top panel) and HSV (bottom panel) color spaces. Note that the x-axis of the CIELAB plot is hue angle centered on the intended color wheel (i.e., centered on a* b* = 20, 38, not on the origin of CIELAB).
Figure 3
 
Reciprocal of circular variance of target-response differences for each color value in CIELAB (top panel) and HSV (bottom panel) color spaces. Note that the x-axis of the CIELAB plot is hue angle centered on the intended color wheel (i.e., centered on a* b* = 20, 38, not on the origin of CIELAB).
Figure 4
 
Estimated κ values fitted by color and observer for each of the two color spaces (a), and estimated κ values fitted by color, collapsed across observers. (b) Solid black curves reflect spline smoothing, showing a pattern of κ variability by color, and dotted black lines show estimated κ when the model was fitted to all responses.
Figure 4
 
Estimated κ values fitted by color and observer for each of the two color spaces (a), and estimated κ values fitted by color, collapsed across observers. (b) Solid black curves reflect spline smoothing, showing a pattern of κ variability by color, and dotted black lines show estimated κ when the model was fitted to all responses.
To characterize these results in terms typical of previous studies, we fit a mixture model comprising a von Mises distribution and a uniform distribution (Zhang & Luck, 2008). The von Mises distribution is a circular analog of the standard normal distribution. The model that we fit included two free parameters: the proportion of target-based (as opposed to lapse) responses (Pm; 0 ≤ Pm ≤ 1) and the concentration (κ; 0 ≤ κ) parameters of the von Mises distribution (equivalent to 1/SD2 of the circular normal):   
Here, Y denotes a response made to a particular stimulus X. The first term in the equation denotes the probability density of a specific response given the parameters (mean and precision) of the von Mises distribution, multiplied by the mixture coefficient Pm. The von Mises precision parameter κ is typically interpreted as the precision of a memory representation since larger values of κ generate narrower distributions (i.e., κ is the inverse of the distribution's variance). The parameter μ, denoting the mean of the von Mises density, was set to the target value (X) in the fitting. The second term in the equation refers to the density of the response according to random guessing (i.e., a uniform density on the circle), weighted by (1 − Pm). The Pm parameter gives the probability of responses that are based on working memory, while (1 − Pm) is the probability of random guesses. We expected guessing rates to be very low in our model fits given the minimal memory load (i.e., Pm was expected to be close to 1). All model fitting was performed by maximum likelihood inference. Parameters were initialized to multiple starting values in an attempt to avoid local maxima. 
With respect to the methods employed by Zhang and Luck (2008) and subsequent work, the only deviation in this study involved fitting the model to the responses elicited by each stimulus separately rather than aggregating responses across all stimuli. The model was also fitted separately to each individual observer's data. In sum, we fitted the model 180 times for each observer, allowing us to identify differences in the von Mises precision (κ) values of the response distributions elicited by different color stimuli. 
Results and discussion
If working memory performance were independent of color value, as is widely assumed in the working memory literature, precision should be comparable for all color values employed. Contrary to this assumption, we found that response precision varied with stimulus value. This is evident in both the direct estimate of precision as 1/SD (Figure 3) and the values of precision estimated from the model (Figure 4a). For each observer (Figure 4a), the fitted precision values obtained for each stimulus clearly varied with hue. This remained true when we collapsed across observers (thus including 30 observations for each individual color). These fitted precision values varied widely from the precision obtained by the typical procedure of fitting the model to all of the colors together (dashed line in Figure 4b). 
Most importantly, stimulus-specific variations in estimated precision show similar patterns across observers, suggesting that our findings are due to stimulus properties rather than random fluctuations. To quantify interobserver similarity, we calculated correlations between the estimated stimulus-specific κ values for the three participants in each condition. Correlations across all pairs of observers were significant for both the CIELAB and HSV wheels, as shown in Figure 5 [t(178) > 3.0, p < 0.01 for all correlations]. The correlations across observers suggest that some target stimuli systematically elicited wider response distributions than others. 
Figure 5
 
Correlation of stimulus-specific κ estimates across observers. CIELAB color space is shown in the top row and HSV color space is shown in the bottom row.
Figure 5
 
Correlation of stimulus-specific κ estimates across observers. CIELAB color space is shown in the top row and HSV color space is shown in the bottom row.
One might worry that the response precision estimate for each stimulus is unreliable due to the relatively small number of observations (10 for each color) included in each stimulus- and participant-specific model fit. Crucially, if model fits had excessive variance, then relative differences among estimates would not be expected to correlate across observers. To make this point quantitatively, we applied a Monte Carlo permutation test (Higgins, 2004, chapter 5). Specifically, for each pair of observers we calculated a null distribution of correlations as follows. Holding the order of the von Mises precision values for one observer fixed, the precision values for the other observer were randomly permuted 10,000 times; a correlation coefficient was calculated for each permutation. The resulting distribution of coefficients indicates that the empirical correlations between observers are highly unlikely to have arisen by chance (p < 0.001): None of the 10,000 simulations yielded a correlation as large as those observed, and none of the simulated correlations were both significant and in the right (positive) direction. 
To further investigate the reliability of our precision estimates, we performed a simulation in which our stimulus-specific model was fit to data generated according to the null hypothesis that the true response precision is independent of color. For each of three simulated observers and each of the 180 stimuli, we generated 10 random responses according to the null hypothesis. Each response was obtained from a von Mises distribution with a κ value that was estimated by collapsing across all stimuli in Experiment 1 (indicated by the black dotted line in Figure 4b). Note that estimating precision from aggregated data is the standard way of fitting the mixture model of Equation 1. We then computed stimulus-specific fits to the responses of each of the simulated observers and calculated pairwise correlations of estimated κ values as above. This entire process was repeated 1,000 times. 
If the correlations between observers in our experiments arise simply as a byproduct of random noise, this simulation should produce correlations similar in magnitude to those of Figure 5. But only one of the simulated across-observer precision correlations was larger in magnitude than the smallest of the correlations found in our experimental data. Thus the probability of obtaining correlations as strong as those found empirically, if the null hypothesis were true, is estimated to be smaller than 1/3,000 (p < 0.001). 
Given both the strong correlations among observers and the simulation results, it seems unlikely that the stimulus-specific variations in precision are an artifact of unreliable estimates. However, the accuracy of the estimates is uncertain because of the large deviations between rendered and nominal stimuli. In later sections we return to this question. For now we note that, regardless of the ultimate cause, stimulus-specific variation in precision is likely to be widespread in color working memory studies, as we used rendering techniques that are standard in the working memory literature. 
We also explored alternative options for analyzing the data, including drawing a new circle in CIELAB space that had minimum distance in color space to the rendered stimuli. This analysis also showed considerable stimulus-specific variability in κ, though the pattern of variation differed from that we found using the nominal stimuli. Importantly, however, the observer-by-observer correlations for κ across colors remained significant (observer 1 vs. observer 2, r2 = 0.49; observer 2 vs. observer 3, r2 = 0.35; observer 1 vs. observer 3, r2 = 0.31; p < 0.001 for all correlations), again suggesting stimulus-specific variability. Here we focus only on the analysis of the nominal stimuli. Given the device specificity of the rending process and the fact that other working memory studies also likely suffer from rendering errors, we emphasize that the important result is not the particular pattern of stimulus-specific variability but the fact that stimulus-specific variability exists and is correlated between observers. 
Possible effects of response method and exposure duration
While much of the relevant literature utilizes procedures like those described here, two important experimental variables that differ across studies are the time of exposure to memory displays and the method for selecting a response. We used 500 ms exposures and a color-wheel response method in the experiments just described. But we also sought to determine whether our findings hold for other exposure durations and a different way of making responses. A new group of two observers was tested in a nearly identical experiment with the CIELAB color space, with two key differences: (1) Memory samples were displayed for only 100 ms and (2) we used a scrolling method for collecting responses (Figure 6; as in van den Berg et al., 2012). At test, the response was made with a single square rather than a wheel. The response square initially appeared in a randomly chosen color. Observers used two keys to scroll through the available colors in either angular direction, and pressed a button when the displayed color matched their memory of the target. 
Figure 6
 
Delayed estimation with the scrolling method.
Figure 6
 
Delayed estimation with the scrolling method.
After fitting the model of Equation 1 in the same way described above, we found a significant correlation between color-specific precision estimates from this experiment and those from the previous one [t(178) = 4.46, r = 0.35, p < 0.001; Figure 7). These results demonstrate that systematic color-specific variability arises for reasons independent of the exposure duration and response method. 
Figure 7
 
(a) Color-by-color κ estimates in the scrolling experiment. (b) Color-by-color correlation between κ estimates obtained in the color wheel and scrolling experiments (collapsed across observers).
Figure 7
 
