To illustrate and validate our visualization tool, we have chosen an eye-tracking data with 40 participants each looking at 48 stimuli of metro maps for study [ 4 ]. This data set is a typical representation of eye-tracking data consisting of fixations and saccades along with their coordinates, and time stamps after a fixation filter was applied on the raw gaze data. It consists of a sequence of fixations (scanpath) for each participant P_i and each stimulus (metro map) S_j. Based on this data, we derived eye-tracking metrics. We have used all stimuli as well as all participants data and considered participants as a variable, since we are interested in grouping participants with common behavior on the basis of their eye-tracking data. However, if an analyst is interested in grouping stimuli based on the performance of participants, then the stimuli are considered as a variable.

We have used metrics from a large range of measures discussed in the book of Holmqvist et al. [ 14 ] and selected typical metrics from three categories of measures, i.e., eye-movement measures based on saccades, numerosity measures by counting events, and statistical (position) measures of the spatial distribution as shown in Table 1. While metrics from the first two categories are often used for evaluation, simple statistical measures could be helpful to get an aggregated view of the positions of fixations. Skewness indicates whether fixations are located on the left or right (top or bottom) side of the stimulus. Standard deviation indicates the general spread of the fixations. Finally, kurtosis tells us whether there is a tendency for multiple clusters of fixations or rather a singular occurrence.

For ease of representation, we call the metrics M_k, where 1 ≤ k ≤ n. For each metric, we calculate the similarity values sv_Mk for each pair of participants P_i,l. There are |P| = p participants in total. Hence all similarity matrices will be of size p ✕ p, with p!/2!(p-2)! = (^p₂) different similarity values, one for each pair of participants, and a total of (^p₂)n for all n metrics. We calculate similarity values sv_i,l,Mk between two participants P_i and P_l based on metric M_k using the Euclidean distance (Equation 1) to find similarities between the reading behavior of participants based on their eye-tracking data.

In our example, sv_i,l,Mk are scalar values, which resulted into n similarity matrices of size p ✕ p. To make it worth for visual inspection, we combined all the similarity values of n metrics into a single matrix as stacked rectangular sub-grid as proposed by Kumar et al. [ 19 ]. Details will be discussed in the section on Similarity Matrix Visualization.

To demonstrate the usefulness and application of our visualization framework for exploration of eye-tracking data, we have computed 16 metrics according to Table 1.

For the visual exploration of eye-tracking data, we use clustering and two different visualization techniques in combination. Due to the large number of metrics, we chose to use parallel coordinates, which are popular for the visualization of high-dimensional data. They are also employed to visually cluster and explore data, which gives an analyst an idea of the metrics that could be used for a grouping of participants. Clustering is then applied to the multi-dimensional stacked matrix as discussed in the section on Similarity Matrix Visualization.

Clustering. Clustering groups of objects or data points based on similarity values helps find structures within data or relations between objects. We have used agglomerative hierarchical clustering of participants based on the metrics used in this study. It is one of the main stream clustering methods with a complexity of 0(N² log N). The popularity of this clustering is due to the fact that it does not need any predefined parameters, which makes it easier to be used for all sorts of realworld data [ 20 ].

The main challenge faced during clustering is choosing the right combination of similarity values to be used for clustering. Even after finding correlation between metrics, it is difficult to identify eye-tracking data metrics that will be beneficial for grouping. Performing clustering for all the combination of metrics for grouping participants could be very tedious.

To solve this problem of finding suitable metrics for clustering, we use parallel coordinates to drill down into the metrics and their properties by various interaction techniques described in the following sections, as parallel coordinate plots can be used for visual clustering, i.e, to find groups based on visual features. We have tried various combinations of metrics through one of the interactive technique of our tool providing varying weights to each of them. It gives us an approximate idea about which metrics we can use in combination for calculating the similarity value matrix. This similarity matrix will then be used to group participants by clustering.

Parallel Coordinates. Understanding complex data has always been a challenging problem, and there are various visualization techniques to understand the data. There are several standard visualization techniques to visualize data such as histograms (1-dimensional) or scatterplots (2-dimensional). The problem becomes even pronounced when we have multi-dimensional data [ 21, 22 ]. In order to visualize data with such techniques, we need to produce several plots, which is not feasible with an increasing number of dimensions.

