Visualizing fuzzy sets using opacity-varying freeform diagrams

Abstract

As an extension of classical sets, fuzzy sets allow their containing elements to have continuous membership, which broadens its applications in various domains. In order to visualize the uncertain owner–member relationship, we design an intuitive diagram to visualize fuzzy sets. The diagram has a freeform boundary, which allows a clear layout of the data. In addition, the opacity of the diagram reveals the uncertainty of the membership. Physical simulation and bi-harmonic interpolation are introduced to generate the proposed diagram. We test our method with different fuzzy sets. The results show that our method is efficient to generate intuitive visualization of fuzzy sets.

Keywords

Fuzzy set set visualization uncertainty visualization mass–spring system bi-harmonic equation

Introduction

Different with classic set theory, in which one element is either inside or outside a set, fuzzy set theory generalizes the membership relation and allows a continuous value from zero to one to characterize to which degree an element belongs to a set.¹ By expanding the binary logic in membership, fuzzy set theory well matches lots of real-world problems and finds its application in various domains such as engineering, management,² and medical sciences.³ For example, by measuring electrical activities in the brain, one can estimate to which degree a subject is in a happy emotion. In other words, samples of electroencephalograms (EEG) form a fuzzy set of happy emotion, in which the membership function characterizes the likelihood that a happy emotion is recognized from the measured EEG. In a general election, we can also model the voters who support one specific elector to be in a fuzzy set, where the membership function indicates the confidence of the voters.

In order to make information from fuzzy sets displayed in an intuitive way, researchers invent various visualization techniques to organize and present data in informative ways. Comparing with the numbers presented in texts or listed in tables, visual languages presented in a picture have higher information density and offer many opportunities for users to understand sets and discover knowledge from data.⁴

Although various techniques are developed for visualizing classic sets, more work on visualizing fuzzy sets is needed.⁴ The uncertainty added to the membership brings more challenges to the visualization task for sets.⁵ In order to interactively visualize general fuzzy sets and their corresponding set operations, Park and Park⁶ invented disk diagram In the proposed design, a fuzzy set is illustrated as a disk and its elements are drawn as dots inside the disk, with the uncertainty in the membership represented by the distance from the element to the disk center. One problem with this design is that the element dots with high membership functions will gather around the disk center and it is too dense to see distinct elements (see Figure 1 (left)). Besides, the diagram with a clear boundary may confuse the users who are used to checking Euler diagrams for visualizing classic sets.

Figure 1.

Visualizing a fuzzy set using its disk diagram (left) and opacity-varying freeform diagram (right). Note that the opacity-varying freeform diagram leaves more space to display the caption of each element.

Inspired by the disk diagrams and other related works, we design an intuitive diagram to visualize fuzzy sets in this work. We improve the disk diagrams in two folds: first, by introducing opacity to the diagrams, the fuzziness of the membership is visually enhanced. Second, by relaxing the shape of the set from a disk to a freeform shape, we optimize the layout for a clear view of distinct elements. We also design a computational framework, which combines physical simulation and geometric interpolation to efficiently generate such diagrams. After testing our method with various fuzzy sets, we show applications of our system and report the user feedbacks. We conclude our work in the final section.

Related work

Visualization of classic sets has been studied for decades. Euler Diagrams⁷ and Venn Diagrams⁸ are popular ways to visualize classic sets and they have been generally used in various fields. Recently, researchers have generalized the shapes of the diagrams from primitives to optimized freeform shapes^9,10 to better display the elements inside the sets.¹¹ In this work, we also take the fuzziness of the relationship into consideration and allow a freeform boundary for a clear visualization. There are other methods for classic set visualization and we refer to Alsallakh et al.⁴ for a comprehensive survey.

Pham and Brown⁵ suggested various requirements and design for visualizing fuzzy systems. In early work, fuzzy sets are visualized as high-dimensional data with uncertainty using, for example, parallel coordinates¹² or multidimensional scaling.¹³ These methods are difficult to recognize the individual membership degree. Cox et al.¹⁴ designed a system to visualize the fuzzy sets formed in clustering. However, they did not discuss about visualizing general fuzzy sets in their work. Later, Park and Park⁶ introduced disk diagrams to visualize general fuzzy sets and Das¹⁵ extended this technique to support variants of fuzzy sets. To the best of our knowledge, it is the latest work which supports visualization of general fuzzy sets and is successfully applied in different fields. In this work, we extend the disk diagrams by allowing the boundary to be freeform and blurred, which directly display the fuzziness. By clustering the elements based on their membership functions, stem-and-leaf plot¹⁶ is another solution to visualize fuzzy sets. We will also use stem-and-leaf plot as a baseline in the evaluation of our method.