(a) Color-by-color κ estimates in the scrolling experiment. (b) Color-by-color correlation between κ estimates obtained in the color wheel and scrolling experiments (collapsed across observers).
Summary
Experiment 1 identified stimulus-dependent response effects in a working memory experiment with standard methods. This undermines the assumption of uniform response variability across colors. 
Experiment 2: Estimation without delay
Experiment 1 showed that the response distributions in a standard delayed estimation task contained stimulus-dependent variability. Next, we investigated the source of this variability. Since there were differences between nominal and rendered colors, it is likely that there was perceptual inhomogeneity in the stimuli. To investigate the relationship between precision in memory and precision in perception, we replicated Experiment 1 but without a memory delay. Stimuli appeared simultaneously with the response color wheel, and the task was simply to identify the color of a cued item by clicking on the wheel. 
Methods
Observers
A new group of 14 Johns Hopkins University undergraduates participated in exchange for course credit. Each observer had normal or corrected-to-normal visual acuity. The protocol for this experiment was approved by the Johns Hopkins University IRB. 
Apparatus and stimuli
The apparatus and stimuli were identical to those used in Experiment 1
Procedure
This experiment was identical to Experiment 1 with the following exceptions. The color target and response wheel were presented simultaneously. The condition with the HSV color space included set sizes of one, two, four, or six objects in a display, distributed randomly over the course of a session; 12 observers completed 60 trials for each set size (240 trials total) in this space. In the CIELAB condition, two observers completed 1,800 trials each, 10 trials for each color presented singly. In the HSV condition, where multiple objects could be presented, one square included a bold black frame identifying it as the item on which the response should be based. 
Analysis
A stimulus-specific mixture model (including precision as a free parameter for each color) was fitted to the results of Experiment 2, collapsing across the responses of all observers in each color space. Although the HSV experiment employed multiple set sizes, we did not expect any specific effect of set size because no delay was introduced and the display remained on the screen until the response. For these reasons, we collapsed the HSV data across set size in the subsequent analyses. (Multiple set sizes were tested for unrelated reasons pertaining to other ongoing research.) 
Results and discussion
Overall, responses were significantly more precise in this experiment than in Experiment 1 [with delay vs. without delay; CIELAB mean κ: 44.10 vs. 66.10, t(179) = −7.17, p < 0.001; HSV mean κ: 52.21 vs. 64.34, t(179) = −2.49, p = 0.017]. This is consistent with findings using calibrated stimuli that discrimination thresholds tend to increase with the addition of a delay (Nemes, Perry, & McKeefry, 2012; Nilsson & Nelson, 1981; Olkkonen & Allred, 2014). 
Importantly, estimated precision continued to vary by stimulus, even in the absence of an explicit memory demand (Figure 8a). Estimated precision without a delay was significantly correlated with the estimates obtained with the delay in Experiment 1 [CIELAB κ: r = 0.37, t(178) = 5.29, p < 0.001; HSV κ: r = 0.49, t(178) = 7.57, p < 0.001; Figure 6b]. 
Figure 8
 
(a) Estimated κ values for CIELAB and HSV color wheels in Experiment 2, and (b) κ correlations between Experiments 1 and 2.
Figure 8
 
(a) Estimated κ values for CIELAB and HSV color wheels in Experiment 2, and (b) κ correlations between Experiments 1 and 2.
Summary
The main finding in this experiment is that stimulus-dependent variability is present even in the absence of a memory delay and that this variability is correlated with that found in the with-delay experiment. Minimally, these results suggest that at least some of the stimulus-dependent variability observed in working memory is already present in perceptual color estimation. Because of the differences between nominal and rendered colors, we cannot be certain of the extent to which this particular pattern of variability would persist even with stimuli rendered properly on a CIELAB circle. Note, however, that color discrimination is known to be inhomogeneous across color space even with careful display calibration, suggesting that at least some of the inhomogeneity found here may not be due to lack of calibration or other rendering issues (Bachy, Dias, Alleysson, & Bonnardel, 2012; Danilova & Mollon, 2012; Witzel & Gegenfurtner, 2013). Determining to what extent the stimulus-dependent variability in working memory is present in perceptual estimation, and to what extent it arises from memory processes, is an important question for future research. 
Experiment 3: Color-by-color fits for varying memory loads
The canonical effect in research on visual working memory is the degradation of performance with increasing memory load, a pattern thought to reveal the limited storage capacity of the system (Cowan, 2001; Luck & Vogel, 1997; Sperling, 1960). With delayed estimation, visual working memory quality—and that of color in particular—has been shown to decline with increasing memory load (though a debate continues about whether it declines further beyond a load of about three or four objects; Anderson & Awh, 2012; Bays et al., 2009; van den Berg et al., 2012). Given the importance of this issue, we sought to examine whether stimulus-dependent differences in response variability persist with increasing memory loads. If they do not, this would surely not settle any debates in working memory research. But we thought it would be important to at least determine whether effects of memory load wash away all stimulus-driven effects. If they do, then the stimulus-dependent variability we identified in Experiments 1 and 2 need not be of central concern to the working memory community. Similarly, if the memory load effect is substantially larger than the stimulus-dependent variability, then it is not functionally important to resolve the methodological problems that led to the differences between nominal and rendered stimuli in our lab (and likely other working memory labs). 
A new group of observers participated in a typical delayed estimation experiment with memory load varying on each trial. Again, we fit a mixture model to response distributions by stimulus. If this experiment also produced stimulus-dependent variability in precision estimates, and this variability is correlated with that found in Experiment 1, this would suggest that the effects of memory load do not completely eliminate stimulus-driven effects. 
Crucially, although the stimuli for Experiment 3 were nominally the same as those in Experiment 1, they were presented on a different monitor. They were generated using the same procedure as in Experiment 1, and the monitor was not calibrated. Thus, the rendered stimuli in Experiment 3 likely differ both from the nominal values and from the rendered stimuli in Experiments 1 and 2. If any stimulus-dependent variability in Experiment 3 remains correlated with that in Experiment 1, this would suggest that the stimulus-specific effects are large enough to survive failures to appropriately calibrate the display. 
Method
Observers
A new group of 24 Johns Hopkins University undergraduates participated in exchange for course credit. Each observer had normal or corrected-to-normal visual acuity. The protocol for this experiment was approved by the Johns Hopkins University IRB. 
Apparatus and stimuli
Stimuli were presented on an iMac computer (Apple, Inc., Cupertino, CA) with a liquid-crystal display monitor. The stimuli were generated using the same procedures as in Experiment 1. However, since the stimuli were displayed on a different monitor, and because this monitor was not calibrated, the stimuli likely differed from both the nominal stimuli and those rendered in Experiment 1. These stimuli were not measured posthoc. 
Procedure
Experiment 3 was identical to Experiment 1 except as noted below. This experiment utilized the CIELAB color space from Experiment 1. The experiment included memory loads of one, two, four, and six distributed randomly over the course of a session. For half of the observers, colors were sampled randomly on each trial. For the other half, colors were sampled with the restriction that no two colors were allowed to be closer than 20° (e.g., Fougnie, Asplund, & Marois, 2010). Each observer completed 60 trials for each memory load (240 trials total). 
Analysis
As in Experiments 1 and 2, a stimulus-specific mixture model was fitted to the results. There were no obvious differences in parameters when the stimuli in a trial were sampled with restrictions and without, so data from the two conditions were collapsed for all subsequent analyses. Because the number of trials for each target stimulus was not equal, a smoothing algorithm was applied to the estimated stimulus-specific precision (κ) values for each memory load; estimates were smoothed by a moving average of ±1 adjacent values weighted by the number of observations for each stimulus. 
Results and discussion
The correlations between stimulus-specific precision estimates in Experiments 1 and 3 are shown in Figure 9. Overall, when collapsing across memory load, the stimulus-specific effects on precision in Experiments 1 and 3 were correlated. In addition, each memory load in Experiment 3 was independently correlated with the single memory load in Experiment 1 [t(178) > 4.53, p < 0.01 for all correlations]. The stimulus dependency of precision estimates survived the effect of higher memory loads. 
Figure 9
 
Correlations between color-specific κ values obtained in Experiment 1 (N = 3) and Experiment 3 (N = 24; CIELAB color wheel). (a) Correlations between the two experiments, collapsing across all memory loads in Experiment 3. (b) Correlations between each individual memory load in Experiment 1 and Experiment 3.
Figure 9
 
