Web-based Scalable Visual Exploration of Large Multidimensional Data Using Human-in-the-Loop Edge Bundling in Parallel Coordinates Wenqiang Cui Girts Strazdins Hao Wang Department of ICT and Natural Department of ICT and Natural Department of Computer Science Sciences Sciences Norwegian University of Science Norwegian University of Science Norwegian University of Science and Technology and Technology and Technology Norway Norway Norway hawa@ntnu.no wenqiang.cui@ntnu.no gist@ntnu.no ABSTRACT data items are mapped to lines (or edges) intersecting the axes at Visual clutter and overplotting are the main challenges for vi- their respective values. The embedding of an arbitrary number of sualizing large multidimensional data in parallel coordinates, parallel axes into the plane allows for the simultaneous display of which greatly hampers the recognition of patterns in the data. many dimensions to provide a good overview of the data, which Although many automatic clustering and edge-bundling methods reveals intrinsic patterns and trends. However, when datasets are have been used in parallel coordinates to reduce visual clutter and large, PCPs create visual clutter and overplotting in which lines overplotting, a scalable, transparent, and interactive approach are crossed and plotted on top of one another, overwhelming that allows analysts to interact with large data and generate the display, and obscuring the underlying patterns. This hides interpretable results of visualization in real time is lacking. To information and hampers the recognition of patterns in the data. solve this problem, we propose an approach, human-in-the-loop Edge bundling [7] and automatic data clustering [10] are two edge bundling, to visually explore and interpret large multidimen- widely used approaches to reduce visual clutter and overplotting sional data in parallel coordinates. This approach combines data in PCPs. Edge bundling bends similar lines to the center of vi- binning-based clustering and density-based confluent drawing, sual clutters in groups to create more informative visualizations. which reduces much data processing time and rendering time. It Automatic data clustering aggregates data points in groups that provides novel interactions, such as splitting, adjusting, and merg- can be visualized in an illustrative fashion using different forms ing clusters, to integrate human judgment into the edge-bundling of edge bundling. process. These interactions make the underlying clustering trans- However, when datasets become large, these methods face parent to users, which allow users to generate interpretable visu- challenges in supporting real-time interactions (limiting the vi- alization without complex data clustering. The scalability of our sual response in a few milliseconds) along with mechanisms for approach was evaluated through experiments on several large information abstraction. Without interactions, these automatic datasets. The results show that our approach is scalable for large methods provide only groups that may contain interesting com- multidimensional data, which supports real-time interactions binations of dimensions and data points, but do not give analysts on millions of data items in web browsers without hardware- control over the data clustering and visualization processes, and accelerated rendering and big data infrastructure-based data pro- do not offer opportunities for analysts to take advantage of their cessing. We used a case study to highlight the effectiveness of judgments and expertise. our approach. The results show that our approach provides an In this study, we propose a web-based visual analytics sys- interpretable way of visually exploring large multidimensional tem that uses data binning-based clustering and density-based data in parallel coordinates. confluent drawing to create a new edge-bundling paradigm in PCPs for large multidimensional data. To the best of our knowl- KEYWORDS edge, this is the first web-based system that supports the HITL (human-in-the-loop) edge-bundling process in PCPs through interactive visualization, human-in-the-loop, visual exploration, specific interactions, such as splitting, adjusting, and merging multidimensional data, big data, parallel coordinates clusters of each dimension, for large multidimensional data. The contribution of this study are as follows: 1 INTRODUCTION A multidimensional dataset contains numerical or categorical • New paradigm for edge bundling in PCP. Our approach dimensions (or features), with n (n > 3) dimensions and m