Since fuzzy sets characterize the uncertainty of the membership, it also relates to uncertainty visualization.¹⁷ MacEachren et al.¹⁸ summarized different visual semiotics and performed an empirical study to evaluate the effects of different visual variables on visualizing uncertainty. Among the visual variables for uncertainty visualization, the effect of opacity is made use in visualizing uncertain numbers¹⁹ and graphs.²⁰ However, as reported in Ibrekk and Granger Morgan²¹ and Mackinlay,²² using varying opacity or density to encode certainty results in poor accuracy. One way to compensate for this problem is adding a channel of geometry in the mapping. A few specific systems for visualizing fuzzy numbers²³ or fuzzy relations²⁴ are also proposed by researchers, in which the shape and color are used as visual variables. In this work, we focus on visualizing fuzzy sets and further exploit the potential of the opacity to display the membership function.

Our work also relates to the information visualization in machine learning. Lawrence²⁵ generated a probabilistic latent space with color maps similar to the visualization in this work. Bishop and Svensén²⁶ introduced the generative topographic mapping to produce the probability density of data. Iwata et al.²⁷ warped a latent mixture of Gaussians to visualize high-dimensional data as nonparametric cluster shapes with color maps. Sivaraksa and Lowe²⁸ used Probabilistic NeuroScale for uncertainty visualization. In their work, data in high-dimensional space are projected to a plane by minimizing a cost function similar to a mass–spring model used in this work. These methods are generally used to visualize high-dimensional data and they are not typically applied to fuzzy sets. In this work, we use similar ideas to design a system for visualizing fuzzy sets.

Opacity-varying freeform diagrams

Overview

We first suppose that $A = {(e_{i}, μ_{A} (e_{i}))}_{i}$ is a fuzzy set to be visualized, where $e_{i}$ is the ith element and $μ_{A} (e_{i}) \in [0, 1]$ is the membership function of A’s element $e_{i}$ . From the definition of fuzzy set, we will visually present what elements belong to the fuzzy set and the degree of each element’s membership, as illustrated in Figure 2.

Figure 2.

The visual features of an opacity-varying freeform diagram (left). The opacity field from a random layout of the elements (right).

In Euler diagrams,⁷ a popular visualization technique for classic sets, sets are mapped to diagrams with a clear boundary and a uniform color. The elements of classic sets are usually drawn as dots placed inside the diagram to indicate the membership. Since fuzzy sets are extended from classic sets, we expect to visualize them as an extension of Euler diagrams in this work. Note that both the clear boundary of the set and the spatial placement of the element help visualize the binary membership in classic sets; therefore, we will design a framework to blur the boundary of the Euler diagrams to present the fuzziness. Similar ideas are also reported in visualizing uncertainty in numbers¹⁹ and graphs.²⁰ Specifically, in visualizing fuzzy sets, we map the membership function to the opacity of the diagram. Suppose the ith element $e_{i}$ is located at the position $(x_{i}, y_{i})$ and the opacity of the point $(x_{i}, y_{i})$ is $α_{i}$ , we set the opacity $α_{i}$ to be the membership function $μ_{A} (e_{i})$ . In this case, lower membership function corresponds to the area with higher transparency, as shown in Figure 2 (left). As an extreme case, an element with zero membership function is located at the area with full transparency, which correctly represents that the element is outside of the set because at its location, it has the same color like the background. Lower accuracy in estimating the membership function is one problem with the opacity field in visualizing fuzzy sets. In order to alleviate this problem, we suggest overlaying the isocurves of opacity on the diagram. In this case, users can better estimate the membership function of one element by checking its distance to neighboring isocurves.

However, only with the mapped opacity and its isocurves, it is not sufficient to create good visualization. As shown in Figure 2 (right), the opacity field may have a complex topology if scattered elements which have different membership functions are randomly placed. Therefore, we need to have a well-ordered placement of the elements. Inspired by the disk diagram, we use the membership function to control each element’s distance to the center of the diagram. Since we have opacity to visualize the membership function, we do not have to strictly make the distance linearly proportional to $1 - μ_{A} (e_{i})$ as disk diagrams suggested, because it may not clearly display the elements with higher membership function, as shown in Figure 1 (left). Instead, we relax the boundary from a circle to a freeform shape and only softly require that elements with higher membership function are close to the diagram center. This design will lead to a more continuous opacity field and leave more space to the layout of the elements, as shown in Figure 2 (left). Note that although freeform diagrams exist for visualizing classical sets,¹⁰ they are not available for visualizing fuzzy sets without any development. Because elements in classical sets all have one membership function, it is more flexible to design the shape of the diagram boundary and then sample the elements inside the diagram. In fuzzy sets, however, elements have different member functions. Sampling of the elements without considering the member functions may not lead to plausible design (Figure 2 (right)). Therefore, we have to design the freeform diagrams according to the member functions of the fuzzy sets.

To sum up, our design combines the benefits of freeform Euler diagrams and disk diagrams. Both informative constraints and esthetical aspects are considered in our design.

