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Generalized Scatter Plots Generalized scatter plots Daniel A. Keim a Abstract Scatter Plots are one of the most powerful and most widely used b techniques for visual data exploration. A well-known problem is that scatter Ming C. Hao plots often have a high degree of overlap, which may occlude a significant Umeshwar Dayal b portion of the data values shown. In this paper, we propose the general­ a ized scatter plot technique, which allows an overlap-free representation of Halldor Janetzko and large data sets to fit entirely into the display. The basic idea is to allow the Peter Baka ,* analyst to optimize the degree of overlap and distortion to generate the best­ possible view. To allow an effective usage, we provide the capability to zoom ' University of Konstanz, Universitaetsstr. 10, smoothly between the traditional and our generalized scatter plots. We iden­ Konstanz, Germany. tify an optimization function that takes overlap and distortion of the visualiza­ bHewlett Packard Research Labs, 1501 Page tion into acccount. We evaluate the generalized scatter plots according to this Mill Road, Palo Alto, CA94304, USA. optimization function, and show that there usually exists an optimal compro­ mise between overlap and distortion . Our generalized scatter plots have been ' Corresponding author. applied successfully to a number of real-world IT services applications, such as server performance monitoring, telephone service usage analysis and financial data, demonstrating the benefits of the generalized scatter plots over tradi­ tional ones. Keywords: scatter plot; overlapping; distortion; interpolation; smoothing; interactions Introduction Motivation Large amounts of multi-dimensional data occur in many important appli­ cation domains such as telephone service usage analysis, sales and server performance monitoring. Analysts want to know how much one attribute of the data set is affected by another. In 1984, William Cleveland et all wrote: The scatter plot is one of our most powerful tools for data analysis. It is still true that scatter plots (sometimes they are also called x-y diagrams) are one of the most common ways to visualize multidimensional data. Using scatter plots, we can identify the relationship between two attributes, clusters of points and outliers. However, todays scatter plots have a high degree of overlap, which obscures the true density of data values. William Cleveland et all already noted: Still, we can add graphical information to scatter plots to make them considerably more powerful. In his article, Cleveland introduced different types of enhancements including a combination with iconic techniques and a superposition of smoothing methods for enhancing the x-y axes dependency and scale-ratio. Clevelands ideas are great enhancements of scatter plots, but they do not solve the overlap problem of scatter plots showing large data sets. In exploring large data sets, the high degree of Distortion plays .an important role in avoiding overlap overlap in scatter plots is one of the most severe draw­ in scatter plots. Bueringll provides two user interac­ backs, which often causes correlations to be hidden or at tion techniques on a small screen: a geometric-semantic least difficult to observe. zoom that smooths transitions between overview and In this article, we address the overlap issue and propose detail; and a fish-eye distortion that displays the focus a generalization of scatter plots where the analyst can and context regions of the scatter plot in a single view. control the degree of overlap and distortion allowing the Keirn 12 uses pixel-based distortion to efficiently generate analyst to generate many different views for revealing cartograms for showing geography-related statistical patterns and relations from the data. In the Section information. 'Applications', we will illustrate the applicability of the Alternatively to distortions, many researchers suggested proposed technique. The potential of the technique alpha-blending, which uses the alpha-transparency of in revealing previously barely visible information will the color system to represent data pOints. As a result, be demonstrated by showing the power of combining highly overplotted areas have high opacity and sparse distortion and pixel placement on real-world data set. areas have higher transparency.13 In the book by Antony Our examples show that the strength of the proposed Unwin et ai,13 a number of interesting visualization methods lies first in the combination of two techniques, . techniques were introduced regarding scatter plots, such and second in the ability of the users to interactively as drawing overlapping points with slightly bigger sizes guide the progress between the original and the general­ and reducing the x and y axes by certain factors. ]MP 8 ized representation of the data. Software14 generates scatter plots with non-parametric density contours and marginal distributions to show where the data is most dense. Each contour line in the Related work curved shape