data provides a novel edge-bundling paradigm (HITL edge items. To avoid confusion, in this paper, a data item is an n- bundling) for the visual exploration of large multidimen- dimensional point, and a data point is the projection of a data sional data in PCPs. With the real-time interactions, such item to a particular dimension. Parallel coordinate plots (PCPs) as splitting, adjusting, and merging clusters, it enables are widely used, and have become a standard tool for visualizing analysts to integrate their judgments and expertise into multidimensional data [6]. In PCPs, axes corresponding to the the data clustering and edge-bundling processes of large number of dimensions are aligned parallel to each other, and multidimensional data. • Fast, scalable, and transparent edge-bundling algo- Copyright © 2020 for this paper by its author(s). Published in the Workshop Proceed- ings of the EDBT/ICDT 2020 Joint Conference (March 30-April 2, 2020, Copenhagen, rithm. To support the real-time interactions of large data Denmark) on CEUR-WS.org. Use permitted under Creative Commons License At- in PCPs, we propose a fast, scalable, and transparent edge- tribution 4.0 International (CC BY 4.0) bundling algorithm that consists of two parts: 1) a data EDBT 2020, March 30-April 2, 2020, Copenhagen, Denmark Wenqiang Cui, Girts Strazdins, and Hao Wang binning-based clustering method, and 2) density-based context visualization in PCPs to represent outliers [9]. In this confluent drawing. study, we use one-dimensional (1D) binning to cluster data points • A web-based visual analytics system. We build a web- for each dimension with the following three considerations: based visual analytics system to support HITL edge bundling • In PCPs, for a single dimension, the clusters must be or- in PCPs for large multidimensional data. dered because the data points are ordered. • Experiments, and a case study. We conducted experi- • A data point belongs to only one cluster. ments and a case study on several datasets to highlight • For large data, to support HITL edge bundling in PCPs, the the benefits of HITL edge bundling in PCPs for large mul- clustering process must be fast, scalable, and transparent tidimensional data. to analysts. The remainder of this paper is organized as follows: Section 2 With the first and second considerations, for each axis, the data presents the proposed approach. Section 3 reports the experi- points are binned into ordered and adjacent clusters, which is ments, a case study, and discusses the result. Section 4 draws shown in Figure 3. Since a data point belongs to only one cluster, the conclusions of this study and discusses directions for future there is no overlaps between clusters. This reduces the overplot- work. ting of clusters in PCPs created by multidimensional clustering methods, such as DBSCAN [5]. As shown in Figure 3, for each 2 SYSTEM AND METHODS axis, the data points are first grouped into the same number of In this section, we first describe the HITL edge-bundling process clusters. For a particular axis, the initial clusters have the same with our system. Then, we introduce the methods used in the initial diameters. Users then use the control points to split, adjust, system and the novel interactions provided by the system. and merge clusters (see Section 2.4), which makes the clustering process transparent for analysts. For an axis with k initial clusters 2.1 System Overview (the initial value of k is configured by users), the initial diameter Figure 1 shows the overview of our system. The system first L is computed as: visualizes multidimensional data in a classic PCP without edge L = (dmax − dmin )/k bundling. For example, in Figure 1 (A), the Cars dataset [1] is visualized in a classic PCP without edge bundling. The system where dmax and dmin are the maxima and minima, respectively, then bundles the edges according to the initial clusters for each of the data points on the corresponding axis. For an axis, the dimension as shown in Figure 1 (B). The system supports HITL initial control points Pi denotes the boundaries of clusters, which edge bundling by allowing analysts to split, adjust, and merge are computed as: clusters for each dimension, which is shown in Figure 1 (C). Pi = dmin + i × L, i = 1, 2, ..., k − 1 During the HITL edge-bundling process, the system can update the visualization according to the corresponding interactions Then, a data point d is grouped into a cluster Ci as: in real time for large multidimensional data. This makes the ( Pi−1 < d < Pi , i = 