Correlations between color-specific κ values obtained in Experiment 1 (N = 3) and Experiment 3 (N = 24; CIELAB color wheel). (a) Correlations between the two experiments, collapsing across all memory loads in Experiment 3. (b) Correlations between each individual memory load in Experiment 1 and Experiment 3.
To be certain that these results are not due to artifacts from data analysis (e.g., our smoothing procedure), correlations were applied over the data with randomly permuted stimuli as in Experiment 1. The Monte Carlo simulations confirmed that the correlations we observed are unlikely to have arisen by chance (zero out of 10,000 empirical correlations were significant in the right direction). Thus properties of individual stimulus values produce large variability in subsequent memory responses, and these differences between stimuli persist as memory load increases. 
Summary
Overall, this experiment makes it clear that the stimulus-dependent effects identified in Experiment 1 persist as memory load increases (at least within the range most typically studied in related experiments). In addition, several specific aspects of the results are worth emphasizing. First, there is a strong interexperiment correlation between stimulus-specific estimates with a memory load of one. Because two different noncalibrated displays were used in the two experiments, it is very likely that the rendered stimuli in the two experiments were somewhat different from each other. Despite this, the stimulus-specific correlation between experiments survived. This suggests both that the stimulus-specific effects are not particular to one display and that they are large enough to be seen despite inaccuracies in color rendering. In light of this, it seems likely that previous working memory experiments have also obtained systematic but unmeasured variation in color precision. 
Second, we note that although stimulus-specific precision estimates were still significantly correlated at higher memory loads of four and six, these correlations were considerably weaker. This could mean that stimulus-specific effects are less important at higher memory loads, but it could also be an artifact of the model. This model has an inherent correlation between the von Mises part (i.e., the estimates of precision, κ) and the uniform part (i.e., guessing, 1 − Pm) of the equations (see also Suchow et al., 2013). If the estimate of guessing rate is low, then the precision estimate must also be lower to accommodate extreme responses; conversely, if a higher guessing rate is estimated, then the von Mises component will have high responsibility for accurate responses only, leading to larger precision estimates. High rates of guessing at higher memory loads could lead the mixture model to overestimate precision for stimuli that tend to elicit noisy responses. Indeed, the specific stimuli that deviated most from the expected correlations seem to be those that were imprecise with a memory load of one; we suspect that the response precision associated with these stimuli at higher memory loads was overestimated. (Note that nontarget responses would be expected to have the same effect on precision as guesses with respect to estimates of precision; consult Bays, Catalao, & Husain, 2009.) 
Experiment 4: A relationship between labeled color regions and response precision
Experiments 1 through 3 demonstrated that the response distributions elicited by some stimuli were more variable than those elicited by other stimuli. Experiment 2, in particular, demonstrated that stimulus-specific effects in a perception experiment correlated with those in the memory experiments. This suggests that the memory effects of color are likely mediated by the perceptual representation of color. To investigate whether this pattern of variability is common to different perceptual tasks, we conducted a color labeling experiment. Observers were shown the same rendered colors that served as the response wheel in Experiments 1 and 2, and they were asked to identify the best example of each of seven color terms. If the observers are consistent in their responses, and if their responses are systematically related to the stimulus-specific precision estimates, it could shed some insight on the origin of the variability. Furthermore, in light of the deviations between nominal and rendered stimuli, a sensible relationship between color-labeling and stimulus effects would provide additional reassurance that the effects we measured survive the inaccuracies of color rendering. 
Method
Observers
A new group of eight Johns Hopkins University undergraduates participated in exchange for course credit. Each had normal or corrected-to-normal visual acuity. The protocol for this experiment was approved by the Johns Hopkins University IRB. 
Apparatus and stimuli
Apparatus and stimuli were identical to those used in Experiment 1
Procedure
At the start of a trial, the 180 stimuli rendered in Experiment 1 were presented in the center of the screen organized around a wheel. To the right of the wheel the seven color terms orange, yellow, lime, green, blue, purple, and pink were vertically presented in a random order. We did not use the basic color terms (Berlin & Kay, 1969) because we did not expect this set of color samples to be labeled by all and only basic terms. Recall that the stimulus set was constructed with constraints unrelated to color terminology. Specifically, we excluded the terms red and brown because of the high luminance values of the sampled colors, and we included the term lime because of the relatively wide spectrum of samples that appeared green to us. 
Observers were asked to find the best example of each of these seven colors by clicking on the color wheel. The color wheel stayed on the screen until an observer made seven responses. The color wheel was rotated randomly on each trial, and each observer completed 20 trials. 
Results and discussion
Combining data from eight observers yielded 160 responses for each color term. Histograms of the results are given in Figure 10a. Response distributions for each term appeared approximately normal. In other words, certain stimuli were noisy attractors for the color labels, with responses for a given label diminishing rapidly with distance from the corresponding attractor. We emphasize that our intent was not to identify definite category boundaries or focal points. More sophisticated methods are available for this purpose and could be employed in future research (e.g., Witzel & Gegenfurtner, 2013), and they would be more appropriate with a set of color samples designed for this purpose and faithfully rendered. Instead, we merely sought to determine whether observers shared even loose intuitions about how to label this stimulus set and whether the labeling structure was related to the stimulus-specific precision effects observed in Experiments 1 through 3
Figure 10
 
Results of Experiment 4. (a) Histogram of responses given as the best example of each of seven color terms, along with seven von Mises distributions fitted to these responses. Overlap points between the distributions designate operational boundaries between regions on the wheel. (b) The relationship between the size of each identified region (in terms of the number of individual colors) and the average κ value obtained for the colors in that region in Experiment 1.
Figure 10
 
Results of Experiment 4. (a) Histogram of responses given as the best example of each of seven color terms, along with seven von Mises distributions fitted to these responses. Overlap points between the distributions designate operational boundaries between regions on the wheel. (b) The relationship between the size of each identified region (in terms of the number of individual colors) and the average κ value obtained for the colors in that region in Experiment 1.
Toward this end, inspection of the histograms in Figure 10a reveals several important features. First, the fact that response distributions are relatively clustered indicates that observers shared intuitions about labels for the color stimuli. Second, the amount of variability in selection of the best instance of a label differs across color space, as seen by the varying width of the peaks in Figure 10a. For example, observers were more consistent in their labeling of purple than lime. The variability could be within observer, between observers, or both. Third, it appeared that there was a relationship between a label's consistency in Experiment 4 and the response precision elicited by those colors in the other experiments. For example, the stimuli labeled as the best exemplars of purple also elicited response distributions with the highest precision in Experiments 1 and 2
To explore this observation quantitatively, we fit seven von Mises distributions to the histograms in Figure 10a, comparing likelihood values to identify operational boundaries in this spatial arrangement of the rendered colors. This in turn afforded a measure of a label's spread: namely, the number of individual colors between boundaries (orange: 40; yellow: 20; lime: 18; green: 23; blue: 19; purple: 15; pink: 45). To relate these findings to the response precision associated with these colors, we averaged together the κ values obtained in Experiment 1 for all of the stimuli falling within each of the seven bounded regions. There was a significant negative correlation between the number of stimuli in a region and the average precision of stimuli within that region [t(5) = −2.87, r = −0.79, p = 0.035; Figure 10b]. 
We emphasize again that these data were not collected for the purpose of understanding color categorization. Rather, they provide some traction in understanding the stimulus-specific effects in perception (Experiment 2, Figure 8) that are propagated to working memory (Experiment 1, Figure 4). Empirically, we have demonstrated that stimuli falling in labeled regions that are narrow tend to produce narrower response distributions. This is of general interest, but it is particularly important in the context of the present experiments because it provides added verification that the stimulus-specific variability observed in perception and memory is not solely an artifact of the deviation between nominal and rendered stimuli. 
General discussion
This study sought to investigate assumptions built into an influential and rapidly growing literature that utilizes the delayed estimation paradigm with color to draw conclusions about the structure and limits of visual working memory (Anderson & Awh, 2012; Bays et al., 2009, 2011; Emrich & Ferber, 2012; Fougnie & Alvarez, 2011; Fougnie et al., 2010, 2012; Gold et al., 2010; van den Berg et al., 2012; Wilken & Ma, 2004; Zhang & Luck, 2008, 2009, 2011). The assumptions are that the colors rendered in the relevant experiments have been sampled from a circle in CIELAB color space, that they are perceptually homogenous, that they have equal lightness values, and that they tend to elicit equally variable response distributions—assumptions that motivate value-independent analysis of response distribution properties. 
In Experiments 1 through 3, we identified effects that undermine these assumptions. First, we found stimulus-specific response variability that correlated across unique observers (Experiment 1). This stimulus-specific variability was present even in an experiment without an explicit memory demand (Experiment 2) and persisted with larger memory loads (Experiment 3). Experiment 4 demonstrated that some of this stimulus-specific variability likely arose during interaction with the stimulus set at test. 
Importantly, we determined posthoc that the colors rendered in Experiments 1, 2, and 4 were different from the nominal colors specified, differing in lightness and nonuniformly distributed in CIELAB space. In this regard, our methods reflect a kind of “worst case” scenario with respect to stimulus display. We did not apply the calibration procedures that ensure faithful rendering of color stimuli. We emphasize that although calibration procedures are standard in studies of color perception (Allen, Beilock, & Shevell, 2012; Olkkonen & Allred, 2014; Witzel & Gegenfurtner, 2013; Xiao, Hurst, MacIntyre, & Brainard, 2012), calibration procedures have not been implemented in delayed estimation reports (see, e.g., Suchow et al., 2013, which provides a tutorial for conducting such experiments and analyses without mention of calibration or color rendering validation). Indeed, we could not identify even one study that reported implementing a full calibration procedure (e.g., that described in Brainard, Pelli, & Robson, 2002). Neither could we find reports that confirmed the properties of color stimuli using posthoc radiometer measurements. 
Taken together, our results undermine the assumption, endemic in the delayed estimation literature, that colored stimuli are represented homogenously in memory. This is not a minor issue: Stimulus homogeneity is central to the parameter fits used to make inferences about the structure of working memory and to compare alternative memory models. 
The assumption of stimulus homogeneity is perhaps most clear in two recent studies, which argued that visual working memory varies stochastically on a moment-to-moment, trial-by-trial, or even item-by-item basis (Fougnie et al., 2012; van den Berg et al., 2012). These inferences are predicated on the assumption that there is no systematic reason for response distributions to vary trial by trial—in particular, no reason for the precision of the distribution to vary according to the trial-specific stimulus. Specifically, one previous model of working memory proposed that memory precision varies from moment to moment in a way that can be described as sampling from a gamma distribution (van den Berg et al., 2012). However, our stimulus-specific precision estimates obtained in Experiments 1 and 2 produced perfect gamma distributions (Figure 11a and b). An important future direction would be investigating how much variability in working memory precision should be attributed to trial-by-trial variability after stimulus-specific variability in perception and working memory has been taken into account. 
Figure 11
 