We map a fuzzy set A to a blurred freeform diagram with varying opacity and map all its elements $e_{i}$ to dots. On the canvas, the location of the ith element $(x_{i}, y_{i})$ has its opacity $α_{i} = μ_{A} (e_{i})$ . These factors completely visualize the information of a fuzzy set and are defined to be hard constraints in our design.

Suppose the center of the diagram A is $(x_{A}, y_{A})$ , the distance $d_{i}$ from the ith element’s location $(x_{i}, y_{i})$ to the diagram center is proportional to $1 - μ_{A} (e_{i})$ . Besides, we hope the elements are evenly sampled on the canvas, that is, the distance between each pair of elements is near a constant value. These two requirements may conflict when placing the elements. Since these factors contribute to the esthetics of the diagram, we regard them as soft constraints in our design. In other words, the requirements on the distances only need to be roughly satisfied.

In order to automatically generate an opacity-varying freeform diagram which satisfies the above design constraints, we further develop a computational framework as shown in Figure 3. We first compute the layout of the elements and then interpolate the opacity from the elements to the entire canvas to complete the design. These two stages are detailed in the following two subsections.

Figure 3.

After (a) the raw data are loaded, we optimize (b) the initial layout of the elements by simulating a mass–spring system, where we illustrate two springs here. (c) Bi-harmonic interpolation of the opacity field on a triangular mesh is then used to create (d) the final design.

Layout computation

In this stage, we convert the elements in fuzzy sets into scattered points and solve for a layout that clearly distributes those points ${(x_{i}, y_{i})}_{i}$ . The set diagram is temporally represented by its center $(x_{A}, y_{A})$ in this stage. As discussed in the previous section, we expect the distance from each element to the set center reflects its membership function. Therefore, the following energy is expected to be minimized

E_{1} = \sum_{i} {(| | (x_{i}, y_{i}) - (x_{A}, y_{A}) | | - (1 - μ_{A} (e_{i})) r)}^{2}

where the parameter r controls the scale of the diagram. Another requirement is that the distribution of the element points should be as even as possible, which yields another energy term to be minimized

E_{2} = \sum_{i, j} w_{i j} (| | (x_{i}, y_{i}) - (x_{j}, y_{j}) | | - d)^{2}

where d is a constant which controls the ideal distance between two points and $w_{i j}$ weights the importance of each pair of elements which is set to be $1$ by default.

In summary, we solve the following optimization problem for the layout of the points

{(x_{A}, y_{A}), {(x_{i}, y_{i})}_{i}} = a r g m i n (E_{1} + ω E_{2})

(1)

where $ω$ weights the second term. In our system, we set $ω = 0.1$ by default. It is because we generally hope that the layout better preserves the distance from the element to the set center comparing with the distance between each element. Users are also allowed to adjust the weights if they have their own preferences.

Note that by regarding $(1 - μ_{A} (e_{i})) r$ in $E_{1}$ and d in $E_{2}$ as the rest length of springs, the formulation of the layout problem well matches the mass–spring system in physical simulation, as shown in Figure 3(b). Therefore, we adopt the projective dynamics^29,30 as the solver to optimize equation (1). Because the energy equation (1) is translational-invariant, that is, the energy does not change if the layout is translated, we fix the point which is closest to the canvas center to avoid drifting in the optimization. In order to enforce a clear layout, we increase the weight $w_{ij}$ by a factor of 10 and run the solver again if we detect that the distance between a pair of elements $(i, j)$ is under a threshold $d_{\min}$ after convergence. The minimal spacing $d_{\min}$ is set as 2.5% of the canvas size throughout this work. It can be set smaller if the diagram is shown in a larger display.

Since the system equation (1) is nonlinear, we need to initialize a layout to start the solver.³⁰ In our implementation, we use the points located in disk diagrams as the initialization. We first place the set center at the center of the canvas as the origin of a local polar coordinate system. Then, we distribute the elements with equal spacing polar angles and set its distance to the set center as $(1 - μ_{A} (e_{i})) r$ . In this article, we generate the diagrams by setting the radius r to be half of the canvas size and the ideal spacing $d = 0.25 r$ . We suggest to decrease r if more fuzzy sets are visualized or decrease d if the number of the elements gets higher.

Opacity computation

After the layout of the element points and set center are found, we are ready to compute the opacity field on the canvas to complete the design. Since the opacity of each element point is set to be the membership function as clarified in section “Overview,” what remains to be solved is the opacity field on the canvas other than the element points and set center. This is a scatter data interpolation problem. Because we hope the opacity field to be as smooth as possible, we introduce bi-harmonic interpolation to improve the quality of the opacity field.