1, 2, ..., k − 1 underlying clustering process transparent to analysts. With the d ∈ Ci if d > Pi−1, i = k interactions, analysts can integrate their judgments and expertise into the edge-bundling process to generate visualizations that To reveal the internal patterns and distribution of data, we com- can be better interpreted. For example, in Figure 1 (C), by creating pute the density of each pair of clusters and use it for density- an empty cluster that ranges from 6 to 8 and a cluster with 0 based confluent drawing (see Section 2.3). For two adjacent axes diameter (ranges from 8 to 8) at 8 on the axis cylinders, we found axisn and axisn+1 , a cluster pair (Caxis i j , Caxisn+1 ) consists of a n that all cars with eight cylinders in the dataset weighted between cluster in axisn and another in axisn+1 , where Caxis i is the i-th 3354 and 5140 kilograms. Moreover, by highlighting the subsets j n that contains cars with eight cylinders in red, the patterns of cluster in axisn , and Caxisn+1 is the j-th cluster in axisn+1 . For other features of these cars are clearly highlighted. two adjacent axes, an edge containing two data points (dn , dn+1 ) The rudiment of our system is the combination of data binning- that belongs to a pair of clusters is defined as: based data clustering and density-based confluent drawing, which i (dn , dn+1 ) ∈ (Caxis j i , Caxisn+1 ) if dn ∈ Caxis j ∧dn+1 ∈ Caxisn+1 supports the real-time interactions for large multidimensional n n data without hardware-accelerated rendering and big data infrast- The density D i,j of a pair of clusters is computed as: ructure-based data processing. Figure 2 shows the workflow of j i N (Caxis , Caxisn+1 ) our system, where the HITL process is highlighted in the dashed D i,j = Í n , n = 1, 2, ... j j line rectangle. The system first uses data binning to cluster data i i=1 j=1 N (C axis n , C axis n+1 ) i Í points for each dimension with the default settings. Then the j density of each pair of clusters on two adjacent axes is computed, where N (Caxis i n , Caxisn+1 ) is the number of edges that belong to and the edges are bundled and rendered through density-based the cluster pair (Caxis i j , Caxisn ). confluent drawing. Finally, users create a more interpretable vi- n The clustering process, including computing the clusters and sualization of edge bundling through the interactions, including the density of cluster pairs, is linearly dependent on the number splitting, adjusting, and merging clusters. of dimensions, the number of data points, and the number of clusters (see Section 3.1). This fast and scalable clustering process 2.2 Data Binning-Based Clustering is the basis of real-time interactions (see Section 2.4), which Data binning groups a number of more or less continuous values supports HITL edge bundling for large multidimensional data in into a smaller number of given data intervals (also called "bins") to PCPs. transform numerical variables into their categorical counterparts Categorical variables are not clustered using the above method. [12]. Multidimensional binning is used to implement focus + Instead, we treat each category as a cluster. Web-based Scalable Visual Exploration of Large Multidimensional Data Using Human-in-the-Loop Edge Bundling in Parallel Coordinates EDBT 2020, March 30-April 2, 2020, Copenhagen, Denmark 47 5140 8 25 455 230 47 5140 8 25 455 230 A 221 C 4577 22 405 212 199 7 37 4258 7 21 358 184 187 37 355 4014 19 331 6 308 3684 145 28 3377 6 16 262 138 3354 5.5 27 5 15 4.5 188 104 18 2495 4 12 165 92 18 2484 4 75 3.5 10 9 9 1613 3 8 68 46 9 1613 3 8 68 46 mpg weight cylinders acceleration displacement horsepower mpg weight cylinders acceleration displacement horsepower 47 5140 8 25 455 230 B 40 4552 7.2 22 391 199 Initial Edge Bundling 34 3964 6.3 19 326 169 Human-in-the-loop Edge Bundling Process 28 3376 5.5 16 262 138 22 2789 4.6 14 197 107 15 2201 3.8 11 133 77 9 1613 3 8 68 46 mpg weight cylinders acceleration displacement horsepower Figure 1: Overview of the system that supports HITL edge bundling in PCPs. A. Visualization of the Cars dataset [1] in a classic PCP. B. Edge bundling of the dataset with 3 initial clusters for each dimension. C. Interpretable edge bundling of the dataset with a subset highlighted (continuous path over axes) in red, which is generated through user interactions. Human in the Loop Edge bundling Cluster Control Point Interpretable Visualization Human Judgment & Expertise Data Interactions Data Point Rendering Diameter Clustering Confluent Drawing Figure 2: The workflow