Data is an essential part of daily life, and many datasets have a degree of uncertainty either in known or unknown form. The uncertainties in data can originate from different sources, and it is important to analyze and measure the amount of uncertainty in the data. The most commonly considered aspect of uncertainty is measures of error, which can emerge from both data collection and processing of collected data [9]. However, a third source of error could be referred to as “use error,” which is associated with the application of data [2].

The first opportunity for error is at the data collection phase, which is also often referred to as source error. Source errors can include errors in the data itself or even in the identification of the data. Missteps on the part of the data collector, time pressure constraints, difficult ambient or environmental conditions, or inherent limitations in instruments used to collect the data are just some of the contributing factors to source errors. But error can also be the result of compromises and tradeoffs since the cost of highly precise data collection may exceed its value. This cost can therefore impact both the accuracy and the completeness of the obtained dataset.

Subsequent modifications to the data collected may include abstraction, scale changes, projections, dimensionality reductions, and analog-to-digital conversions [2] as well as many types of errors resulting from modeling including machine learning. We refer to these as process or modeling errors, and the potential for such errors is ever present especially when data is subjected to a chain of multiple manipulations, each of which can contribute to compounding errors into the data.

In a non-domain specific way, we might define error as the discrepancy between measurement and true value. However, it has been noted that a universally complete definition of data quality may be difficult to define as the particular application area can be a factor [9]. Nevertheless, a variety of common measures have been developed as a metric for reliability or confidence and used across many disciplines [3].

Uncertainty visualization is an ongoing area of research [39, 49] but a topic that many commercial practitioners avoid due to the additional complexity that it introduces. However, Greis et al. [18] explored game-like experimental tasks and compared representations that communicate different amounts of uncertainty information to the user, and results showed that participants did not favor representations with no uncertainty as they valued the additional information. Deitrick et al. [14] also studied whether uncertainty visualization influences, or results in, different decisions and found through a human-subject experiment that it can be a factor.

There are several traditional approaches to handling uncertainty, including error bars [3]; however, these can often be difficult to integrate directly into more general visualizations [20]. In addition, there are various studies conducted for uncertainty representations; for example, textual representation such as captions or tooltips [29], graphical representations such as glyphs [29, 39], custom color palettes such as value-suppressing uncertainty palette (VSUP) [12], bivariate choropleth maps [35], and texture patterns [4]. But to our knowledge, no uncertainty representation has made use of chromatic aberration (CA).

Chromatic aberration is a well-known phenomenon of color distortion or alteration that is sometimes seen around high contrast edges of objects in photographs and can also result from impaired vision. Since different colors of light refract to different angles upon traveling through materials with refractive indices [47], the resulting images may appear to be distorted [24]. Since CA is an image quality problem, most of the research concerning CA is conducted to fix the problem and improve image quality. On the other hand, uncertainty is the problem of data quality, and relevant research is most often conducted to reduce it to improve data certainty. Our goal is neither to improve image quality nor data quality; our aim is to utilize simulated and approximate chromatic aberration as a novel representation of uncertainty visualization.

To evaluate this novel approach, we first collected relevant data from well-known sources and generated uncertainty data from model predictions using machine learning models. Uncertainties were then calculated from the resultant forecasts [16]. We then visualized the uncertainty in the data using CA as well as a recent state-of-the-art competing method, VSUP. We then conducted a controlled human–computer interaction experiment to evaluate whether our new visual representation was more beneficial than the existing approach.

We chose to compare our method with VSUP [12] as it is a recent and novel visualization technique that emphasizes uncertainty over specific data values. By using color gradients, VSUP palettes help users focus on the range and impact of uncertainty rather than on precise numerical details. As with our method, VSUP is intended for situations in which understanding the variability and confidence in the data is essential for making well-informed decisions. We further chose VSUP as a comparator since they also conducted a user study which showed that it offers benefits over traditional bivariate mappings of uncertainty and value. This is due to the non-uniform budgeting of visual channels, which allows VSUP to make more efficient use of the limited visual encoding space. Our own study will test for improvements from our CA approach over VSUP itself.

From a vision perspective, chromatic aberration leads to various forms of color imperfections in the image. Koh et al. [30] presented a user study to observe the effect of CA on users’ judgment with lateral chromatic aberration for chart reading in information visualization on display devices and suggested some guidelines for designers to avoid such issues. Other work [5, 25] proposed image warping techniques to resolve the problem. Real cameras have an aperture through which light falls on an image plane to register the image, but diffraction is an issue in this process. Therefore, [33] presents a novel rendering system for defocus blur and lens effects by approximating optical aberrations. But our purpose is quite different from these prior works as we explore the use of an approximate and simulated CA as a means to represent uncertainty.