Specifically, we first use Delaunay triangulation to triangulate the domain.³¹ The constraints in the triangulation are the boundary of the canvas, the set center, and the element points. After the triangulation, it is already possible to create an opacity field that satisfies

{\begin{matrix} α (x_{i}, y_{i}) = μ_{A} (e_{i}), i = 1, 2, 3, \dots \\ α (x_{A}, y_{A}) = 1 \end{matrix}

(2)

However, simply using linear interpolation in each triangle may introduce observable discontinuous artifacts if the triangulation is too coarse, as shown in Figure 4(a). Therefore, we add area constraints in the triangulation to make the mesh dense. For a dense mesh, linear interpolation does not guarantee a globally smooth opacity field. So, we use bi-harmonic interpolation to generate a smooth opacity field on the canvas. Formally, we solve a bi-harmonic equation

Δ^{2} α = 0

Figure 4.

(a) The opacity field from linear interpolation on a coarse mesh. (b) The bi-harmonic opacity field obtained with the constraints (2). (c) The bi-harmonic opacity field obtained with the constraints (2) and (3).

subject to the constraints (2), where $Δ$ is the Laplacian operator defined on the mesh. In the discrete case, we use the cotangent Laplacian L³² to discretize the bi-harmonic equation. In this case, the bi-harmonic equation turns to be a linear equation

L^{2} α = 0

where the opacity values of all the vertices on the canvas mesh are stacked into a vector $α$ . Combining the linear constraints (2), we only need to solve a least-square sparse linear system to obtain the smooth opacity field. We refer to Jacobson et al.³³ for more details about bi-harmonic equations defined on irregular meshes.

In our test, we find that the bi-harmonic opacity field from the previous definition usually looks unbounded as shown in Figure 4(b). This is because we only have nonzero constraints in equation (2) and the smooth field will not vanish even at the boundary of the canvas. Therefore, in addition to the constraints from membership functions, we introduce more points which are not inside the fuzzy set to enforce a compact opacity-varying diagram. Specifically, we extrapolate each element from the set center and define the point

({\tilde{x}}_{i}, {\tilde{y}}_{i}) = (x_{A}, y_{A}) + \frac{(x_{i} - x_{A}, y_{i} - y_{A})}{(1 - μ_{A} (e_{i}))}

to represent an element outside of the fuzzy set. By adding this set of interpolatory constraints

α ({\tilde{x}}_{i}, {\tilde{y}}_{i}) = 0, i = 1, 2, 3, \dots

(3)

to the previous system, we finally get the opacity-varying freeform diagrams as shown in Figure 4(c).

As reported by MacEachren et al.,¹⁸ it is hard to acquire the exact value from the opacity channel. In order to help perceive the membership function, we suggest overlaying the isocurves of opacity on the generated diagrams, with their labeled isovalues. We propose two optional strategies to show the isocurves. First, we can interactively overlay an isocurve passing through a specified element. User selects a point of interest i, and our system interactively displays an isocurve $μ_{A} = μ_{A} (e_{i})$ . By checking the points inside or outside this isocurve, one can also tell the elements with higher or lower membership functions than the element i. This strategy is valid if the fuzzy set is visualized in a display with our system running in the background. The other strategy is to simultaneously show several isocurves with equal steps ${μ_{A} = i / n, i = 0, 1, \dots, n - 1}$ . For example, when we set $n = 10$ , user can find elements with their member functions between 30% and 40% by checking the points between the isocurve $μ_{A} = 0.3$ and $μ_{A} = 0.4$ . This strategy allows users to estimate the membership function of one element by checking its distance to the neighboring isocurves, and the accuracy is usually better than the step 1/n. Note that users would be hard to read the diagram if too many isocurves are drawn. Therefore, we suggest drawing less than 10 isocurves if the canvas is small or in a low resolution.

Discussion

There are also other possible solutions to generate an opacity-varying diagram that satisfies the interpolatory constraints on opacity and the soft constraints on distances. For example, one can start from a disk diagram and first generate an opacity field that satisfies $α (x, y) = \max (1 - | | (x, y) - (x_{A}, y_{A}) | | / r, 0)$ . Then, by warping³⁴ the opacity-varying disk diagram, the distances between pairs of elements can be adjusted to be near a constant value. However, we need to carefully formulate the warping process by specifying the local scaling factors of each element. It is not easy because the soft constraints on distances form a global system and the scaling factors chosen from local greedy strategies are usually not sufficient. Another alternative solution is to deform an opacity-varying disk diagram using points from the layout computation as the control points,³⁵ instead of computing the opacity field using bi-harmonic interpolation. This method is also not simpler than our method. In order to avoid fold-overs, we need to solve a complex constrained nonlinear system or update the triangulation of the domain, which leads to an expensive solution in the end. After rounds of exploration and testing, we stop at the proposed two-step method (layout optimization and opacity field interpolation). Thanks to the efficient solver for mass–spring systems and linear formulation of the bi-harmonic interpolation, our method works well and it is computationally efficient (see Table 1 for the timing statistics). Note that our method is still not the only optimal solution to generate the designed diagrams. We will keep searching for other methods to improve the computational design.