of the system. Figure 3: Using 1D binning to cluster data points for each 2.3 Density-based Confluent Drawing axis in PCPs. The blue points are data points and the red points Confluent drawing is a technique for bundling links in node- are control points. An edge between the axes represents two data link diagrams. It coalesces groups of lines into common paths points that belong to two clusters respectively. Elliptical areas or bundles based on network connectivity to reduce edge clut- represent clusters in an axis. The initial k is 2. For each axis, the ter in node-link diagrams [2, 4]. In this study, we use confluent two initial clusters have the same diameter. The two red clusters drawing to coalesce edges that belong to a pair of clusters to form a pair of clusters. Its density is 0.4. reduce visual clutter in PCPs, where we use the clusters as nodes and edges between them as links. Each pair of clusters then has only one bundled edge, which is shown in Figure 4. This elimi- nates the occlusion and ambiguity near the bundle joints created by bundling techniques that bundle edges by spatial proximity. where Wmax is the width of a bundle with the density of one. More importantly, it reduces rendering time by coalescing edges, Wmax is a constant and is configured by users. which supports real-time interactions for HITL edge bundling of To guarantee C 1 -continuity across axes, we draw bundles as large multidimensional data in PCPs. Bézier curves. Figure 4 shows the bundled edge of a pair of clus- To reveal the information hidden by coalescing of the edges ters. Between two adjacent axes, the width of a bundle represents and the distribution of the data points between axes, we use the the proportion of the data points (coalesced edges) that belong to j density D i,j of a pair of clusters (Cax i is n , C ax is n+1 ) to define the the corresponding cluster pair. This reveals the trend and distribu- width Wi,j of the coalesced bundle as follow: tion of the data items as well as outliers in large multidimensional Wi,j = D i,j × Wmax data in PCPs (see Section 3.2). EDBT 2020, March 30-April 2, 2020, Copenhagen, Denmark Wenqiang Cui, Girts Strazdins, and Hao Wang Cluster Center two new clusters. In Figure 5, the red dashed line circle on Axis A is a newly added control point by double-clicking. • Adjust clusters. All control points can be dragged along Cluster Control Point the axes. Dragging a control point to a new position ad- justs the boundaries and the diameters of the two adjacent clusters. Figure 5 shows dragging the control point on Axis B to a new position (red dashed line circle on Axis B). Bé zie • Merge clusters. All control points can be double-clicked rC ur to be deleted. The two adjacent clusters of the deleted ve Width control point are merged into a new cluster. • Highlight bundles over axes. Hovering the pointer over a bundle highlights it and its related bundles in red. Only bundles with a density greater than a threshold will be Figure 4: Using the density-based confluent drawing to highlighted. The threshold is a constant and is configured bundle the edges that belong to a pair of clusters. For a by users. pair of clusters, the bundled edge is rendered as a Bézier curve • Re-order axes. The labels of axes can be dragged to the that starts from the center of a cluster and ends at the center of front or back of other labels to re-order them to the corre- another. Its width represents the density of the cluster pair. sponding positions. 3 EVALUATION In this section, we evaluate the scalability and the effectiveness Bundle of our system through experiments and a case study on the Office Mouseover Axis Area Occupancy Detection dataset [3] and the Cars dataset [1]. Double Click 3.1 Experiments Control Point To examine the scalability of our system, we synthesized several Double Click & Drag large datasets based on the office dataset. All experiments were conducted on the same laptop without big data infrastructure- based data processing and hardware-accelerated rendering. In our system, the HITL edge-bundling process contains two time-consuming processes: the data binning-based clustering and Label Area Drag the density-based confluent drawing (rendering process). We first performed a run time analysis of the clustering process. Table Axis A Axis B Axis A 1 shows the run times (measured by the second) of the cluster- ing process on large multidimensional datasets (with different number of dimensions, data points, and clusters). According to Figure 5: Interactions provided by our