Distribution of estimated precisions (κ) obtained in Experiment 1 and Experiment 2. (a) Histogram of κ values for all colors estimated in the with-delay experiment (Experiment 1, CIELAB). The black curve represents the best fit gamma distribution. The smaller graphs show von Mises densities for three example colors. (b) The same graph with the data from the without-delay experiment (Experiment 2).
Figure 11
 
Distribution of estimated precisions (κ) obtained in Experiment 1 and Experiment 2. (a) Histogram of κ values for all colors estimated in the with-delay experiment (Experiment 1, CIELAB). The black curve represents the best fit gamma distribution. The smaller graphs show von Mises densities for three example colors. (b) The same graph with the data from the without-delay experiment (Experiment 2).
Similar points extend to what is perhaps the central debate in this area: whether or not there is a discrete capacity limit in addition to a more continuous precision limit on memory for smaller memory loads. This debate has often hinged on whether the estimate of memory precision plateaus at larger memory loads and on whether guessing rates increase in a way that suggests frequent guesses when memory load exceeds some fixed quantity. But these parameter values are obtained as the best fits of models that assume that responses to a memory target (as opposed to guesses) are drawn from a distribution whose characteristics are independent of the target color. Similarly, some studies have suggested that observers make responses that are not random but that also are not drawn from a target representation. Instead, responses may be based on a nontarget display item (Bays et al., 2009; Emrich & Ferber, 2012). Correctly estimating the probability of such nontarget responses depends on accurate expectations about the response distributions for specific target and nontarget colors. 
Conclusions
We draw two broad conclusions from our results. The first is prescriptive: Replications and extensions of previous work should be conducted with CIELAB specified color stimuli that are rendered faithfully following standardized calibration procedures (Brainard, Pelli, & Robson, 2002; Gegenfurtner & Kiper, 2003). 
Second, it seems likely that the assumption of perceptual and memory homogeneity among stimuli in studies of color working memory is unwarranted. Our results indicate clearly that stimulus-specific variation in precision exists, even though the deviations between nominal and rendered stimuli do not allow us to definitively establish the cause of those stimulus-specific effects. More broadly, our results motivate examination of implicit homogeneity assumptions for other stimulus classes. For example, in the case of orientation a variety of phenomena suggest differences in the fidelity of the representation of oblique and cardinal values (Girshick, Landy, & Simoncelli, 2011; Wolfe, Klempen, & Shulman, 1999)—differences that have not been incorporated into the modeling of delayed estimation experiments focused on orientation (Fougnie & Alvarez, 2011; Keshvari, van den Berg, & Ma, 2013; van den Berg et al., 2012). 
Ultimately, we suggest that a complete understanding of the structure of visual working memory and its capacity limits will require a stimulus-specific understanding of both perceptual and memory representations. For this reason, despite the caution provided here, color remains a good candidate for a stimulus class with which to investigate working memory. A great deal is known about color perception, including its neurophysiological basis, the computations that support color adaptation and constancy, and its relationship to higher-level reasoning and language (for reviews of color vision see, e.g., Gegenfurtner & Kiper, 2003; Solomon & Lennie, 2007). This creates a unique opportunity for combining expertise across areas to relate visual working memory to visual perception and cognition more broadly.1534 
Acknowledgments
We thank George Alvarez, Tim Brady, Daryl Fougnie, and Weiwei Zhang for sharing details of their experimental procedures with us. This work was supported by grant NSF CAREER BCS-0954749, awarded to SA. 
Commercial relationships: none. 
Corresponding author: Jonathan Isaac Flombaum. 
Email: flombaum@jhu.edu. 
Address: Department of Psychological and Brain Sciences, Johns Hopkins University, Baltimore, MD, USA. 
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Appendix A
Table A1
 
CIELAB values and corresponding CIE xyY values for stimulus color values used in experiments. The xyY values were obtained by measuring light emission spectrum using a spectroradiometer (PR-655 SpectraScan, Photo Research). Unit of Y is candelas per square meter.
Table A1
 