Table 1.

Statistics and timings of our method applied on the data shown in section “Opacity-varying freeform diagrams.”

Data	#E	#S	#V	#F	Layout computation		Opacity computation		Render
Data	#E	#S	#V	#F	Mean (ms)	95% CI (ms)	Mean (ms)	95% CI (ms)	Mean (ms)	95% CI (ms)
Figure 5(a)	9	1	3391	6585	2	1–2	106	104–109	2	1–2
Figure 5(b)	22	1	3408	6619	2	2–2	158	157–161	2	1–2
Figure 8(a)	25	1	3370	6543	2	2–2	167	166–168	2	2–3
Figure 8(b)	50	1	3358	6512	4	4–6	273	272–275	3	3–4
Figure 8(c)	150	1	3439	6635	28	27–28	675	674–677	5	5–6
Figure 6(a)	10	2	3387	6579	1	1–2	175	172–178	3	2–3
Figure 6(b)	10	3	3375	6553	1	1–2	230	229–232	4	3–4

#E and #S represent the number of elements and fuzzy sets, respectively. #V and #F represent the number of the vertices and faces, respectively, of the mesh after the canvas is triangulated. We report the time of layout computation, opacity computation, and rendering in milliseconds (ms) as listed in the last three columns, including the average time (mean) and its 95% confidence interval (95% CI). The time for loading data is less than 1 ms and we omit it in Table 1.

Results

We implement our system and test it on a laptop with an Intel i3-3120M 2.5 HZ CPU and 4.0 GB RAM. We list the statistics in Table 1. We warm up the system first and then run our algorithm for a hundred times. The last 80 samples are used to compute the average computational time and its 95% confidence interval. In our experiments, each single diagram is automatically generated within 0.7 s on a single thread, the CPU rendering time is around 3 ms, and the data loading time is less than 1 ms. For visualizing multiple fuzzy sets, even if we need to compute the opacity field several times, the processing time increases only a little. This is because different diagrams share the same triangulated canvas, and we can reuse the factorization of the linear systems to accelerate the opacity computation. The time for computing the isocurves is less than 2 ms, and users can check isocurves of different elements in real time. In our implementation, we do not display the frequency of data as the disk diagrams show. But it is not hard to generate such bar charts from the data.

In Figure 5, we show the generated diagrams of two different fuzzy sets. In Figure 5(a), five elements have their membership functions equal to 1. In this example, the layout optimization creates a clear layout of the elements and the region around the center has full opaque to indicate the membership. It demonstrates that our technique well supports such cases, while the elements may be overlapped in disk diagrams. In Figure 5(b), a fuzzy set with more elements is visualized. We can overlay isolines to help visualize the membership function, as shown in Figure 7(a).

Figure 5.

Visualizing a single fuzzy set with 9 elements (a) and 22 elements (b) using our method.

We also try our system with multiple sets on one canvas, although our method is not specially designed for visualizing multiple fuzzy sets. We assign different colors to different sets. For multiple sets, if one element is only specified to belong to set A, its membership function to set B is set to $0$ . With this definition, we simply add more terms to the mass–spring system to solve for the layout. In the opacity field computation, we note that each fuzzy set has its own opacity field and we solve sparse linear systems several times. With this method, we generate overlapped diagrams similar to Euler diagram except that our diagram has varying opacity and blurred boundary, as shown in Figure 6. The result roughly reveals the relationship of different fuzzy sets. If users are interested in a better visualization of the union or intersection of different sets, we also recommend to first compute the fuzzy set union or intersection and then use our method to visualize the results. In Figure 6, we simply overlay the diagram of each fuzzy set. One problem with this method is that users find it difficult to assess the value in the overlapping region with the blended color and opacity.³⁶ In order to alleviate this problem, we overlay the isocurves on each diagram. We show a snapshot in Figure 7(b), in which one user is checking the membership function of the seventh element in Figure 6(b).

Figure 6.

Visualizing two fuzzy sets (a) and three fuzzy sets using our method (upper row) and disk diagrams (lower row)

Figure 7.

Highlighting the membership function by overlaying isocurves on the diagrams. (a) Overlaying multiple isocurves with equal steps. (b) Overlaying isocurves passing through a specific element.

We test our system using fuzzy sets with different cardinalities to show how our method scales. The results are shown in Figure 8 and the timing statistics are listed in Table 1. With growing cardinality, our method still attempts to evenly layout the elements. However, it becomes challenging to clearly layout each element in a limited space if the number of elements goes extremely high. Regarding the performance, the statistics verify that the cost of layout optimization depends on the number of the elements. For the opacity computation, the cost of the linear system depends both on the triangulation of the canvas and the number of elements because more linear constraints will be introduced to the system if the set cardinality is high.

