1.2. Results and discussion
The average trial duration was 2,319 msec, and the mean duration of fixations was 207 msec. In 32.5% of all trials, at least one of the last two fixations was located inside the target rectangle. Fig. 1 shows a sample scanpath generated by one of the participants. Fixations from all trials were included in the analysis of saccadic selectivity if they started after the onset of the search display, ended before the manual response or timeout, and were not located on a rectangle of the target color. For each target color T, we computed the average distribution of saccadic endpoints across the 64 display colors, using a scale from 0 to 100. A display color D would receive a value of 0 if no saccadic endpoints landed on a rectangle of color D, given target color T; and 100 would mean that for target T all saccadic endpoints from all participants landed on color D. Consequently, the selectivity data formed a 64 × 64 matrix with zeros on its diagonal because we excluded fixations on the target color. This saccadic selectivity matrix was roughly symmetrical, which is in line with the premise that saccadic selectivity for a display color D increases with greater similarity—a symmetrical concept—between D and the target color T. Therefore, we computed the arithmetic mean of all symmetrical pairs in the matrix, which resulted in 2,016 values indicating the mutual saccadic selectivity between any two colors from the chosen set.
To get a rough sketch of the color space of saccadic selectivity (i.e., a representation in which proximity of two colors indicates their mutual saccadic selectivity), we employed the technique of multidimensional scaling (e.g., Cox & Cox, 2001). This technique maps a similarity or distance matrix for a set of objects onto a multidimensional (usually 2-dimensional or 3-dimensional) abstract space in which the objects are placed in such a way that more similar ones are separated by a smaller Euclidean distance. Fig. 2 shows a stereoimage pair visualizing the result of 3-dimensional multidimensional scaling of the present selectivity data (PROXSCAL algorithm, spline transformation degree 3, one interior knot, simplex start configuration, 100 iterations, resulting normalized raw stress 0.11). By crossing one's visual axes to fuse the two images into one, it can be seen that the 64 colors are roughly placed on the surface of a sphere. Notice that this spherical shape is not a consequence of the specific algorithm used, but solely reflects the pattern of mutual saccadic selectivity between colors.
Figure 2. Stereo image pair for convergent viewing of the abstract three-dimensional color space. Note: To perceive the depth information, please cross your eyes to fuse the two panels into one. It should become visible that the color markers roughly form a hollow sphere. In this three-dimensional arrangement, greater proximity of two markers represents greater mutual saccadic selectivity between their colors.
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In Fig. 2, hues fall about the approximately vertical axis much as they do in the HSI color space. It also seems that along this axis, from the bottom to the top of the sphere, the intensity of colors tends to increase. Saturation, however, does not seem to play an important part, as there are no clusters of high- or low-saturation colors. Probably the most conspicuous exception from this rough picture is the color white, which is not located at the top of the sphere but close to the group of light blue colors. It is likely that the high color temperature of the monitor (9,300 K) contributed to this result. Regarding the hypothesis of color categorization (Yokoi & Uchikawa, 2005), Fig. 2 suggests a continuum of colors rather than strict categories, which would be indicated by tight clusters around basic colors (Berlin & Kay, 1969). Although the spatial distribution of colors on the sphere is not exactly homogeneous, there are transitions such as from the purple colors toward bluish ones (leftward) or reddish ones (rightward). Although categorization, possibly through verbal memorization of colors, cannot be ruled out, it does not seem to be the predominant factor determining saccadic selectivity.