Uncertainty is an unavoidable part of data, and due to the complexity it introduces, practitioners often avoid it in their visualizations. The term uncertainty can refer to data quality, errors in data, or the accuracy of predictions. Given that errors are inherent in many types of data, improper or absent uncertainty representations can mislead decision-making for data analysts. Prediction generation is becoming increasingly important when using machine learning models such as neural networks (multilayer perceptron [MLP], LSTM, and GRU) [40] for performance evaluation, ARIMA or PROPHET [16, 46] for time series analysis, and XGBoost for epidemic predictions [34]. For example, a decision-support tool [38] for medical centers and health-care services has been proposed for influenza prediction and liver disease predictive analysis [47]. All these works have been conducted without specific concern for uncertainty visualization.

Botchen et al. [4] focus on the uncertainty that occurs during data acquisition and utilize texture-based techniques to visualize uncertainty in time-dependent 2D flow fields. In their system, the user can interactively manipulate aspects of the system such as particle density, error influence, or dye injection to affect the visualization of the uncertainty within the flow field. They cite several potential sources of uncertainty but focus on those resulting from data acquisition. Their solution is to use semi-Lagrangian texture advection to show flow direction by streaklines and convey uncertainty by blurring these streaklines. However, unlike our more abstract visualization approach, their solution is tailored to a specific problem, namely the representation of flow fields.

A common task in medical visualization is the partitioning of images or volumetric data into salient regions that correspond to a variety of structures, materials, or pathologies. Medical data collection often includes noise, which is a source of data uncertainty. Quite often, the segmentation task employs sophisticated computational models, which can also introduce a second layer of model uncertainty. Lundstrom et al. [36] propose probabilistic transfer functions in order to assign material probabilities to model cases. This produces a distribution of materials at every 3D location, for which animation is used where each material is shown for a duration that is proportional to its probability. There are interesting ideas introduced in this paper; however, this is also a specialized solution to a particular problem. In addition, we did not pursue an animated solution due to the well-known issues of limited short-term memory and change blindness, which can be a factor to consider for users [10].

Finger et al. [15] describe two studies in which blended icons were used to convey uncertainty regarding the identity of a radar contact as hostile or friendly. A classification study first showed that participants could sort, order, and rank icons from five sets intended to represent different levels of uncertainty. Three icon sets were selected for further study in an experiment in which participants had to identify the status of contacts as either hostile or friendly. Contacts and probabilistic estimates of their identities were depicted on a simulated radar screen in one of three ways: with degraded icons and probabilities, with non-degraded icons and probabilities, and with degraded icons only. Results showed that participants using displays with only degraded icons performed better on some measures than the other tested conditions.

Kay et al. [29] present a novel mobile interface design and visualization of uncertainty for transit predictions on mobile phones based on discrete outcomes. In a controlled experiment, they found that quantile dot plots reduce the variance of probabilistic estimates compared to density plots and facilitate more confident estimation by end users in the context of real-time transit prediction scenarios. Other researchers [31, 35] investigated how data uncertainty visualized in maps might influence the process and outcomes of spatial decision-making, especially when made under time pressure. According to researchers, the limitations of this research are that they did not consider the effect of stress along with time constraints and it was limited to a cartographic display.

Bubble treemaps [17] combine the representation of treemaps with the quantitative encoding of bubble charts, offering a visualization method for hierarchical datasets with an element of uncertainty. It maintains the hierarchical structure of data by representing nodes as bubbles within hierarchically nested areas as represented and bounded by contoured lines. The size of the bubbles represents the magnitude of a particular measure, while the wiggles of the contours can indicate uncertainty. This method is designed for exploring hierarchical structures and understanding how uncertainty varies across different levels of the hierarchy. They present some interesting use cases but, unlike for example VSUP [12], do not evaluate its effectiveness with a user study. And while interesting and novel, it is worth noting that these bubble treemaps are a custom type of visualization for a particular type of data, namely hierarchical data, whereas our aim is to introduce a new basic visualization element that may have applicability to a variety of visual designs and data types.

We can also consider uncertainty handling in widely used software packages such as ggdist [26], which is an R package that enhances data distribution visualization by extending ggplot2 with specialized tools. This also extends to tidybayes [21], which is an R package that integrates Bayesian statistical methods and provides a set of functions that make it easier to interpret Bayesian inferences. The package tidybayes builds on top of (and re-exports) several functions for visualizing uncertainty from its sister package ggdist [26]. One of these functions is to create curves to represent time series data, incorporating uncertainty bands to show confidence intervals. This type of plot aims to help users understand the variability and uncertainty for this type of data, providing a view of the range of possible values. However, as with our previous note regarding bubble treemaps, with the introduction of our approach, our initial aim is first to assess CA as a new basic visualization element that may have applicability to a variety of visual designs and data types rather than focusing on any particular type of data such as time series or hierarchical data. Once we have established CA’s potential as a basic visual element, further work may explore how it might be effectively incorporated into particular visualization designs, a topic that we will revisit in future work.