Figure 8.

Visualizing fuzzy sets with different cardinalities using our method. #E denotes the number of the elements in the fuzzy set: (a) #E = 25, (b) #E = 50, and (c) #E = 150.

In Figure 9, we show the generated diagrams from different weights $ω$ in equation (1). From the results, we see that the elements are widely spread on canvas with the increasing weight $ω$ . Large $ω$ indicates that the system is penalized more on the distances between pairs of elements, which leads to a smaller energy $E_{2}$ .

Figure 9.

The visualization generated from different weights $ω$ in equation (1). We also list the energy $E_{1}$ and $E_{2}$ of the optimized layout: (a) ω = 0.01, (b) ω = 0.1, and (c) ω = 1.

Application and evaluation

Fuzzy healthy status of machines

In Figure 10, we demonstrated how our method can be applied to visualize the fuzzy healthy status of machines. The data list in Figure 10(a) was collected from a production workshop. It recorded the healthy status of 52 machines, where 100% indicated that the machine was brand new and 0% meant the machine had to be scrapped. We generated an opacity-varying freeform diagram from the data and overlaid isocurves on it, as shown in Figure 10(c). In this application, we found that it was not clear to display a fuzzy set with so many elements using a disk diagram. Instead, as a baseline, we used a stem-and-leaf plot¹⁶ shown in Figure 10(b) to visualize the data. In the stem, we set 10 ranges and the accuracy was within 10% in each branch. Note that we could set more ranges in the plot to increase the accuracy, at the cost that the visualization went complicated. In the extreme case, it might degenerate to a sorted data list. Therefore, for a fair comparison, we overlaid 10 isocurves on the opacity-varying freeform diagram, in which the number of the isocurves was the same as the number of the ranges in Figure 10(b).

Figure 10.

Visualizing the healthy status of machines using a (a) list, (b) stem-and-leaf plot, and our (c) method.

Study design

In order to quantitatively verify the efficiency of our method, we designed a user study to check whether users could quickly and concisely convey the information from the visualization. Specifically, we designed two tasks. In each task, we showed one of the subfigures in Figure 10 on a 13.3-in display with $1920 \times 1080$ pixels.

In the first task, a user was asked to find a specific machine and estimate its healthy status. At the background, we also recorded the response time of the user, starting from the time when a subfigure was displayed and ending at the time when the user entered the estimation. Since the data list was the most precise description of the data, the healthy status could be directly read from the list. For the data shown in the stem-and-leaf plot, users could not estimate the actual healthy status because they only knew the range. Therefore, the error of the estimation was expected to be 5% under the assumption that the membership functions were uniformly distributed in the range. The hypothesis of the first task is that the response time in reading the stem-and-leaf plot and our proposed diagram is similar, but the accuracy of the estimated healthy status from our method is higher.

In the second task, a user was asked to enumerate the machines whose healthy status was in a given range [a%, b%]. There was no means to enumerate the machines from the stem-and-leaf plot, if the given range did not fit the data in the stem. For example, users complained that it did not make sense to ask them enumerate the machines whose healthy status was in [74%, 82%] from Figure 10(b), because they could only guess from the machines listed as the leaves of the range [70%, 80%) and [80%, 90%). Therefore, we randomly chose one of the ranges in the stem for the study. In this task, users read the elements in a specified range and pressed a button after they finished. We recorded the response time only if the machines were correctly enumerated. The hypothesis of the second task is that the response time in reading our proposed diagram is a bit longer than that in reading a stem-and-leaf plot, but it is shorter than that in reading the data list.

Study procedure

We recruited seven participants (3 males and 4 females) from the students who studied engineering in our university (average age 23 years, median age 24 years) on a voluntary basis. All of them had experience in reading tables and stem-and-leaf plots. So, we only gave them a 5-min session to introduce our proposed diagram and the tasks. We randomly assigned 10 trials for each visualization in the first task and 5 trials for each visualization in the second task, yielding $10 \times 3 + 5 \times 3 = 45$ trials for each participant. We also set a gap of 5 s between trials in the study.

Study results

With 7 participants and 45 trials per participants, we collected a total of 210 responses for the first task and 105 responses for the second task. After checking the enumerated machines in the second task, we discarded the samples in which the enumeration was not correct, yielding 86 valid responses. Most participants finished the study within 10–15 min.

For the first task, we listed the average response time and the accuracy of the estimation from each participant in Table 2. From the results, the response time in reading the data list was the fastest because it was ranked by the ID of the machine. By running a t-test for the response time in reading the stem-and-leaf plot $t_{S}$ and the proposed diagram $t_{O}$ , we got the result $(t (6) = 0.405, p = 0.6993)$ . At the significance level 0.05, the absolute of the t-value was smaller than $t_{0.05} (6) = 2.447$ and the p value was greater than 0.05. Therefore, the hypothesis that the response time $t_{S}$ and $t_{O}$ are similar was not rejected. Besides, the average accuracy of the estimated healthy status was no larger than 1.6%, which was better than the expected accuracy 5% of the stem-and-leaf plot.