system for support- Table 1, the computation time of data binning-based clustering is ing HITL edge bundling. Double click on the axis area to add linearly dependent on the number of dimensions, the number of a control point to split a cluster. Double click on a control point data points, and the number of clusters. More importantly, this to delete it to merge two clusters. Drag a control point along an data binning-based clustering is much faster than other cluster- axis to adjust the adjacent clusters. Mouseover on a bundle to ing algorithms used for bundling edges in PCPs. For example, highlight a subset with color. Drag an axis label to re-order the Palmas et al. [10] used a density-based clustering method for axes. each dimension independently to bundle edges in PCPs, which takes approximately 60 seconds to cluster 105 data points for one dimension. By contrast, our clustering method takes approx- 2.4 Interactions for HITL Edge Bundling imately 1 seconds to cluster 106 data points for four dimensions. In our system, in addition to common interactions in PCPs such as re-ordering the axes and brushing (highlighting) [11], we use We then examined the efficiency of the rendering process by specifically designed interactions to allow users to split, adjust, comparing the rendering time of our method with both the clas- and merge clusters. Our system updates the visualization ac- sic PCP and Lima et al.’s edge-bundling PCP [5] that also uses cording to user interactions in real time, which is the key to confluent drawing to coalesce edges. To compare the rendering implement the HITL edge bundling process. These interactions time, all three PCPs were implemented with the same JavaScript are supported by the combination of the data binning-based clus- library (D3.js) and rendered in Chrome. The times needed for tering and the density-based confluent drawing. Figure 5 shows rendering the axes, labels, and stickers were not included, which the interactions provided by our system, which are described as are constant regardless of the number of data points. Table 2 follows: shows the rendering time of the three methods (measured by the • Split a cluster. Each axis has a clickable area (called axis second) on the datasets that has six dimensions and the different area) around it, which is shown as gray rectangle area numbers of data items. For our method and [5], each dimension around Axis A in Figure 5. Double-clicking on this area has 3 clusters. According to Table 2, the classic PCP and [5] take adds a new control point to the corresponding position on 1.7672 and 3.6989 seconds to visualize 105 data points. The clas- the axis. This control point splits the original cluster into sic PCP takes 8.7183 seconds to visualize 5 × 105 data points Web-based Scalable Visual Exploration of Large Multidimensional Data Using Human-in-the-Loop Edge Bundling in Parallel Coordinates EDBT 2020, March 30-April 2, 2020, Copenhagen, Denmark Table 1: Run-time analysis of the data binning-based clus- 40 2077 24 1697 Occupied tering Dimensions Data Points Clusters Run-time 34 1661 23 1273 2 104 3 0.0169 2 104 4 0.0167 28 1245 22 849 3 104 3 0.0230 3 104 4 0.0277 2 105 3 0.0505 22 829 20 424 2 105 4 0.0554 3 105 3 0.0937 17 413 19 0 Not occupied 3 105 4 0.0996 humidity CO2 temperature light occupancy 4 105 3 0.1175 (a) 4 105 4 0.1404 4 105 10 0.2574 40 2077 24 1697 4 105 20 0.4139 37 1869 24 1485 4 105 30 0.5495 34 1661 23 1273 Occupied 4 105 40 0.6892 4 105 50 0.8872 31 1453 22 1061 4 106 3 0.8211 28 1245 22 849 4 106 4 0.9398 25 1037 21 636 22 829 20 424 Not occupied Table 2: Comparison of the rendering time 20 621 20 212 17 413 19 0 humidity CO2 temperature light occupancy Data Points Our Method Classic PCP [5] (b) 103 0.00243 0.0273 0.0503 104 0.00231 0.1916 0.3740 40 2077 24 1697 105 0.00230 1.7672 3.6989 37 5 × 105 0.00229 8.7183 N/A 1843 1414 106 23 0.00248 N/A N/A 34 33 1609 Occupied 1131 31 1459 29 22 1308 and crashes the browser when visualizing 106 data points. The 27 743 1110 method [5] crashes the browser when visualizing 5 × 105 data 912 21 points. By contrast, the rendering process of our method is inde- 22 354 Not occupied pendent of the number of data points, which takes approximately 662 177 0.002 seconds for each dataset. 