One approach of uncertainty visualization is to encode data values and uncertainty values independently, using two visual attributes in a bivariate map. But these resulting bivariate maps can be difficult to interpret, and the discriminability of marks can be reduced due to the interference between visual channels. To address this issue, Correll et al. [12] introduce VSUPs, and we highlight this approach as it is the comparator approach of our user study. The VSUP allocates smaller color ranges of the visual channel to data when uncertainty is high and larger ranges when uncertainty is low. This allocation of visual variables promotes patterns of decision-making that make use of uncertainty information, discouraging comparison of values in unreliable regions of the data and promoting comparison in regions of high certainty. In traditional approaches, the outputs for each combination of value and uncertainty might be represented as a 2D grid of colors whereas the VSUP approach produces a grid of pie-shaped arcs, mapping data to a smaller set of colors for higher levels of uncertainty.

However, the main limitation of VSUPs [12] results from their key design decision to filter out higher uncertainty values by grouping them altogether, which suppresses the values for decision-making when uncertainties are high. Due to this elimination of uncertainty information, designers need to carefully consider whether this representation is suitable for specific systems. Another limitation is that both uncertainty and value are represented by color, and the perceptual challenges associated with color channels are well-known. The limited ability of users to distinguish fine differences of hue means that users may struggle to match an array of hues that are simultaneously mapped to both value and uncertainty. This requires the concept of a limited “budget” of distinguishable colors. Given the limited budget of discernible hues, it necessitates that the data be quantized; due to this quantization, uncertainty visualization for continuous values is not possible.

We require data with uncertainty to proceed with our study. Although the novelty of this work does not lie in the chosen method of data prediction, for clarity, we now discuss the collected dataset, data manipulation and pre-processing, a brief description of the predictive model, and the generation of uncertainty from predicted data.

Chromatic aberration occurs when light of different wavelengths does not focus to the same convergent point because light with shorter wavelengths refracts more than light with longer wavelengths. Inspired by this phenomenon, we can consider a circle that represents the predicted number of new cases for a country on a specific day. But since there is an associated uncertainty with each prediction, a single circle will not be sufficient to represent the bivariate (number of cases and uncertainty) distribution. Instead of a single circle, we use three circles with separated RGB color channels, applying lateral shifts from the center of the circle by the amount of uncertainty, and blend them together in the center. Figure 3 shows this geometric arrangement on a unit radius circle.

To draw a circle representing uncertainty, we can draw three chromatic circles. We first set the center of the target circle at (x, y) and let the radial offset r represent the uncertainty. This can be implemented using many available visualization toolkits. For example, in the widely used D3 (d3js.org) library, this can be easily achieved with a commonly available blending mode (such as the CSS mix-blend-mode “darken”) to blend all three circles. Using the normalized radial displacements rc calculated in Section 3.2, we then draw the first chromatic circle C1 with color (R, 255, 255) at a shifted location of the second chromatic circle C2 with color (255, G, 255) at a shifted location of and the third chromatic circle C3 with color (255, 255, B) at a shifted location of Using this approach, the resultant aberration is presented with the uncertainty for a country in Figure 4. The center area’s color represents the predicted number of new cases, and the color separated edges represent the amount of uncertainty in that prediction. In this way, each of the items in the figure represents both the predicted value and the amount of uncertainties.

Uncertainty visualization presents a complex challenge that requires careful design of user studies. Various study types include experimental, descriptive, observational, and within-/between-subject studies. Our research involved a within-subject comparative study evaluating uncertainty visualizations using metrics such as task time, error rate, and subjective assessments (NASA-TLX, System Usability Scale [SUS]). We aim to understand how well chromatic aberration performs compared to the state-of-the-art VSUP visualization in terms of accuracy, efficiency (user response times), and user preference. This helps in assessing how well the visualization supports the intended tasks and identifying any additional needs or requirements.

We recorded several sources of data from the user study, which include (i) quantitative questionnaire results; (ii) time utilization data for each component; (iii) SUS data for CA and VSUP; and (iv) NASA-TLX for CA and VSUP. We analyzed these data and present the results in the following sections.

There are several limitations of this work that we wish to highlight. Although we did not have a prerequisite for participants to be university students, based on the responses we received, most were from universities (undergraduate and graduate students) with only a subset of the population being working professionals. Based on this, one must be cautious not to generalize to significantly younger or older demographics when considering issues such as fatigue. We believe the prior VSUP study [12] also had participants with similar backgrounds.