Table 2.

In the task of finding a specific machine and estimating its healthy status, we report the average response time of the seven participants (P1–P7) in reading the data list $t_{L}$ , the stem-and-leaf plot $t_{S}$ , and our proposed diagram $t_{O}$ . The average error $e_{O} = | e - g |$ is listed at the last column, where e is the estimated healthy status from the proposed diagram and g is the actual healthy status.

P	$t_{L}$ (s)	$t_{S}$ (s)	$t_{O}$ (s)	$e_{O}$ (%)
1	1.694	7.855	9.484	1.3
2	1.208	6.651	2.597	1.6
3	1.264	3.957	3.257	0.4
4	1.405	4.578	6.161	1.1
5	1.742	4.798	5.652	0.5
6	1.566	5.232	3.347	1.6
7	1.87	3.289	3.639	0.2

We report the time in seconds (s).

In Table 3, we listed the average response time in the second task. Because in the study the range was designed to well fit the stem value in the stem-and-leaf plot, the corresponding response time $t_{S}$ was shorter. We compared the average response time $t_{L}$ and $t_{O}$ then. As reported from the t-test for the paired samples, we got the t value and p value $(t (6) = 10.242, p = 0.0001)$ , which indicated that the average response time $t_{O}$ was different than $t_{L}$ at the significance level 0.05. We also ran a t-test for the paired samples of $t_{O}$ and $t_{S}$ and it returned $(t (6) = 5.478, p = 0.0015)$ . In other words, $t_{O}$ was still different than $t_{S}$ at the significance level 0.05. Note that the stem-and-leaf plot was not suitable for a good enumeration of the elements in arbitrary range, but our design gave a slightly better way for this task.

Table 3.

In the task of enumerating machines whose healthy status is in a given range, we report the average response time of the seven participants (P1–P7) in reading the data list $t_{L}$ , the stem-and-leaf plot $t_{S}$ , and our proposed diagram $t_{O}$ .

P	$t_{L}$ (s)	$t_{S}$ (s)	$t_{O}$ (s)
1	18.022	6.05	7.466
2	14.238	3.706	6.95
3	13.95	1.906	5.226
4	14.646	4.03	6.01
5	11.324	2.378	5.924
6	13.72	4.464	5.004
7	11.824	3.2	6.866

We report the time in seconds (s).

Fuzzy diseases of children

As another application of our system, we visualized fuzzy diseases of children using opacity-varying freeform diagrams, as shown in Figure 11. The data set was summarized by a pediatrician, in which different symptoms were used to determine a fuzzy diagnosis of common diseases. We could use our method to visualize the related fuzzy sets separately. As shown in the lower-left subfigures, both the distribution of the elements and the opacity well presented the fuzziness in the diagnosis. We also created a visualization which simultaneously showed three sets of diseases to the right of Figure 11. The visualization gave an intuitive view to the data and set. For example, cough is not related to hand, foot, and mouth disease (HFMD). This fact could be observed in the figure that the element with the caption “cough” was far from the red diagram, which represented the set “HFMD.” Comparing to pneumonia, children with rash have higher probabilities to get a disease of upper respiratory tract infections (URI) or HFMD. This factor corresponded to the observation that the element “rash” lay on the blurred boundary between the set “URI” and “HFMD” and it was far from the blue diagram “pneumonia.”

Figure 11.

Visualizing a fuzzy set of children’s disease using our method.

We also made a quick survey by collecting user preferences in this application. Note that this study was preliminary because it only compared our method with other methods using the specific fuzzy diseases. Strictly speaking, the conclusions were limited to this specific application. We reported it to provide feedbacks from this application, regarded as an evidence to evaluate our method.

In this study, we chose the numbers listed in a table and disk diagrams as baselines, as shown in the upper-left subfigure of Figures 11 and 12(a), respectively. Note that disk diagrams were originally designed for interactive visualization of fuzzy set operations. We showed the disk diagram of each individual fuzzy set in the survey, because we printed the charts and questionnaire using color printers and preferred static charts in this study. Inspired by the work,³⁷ we designed a Likert-scale survey and used six dimensions to assess the visualization, including the informativity, compactness, discriminability, intuitivity, esthetic quality, and overall preference, as detailed in Figure 13.

Figure 12.

(a) The disk diagrams generated for the fuzzy diseases of children shown in Figure 11. The scores from (b) the survey are illustrated in (c) the box-and-whisker plot.

Figure 13.

The questionnaire used in the survey.

