# Multidimensional Data Exploration by Explicitly Controlled Animation

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## Abstract

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## 1. Introduction

- a novel representation of the view space based on a small multiple metaphor
- a set of interaction techniques to continuously navigate the view space and combine partial insights obtained from different views

## 2. Related Work

#### 2.1. Element-Based Plots

**Multidimensional projections**(MPs) are a particularly important type of element-based plots. Here, ${d}_{i}$ are nD points (as for table lenses, parallel coordinate plots (PCPs) [8,9], or scatterplots [10]), and $V\left({d}_{i}\right)$ are 2D points (as for scatterplots). Hence, MPs are as visually scalable and clutter-free as scatterplots, and more visually scalable and clutter-free than table lenses and PCPs. Moreover, MPs improve upon scatteplots since scatterplots are constructed by using only two of the n dimensions of D, whereas MPs are constructed by considering all n dimensions. As such, scatterplots visually encode the similarity of points ${d}_{i}\in D$ according to only two of their n dimensions, whereas MPs encode the similarity according to all n dimensions.

**Graph drawings**(GDs) are a second important type of element-based plots. In detail, a graph is a dataset $D=(N,E)$ with nodes $n\in N$ and edges $e\in E$. Both nodes and edges can have data attributes, thereby making D a multidimensional dataset. A graph drawing is a visualization $V\left(D\right)=\left(V\right(N),V(E\left)\right)$, where $V\left(D\right)$ is typically a 2D scatterplot of points $V\left(n\right)|n\in N$. $V\left(D\right)$ is typically created by embedding methods such as force-directed layouts [25] but also, as shown recently, multidimensional projections [26]. $V\left(E\right)$ can be a set of straight line segments $V\left(e\right)|e\in E$. However, $V\left(e\right)$ can also be 2D curves, as follows. Drawing large graphs (thousands of edges or more) with straight lines easily creates massive clutter which renders such drawings close to useless. One prominent method to simplify, or tidy up, such drawings is to bundle their edges, thereby trading clutter for overdraw—that is, many bundled edges will overlap in $V\left(E\right)$. This creates empty space between the edge bundles which allows one to easily follow them visually, thereby assessing the coarse-scale connections between groups of related nodes in $V\left(N\right)$ more easily than in an unbundled drawing. The drawback of bundling is that individual edges in $V\left(E\right)$ are harder to distinguish due to the overlap. Numerous edge bundling (EB) methods exist [27,28,29,30,31,32]. However, as a recent survey points out [7], no such method is optimal from all perspectives. For instance, some methods offer a very precise control of the shape and positions of the bundled edges, but only handle particular types of graphs [27,32]. Other methods can handle graphs of any kind and are very scalable but offer far less control on the resulting look-and-feel of the bundled drawing [28,29]. Yet other methods fall in-between the above two extremes but generate a very wide set of drawing styles [33].

#### 2.2. Navigating Multidimensional Data Visualizations

**Animation**is a prominent technique that supports exploring the parameter space $\mathcal{P}$ and the related view space $\mathcal{V}$. Animation has a long history in data exploration [34,35]. In this context, animation corresponds to a (smooth) change of the visual variables used to encode the input data $V\left(D\right)$ [1,36].

**Small multiples:**Explicitly showing a sampling of $\mathcal{V}$ partially addresses this issue. One can show a history of views $V\left({P}_{i}\right)$ obtained for various parameter settings ${P}_{i}$ used in the exploration so far. The history can be shown as a linear, grid-like, or hierarchical set of thumbnails depicting $V\left({P}_{i}\right)$, a metaphor also called ‘projection board’ [44]. By clicking on the desired thumbnail, the user can go back to the corresponding state ${P}_{i}$ and associated view $V\left({P}_{i}\right)$, and continue exploration from there. Approaches that utilize such a methodology are Ma’s Image Graphs [45] and Elmqvist et al.’s Data Meadow [37]. Most projection pursuit variants also fall into this category [42,46], every view ${V}_{i}$ being one deemed ‘interesting’ from the perspective of the projection pursuit index, a metric which typically measures the distance from a given projection to an uninteresting generic Gaussian-like scatterplot. However, a limitation of most projection pursuit variants is that, just like grand tour variants, they require attribute-based projections [44]. Projection pursuit methods have been recently enhanced by limiting the number of interesting views to $n/2$ [47].