CIELAB values and corresponding CIE xyY values for stimulus color values used in experiments. The xyY values were obtained by measuring light emission spectrum using a spectroradiometer (PR-655 SpectraScan, Photo Research). Unit of Y is candelas per square meter.
Color Intended CIELAB coordinates Rendered CIELAB coordinates Measured xyY coordinates
L* a* b* L* a* b* x y Y
1 70 80.000 38.000 62.693 71.424 27.188 0.518 0.311 23.917
2 70 79.963 40.094 62.605 71.274 29.222 0.528 0.314 23.229
3 70 79.854 42.185 62.541 71.066 31.297 0.535 0.319 23.192
4 70 79.671 44.272 62.503 70.799 33.410 0.542 0.324 23.052
5 70 79.416 46.350 62.491 70.474 35.557 0.548 0.328 22.955
6 70 79.088 48.419 62.503 70.092 37.737 0.554 0.332 22.909
7 70 78.689 50.475 62.539 69.653 39.944 0.559 0.337 22.938
8 70 78.218 52.515 62.601 69.158 42.176 0.566 0.340 22.818
9 70 77.676 54.538 62.686 68.610 44.428 0.570 0.345 22.970
10 70 77.063 56.541 62.794 68.010 46.696 0.574 0.349 22.948
11 70 76.382 58.521 62.925 67.360 48.977 0.578 0.354 23.137
12 70 75.631 60.476 63.078 66.661 51.264 0.581 0.358 23.255
13 70 74.813 62.404 63.253 65.916 53.554 0.584 0.361 23.296
14 70 73.928 64.302 63.447 65.129 55.841 0.585 0.364 23.449
15 70 72.977 66.168 63.661 64.300 58.121 0.587 0.367 23.501
16 70 71.962 68.000 63.893 63.433 60.387 0.588 0.370 23.753
17 70 70.883 69.795 64.170 62.612 61.015 0.588 0.373 23.994
18 70 69.742 71.552 64.469 61.775 61.251 0.587 0.375 24.253
19 70 68.541 73.267 64.780 60.904 61.497 0.587 0.377 24.406
20 70 67.281 74.940 65.103 60.001 61.752 0.585 0.379 24.697
21 70 65.963 76.567 65.437 59.071 62.016 0.583 0.381 24.957
22 70 64.589 78.148 65.781 58.116 62.288 0.580 0.382 25.316
23 70 63.160 79.680 66.134 57.139 62.567 0.579 0.384 25.638
24 70 61.680 81.160 66.496 56.143 62.852 0.576 0.386 25.987
25 70 60.148 82.589 66.864 55.130 63.143 0.573 0.388 26.291
26 70 58.567 83.963 67.239 54.104 63.438 0.571 0.389 26.632
27 70 56.940 85.281 67.619 53.067 63.738 0.569 0.391 27.009
28 70 55.267 86.541 68.004 52.022 64.041 0.566 0.394 27.419
29 70 53.552 87.742 68.392 50.971 64.347 0.563 0.395 27.815
30 70 51.795 88.883 68.783 49.916 64.655 0.561 0.397 28.113
31 70 50.000 89.962 69.177 48.859 64.965 0.559 0.399 28.573
32 70 48.168 90.977 69.571 47.803 65.275 0.556 0.401 29.016
33 70 46.302 91.928 69.966 46.750 65.586 0.553 0.403 29.329
34 70 44.404 92.813 70.326 45.629 65.853 0.551 0.405 29.829
35 70 42.476 93.631 70.335 43.761 65.679 0.548 0.407 30.294
36 70 40.521 94.382 70.344 41.865 65.504 0.546 0.409 30.800
37 70 38.541 95.063 70.351 39.943 65.328 0.544 0.410 30.968
38 70 36.538 95.676 70.359 37.998 65.152 0.541 0.412 31.559
39 70 34.515 96.218 70.366 36.033 64.976 0.537 0.416 31.378
40 70 32.475 96.689 70.372 34.050 64.801 0.533 0.418 31.454
41 70 30.419 97.088 70.378 32.051 64.625 0.530 0.422 31.129
42 70 28.350 97.416 70.383 30.039 64.451 0.524 0.425 31.022
43 70 26.272 97.671 70.388 28.016 64.278 0.519 0.430 31.089
44 70 24.185 97.854 70.392 25.985 64.106 0.514 0.434 31.015
45 70 22.094 97.963 70.396 23.948 63.936 0.510 0.436 30.760
46 70 20.000 98.000 70.399 21.907 63.767 0.504 0.441 30.784
47 70 17.906 97.963 70.402 19.865 63.600 0.500 0.444 30.655
48 70 15.815 97.854 70.404 17.825 63.435 0.494 0.448 30.668
49 70 13.728 97.671 70.406 15.788 63.272 0.490 0.451 30.439
50 70 11.650 97.416 70.407 13.757 63.112 0.484 0.456 30.526
51 70 9.581 97.088 70.408 11.734 62.954 0.479 0.460 30.480
52 70 7.525 96.689 70.408 9.721 62.798 0.475 0.463 30.230
53 70 5.485 96.218 70.408 7.721 62.645 0.469 0.468 30.318
54 70 3.462 95.676 70.407 5.736 62.495 0.463 0.472 29.992
55 70 1.459 95.063 70.405 3.768 62.347 0.458 0.476 29.817
56 70 −0.521 94.382 70.403 1.818 62.202 0.454 0.479 29.971
57 70 −2.476 93.631 70.400 −0.111 62.060 0.449 0.483 29.728
58 70 −4.404 92.813 70.397 −2.018 61.920 0.441 0.489 30.346
59 70 −6.302 91.928 70.393 −3.900 61.783 0.436 0.494 30.302
60 70 −8.168 90.977 70.387 −5.756 61.649 0.431 0.497 30.121
61 70 −10.000 89.962 70.381 −7.585 61.518 0.426 0.500 30.021
62 70 −11.795 88.883 70.375 −9.384 61.389 0.420 0.506 29.929
63 70 −13.552 87.742 70.367 −11.152 61.263 0.415 0.509 29.917
64 70 −15.267 86.541 70.358 −12.888 61.139 0.411 0.512 29.588
65 70 −16.940 85.281 70.348 −14.590 61.018 0.405 0.517 29.483
66 70 −18.567 83.963 70.337 −16.258 60.899 0.400 0.520 29.442
67 70 −20.148 82.589 70.324 −17.890 60.783 0.395 0.524 29.516
68 70 −21.680 81.160 70.311 −19.485 60.669 0.390 0.528 29.297
69 70 −23.160 79.680 70.296 −21.042 60.557 0.386 0.531 29.123
70 70 −24.589 78.148 70.280 −22.560 60.447 0.382 0.533 29.161
71 70 −25.963 76.567 70.262 −24.039 60.339 0.378 0.537 29.154
72 70 −27.281 74.940 70.242 −25.477 60.232 0.374 0.540 28.812
73 70 −28.541 73.267 70.221 −26.874 60.127 0.369 0.544 29.010
74 70 −29.742 71.552 70.198 −28.230 60.024 0.365 0.547 29.076
75 70 −30.883 69.795 70.174 −29.543 59.923 0.362 0.549 28.925
76 70 −31.962 68.000 70.147 −30.814 59.822 0.358 0.552 28.911
77 70 −32.977 66.168 70.119 −32.042 59.723 0.354 0.555 28.712
78 70 −33.928 64.302 70.088 −33.227 59.625 0.351 0.557 28.974
79 70 −34.813 62.404 70.056 −34.367 59.527 0.347 0.558 28.821
80 70 −35.631 60.476 70.021 −35.464 59.431 0.344 0.558 28.722
81 70 −36.382 58.521 70.000 −36.382 58.521 0.339 0.557 29.004
82 70 −37.063 56.541 70.000 −37.063 56.541 0.334 0.555 28.750
83 70 −37.676 54.538 70.000 −37.676 54.538 0.330 0.551 28.663
84 70 −38.218 52.515 70.000 −38.218 52.515 0.326 0.547 28.965
85 70 −38.689 50.475 70.000 −38.689 50.475 0.321 0.542 29.055
86 70 −39.088 48.419 70.000 −39.089 48.419 0.317 0.537 28.913
87 70 −39.416 46.350 70.000 −39.416 46.350 0.312 0.530 28.751
88 70 −39.671 44.272 70.000 −39.671 44.272 0.307 0.521 28.857
89 70 −39.854 42.185 70.000 −39.854 42.185 0.303 0.514 28.798
90 70 −39.963 40.094 70.000 −39.963 40.094 0.298 0.508 29.134
91 70 −40.000 38.000 70.000 −40.000 38.000 0.293 0.497 29.186
92 70 −39.963 35.906 70.000 −39.963 35.906 0.289 0.488 29.205
93 70 −39.854 33.815 70.000 −39.854 33.815 0.285 0.479 29.304
94 70 −39.671 31.728 70.000 −39.671 31.728 0.281 0.469 29.384
95 70 −39.416 29.650 70.000 −39.416 29.650 0.277 0.460 29.394
96 70 −39.088 27.581 70.000 −39.089 27.581 0.273 0.448 29.137
97 70 −38.689 25.525 70.000 −38.689 25.525 0.270 0.438 29.330
98 70 −38.218 23.485 70.000 −38.218 23.485 0.266 0.428 29.331
99 70 −37.676 21.462 70.000 −37.676 21.462 0.262 0.418 29.482
100 70 −37.063 19.459 70.000 −37.063 19.459 0.259 0.408 29.649
101 70 −36.382 17.479 70.000 −36.382 17.479 0.257 0.401 29.698
102 70 −35.631 15.524 70.000 −35.631 15.524 0.254 0.389 29.588
103 70 −34.813 13.596 70.000 −34.813 13.596 0.251 0.379 29.639
104 70 −33.928 11.698 70.000 −33.928 11.698 0.249 0.372 29.937
105 70 −32.977 9.832 70.000 −32.977 9.832 0.247 0.361 29.638
106 70 −31.962 8.000 70.000 −31.962 8.000 0.245 0.353 29.795
107 70 −30.883 6.205 70.000 −30.883 6.205 0.243 0.346 30.062
108 70 −29.742 4.448 70.000 −29.742 4.448 0.241 0.336 29.855
109 70 −28.541 2.733 70.000 −28.541 2.733 0.240 0.330 30.120
110 70 −27.281 1.060 70.000 −27.281 1.060 0.239 0.322 30.023
111 70 −25.963 −0.567 70.000 −25.963 −0.567 0.237 0.314 30.341
112 70 −24.589 −2.148 70.000 −24.589 −2.148 0.236 0.306 30.461
113 70 −23.160 −3.680 70.000 −23.160 −3.680 0.235 0.298 30.388
114 70 −21.680 −5.160 70.000 −21.680 −5.160 0.234 0.293 30.643
115 70 −20.148 −6.589 70.000 −20.148 −6.589 0.235 0.288 30.611
116 70 −18.567 −7.963 70.000 −18.567 −7.963 0.234 0.281 30.589
117 70 −16.940 −9.281 70.000 −16.940 −9.281 0.234 0.278 30.932
118 70 −15.267 −10.541 70.000 −15.267 −10.541 0.234 0.271 30.985
119 70 −13.552 −11.742 70.000 −13.552 −11.742 0.235 0.267 30.890
120 70 −11.795 −12.883 70.000 −11.795 −12.883 0.236 0.262 30.955
121 70 −10.000 −13.962 70.000 −10.000 −13.962 0.236 0.258 31.013
122 70 −8.168 −14.977 70.000 −8.168 −14.977 0.237 0.254 31.128
123 70 −6.302 −15.928 70.000 −6.302 −15.928 0.239 0.250 31.112
124 70 −4.404 −16.813 70.000 −4.404 −16.813 0.240 0.248 31.150
125 70 −2.476 −17.631 70.000 −2.476 −17.631 0.242 0.244 31.318
126 70 −0.521 −18.382 70.000 −0.521 −18.382 0.243 0.242 31.338
127 70 1.459 −19.063 70.000 1.459 −19.063 0.245 0.239 31.575
128 70 3.462 −19.676 70.000 3.462 −19.676 0.247 0.235 31.295
129 70 5.485 −20.218 70.000 5.485 −20.218 0.249 0.233 31.540
130 70 7.525 −20.689 70.000 7.525 −20.689 0.252 0.230 31.533
131 70 9.581 −21.088 70.000 9.581 −21.089 0.254 0.229 31.686
132 70 11.650 −21.416 70.000 11.650 −21.416 0.258 0.228 31.798
133 70 13.728 −21.671 70.000 13.728 −21.671 0.261 0.226 31.699
134 70 15.815 −21.854 70.000 15.815 −21.854 0.264 0.226 31.814
135 70 17.906 −21.963 70.000 17.906 −21.963 0.268 0.225 32.178
136 70 20.000 −22.000 70.000 20.000 −22.000 0.270 0.223 32.179
137 70 22.094 −21.963 70.000 22.094 −21.963 0.274 0.223 32.531
138 70 24.185 −21.854 70.000 24.185 −21.854 0.278 0.223 32.236
139 70 26.272 −21.671 70.000 26.272 −21.671 0.282 0.222 32.348
140 70 28.350 −21.416 70.000 28.350 −21.416 0.287 0.222 32.621
141 70 30.419 −21.088 70.000 30.419 −21.089 0.292 0.223 32.669
142 70 32.475 −20.689 70.000 32.475 −20.689 0.295 0.221 32.596
143 70 34.515 −20.218 70.000 34.515 −20.218 0.301 0.223 32.921
144 70 36.538 −19.676 70.000 36.538 −19.676 0.305 0.223 32.854
145 70 38.541 −19.063 70.000 38.541 −19.063 0.311 0.224 32.944
146 70 40.521 −18.382 70.000 40.521 −18.382 0.315 0.224 33.003
147 70 42.476 −17.631 70.000 42.476 −17.631 0.321 0.225 33.193
148 70 44.404 −16.813 70.000 44.404 −16.813 0.327 0.227 33.344
149 70 46.302 −15.928 70.000 46.302 −15.928 0.333 0.228 33.543
150 70 48.168 −14.977 70.000 48.168 −14.977 0.339 0.230 33.561
151 70 50.000 −13.962 70.000 50.000 −13.962 0.345 0.232 33.509
152 70 51.795 −12.883 70.000 51.795 −12.883 0.352 0.234 33.769
153 70 53.552 −11.742 70.000 53.552 −11.742 0.355 0.234 33.458
154 70 55.267 −10.541 70.000 55.267 −10.541 0.360 0.236 32.979
155 70 56.940 −9.281 70.000 56.940 −9.281 0.363 0.236 32.415
156 70 58.567 −7.963 70.000 58.567 −7.963 0.366 0.237 31.910
157 70 60.148 −6.589 70.000 60.148 −6.589 0.370 0.237 31.386
158 70 61.680 −5.160 70.000 61.680 −5.160 0.374 0.239 30.824
159 70 63.160 −3.680 69.651 62.606 −4.234 0.378 0.240 30.336
160 70 64.589 −2.148 69.204 63.353 −3.409 0.384 0.241 29.642
161 70 65.963 −0.567 68.763 64.081 −2.525 0.389 0.243 29.187
162 70 67.281 1.060 68.327 64.789 −1.583 0.392 0.244 28.945
163 70 68.541 2.733 67.898 65.475 −0.583 0.399 0.248 28.406
164 70 69.742 4.448 67.477 66.136 0.476 0.405 0.250 27.815
165 70 70.883 6.205 67.065 66.770 1.594 0.411 0.252 27.362
166 70 71.962 8.000 66.664 67.376 2.771 0.417 0.254 26.965
167 70 72.977 9.832 66.275 67.949 4.006 0.423 0.257 26.599
168 70 73.928 11.698 65.900 68.490 5.299 0.431 0.260 26.149
169 70 74.813 13.596 65.538 68.994 6.651 0.437 0.264 25.817
170 70 75.631 15.524 65.192 69.459 8.061 0.443 0.266 25.472
171 70 76.382 17.479 64.862 69.883 9.528 0.451 0.271 25.190
172 70 77.063 19.459 64.550 70.265 11.052 0.458 0.273 24.838
173 70 77.676 21.462 64.257 70.600 12.633 0.465 0.277 24.622
174 70 78.218 23.485 63.984 70.888 14.270 0.473 0.281 24.300
175 70 78.689 25.525 63.731 71.127 15.961 0.481 0.285 24.069
176 70 79.088 27.581 63.500 71.314 17.706 0.487 0.289 23.889
177 70 79.416 29.650 63.291 71.448 19.504 0.496 0.293 23.756
178 70 79.671 31.728 63.106 71.527 21.353 0.502 0.297 23.534
179 70 79.854 33.815 62.944 71.550 23.251 0.510 0.302 23.316
180 70 79.963 35.906 62.806 71.516 25.196 0.518 0.307 23.217
Background 100 0 0 100 0 0 0.286 0.294 37.530
Figure 1
 