encloses 5 per cent of the data. CarrIS uses a hexagonal-shaped symbol whose size increases mono­ To create a representation of an entire high-density scatter tonically as the number of observations in the associated plot without overlap, we need a visualization technique bin increases, and HexBin scatter plots1S determine the that places a large volume of data in the limited size brightness value of each HexBin cell depending on the of the display screen. Early successful high information number of data points in the cell. All three techniques, density displays were for example pioneered by Eicks Unwins distortion, Carrs binning and the HexBin visu­ SeeSoft system.2 Eick allows users to analyze up to 50 alization techniques, are close to the method presented 000 lines of code simultaneously by mapping each line of in this article. However, using these techniques analysts code into a thin row for finding interesting patterns. To are not able to see and access all data points, especially if explore the high-density display, Eick also provided Data the third variable mapped to color is of high importance. Visualization Slides3 to filter the information displayed In order to overcome this problem, scientists suggested on the screen. Users can use the slider to prune the visual to use jitter or minimal random noise, which aim at clutter and explore the display from several perspectives. avoiding overplotting, but result in random distribution In the meantime, many interesting approaches for large of data points, and the heterogeneous patterns of colored high-density displays have been developed. In ]erding data pOints might bias users' perception. IS and Staskos4 Information Mural developed a technique Also different interaction techniques are suggested to for displaying and navigating large information spaces solve the overplotting problem. Different zooming tech­ without filtering. Information Murals use gray-scale niques are commonly used. With these methods, users are shading and color along with anti-aliasing techniques to able to request more details by increasing the magnifica­ create a miniature version of the entire data set. ]erding tion of representation. The question of determining the and Stask04 are able to plot over 52000 sunspot values in optimal level of the resolution still remains a challenge. 13 a small display window. Users can navigate the detailed Additionally, scaling of the axis has been applied in many subset of the data from the miniature overview window fields to endow higher density areas with a larger space of the entire data set. and sparse areas with a lower space. Square root and Loga­ Later, Fish-Eye views and Degree of Interest techniques rithmic scaling to x- and/or y-axis is certainly very popular. try to integrate overview and detail windows. SpenceS and This method, however, is only applicable to the visualiza­ Furnas6 for example, enable users to focus on something tion when it represents an explicit feature of the data. The interesting in a large graph. Bertini and Santucci? use applicability of such scaling techniques is therefore very sampling techniques to localize a particular region of the limited. Figure 1 illustrates some examples of the methods display. Bertini and SantUCCi? allow the user to examine mentioned. The data used for the representations is the small data items in detail, avoiding a loss of informa­ Telephone Conference Data (N = 37788), described in tion in low-density areas while reducing overplotting in the Application section in more detail. The data set shows high-density areas. Ellis8 uses auto-sampling to reduce the the duration of the conference call on the x-axis, the overplotting in parallel coordinates and scatter plots. Simi­ charges for the call on the y-axis and the number of partic­ larly, Fua et ai9 uses hierarchical clustering techniqueslO ipants are mapped to color. First the (a) original data set to reduce clutter in several visualizations in the system. with a linear scaling, (b) logarithmic scaling on the x- and 302 , 0908 0 - - - ,- ~ - ._,- 19179 lr~r 79 111 f-- $988 .. lr ~f - I - -,.- r - .\ Ir - '9989 , . , 1 .-:-- -- - I . .' 100 •.' ;,r " U k - ~; ' -. i--- .-1-- -c- - - - ~.(,~ . t -- l- \'!1. 2.. "" "e- rr ·1U I ~( - - r - 'lD 1]"4.8 2G8615 401SO . 5311932 1171 080 ·815 1 11 191C 26 4&k 40 1l. 53 811 87412. 7 Counts ••••••••••••• eoo 1005 10000 10000 eooo eooo 6000 6000 •. • 4000 eo 4000 • ••I, • 2000 2000 ",-- ~/ 0 0 42.0 750.0 10000 20000 30000 40000 50000 60000 0 10000 2000030000400005000060000 Figure 1: Traditio nal techniq ues aiming to overcome the overplottin g problem appli ed on a Te lephone Service Usage data set. X-ax is represents the duration, y-axis represents the charges fo r the ca ll and number of participants are mapped to colo r. y-axis, (c) jitter enhanced, (d) zooming, (e) alpha-blending In addition, we incorporate the intelligent visual and finally the (f) HexBin approach are shown in Figure l. analytics queries 16 to perform root-cause visual analyzes.
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