**Preset controller:**In contrast to small multiples, Van Wijk et al. [57] show several values of specific parameter-sets ${P}_{i}$ by a 2D scatterplot $S=\left\{x\left({P}_{i}\right)\right\}$, where $x\left({P}_{i}\right)\in {\mathbb{R}}^{2}$ is the projection of the point ${P}_{i}$. Next, one can manipulate a point of interest $x\in {\mathbb{R}}^{2}$, or the scatterplot points $x\left({P}_{i}\right)$, and generate a corresponding parameter-set value $p\in \mathcal{P}$ by using Shepard interpolation of the values ${P}_{i}$ based on the distances $\parallel x-x\left({P}_{i}\right)\parallel $. While the preset controller is very simple to use and is scalable in the number of parameters $\left|P\right|$, it does not explicitly depict the view space $\mathcal{V}$ but, rather, only an abstract view of the parameter space $\mathcal{P}$.

**Direct manipulation:**Animation can be also controlled directly in the view space $\mathcal{V}$ rather than in the parameter space $\mathcal{P}$. For this, the user directly manipulates the depicted visual elements in $V\left(P\right)$ to modify the parameters P. Examples of such manipulations are deformation, focus-and-context, and semantic lens techniques, all of which typically linearly interpolate between two parameter-set values ${P}_{1}$ and ${P}_{2}$ and show the corresponding animation of $V\left({P}_{1}\right)$ to $V\left({P}_{2}\right)$. Such animations have been applied to large element-based plots such as bundled graphs and scatterplots [6,58]. Smooth real-time animations of large datasets have been made possible by using GPU-based techniques [59]. For multidimensional projections, we have the following direct manipulation techniques. Control-point-based projections, such as LSP [60], PLMP [61], generalized Sammon mapping [62], hybrid MDS [63], and LAMP [15], allow users to interactively (dis)place a small subset of $V\left(D\right)$, called control points, on the 2D view plane, after which they arrange the remaining points around these controls so as to best preserve the nD data structure. This effectively allows users to customize their projections, at the risk of creating visual structures that do not relate well to the data. Targeted projection pursuit (TPP) [64] allows users to drag elements in a multidimensional projection plot $V\left(D\right)$ to, for example, better separate classes. From the resulting scatterplot ${V}^{user}\left(D\right)$, it seeks the parameters P for an actual projection $V(P,D)$ that is close to ${V}^{user}\left(D\right)$. While powerful as an interaction mechanism, TPP limits itself to only linear projections. Recently, ProjInspector use a preset controller, where the k presets correspond to user-chosen DR projections ${V}_{i}\left(D\right)$, $1\le i\le k$. When one drags the point of interest in the controller, the tool generates a view $V\left(D\right)$ that blends all ${V}_{k}\left(D\right)$ by means of mean values coordinates interpolation [44]. ProjInspector is arguably the closest technique to the one we present here, as such, we will discuss the similarities and differences in detail in Section 5.

**Interaction techniques:**All the above-mentioned visualization techniques use a mix of interaction techniques to enable exploration. While these are not specific to multidimensional data exploration, it is worth mentioning them here, as they next allow us better placing our contribution. First, as already explained, the parameters P can be changed by means of classical GUIs: the elements of the view $V\left(D\right)$, the control points of a preset controller [44,57], and the control points of a projection [15,60,61,62,63] are changed by simple mouse-based click-and-drag. Techniques can be next classified as single-view or multiple-view. Single-view techniques are either interaction-less, e.g., some grand tour and projection pursuit variants, or require direct manipulation in the single view, as explained earlier. Multiple-view techniques display several views ${V}_{i}\left(D\right)$ of the data. If direct manipulation is implemented in one view ${V}_{i}\left(D\right)$, then the elements undergoing change should be updated in all other views ${V}_{j}\left(D\right)$ – a well-known technique under the name of linked views [65] or coordinated views [66]. Depending on the exact semantics of the views, either unidirectional or bidirectional linking can be used [67]. For instance, a preset controller is unidirectionally linked to the data view(s) it controls. Finally, brushing and selection are ubiquitous techniques for exploring the view space, by showing details of the data element under the mouse and click-and-drag (typically) to select a subset of $V\left(D\right)$ for special treatment, respectively [68]. We will use all these techniques in the design of our exploratory visualization in Section 2.2.