Schematic display of the delayed estimation procedure (with a memory load of one). Note that the display background, which is shown as white here, was a neutral gray in the experiments reported below, and its color has varied in the published literature.
Figure 1
 
Schematic display of the delayed estimation procedure (with a memory load of one). Note that the display background, which is shown as white here, was a neutral gray in the experiments reported below, and its color has varied in the published literature.
Figure 2
 
Comparison between intended and measured color values. (a) Distortions in a* and b* dimensions. Black circle represents a* and b* coordinates for intended color values. Colored outline represents measured color values. Gray dot in the middle represents the background color. Note that rendered colors were not circularly arranged and were not equally distant from the background. (b) Distortions in L* value. Plotted here is a* versus L* for measured (colored shape) colors. Rendered colors had different L* values, violating the isoluminance assumption. (c) Measured xy values (colored shape) and nominal xy values (black shape). Nominal xy values were calculated from nominal CIELAB values using the measured background xyY of the monitor as the white point in the conversion. The triangle in the figure represents the gamut of the monitor used in the study. Colors outside the triangle cannot be rendered correctly on the monitor, and some of the nominal colors lay outside this gamut. (d) Measured color values in xyY dimensions. Again, rendered colors were not isoluminant. Note: Although rendered colors deviated from nominal colors, the posthoc radiometer measurements demonstrate that the xy coordinates of all rendered stimuli were measurably different from each other.
Figure 2
 
Comparison between intended and measured color values. (a) Distortions in a* and b* dimensions. Black circle represents a* and b* coordinates for intended color values. Colored outline represents measured color values. Gray dot in the middle represents the background color. Note that rendered colors were not circularly arranged and were not equally distant from the background. (b) Distortions in L* value. Plotted here is a* versus L* for measured (colored shape) colors. Rendered colors had different L* values, violating the isoluminance assumption. (c) Measured xy values (colored shape) and nominal xy values (black shape). Nominal xy values were calculated from nominal CIELAB values using the measured background xyY of the monitor as the white point in the conversion. The triangle in the figure represents the gamut of the monitor used in the study. Colors outside the triangle cannot be rendered correctly on the monitor, and some of the nominal colors lay outside this gamut. (d) Measured color values in xyY dimensions. Again, rendered colors were not isoluminant. Note: Although rendered colors deviated from nominal colors, the posthoc radiometer measurements demonstrate that the xy coordinates of all rendered stimuli were measurably different from each other.
Figure 3
 
Reciprocal of circular variance of target-response differences for each color value in CIELAB (top panel) and HSV (bottom panel) color spaces. Note that the x-axis of the CIELAB plot is hue angle centered on the intended color wheel (i.e., centered on a* b* = 20, 38, not on the origin of CIELAB).
Figure 3
 
Reciprocal of circular variance of target-response differences for each color value in CIELAB (top panel) and HSV (bottom panel) color spaces. Note that the x-axis of the CIELAB plot is hue angle centered on the intended color wheel (i.e., centered on a* b* = 20, 38, not on the origin of CIELAB).
Figure 4
 
Estimated κ values fitted by color and observer for each of the two color spaces (a), and estimated κ values fitted by color, collapsed across observers. (b) Solid black curves reflect spline smoothing, showing a pattern of κ variability by color, and dotted black lines show estimated κ when the model was fitted to all responses.
Figure 4
 
Estimated κ values fitted by color and observer for each of the two color spaces (a), and estimated κ values fitted by color, collapsed across observers. (b) Solid black curves reflect spline smoothing, showing a pattern of κ variability by color, and dotted black lines show estimated κ when the model was fitted to all responses.
Figure 5
 
Correlation of stimulus-specific κ estimates across observers. CIELAB color space is shown in the top row and HSV color space is shown in the bottom row.
Figure 5
 
Correlation of stimulus-specific κ estimates across observers. CIELAB color space is shown in the top row and HSV color space is shown in the bottom row.
Figure 6
 
Delayed estimation with the scrolling method.
Figure 6
 
Delayed estimation with the scrolling method.
Figure 7
 
(a) Color-by-color κ estimates in the scrolling experiment. (b) Color-by-color correlation between κ estimates obtained in the color wheel and scrolling experiments (collapsed across observers).
Figure 7
 
(a) Color-by-color κ estimates in the scrolling experiment. (b) Color-by-color correlation between κ estimates obtained in the color wheel and scrolling experiments (collapsed across observers).
Figure 8
 
(a) Estimated κ values for CIELAB and HSV color wheels in Experiment 2, and (b) κ correlations between Experiments 1 and 2.
Figure 8
 
(a) Estimated κ values for CIELAB and HSV color wheels in Experiment 2, and (b) κ correlations between Experiments 1 and 2.
Figure 9
 
Correlations between color-specific κ values obtained in Experiment 1 (N = 3) and Experiment 3 (N = 24; CIELAB color wheel). (a) Correlations between the two experiments, collapsing across all memory loads in Experiment 3. (b) Correlations between each individual memory load in Experiment 1 and Experiment 3.
Figure 9
 
Correlations between color-specific κ values obtained in Experiment 1 (N = 3) and Experiment 3 (N = 24; CIELAB color wheel). (a) Correlations between the two experiments, collapsing across all memory loads in Experiment 3. (b) Correlations between each individual memory load in Experiment 1 and Experiment 3.
Figure 10
 
Results of Experiment 4. (a) Histogram of responses given as the best example of each of seven color terms, along with seven von Mises distributions fitted to these responses. Overlap points between the distributions designate operational boundaries between regions on the wheel. (b) The relationship between the size of each identified region (in terms of the number of individual colors) and the average κ value obtained for the colors in that region in Experiment 1.
Figure 10
 