To show a lot of data dimensions, we chose parallel coordinates for visualizing eye-tracking metrics for the second part of our visual analytics workflow. Parallel coordinates made their first-time appearance in 1885 [ 23 ]. However, they became more popular as a tool for multi-dimensional data exploration after the work of Inselberg [ 15 ] and Wegman [ 24 ]. Parallel coordinates are based on the concept of point-line duality, where data points in Cartesian space are plotted in the form of lines in parallel coordinates. In parallel coordinates, two or more axes are plotted parallel to each other, where each axis serves as one dimension of the multi-dimensional data. Data points are then plotted on the parallel axes, which results into intersection of polylines at respective data values, as shown in Figure 2(a). Multiple data points can be plotted similarly, as shown in Figure 2(b).

Parallel coordinates plots serve the purpose of visual exploration, as finding the relation among the eyetracking metrics in our case. Of all the three possible correlations (positive, negative, and none), negative correlation has the most striking pattern. The intersection of data points of higher values on one axis are mapped to lower values on the neighboring axis and vice versa, resulting into an accumulation point that is easily spotted. A sample case for perfect negative correlation is shown in Figure 3(a). Positive correlation shows properties opposite of negative correlation, thus resulting in straight lines, as depicted in Figure 3(b).

There are several interaction techniques that support visual exploration. Our parallel coordinates framework, as shown in Figure 4, provides a collection of such interaction techniques. Brushing, being one of the basic action techniques for parallel coordinates plots, allows the user to select data points on the axis. The selection is typically done in an interactive fashion; often, the user marks interesting areas in the plot by mouse interaction as shown in annotation (a) of Figure 4. There are generally three types of brushes, such as axis-aligned brush, polygonshaped brush, and angular brush [ 25 ]. We use axisaligned brushing, which enables us to select points in a vertical fashion on axes at a time.

Selected data points are then used for further exploration. We can color each axis individually based on the data points plotted on each axis, as for an example case SkewY is color-coded in annotation (g) of Figure 4. The user can apply coloring according to the metric of their choice by simply clicking on that axis. To find correlations between metrics, we need to order the axes representing metrics in a meaningful manner. We can reorder an axis by simple dragging and dropping it at the desired position. Sometimes correlation can also be seen by inverting the orientation of an axis, which is supported by double clicking on an axis.

There are several problems with the polylines used in the traditional parallel coordinates plots. They result in loss of visual continuation if two or more data points have similar values, which makes it difficult to follow the line throughout the plot. To solve this problem, we optionally replace the polylines with smooth curves controlled by a slider (see annotation (c) of Figure 4) as done by Graham et al. [ 26 ]. Smoothing the polylines is beneficial for bundling as it allows analyst to take advantage of the entire plot area because it separates distinct components of data into distinct regions by assigning a centroid as shown in Figure 5(b). Curve bundling is also highlighted in our framework, which supports the formation of curve bundles for clustered data plotted on parallel coordinates, as shown in Figure 5. The slider used in the framework (see annotation (d) of Figure 4), helps the user tune the tightness of the bundled curve, which enables them to have a different view of the same data [ 27 ].

There could be a case where two or more metrics used for analysis are of similar nature. Our framework allows the analyst to remove axes that are not of interest by selecting check-boxes as shown in annotation (b) of Figure 4. There could be a scenario where the analyst wants to merge two or more axes, assigning desirable weight to each one of them for the metric analysis in order to group the data based on the new combined metric. To support this feature, we have a section in our framework (see. annotations (e) in Figure 4) that enables user to select the metrics they want to combine. The weight can be adjusted with a slider below the checkbox.

Similarity Matrix Visualization. The third part of our visual analytics process deals with the participant group analysis and uses a matrix-based approach for visualization. It enables users to distinguish groups of participants based on various metrics of eye-tracking data. A matrixbased representation is one of the most basic ways of visualizing two-dimensional data. However, it is difficult to visualize multi-dimensional data using a matrix. Therefore, we adopt the concept of dimensional stacking: embedding dimensions within other dimensions [ 21 ]. Each matrix cell is divided into multiple sub- grids depending on the number of metrics to be stacked as used by Kumar et al. [ 19 ]. In this paper, each grid is divided into 16 subgrids since 16 metrics are supposed to be embedded into single grid.