17 413 19 0 humidity CO2 temperature light occupancy 3.2 Case Study (c) To assess the effectiveness of our system, we compared our method with the classic PCP and several algorithmic analysis Figure 6: The visualization of the office dataset in the clas- methods with the office dataset. The office dataset uses the data sic PCP and our system. (a) Visualization of the office dataset on temperature, humidity, light, and CO2 to detect the occupancy in the classic PCP. (b) Visualization of the office dataset in our of an office room. It has five dimensions and 20,560 data points system with 4 initial clusters for each dimension. (c) Visualiza- for each dimension. tion of the office dataset in our system generated by a user who Figure 6 shows the visualization of the office dataset in the does not have knowledge of the dataset. classic PCP and our system. Figure 6c shows the visualization in our system, which is generated by a user who does not have knowledge of the dataset. In Figure 6b and Figure 6c, the red Moreover, by integrating human judgments into the edge- bundles are the subsets highlighted by hovering the pointer on bundling process, our method creates a interpretable visualization the widest bundle between the axes of light and occupancy. The in PCPs for the office dataset. For example, during the HITL edge- extreme narrow bundles (data points with extreme low densi- bundling process (from Figure 6b to Figure 6c), the user obtained ties) are visualized as the dashed lines to detect and highlight the following findings: the outliers (rare data points that raise suspicions by differing • Finding 1. The dataset contains outliers which are high- significantly from the majority of the data [8]) in the dataset. By lighted by the dashed lines in Figure 6c. comparing Figure 6a and and Figure 6c, it is clear that for large • Finding 2. When the value of light is smaller than 354 multidimensional datasets, our method reduces the visual clutter Lux, the room is considered unoccupied. When it is be- and overplotting in the classic PCP and reveals the patterns in tween 354 and 1131 Lux, the room is considered occupied. the data. The accuracy of this estimation is higher than 90% (the EDBT 2020, March 30-April 2, 2020, Copenhagen, Denmark Wenqiang Cui, Girts Strazdins, and Hao Wang Table 3: The comparison our system with the algorithmic this process, users can continuously gain insights from data and methods in [3]. visualization. Criteria Our Method [3] 4 CONCLUSION AND FUTURE WORK Finding 1 Yes No In this study, we proposed HITL edge bundling and built a system Finding 2 Yes Yes based on it to support the visual exploration of large multidi- Finding 3 Yes Yes mensional data in PCPs. The system provides an interpretable Finding 4 Yes Yes visualization, which reduces the visual clutters and overplotting, Interpretability Interpretable Black-box process and eliminates the occlusion and ambiguity of large multidimen- visualization of training the mod- sional data in PCPs. More importantly, the system provides the with transparent els. specifically designed interactions, including splitting, adjusting, clustering process. and merging clusters, to integrate human judgments into the Processing time Real-time. Time for training edge-bundling process in real time. We evaluated the scalability and selecting mod- and effectiveness of the system through experiments and a case els. study. We compared our system with the classic PCP and the algorithmic analysis methods. The results show that our system provides a scalable and interpretable way of visually exploring large multidimensional data in PCPs. estimated sum of the densities of the two widest bundles Anchoring bundled edges in different positions, such as the between the axes of light and the occupancy). mean/centroid position of all data points in a cluster, could be • Finding 3. When the temperature is between 19 and 22 ℃, investigated in the future to improve the continuity across axes the room is considered unoccupied. When the temperature and reveal more information of clusters. This requires more com- is higher than 22 ℃, the room is considered occupied. putation and may delay the visual response of the interactions. The accuracy of this estimation is higher than 80% (the The interactions and color effects (highlighting subsets in dif- estimated sum of the densities of the two widest bundles ferent colors) of the system are not fully evaluated. This can be between the axes of temperature and light). done in a qualitative user study in future work. • Finding 4. Using all features may reduce the accuracy of prediction. Humidity has a much weaker correlation with REFERENCES occupancy than other features. [1] 2005. Cars DataSet. Retrieved September 20, 2019 from http://davis.wpi.edu/ xmdv/datasets/cars.html Candanedo and Feldheim tested linear discriminant analysis, [2] B. Bach, N. H. Riche, C. Hurter, K. Marriott, and T. Dwyer. 