Results of Experiment 4. (a) Histogram of responses given as the best example of each of seven color terms, along with seven von Mises distributions fitted to these responses. Overlap points between the distributions designate operational boundaries between regions on the wheel. (b) The relationship between the size of each identified region (in terms of the number of individual colors) and the average κ value obtained for the colors in that region in Experiment 1.
Figure 11
 
Distribution of estimated precisions (κ) obtained in Experiment 1 and Experiment 2. (a) Histogram of κ values for all colors estimated in the with-delay experiment (Experiment 1, CIELAB). The black curve represents the best fit gamma distribution. The smaller graphs show von Mises densities for three example colors. (b) The same graph with the data from the without-delay experiment (Experiment 2).
Figure 11
 
Distribution of estimated precisions (κ) obtained in Experiment 1 and Experiment 2. (a) Histogram of κ values for all colors estimated in the with-delay experiment (Experiment 1, CIELAB). The black curve represents the best fit gamma distribution. The smaller graphs show von Mises densities for three example colors. (b) The same graph with the data from the without-delay experiment (Experiment 2).
Table A1
 
CIELAB values and corresponding CIE xyY values for stimulus color values used in experiments. The xyY values were obtained by measuring light emission spectrum using a spectroradiometer (PR-655 SpectraScan, Photo Research). Unit of Y is candelas per square meter.
Table A1
 