Each sub-grid is used to stack the similarity values sv_i,l,Mk calculated in Equation 1 for each metric. However, it is not easy to arrange the data with n metrics into a ⌈√n ⌉ ✕ ⌈√n ⌉ grid and to make it visually appealing, so that it becomes easier to visually group participants on the generated clustered matrix representation. We solve this issue by computing correlation coefficients between all pairs of n metrics, which results into n!/2!(n-2)! = (ⁿ₂) values. We then clustered all (ⁿ₂) combinations using hierarchical clustering and plot them in the form of a dendrogram as shown in Figure 6(a). This clustering is now, used to arrange similarity metrics next to each other. However, the problem of ordering these metrics in a sub-grids is still challenging. So, we followed the Hilbert space-filling curve to fill the sub-grids, as shown in Figure 6(b). This space-filling representation keeps nearby values close to each other, which helps preserves locality [ 28 ]. For our study, we chose 16 different metrics preprocessed from the raw gaze data in the first phase. Therefore, we chose 16 different colors in the CIELAB color space that represent all 16 similarity values sv_i,l,Mk in each sub-grid as base colors, as shown in Figure 6(c). The selected 16 colors are saturated colors from the CIELAB color circle with uniformly distributed hue values, with a hue difference of 20 between neighboring hues. We chose hue to encode metric classes because hue is effective in perceptual grouping of categorical data [ 29 ]. The CIELAB color space is well suited since colorimetric distances between any two colors in this space corresponds to perceived color differences due to its threedimensional in nature where each a∗, b∗ and L∗ respectively serves as an individual dimension. While a∗ and b∗ being chromatic channels of the CIELAB color space is fixed according to the class of metric, lightness L∗ varies to encode the similarity values for the respective metric. We assigned these colors to metrics in sub-grids in the order of the Hilbert space-filling curve, which resulted in Figure 6(d). An example of eight participants with 16 metrics is shown in Figure 6(e).

The first step in most evaluations is to generate an overview of the data that should be analyzed. Here, the data corresponds to metrics derived from recorded eyegaze data, i.e., we first focus on the Metric Analysis part of the visual analytic workflow (see Figure 1). This can be shown in Figure 7, as an example of metrics data visualization using parallel coordinates. By rearranging the axes (metrics), it is possible to swiftly find those, that exhibit similar or identical behavior indicated through parallel line segments between neighboring axes. This is the case for FixRate/ SacRat, AvgSacDur/ AvgSac, and also FixNum/ SacNum (annotations (b) in Figure 7), enabling an analyst to discard metrics that convey the same kind of information. Furthermore, we are able to find metrics that exhibit an inverse behavior, which is the case for, e.g., SacRat/ AvgFix, FixRate/ AvgFix, AvgFix/ AvgSacDur, and AvgSac/ FixNum. An increase of the first attribute of each tuple leads to a decrease of the second attribute. These findings reflect the well-known relationships between the previously described metrics. Another way to find similarities between attributes is possible by selecting a specific axis that will cluster the data based on the selected metric and a color will be assigned to each data instance. Considering the color of the lines as well as the order of the color at other axes, we can see similar behavior. This is shown in Figure 7, where clustering was performed based on AvgFix (see annotation (a)), resulting in the same color gradient on the axes FixRate and SacRate.

Using clustering based on a single metric, we can find an inverse relationship between Ҡ Coef and FixRate (see Figure 8(a)). This indicates that the ambient/focal attention changes inversely to the fixation rate. Furthermore, ambient/focal attention varies with the scanpath length. As seen in Figure 8 for a length of more than about 15,000 px, the distribution of the Ҡ coefficients is centered around zero (ambiguous cases), whereas for shorter scanpaths the distribution is more widespread and tends toward negative values (ambient attention). The Ҡ coefficient is also related to the number of fixations in a similar fashion. Here, we can observe the same kind of distributions for more than 100 fixations (ambiguous cases) and fewer fixations (ambient attention). If we consider the scanpath length, we can see that the separation into two groups having more and less than 100 fixations yields roughly the same separation as splitting the subjects according to a scanpath length of more or less than 15,000 px. This is shown in Figure 9.