Related to the sample demographics, in order to maintain coherence with the prior work of Correll et al. [12], and more generally, based on standard practice in such experiments, users with color blindness did not participate in the experiment. However, with regard to color vision impairment, a point of note is that although we did not design or test the visualization for color blind support, when compared with an alternative such as VSUP, which is purely based on very subtle differences of color, our approach at least offers a spatial encoding of the uncertainty information, namely when the R, G, and B elements separate. The limited discernible color palette of someone with limited color vision might be solely applied to the value, leaving uncertainty to be encoded with spatial separation. We speculate that because we are not trying to encode both uncertainty and the data values into color, our method may be more adaptable for color impaired applications than VSUP, but this would need to be tested with further studies.

However, if one considers other visual impairments such as myopia, presbyopia, or astigmatism, which affect the sharpness of the image, then other issues may come into play such as fatigue, eye strain, or general effectiveness. This also warrants further study. A general countermeasure worth mentioning is providing some choice for users with visual impairments, whether that be VSUP, CA, or CA with a limited color palette for value.

In addition, in the CA representation, one needs to be careful so that chromatic objects and adjacent objects do not overlap. An additional level of care must be applied in the case of implementing zooming based on the zoom scale of the visualization in order to keep them consistent. This relates to the potential use of CA in geographic information system (GIS) applications, which we will return to in the following section. Another potential limitation is how CA may interact with other glyph shapes.

Lastly, although our approximation of CA is computationally inexpensive and very accessible for use in widely used web-based visualizations, if one were to implement a more complex CA rendering method, then further study with participants may be required. In particular, we note that in real-world chromatic aberration, chromatic blurring appears continuously from the inner edge to the outer edge.

In this paper, we proposed a novel approach for uncertainty visualization, namely chromatic aberration. We conducted a within-subject comparative user study between the state-of-the-art alternative, VSUP, and our system to assess user performance accuracy/error rate, task completion time, and subjective assessment (with NASA-TLX and SUS). From the numerical analysis of the results, we see that the user performance of CA is both statistically more accurate (10%) and faster (11%) when compared to VSUP, whereas in the subjective assessment, the two methods do not vary significantly. Future studies could look at further task-specific uses of CA, which might shed more light on the lack of statistical differences with regard to subjective preference.

We noted that in real chromatic aberration, blurring appears continuously from the inner edge to the outer edge. However, our simplified implementation allows us to reduce the aberration to both double and/or single parameter(s), which facilitates the representation of uncertainty. It also allows one to implement the approach relatively easily using standard and widely available graphical operations. However, additional research could be conducted that examines more sophisticated chromatic aberration effects. In addition, further research could be carried out with more levels of uncertainties than were tested by Correll et al. [12].

Related to this, one might look more closely at the mapping of uncertainty to the scale of CA displacements. Additional research would be required to determine the optimal or minimal displacement required to convey uncertainty. More broadly speaking, as this is the first attempt at using chromatic aberration for this novel purpose, there likely remains significant scope to optimize and adapt its use with further refinements and ideas.

Other possibilities include studying the use of CA in animated and interactive visualizations, using CA for contextual highlighting, and testing the effectiveness of CA for use with more complex glyphs, as well as the potential intersection between CA and known pre-attentive visual effects. Another aspect that could be explored is using CA with alternative data types such as categorical, hierarchical, and time series data, which might then relate more specifically to other existing work such as bubble treemaps [17], ggdist [26], and tidybayes [21].

It would also be interesting to explore the limits of CA in applications requiring choropleth maps and other GIS-related visualizations. Despite the spatial requirement that the CA separation requires, a choropleth map might be possible using a buffer between each element to allow for some degree of chromatic aberration at the boundaries. Alternatively, the R, G, and B chromatic displacements for each element in the map might instead maintain the outer boundary shape as a maximum area. Then each R, G, and B displacement would need to shrink somewhat and move away from the center internally, within that original boundary of each map element. This may offer a stronger overall Gestalt of the uncertainty in the map, but the tradeoff might be a tighter limit on how small individual elements could become.

Beyond GIS maps, one might speculate on the breadth of applicability to other types of data, such as temporal data, and other forms of visualization. With some graphs that are defined by lines and curves, the CA would may need to be adapted and applied locally along the line or curve, rather than uniformly displacing the entire curve into three renderings for R, G and B. Other graphs are primarily defined with their silhouettes, such as violin charts, density charts, and ridgeline charts. And as with curve and line-based charts, for these charts that provide most of their information along silhouettes, the CA would likely need to be applied locally along those boundary curves. Some other types of graphs, such as stream graphs, may often contain large flat areas of color and these might benefit from the overlay of an artificial texture prior to the local application of simulated CA. This might allow the CA to differentiate that texture locally based on uncertainty. But given that our method has purposely been designed for implementation in widely used online visualization libraries, such as D3, it is our hope that other practitioners begin to experiment with possibilities that we are as yet unable to anticipate.