**Other approaches:**Visualization presets have also been investigated for graph datasets [69]. Separately, exploratory visualization approaches have used the view space in a foresighted manner to sketch possible next steps along the visual exploration path. Several such approaches exist, which can be subsumed by the term ‘visualization by example’ [70]. All such approaches allow one to select a desired view $V\left({P}_{i}\right)$ from a range of candidates, the main distinction being how these candidates are picked from the view space $calV$ and how $\mathcal{V}$ is presented to the user.

**Our proposal:**We combine several of the advantages, and reduce some of the limitations, of the above-mentioned techniques for navigating a view space $\mathcal{V}$ constructed from a high-dimensional dataset D by means of a parameter space $\mathcal{P}$, as follows:

- Genericity: We handle all types of element-based plots (Section 2.1), e.g., scatterplots, graph/trail drawings, and DR projections, in an uniform way and by a single implementation.
- View by example: We provide an explicit small-multiple-like depiction $V\left({P}_{i}\right)$ of the view space $\mathcal{V}$.
- Continuity: We allow a continuous change of the current view based on smooth interpolation between the small-multiple views $V\left({P}_{i}\right)$ without having to bother about understanding the explicit abstract parameter space $\mathcal{P}$. This allows generating an infinite set of intermediary views in $\mathcal{V}$.
- Free navigation: The view generation is in the same time controlled by the user (one sees along which existing views one navigates) and unconstrained (one can freely and fully control the shape of the navigation path).
- Ease of use and scalability: We generate our intermediary views by simple click-and-drag of a point in the view space; these views are generated in real-time for large datasets D (millions of elements).
- Control: Most importantly, and novel with respect to all approaches discussed so far, we propose a simple mechanism for changing only parts of the current view, while keeping other parts fixed. This enables us to combine insights from different views $V\left({P}_{i}\right)$ on-the-fly, to accumulate insights on the input dataset D.

## 3. Proposed Method

**Guided:**The set of preset views between which the user can choose $V\left({P}_{i}\right)$ is limited by construction, and depends on the dimensionality of the input dataset D.**Free:**The set of preset views is fully configurable by the user, who can choose any number and type of views in $\mathcal{V}$ to animate between.

**Implicit:**Once the transition (animation) between ${V}_{1}$ and ${V}_{2}$ is triggered by the user, the generation of intermediate views between ${V}_{1}$ and ${V}_{2}$ happens automatically (usually via some type of linear interpolation). The user can specify ${V}_{1}$ and ${V}_{2}$, but not the path in the view-space $\mathcal{V}$ along which the animation evolves nor can he slow/accelerate/pause the animation.**Explicit:**The user can choose the path along which the animation evolves, and also the speed thereof.

#### 3.1. Our Proposal

- Depict the view presets $V\left({P}_{i}\right)$ by a simple grid-like small-multiple metaphor (as in [55]). This way, users see directly which visualizations $V\left({P}_{i}\right)$ they can interpolate between, rather than seeing the more abstract parameters ${P}_{i}$ that would generate these visualizations (as in [57]).
- Allow one to freely sketch the interpolation path between two such view presets $V\left({P}_{i}\right)$, in an interactive and visual way (as in [57]), rather than automatically controlling the interpolation via a linear formula.