Another interesting question is how the Ҡ coefficient is related to the completion time and scanpath length, since there could be, e.g., four subgroups of participants using a categorization of fast/slow for the completion time and short/long for the scanpath length. Previously, we had already a closer look at the scanpath length and the Ҡ coefficient. Therefore, we now want to focus more on the influence of the completion time, assigning a weight of 0.7, while we assign a weight of 0.3 to the scanpath length. The results are shown in Figure 10. Based on this affine combination of metrics, we achieve a similar result compared to utilizing the scanpath length alone. One group of participants exhibits ambiguous eye movements, whereas the other groups show more ambient eye movements.

Besides the so-far-used eye-tracking metrics that characterize user behavior, we are also interested in identifying spatial behavior. Therefore, we have a closer look at metrics that give us an impression about where the participants were looking at. The skewness of the distribution of positions, projected onto the x- and y-axis, can be used for getting a rough impression. Figure 11 shows the combination of SkewX and SkewY with an equal weight of 0.5 for each metric. Overall, we can see a negative correlation between the metrics, which is also the case, if we have a closer look at different intervals of values of the combined metric (see Figure 11(b)).

To further investigate the spatial distribution, we can include kurtosis into the analysis. Based on the skewness, we have derived rough areas where participants were looking at. The kurtosis of the distributions describes now whether there are frequent small deviations of mean values or frequent larger dispersed data. The first case would indicate the existence of focus attention areas, whereas the later a wider spread of attention on possibly multiple locations. Since we are more interested in the effects of kurtosis, this metrics got a higher weight (0.7). Skewness got lower weights (0.3). Based on the new combined metric, we could roughly separate participants into four groups. Each group shows distinct visual patterns. The first one (Figure 12(a)) contains only a minority of the data that shows large changes between neighboring axes. The second group (Figure 12(b)) exhibits a drop-off of changes from left to right. The third (Figure 12(c)) and fourth (Figure 12(d)) group show similar values for the kurtosis in both spatial directions and also for the skewness in x-direction, but differ visually in the skewness in y-direction. With this kind of analysis, using parallel coordinates, we were able to investigate characteristics of each scanpath of each participant during the Metric Analysis step of our workflow (see Figure 1) and to form clusters based on color encoding or similar visible line structures.

During the following Participant Group Analysis, our multi-dimensional stacking approach allows us to compare all participants to each other in much detail. Standard approaches generate therefore a similarity matrix where the columns and rows represent the participants, which are order alphanumerically. Each cell of the matrix contains a similarity value. In our case, a matrix cell contains several color-encoded elements, which correspond to the similarity values of the used metrics. An example of this is shown in Figure 13. Here, we can see some isolated group structures, but in general the visualization looks rather chaotic. By performing a reordering of the participants based on the visual inspection in parallel coordinate plots, we are able to introduce, to some degree, order into the chaos: getting similar participants spatially closer reveals more group structures than before. Figure 14(a) depicts the similarity matrix with reordered participants, whereas Figure 14(b) highlights a possible grouping. The new order of participants is the result of hierarchical clustering based on a combined metric (skewness in x- and y- direction with a weight of 0.3; kurtosis in x- and y- direction with a weight of 0.7). By using the new ordering, we identified six different groups exhibiting same visually characteristics.

Inspecting the different groups in more detail using scanpaths as part of a qualitative evaluation, we can see that, e.g., group (a) and (b) differ in defining characteristics of scanpaths that were investigated by the participants. Selecting scanpaths of participants from the horizontal axis that are within group (b) (see Figure 15 (a)) shows that there is a designated path that was investigated, while scanpaths of participants in group (a) (see Figure 15 (b)) are more diverse. Comparing these scanpaths with those of participants selected from the vertical axis that are part of both groups (see Figure 15 (c) and (b)), we can see that the scanpaths share similar characteristics to the scanpaths of participants chosen from the horizontal axis, thus the scanpaths are separated into two groups exhibiting two defining characteristics: a more focused versus a more diverse or widespread investigation.

The author(s) declare(s) that there is no conflict of interest regarding the publication of this paper.

This research was partially supported by NSF grant IIS 1527200 and the MSIP (Ministry of Science, ICT and Future Planning), Korea, under the "ITCCP Program" (NIPA-2013- H0203-13-1001), directed by NIPA. We also thank the German Research Foundation (DFG) for financial support within project B01 of SFB/Transregio 161. Metro maps in Figure 15 are designed by Robin Woods ( www.robinworldwide.com ) and licensed from Communicarta Ltd ( www.communicarta.com ).