What is the most precise and useful mathematical description of this color space? To have a baseline for evaluating a variety of mathematical models, we first devised an overly simple model (Constant model), which assumes that color has no effect on saccadic selectivity at all. In other words, this model maintains that all display colors receive the same amount of saccadic endpoints, regardless of the target color. For the actual modeling, our aim was to find functions with concise mathematical descriptions that estimate the mutual saccadic selectivity of two given colors as accurately as possible. As first approaches, we modeled the mutual saccadic selectivity m between two colors c1 and c2 as linearly decreasing with the colors' weighted Euclidean distance in the four standard color spaces RGB, HSI, CIE XYZ, and CIE Lab (see Table 1, rows 2 to 5). For each color space, the free model parameters were numerically determined to minimize the mean square error (MSE) between the computed and the empirical saccadic selectivity across all 2,016 color pairings. In particular, for the HSI space, we evaluated three different ways of computing the hue variable for a given color: its polar angle in CIE XYZ relative to the point (1/3, 1/3), its polar angle in CIE Lab relative to the point (0, 0), and its standard definition (derived from the RGB space). For these and all following HSI-based models, the CIE XYZ version of hue computation was found to be the most accurate. Thus, throughout the remainder of this text, this version is used.
Table 1. Equations with fitted parameters and resulting mean square errors for some of the evaluated models. Notice that some of these variables, such as the hue variable in the HSI space, correspond to angles. These angles are given in radians, and differences between them are measured in such a way that they never exceed π. The values of R, G, and B range from 0 to 1
Although the HSI model produced the smallest MSE, it did not differ significantly from the other three linear approaches: all ts(2,015) < 2.52, ps > .10. All of these linear models, however, outperformed the Constant model: all ts(2,015) > 8.46, ps < .001. Notice that, due to the low number of data points per cell, only the cumulative saccadic selectivity for all participants was analyzed, and the standard error was computed across the 2,016 color pairings. All results from t tests were Bonferroni adjusted.
To devise models with better fit to the empirical data, we tested a large number of linear, logarithmic, polynomial, and exponential functions and their combinations in all of the four color spaces. For all of these functions, computations in the HSI space either outperformed those in the other three color spaces or were statistically identical to them. Moreover, it was found that Gaussian models, which simply apply a Gaussian function to the weighted Euclidean distance, provided better fits than all other approaches. These models are of the form shown in Table 1 for the HSI Gauss model.
Accordingly, the HSI Gauss model achieved the best fit in this competition. Fig. 3 shows the significant improvement of the HSI Gauss model over the linear HSI model: t (2,015) = 5.89, p < .001. Because the difference between two colors ranges from 0 to π in their hue, but only from 0 to 1 in their saturation and intensity, the fitted parameters shown in Table 1 suggest that hue is dominant in guiding attention, followed by intensity, whereas saturation is much less important. This finding is in line with the results of the multidimensional scaling shown in Fig. 2.
Figure 3. Mean square error produced by the different saccadic selectivity models for the data obtained in Experiments 1 and 2. Note: The error bars indicate standard error.
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The aforementioned evidence for the small impact of the saturation dimension on saccadic selectivity raises the question whether saturation can be completely disregarded without losing significant predictive accuracy. To answer this question, we implemented and evaluated the HI Gauss model as defined in Table 1. Fig. 3 shows that the HI Gauss model is only slightly, but significantly, less accurate than the HSI Gauss model that accounts for all three dimensions, t (2,015) = 3.62, p < .01; and outperforms the linear HSI model, t (2,015) = 5.54, p < .001. Finally, motivated by the results of the multidimensional scaling, we implemented the Sphere model, which computes saccadic selectivity based on the Euclidean distance between colors arranged on a spherical surface according to their hue and intensity. The most accurate approach for this model is given in the bottom row of Table 1, where the variable d—set to 1.68 in the fitted model—determines whether brightness ranges from pole to pole (d = 2) or is restricted to a maximum distance of d/2 from the equator (0 ≤ d < 2). As shown in Fig. 3, the MSE of the Sphere model was slightly greater than the one for the HSI Gauss model, t (2,015) = 3.24, p < .05; and minimally, but not significantly, smaller in comparison to the HI Gauss model, t (2,015) = 1.71, p > .50.
In summary, the HSI Gauss model yields the best fit among the three-dimensional models, whereas the Sphere and HI Gauss models are the best fitting two-dimensional ones. To test whether this result generalizes for different color sets than the one used in Experiment 1, we conducted Experiment 2, which employed the same experimental paradigm as Experiment 1 but a different set of colors.