2017. Towards classification and regression trees, and random forest on the Unambiguous Edge Bundling: Investigating Confluent Drawings for Network Visualization. IEEE Transactions on Visualization and Computer Graphics 23, 1 office dataset to detect the occupancy of rooms [3]. In Table 3, we (Jan 2017), 541–550. https://doi.org/10.1109/TVCG.2016.2598958 compared the findings obtained in our system with that obtained [3] Luis M. Candanedo and Véronique Feldheim. 2016. Accurate occupancy detection of an office room from light, temperature, humidity and CO2 mea- in [3] of the office dataset. It shows that our system obtained surements using statistical learning models. Energy and Buildings 112 (2016), more findings of the data than the algorithmic methods in [3]. We 28 – 39. https://doi.org/10.1016/j.enbuild.2015.11.071 also compared the interpretability of our system with that of the [4] Matthew Dickerson, David Eppstein, Michael T. Goodrich, and Jeremy Y. Meng. 2005. Confluent Drawings: Visualizing Non-planar Diagrams in a algorithmic methods in [3]. It shows that without the black-box Planar Way. Journal of Graph Algorithms and Applications 9, 1 (2005), 31–52. process of training the models, our system is more interpretable https://doi.org/10.7155/jgaa.00099 with the visualization by integrating human judgments into the [5] Rodrigo Santos do Amor Divino Lima, Carlos Gustavo Resque dos Santos, San- dro de Paula Mendonça, Jefferson Magalhães de Morais, and Bianchi Serique edge-bundling process. Moreover, our system can obtain the Meiguins. 2018. Understanding Data Dimensions by Cluster Visualization result faster by eliminating the time to train the models. Using Edge Bundling in Parallel Coordinates (SAC ’18). ACM, New York, NY, USA, 640–647. https://doi.org/10.1145/3167132.3167203 [6] Julian Heinrich and Daniel Weiskopf. 2013. State of the Art of Parallel Coordi- 3.3 Discussion nates. In Eurographics 2013 - State of the Art Reports, M. Sbert and L. Szirmay- Kalos (Eds.). The Eurographics Association. https://doi.org/10.2312/conf/ Our approach uses data binning to create initial clusters for each EG2013/stars/095-116 dimension. For a particular dimension, it divides the entire range [7] D. Holten. 2006. Hierarchical Edge Bundles: Visualization of Adjacency Rela- tions in Hierarchical Data. IEEE Transactions on Visualization and Computer of values into a series of consecutive, non-overlapping and equal- Graphics 12, 5 (Sep. 2006), 741–748. https://doi.org/10.1109/TVCG.2006.147 size intervals (clusters/bins). By computing the density of cluster [8] Ling Liu and M. Tamer Zsu. 2009. Encyclopedia of Database Systems (1st ed.). pairs, our approach counts the number of data points for each Springer Publishing Company, Incorporated. [9] M. Novotny and H. Hauser. 2006. Outlier-Preserving Focus+Context Visualiza- cluster, which is represented by the total width of the bundled tion in Parallel Coordinates. IEEE Transactions on Visualization and Computer edges starting from the cluster. Therefore, the initial clustering re- Graphics 12, 5 (Sep. 2006), 893–900. https://doi.org/10.1109/TVCG.2006.170 sults in our approach is an adapted histogram for each dimension. [10] G. Palmas, M. Bachynskyi, A. Oulasvirta, H. P. Seidel, and T. Weinkauf. 2014. An Edge-Bundling Layout for Interactive Parallel Coordinates. In 2014 IEEE With the appropriate initial number of clusters, it can capture Pacific Visualization Symposium. 57–64. https://doi.org/10.1109/PacificVis. the accurate distribution of data points for each dimension. This 2014.40 [11] R. C. Roberts, R. S. Laramee, G. A. Smith, P. Brookes, and T. D’Cruze. 2019. is the basis for users to use their judgments and expertise in the Smart Brushing for Parallel Coordinates. IEEE Transactions on Visualization edge bundling process and generate interpretable visualization. and Computer Graphics 25, 3 (March 2019), 1575–1590. https://doi.org/10. With HITL edge bundling, to obtain the final interpretable visual- 1109/TVCG.2018.2808969 [12] Bernard W Silverman. 2018. Density estimation for statistics and data analysis. ization, for example, from Figure 6b to Figure 6c, users may need Routledge. several iterations to adjust the initial clusters for each dimension, such as merging a cluster with small density to an adjacent clus- ter, or splitting a cluster with large density to obtain more details of data. This process may take 1 or 2 minutes. However, during