#### 3.2. Implementation Details

## 4. Applications

#### 4.1. Multidimensional Projections

#### 4.1.1. Software Dataset

#### 4.1.2. Segmentation Dataset

#### 4.2. Bundled Graph Drawings

## 5. Discussion

**Generality:**Our technique can be applied to any so-called element-based plots, i.e., visualizations that consist of a (large) number of simple geometric objects such as points or lines. Scatterplots, projections, graph drawings, Cartesian uniformly-sampled fields, and parallel coordinate plots (the latter two not illustrated in this paper) fall into this class. All that one needs is a number of such plots expressed as a set of 2D primitives.

**Scalability and simplicity:**Our technique is simple to implement, requiring only the application of Equtaion (1) on the sets of 2D primitives corresponding to the preset views. Our current CUDA-based implementation can handle interpolation of datasets of roughly 10 preset views, each having about one million points, in real time on a modern GPU.

**Ease of use:**The technique is fully automatic, requiring no user parameter setting, apart from organizing the available preset views in the thumbnails matrix—which can be done by a simple click-to-place process. Apart from that, all interactions are done via mouse dragging (to control the animation) and brushing (to lock or unlock elements in the current view).

**Related techniques:**We have discussed several related techniques in Section 2. It is insightful to visit these after the presentation of our own method to pinpoint similarities and differences.

**Limitations and open points:**Our technique has, however, several limitations, as follows.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Animation design space between two data views ${V}_{1}$ and ${V}_{2}$. The user can control or not the transition (controlled vs. automatic), and the transition is defined by the user or predefined by the tool (explicit vs. implicit). The presented tools are Histomages [71], Rolling-the-Dice [39], FromDaDy [58], and our proposal.

**Figure 2.**Proposed animation-based exploration pipeline. See Section 3.1.

**Figure 3.**Thumbnail grid and view interpolation. Each image shows a set of five preset views ${V}_{1}\dots {V}_{5}$ arranged on a five by five grid. The views are simple color pixels for illustration purposes. Each grid pixel is colored to reflect what the interpolated view would be at that position. (

**a**) Initial placement of the preset views; (

**b**) Effect of dragging the views along the arrows shown in (

**a**); (

**c**,

**d**) Effect of setting $p=1$ and $p=0.5$, respectively. The grids (

**a**,

**b**) use the default $p=2$.

**Figure 5.**Software quality dataset (Section 4.1.1). No projection in the view space is able to separate well repositories of one kind from those of other kinds.

**Figure 6.**Segmentation dataset (Section 4.1.2). Different views can separate well one or more of the classes, but no view succeeds this for all classes.

**Figure 7.**Creating a mixed view which separates well the six classes present in the segmentation dataset.

**Figure 8.**Extracting relevant structures from four types of bundling algorithms to create a new bundling view for a graph dataset.

**Figure 9.**Understanding differences between several EB techniques. The transition between KDEEB-cluster and any other technique produces fuzzy intermediate states (1 and 2), while the transition between SBEB and the othe techniques produces sharper images (3 and 4). This shows that KDEEB-Cluster is visually very different from the other clustering techniques considered.

**Figure 10.**Bundling relaxation provided for four bundling techniques (presets in the corners of the grid) by our animation technique.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kruiger, J.F.; Hassoumi, A.; Schulz, H.-J.; Telea, A.; Hurter, C. Multidimensional Data Exploration by Explicitly Controlled Animation. *Informatics* **2017**, *4*, 26.
https://doi.org/10.3390/informatics4030026

**AMA Style**

Kruiger JF, Hassoumi A, Schulz H-J, Telea A, Hurter C. Multidimensional Data Exploration by Explicitly Controlled Animation. *Informatics*. 2017; 4(3):26.
https://doi.org/10.3390/informatics4030026

**Chicago/Turabian Style**

Kruiger, Johannes F., Almoctar Hassoumi, Hans-Jörg Schulz, AlexandruC Telea, and Christophe Hurter. 2017. "Multidimensional Data Exploration by Explicitly Controlled Animation" *Informatics* 4, no. 3: 26.
https://doi.org/10.3390/informatics4030026