We found 15 volunteers in a children’s hospital, including 9 pediatricians and 6 parents of the patients. None of them had experience with the disk diagrams before. Therefore, we first gave them a briefing session for explaining the data sets and functions of our study. Then, we introduced the three different visualization techniques to them. This session took around 10 min. After that, they were asked to rate the above charts by filling the printed survey form. The rating process usually took 3–6 min. After we got their ratings, we also interviewed the participants in person to collect other feedbacks, mostly about the reasons why they gave such scores.

Figure 12(b) showed the scores rated from 0 to 10 by the participants. We then ran a Kruskal–Wallis test on the data and showed the box-and-whisker plot in Figure 12(c). From the results, we found that the proposed visualization was generally more friendly to users, at the significance level around 0.0391. All visual maps of the fuzzy sets were thought less informative comparing with the representation of numbers in a table. From a follow-up interview, the participant (P5) replied that it was hard to tell the exact values solely from the geometry, color, and opacity. However, the visual designs were more intuitive and visually plausible than the numbers in a table. From the feedback of the participants, our method was thought to be more intuitive comparing with disk diagrams. The participants (P7 and P13) said that they did not need to learn the definition of the orbits and they could directly assess the membership function by estimating its opacity and color. Another intuition is that by composing three fuzzy sets in one canvas, our design is more compact in this application. However, from the Kruskal–Wallis test, the significance level was not high. The participants (P2 and P5), who gave lower score to our method in the compactness, explained that it was because most of the canvas was filled by the background and the visualization looked not so compact. We also got similar feedbacks in terms of discriminability. The participant (P5) thought the blended background color was harmful in clearly showing the elements and that was why he gave a lower score to our method.

We also interviewed the participants for the explanation of their ratings to the overall preference. Subjects (P10 and P13) who favored our design said that they could quickly recognize and compare the fuzzy membership between the symptoms and the diagnosis by checking our designed chart. They also reported the weakness that it was hard to quantitatively estimate the membership function, which is consistent with the observation in MacEachren et al.¹⁸ Since they cared more about the relative and fuzzy membership between the symptoms and diagnosis, they found it was efficient and convenient to check the membership by looking up the position of the elements in the chart. A few pediatricians (P5 and P14) gave their highest score to the numbers in a table. The reason was that they did not like the fuzziness in the diagnosis and the numbers were more definitive although they only represented the probability. They (P5 and P14) also preferred disk diagrams to our method, explaining that the disk diagrams looked more definitive compared to the blurred diagrams. But they also agreed that our opacity-varying diagrams better revealed the uncertainty in the membership. When they were told that the probability in the table was only a rough estimation, they acknowledged the efficiency of the opacity-varying diagrams. One participant (P15) was interested in fuzzy set operations such as set union and intersection. In this case, disk diagrams fit their requirements better if user interactions were allowed. In this work, we only focus on generating static charts for visualizing fuzzy sets.

Conclusion

In this work, we focus on visualizing general fuzzy sets. We design opacity-varying freeform diagrams, which extends Euler diagrams and disk diagrams. Both the layout of the elements and the opacity field on the canvas help visualize the fuzzy sets in our design. We also propose a computational framework to automatically generate our design from input fuzzy sets. Experimental results show that our method is efficient at creating visualization for fuzzy sets.

In the future, we plan to study the union, intersection, and complement operations of fuzzy sets. By constraining the area of the diagrams, we can further improve our design to better support visualization of these fuzzy set operations. We also plan to support user interactions in the visualization. We only visualize two or three sets simultaneously in this work. Similar to Euler diagrams for classic sets, the visualization gets challenging if the number of sets drastically increases. To scale our method for visualizing more sets is another direction for future research. Besides, it is also challenging to scale our method to support visualizing large amount of elements in a limited size of canvas. In this extreme case, the soft constraints on distances may significantly conflict and the minimal spacing may not be guaranteed. We will also keep working on other algorithms to generate a layout with better space utilization. Another possible solution is to develop a multi-scale framework to visualize large amount of elements in fuzzy sets.

We focus on visualizing traditional Zadehian fuzzy sets¹ in this work. It is also possible to extend our design to support other variants of fuzzy sets³⁸ as what Das¹⁵ proposed to disk diagrams. In this work, we extend the freeform Euler diagrams by introducing an opacity channel to support the visualization of fuzzy sets. It is also possible by extending other techniques listed in Alsallakh et al.⁴ and we will keep working on this understudied area. As another future work, more formal user study will be conducted to improve our design. Finally, we use our method to visualize general fuzzy sets modeled from different domains in this work. We can tailor our system for a more friendly visualization to different users, which may better support domain-specific applications.

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work has been supported by the National Science Foundation of China(NSFC) under Grants No.61502096, the Natural Science Foundation of Jiangsu Province of China under Grants No.BK20150634, the National Key Technologies R&D Program under Grants No.2016YFB1001300, the Fundamental Research Funds for the Central Universities and the Open Fund of Jiangsu Province Key laboratory of Remote Measuring and Control.

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