CIELAB values and corresponding CIE xyY values for stimulus color values used in experiments. The xyY values were obtained by measuring light emission spectrum using a spectroradiometer (PR-655 SpectraScan, Photo Research). Unit of Y is candelas per square meter.
Color Intended CIELAB coordinates Rendered CIELAB coordinates Measured xyY coordinates
L* a* b* L* a* b* x y Y
1 70 80.000 38.000 62.693 71.424 27.188 0.518 0.311 23.917
2 70 79.963 40.094 62.605 71.274 29.222 0.528 0.314 23.229
3 70 79.854 42.185 62.541 71.066 31.297 0.535 0.319 23.192
4 70 79.671 44.272 62.503 70.799 33.410 0.542 0.324 23.052
5 70 79.416 46.350 62.491 70.474 35.557 0.548 0.328 22.955
6 70 79.088 48.419 62.503 70.092 37.737 0.554 0.332 22.909
7 70 78.689 50.475 62.539 69.653 39.944 0.559 0.337 22.938
8 70 78.218 52.515 62.601 69.158 42.176 0.566 0.340 22.818
9 70 77.676 54.538 62.686 68.610 44.428 0.570 0.345 22.970
10 70 77.063 56.541 62.794 68.010 46.696 0.574 0.349 22.948
11 70 76.382 58.521 62.925 67.360 48.977 0.578 0.354 23.137
12 70 75.631 60.476 63.078 66.661 51.264 0.581 0.358 23.255
13 70 74.813 62.404 63.253 65.916 53.554 0.584 0.361 23.296
14 70 73.928 64.302 63.447 65.129 55.841 0.585 0.364 23.449
15 70 72.977 66.168 63.661 64.300 58.121 0.587 0.367 23.501
16 70 71.962 68.000 63.893 63.433 60.387 0.588 0.370 23.753
17 70 70.883 69.795 64.170 62.612 61.015 0.588 0.373 23.994
18 70 69.742 71.552 64.469 61.775 61.251 0.587 0.375 24.253
19 70 68.541 73.267 64.780 60.904 61.497 0.587 0.377 24.406
20 70 67.281 74.940 65.103 60.001 61.752 0.585 0.379 24.697
21 70 65.963 76.567 65.437 59.071 62.016 0.583 0.381 24.957
22 70 64.589 78.148 65.781 58.116 62.288 0.580 0.382 25.316
23 70 63.160 79.680 66.134 57.139 62.567 0.579 0.384 25.638
24 70 61.680 81.160 66.496 56.143 62.852 0.576 0.386 25.987
25 70 60.148 82.589 66.864 55.130 63.143 0.573 0.388 26.291
26 70 58.567 83.963 67.239 54.104 63.438 0.571 0.389 26.632
27 70 56.940 85.281 67.619 53.067 63.738 0.569 0.391 27.009
28 70 55.267 86.541 68.004 52.022 64.041 0.566 0.394 27.419
29 70 53.552 87.742 68.392 50.971 64.347 0.563 0.395 27.815
30 70 51.795 88.883 68.783 49.916 64.655 0.561 0.397 28.113
31 70 50.000 89.962 69.177 48.859 64.965 0.559 0.399 28.573
32 70 48.168 90.977 69.571 47.803 65.275 0.556 0.401 29.016
33 70 46.302 91.928 69.966 46.750 65.586 0.553 0.403 29.329
34 70 44.404 92.813 70.326 45.629 65.853 0.551 0.405 29.829
35 70 42.476 93.631 70.335 43.761 65.679 0.548 0.407 30.294
36 70 40.521 94.382 70.344 41.865 65.504 0.546 0.409 30.800
37 70 38.541 95.063 70.351 39.943 65.328 0.544 0.410 30.968
38 70 36.538 95.676 70.359 37.998 65.152 0.541 0.412 31.559
39 70 34.515 96.218 70.366 36.033 64.976 0.537 0.416 31.378
40 70 32.475 96.689 70.372 34.050 64.801 0.533 0.418 31.454
41 70 30.419 97.088 70.378 32.051 64.625 0.530 0.422 31.129
42 70 28.350 97.416 70.383 30.039 64.451 0.524 0.425 31.022
43 70 26.272 97.671 70.388 28.016 64.278 0.519 0.430 31.089
44 70 24.185 97.854 70.392 25.985 64.106 0.514 0.434 31.015
45 70 22.094 97.963 70.396 23.948 63.936 0.510 0.436 30.760
46 70 20.000 98.000 70.399 21.907 63.767 0.504 0.441 30.784
47 70 17.906 97.963 70.402 19.865 63.600 0.500 0.444 30.655
48 70 15.815 97.854 70.404 17.825 63.435 0.494 0.448 30.668
49 70 13.728 97.671 70.406 15.788 63.272 0.490 0.451 30.439
50 70 11.650 97.416 70.407 13.757 63.112 0.484 0.456 30.526
51 70 9.581 97.088 70.408 11.734 62.954 0.479 0.460 30.480
52 70 7.525 96.689 70.408 9.721 62.798 0.475 0.463 30.230
53 70 5.485 96.218 70.408 7.721 62.645 0.469 0.468 30.318
54 70 3.462 95.676 70.407 5.736 62.495 0.463 0.472 29.992
55 70 1.459 95.063 70.405 3.768 62.347 0.458 0.476 29.817
56 70 −0.521 94.382 70.403 1.818 62.202 0.454 0.479 29.971
57 70 −2.476 93.631 70.400 −0.111 62.060 0.449 0.483 29.728
58 70 −4.404 92.813 70.397 −2.018 61.920 0.441 0.489 30.346
59 70 −6.302 91.928 70.393 −3.900 61.783 0.436 0.494 30.302
60 70 −8.168 90.977 70.387 −5.756 61.649 0.431 0.497 30.121
61 70 −10.000 89.962 70.381 −7.585 61.518 0.426 0.500 30.021
62 70 −11.795 88.883 70.375 −9.384 61.389 0.420 0.506 29.929
63 70 −13.552 87.742 70.367 −11.152 61.263 0.415 0.509 29.917
64 70 −15.267 86.541 70.358 −12.888 61.139 0.411 0.512 29.588
65 70 −16.940 85.281 70.348 −14.590 61.018 0.405 0.517 29.483
66 70 −18.567 83.963 70.337 −16.258 60.899 0.400 0.520 29.442
67 70 −20.148 82.589 70.324 −17.890 60.783 0.395 0.524 29.516
68 70 −21.680 81.160 70.311 −19.485 60.669 0.390 0.528 29.297
69 70 −23.160 79.680 70.296 −21.042 60.557 0.386 0.531 29.123
70 70 −24.589 78.148 70.280 −22.560 60.447 0.382 0.533 29.161
71 70 −25.963 76.567 70.262 −24.039 60.339 0.378 0.537 29.154
72 70 −27.281 74.940 70.242 −25.477 60.232 0.374 0.540 28.812
73 70 −28.541 73.267 70.221 −26.874 60.127 0.369 0.544 29.010
74 70 −29.742 71.552 70.198 −28.230 60.024 0.365 0.547 29.076
75 70 −30.883 69.795 70.174 −29.543 59.923 0.362 0.549 28.925
76 70 −31.962 68.000 70.147 −30.814 59.822 0.358 0.552 28.911
77 70 −32.977 66.168 70.119 −32.042 59.723 0.354 0.555 28.712
78 70 −33.928 64.302 70.088 −33.227 59.625 0.351 0.557 28.974
79 70 −34.813 62.404 70.056 −34.367 59.527 0.347 0.558 28.821
80 70 −35.631 60.476 70.021 −35.464 59.431 0.344 0.558 28.722
81 70 −36.382 58.521 70.000 −36.382 58.521 0.339 0.557 29.004
82 70 −37.063 56.541 70.000 −37.063 56.541 0.334 0.555 28.750
83 70 −37.676 54.538 70.000 −37.676 54.538 0.330 0.551 28.663
84 70 −38.218 52.515 70.000 −38.218 52.515 0.326 0.547 28.965
85 70 −38.689 50.475 70.000 −38.689 50.475 0.321 0.542 29.055
86 70 −39.088 48.419 70.000 −39.089 48.419 0.317 0.537 28.913
87 70 −39.416 46.350 70.000 −39.416 46.350 0.312 0.530 28.751
88 70 −39.671 44.272 70.000 −39.671 44.272 0.307 0.521 28.857
89 70 −39.854 42.185 70.000 −39.854 42.185 0.303 0.514 28.798
90 70 −39.963 40.094 70.000 −39.963 40.094 0.298 0.508 29.134
91 70 −40.000 38.000 70.000 −40.000 38.000 0.293 0.497 29.186
92 70 −39.963 35.906 70.000 −39.963 35.906 0.289 0.488 29.205
93 70 −39.854 33.815 70.000 −39.854 33.815 0.285 0.479 29.304
94 70 −39.671 31.728 70.000 −39.671 31.728 0.281 0.469 29.384
95 70 −39.416 29.650 70.000 −39.416 29.650 0.277 0.460 29.394
96 70 −39.088 27.581 70.000 −39.089 27.581 0.273 0.448 29.137
97 70 −38.689 25.525 70.000 −38.689 25.525 0.270 0.438 29.330
98 70 −38.218 23.485 70.000 −38.218 23.485 0.266 0.428 29.331
99 70 −37.676 21.462 70.000 −37.676 21.462 0.262 0.418 29.482
100 70 −37.063 19.459 70.000 −37.063 19.459 0.259 0.408 29.649
101 70 −36.382 17.479 70.000 −36.382 17.479 0.257 0.401 29.698
102 70 −35.631 15.524 70.000 −35.631 15.524 0.254 0.389 29.588
103 70 −34.813 13.596 70.000 −34.813 13.596 0.251 0.379 29.639
104 70 −33.928 11.698 70.000 −33.928 11.698 0.249 0.372 29.937
105 70 −32.977 9.832 70.000 −32.977 9.832 0.247 0.361 29.638
106 70 −31.962 8.000 70.000 −31.962 8.000 0.245 0.353 29.795
107 70 −30.883 6.205 70.000 −30.883 6.205 0.243 0.346 30.062
108 70 −29.742 4.448 70.000 −29.742 4.448 0.241 0.336 29.855
109 70 −28.541 2.733 70.000 −28.541 2.733 0.240 0.330 30.120
110 70 −27.281 1.060 70.000 −27.281 1.060 0.239 0.322 30.023
111 70 −25.963 −0.567 70.000 −25.963 −0.567 0.237 0.314 30.341
112 70 −24.589 −2.148 70.000 −24.589 −2.148 0.236 0.306 30.461
113 70 −23.160 −3.680 70.000 −23.160 −3.680 0.235 0.298 30.388
114 70 −21.680 −5.160 70.000 −21.680 −5.160 0.234 0.293 30.643
115 70 −20.148 −6.589 70.000 −20.148 −6.589 0.235 0.288 30.611
116 70 −18.567 −7.963 70.000 −18.567 −7.963 0.234 0.281 30.589
117 70 −16.940 −9.281 70.000 −16.940 −9.281 0.234 0.278 30.932
118 70 −15.267 −10.541 70.000 −15.267 −10.541 0.234 0.271 30.985
119 70 −13.552 −11.742 70.000 −13.552 −11.742 0.235 0.267 30.890
120 70 −11.795 −12.883 70.000 −11.795 −12.883 0.236 0.262 30.955
121 70 −10.000 −13.962 70.000 −10.000 −13.962 0.236 0.258 31.013
122 70 −8.168 −14.977 70.000 −8.168 −14.977 0.237 0.254 31.128
123 70 −6.302 −15.928 70.000 −6.302 −15.928 0.239 0.250 31.112
124 70 −4.404 −16.813 70.000 −4.404 −16.813 0.240 0.248 31.150
125 70 −2.476 −17.631 70.000 −2.476 −17.631 0.242 0.244 31.318
126 70 −0.521 −18.382 70.000 −0.521 −18.382 0.243 0.242 31.338
127 70 1.459 −19.063 70.000 1.459 −19.063 0.245 0.239 31.575
128 70 3.462 −19.676 70.000 3.462 −19.676 0.247 0.235 31.295
129 70 5.485 −20.218 70.000 5.485 −20.218 0.249 0.233 31.540
130 70 7.525 −20.689 70.000 7.525 −20.689 0.252 0.230 31.533
131 70 9.581 −21.088 70.000 9.581 −21.089 0.254 0.229 31.686
132 70 11.650 −21.416 70.000 11.650 −21.416 0.258 0.228 31.798
133 70 13.728 −21.671 70.000 13.728 −21.671 0.261 0.226 31.699
134 70 15.815 −21.854 70.000 15.815 −21.854 0.264 0.226 31.814
135 70 17.906 −21.963 70.000 17.906 −21.963 0.268 0.225 32.178
136 70 20.000 −22.000 70.000 20.000 −22.000 0.270 0.223 32.179
137 70 22.094 −21.963 70.000 22.094 −21.963 0.274 0.223 32.531
138 70 24.185 −21.854 70.000 24.185 −21.854 0.278 0.223 32.236
139 70 26.272 −21.671 70.000 26.272 −21.671 0.282 0.222 32.348
140 70 28.350 −21.416 70.000 28.350 −21.416 0.287 0.222 32.621
141 70 30.419 −21.088 70.000 30.419 −21.089 0.292 0.223 32.669
142 70 32.475 −20.689 70.000 32.475 −20.689 0.295 0.221 32.596
143 70 34.515 −20.218 70.000 34.515 −20.218 0.301 0.223 32.921
144 70 36.538 −19.676 70.000 36.538 −19.676 0.305 0.223 32.854
145 70 38.541 −19.063 70.000 38.541 −19.063 0.311 0.224 32.944
146 70 40.521 −18.382 70.000 40.521 −18.382 0.315 0.224 33.003
147 70 42.476 −17.631 70.000 42.476 −17.631 0.321 0.225 33.193
148 70 44.404 −16.813 70.000 44.404 −16.813 0.327 0.227 33.344
149 70 46.302 −15.928 70.000 46.302 −15.928 0.333 0.228 33.543
150 70 48.168 −14.977 70.000 48.168 −14.977 0.339 0.230 33.561
151 70 50.000 −13.962 70.000 50.000 −13.962 0.345 0.232 33.509
152 70 51.795 −12.883 70.000 51.795 −12.883 0.352 0.234 33.769
153 70 53.552 −11.742 70.000 53.552 −11.742 0.355 0.234 33.458
154 70 55.267 −10.541 70.000 55.267 −10.541 0.360 0.236 32.979
155 70 56.940 −9.281 70.000 56.940 −9.281 0.363 0.236 32.415
156 70 58.567 −7.963 70.000 58.567 −7.963 0.366 0.237 31.910
157 70 60.148 −6.589 70.000 60.148 −6.589 0.370 0.237 31.386
158 70 61.680 −5.160 70.000 61.680 −5.160 0.374 0.239 30.824
159 70 63.160 −3.680 69.651 62.606 −4.234 0.378 0.240 30.336
160 70 64.589 −2.148 69.204 63.353 −3.409 0.384 0.241 29.642
161 70 65.963 −0.567 68.763 64.081 −2.525 0.389 0.243 29.187
162 70 67.281 1.060 68.327 64.789 −1.583 0.392 0.244 28.945
163 70 68.541 2.733 67.898 65.475 −0.583 0.399 0.248 28.406
164 70 69.742 4.448 67.477 66.136 0.476 0.405 0.250 27.815
165 70 70.883 6.205 67.065 66.770 1.594 0.411 0.252 27.362
166 70 71.962 8.000 66.664 67.376 2.771 0.417 0.254 26.965
167 70 72.977 9.832 66.275 67.949 4.006 0.423 0.257 26.599
168 70 73.928 11.698 65.900 68.490 5.299 0.431 0.260 26.149
169 70 74.813 13.596 65.538 68.994 6.651 0.437 0.264 25.817
170 70 75.631 15.524 65.192 69.459 8.061 0.443 0.266 25.472
171 70 76.382 17.479 64.862 69.883 9.528 0.451 0.271 25.190
172 70 77.063 19.459 64.550 70.265 11.052 0.458 0.273 24.838
173 70 77.676 21.462 64.257 70.600 12.633 0.465 0.277 24.622
174 70 78.218 23.485 63.984 70.888 14.270 0.473 0.281 24.300
175 70 78.689 25.525 63.731 71.127 15.961 0.481 0.285 24.069
176 70 79.088 27.581 63.500 71.314 17.706 0.487 0.289 23.889
177 70 79.416 29.650 63.291 71.448 19.504 0.496 0.293 23.756
178 70 79.671 31.728 63.106 71.527 21.353 0.502 0.297 23.534
179 70 79.854 33.815 62.944 71.550 23.251 0.510 0.302 23.316
180 70 79.963 35.906 62.806 71.516 25.196 0.518 0.307 23.217
Background 100 0 0 100 0 0 0.286 0.294 37.530
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