Pascal Goffin
email: pas-cal.goffin@inria.fr

Jeremy Boy

Wesley Willett
email: wesley.willett@ucalgary.ca

Petra Isenberg
email: pe-tra.isenberg@inria.fr

OF VISUALIZATION AND COMPUTER GRAPHICS

Keywords: scale visualization, word-scale graphic, text visualization, sparklines, authoring tool, information visualization. !

We contribute an investigation of the design and function of word-scale graphics and visualizations embedded in text documents. Word-scale graphics include both data-driven representations such as word-scale visualizations and sparklines, and non-data-driven visual marks. Their design, function, and use has so far received little research attention. We present the results of an open ended exploratory study with 9 graphic designers. The study resulted in a rich collection of different types of graphics, data provenance, and relationships between text, graphics, and data. Based on this corpus, we present a systematic overview of word-scale graphic designs, and examine how designers used them. We also discuss the designers' goals in creating their graphics, and characterize how they used word-scale graphics to visualize data, add emphasis, and create alternative narratives. Building on these examples, we discuss implications for the design of authoring tools for word-scale graphics and visualizations, and explore how new authoring environments could make it easier for designers to integrate them into documents.

INTRODUCTION

I N this article, we explore a variety of applications for word- scale graphics (e. g. , , , Ca) embedded in text documents. We focus on how authors can use these graphics to emphasize elements in the text and to present data. Word-scale graphics can help support many forms of written discourse and can be used in a variety of different media. Some are data-driven; these have been referred to as sparklines [START_REF] Tufte | Beautiful Evidence[END_REF], micro visualizations [START_REF] Parnow | Micro visualizations: Data-driven typography and graphical text enhancement[END_REF], or word-scale visualizations [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF]. Others are not. As graphical supplements to written text, word-scale graphics can be used to emphasize specific elements of a document (like words or sentences), or to provide additional information. They can also enable visual comparison of data points, communicate complex relationships between elements of the text, or simply highlight values in the text to facilitate understanding.

In contrast to larger graphics and visualizations, word-scale graphics allow the reader to remain focused on the text, as they are set within the text, rather than alongside it. As such, word-scale graphics have the potential to deeply change the form and meaning of a written document, and designing and integrating them requires important editorial, rhetorical, and design decisions.

Previous examinations have focused mostly on contrasting data-driven word-scale graphics (or word-scale visualizations) with larger, more-fully featured data graphics [START_REF] Tufte | Beautiful Evidence[END_REF], [START_REF] Parnow | Micro visualizations: Data-driven typography and graphical text enhancement[END_REF], [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF], [START_REF] Victor | Explorable explanation[END_REF]. However, researchers have not explored the broader space of possible wordscale designs or potential uses for them. The rhetorical impact of word-scale graphics remains similarly unexplored, as is the process of authoring and integrating them with text. As a result, most guidelines on where and how to use word-scale graphics are world's population was 25%. [10] including carbon, calcium and phosphorus. based only on a very limited, curated, and strictly data-driven set of examples (such as Tufte's [START_REF] Tufte | Beautiful Evidence[END_REF]). We argue that these represent a mere fraction of the possibilities offered by word-scale graphics, and that many practical questions regarding their design and use remain unanswered.

To address this issue, we present the results of a study we conducted with nine graphic designers, in which we asked participants to annotate Wikipedia articles with word-scale graphics of their own design. We provide a rich categorization and analysis of the 200 resulting graphics 1 (see Fig. 1 for a subset), and we characterize the goals and rationales that informed these designs. We also report a number of challenges participants faced when 1. All designs with their associated text can either be found in the supplemental material, or at http://aviz.fr/wsv/. creating word-scale graphics. We then derive from our study three typical use-cases for word-scale graphics, which we propose as high-level inspiration for authors and designers. Finally, we discuss a set of implications for future word-scale graphics authoring tools.

RELATED WORK

Research on the use, function, and design of word-scale graphics is still relatively rare. In this section, we review related work on small scale graphics, and we highlight existing use-cases.

Small Contextual Visualizations

Edward Tufte's sparklines [START_REF] Tufte | Beautiful Evidence[END_REF] are probably the best-known example of data-driven word-scale graphics. Tufte describes sparklines as small, intense, simple, word-sized graphics with typographic resolution. He provides several different designs, including line charts depicting financial or health data. Small "cellsized" sparklines modeled after Tufte's examples have been available in Microsoft Excel since 2010 [START_REF]Use sparklines to show data trends[END_REF].

Expanding on Tufte's definition, we have previously proposed the term word-scale visualizations [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF] to define a wider variety of small embedded data-driven graphics. Typically, word-scale visualizations are open to greater variations in size than sparklines, ranging from e. g. as small as a single letter to as large as a whole sentence or paragraph. In our previous work, we also introduced the notion of an entity-a specific word or word combination to which a word-scale visualization is associated-which we re-evaluate in this paper. We further discussed different placement options for word-scale visualizations within a text document [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF] and how word-scale visualizations can be augmented with interaction [START_REF] Goffin | Design considerations for enhancing word-scale visualizations with interaction[END_REF].

Recently, Parnow and Dörk introduced the term micro visualization [START_REF] Parnow | Micro visualizations: Data-driven typography and graphical text enhancement[END_REF], [START_REF] Parnow | Micro visualizations: How can micro visualisations enhance text comprehension, memorability, and exploitation?[END_REF] to describe data-driven word-scale graphics that are small in physical space and represent few data dimensions or points. This definition reduces the concept of word-scale visualizations to those with few data dimensions or points. The authors further discuss micro visualizations in the context of typography and typographic data visualizations such as FatFonts [START_REF] Nacenta | Fatfonts: Combining the symbolic and visual aspects of numbers[END_REF]. Similar to our current and previous work [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF], the authors describe characteristics of micro visualizations such as their placement in or next to text, and their connection to data.

While all three existing definitions vary slightly, the authors of the respective works all postulate that small data-driven graphics can "enhance" text content in particular through the addition of detail in context that could not otherwise be reported. Parnow and Dörk even postulate that micro visualizations can enhance the comprehension of text by making it more accessible, comprehensible, and memorable.

Word-Scale Visualization Designs

While evaluations on the benefits of word-scale visualizations are still relatively rare, researchers have previously proposed and tested some specific word-scale visualization designs. SportLines [START_REF] Perin | SoccerStories: A kick-off for visual soccer analysis[END_REF] , for example, show phases of player actions across a soccer field. Brandes et al. introduced the concept of Gestaltlines [START_REF] Brandes | Gestaltlines[END_REF] which are sparkline-sized data-intensive graphics designed to convey a general gestalt or overall form. The authors tested one Gestaltline design ( ) and found that untrained readers could quickly detect holistic patterns, outliers, and breaks in the data. ) can help readers assess the predictive power of models with binary outcomes.

A perfect model would correctly predict all actual outcomes of an event and produce a plot with all dark red stripes clustered to the right side (as in this example). Greenhill et al. discuss several examples and potential problems with the interpretation of their plots, but they did not conduct a user study.

While the first three discussed above are meant to be used within a text document, other designs have been proposed for noncontinuous documents such as tag clouds or tables. For example, SparkClouds [START_REF] Lee | SparkClouds: Visualizing trends in tag clouds[END_REF] ( medical medical data ) convey temporal trends for words in a tag cloud by attaching small line charts to each word. In their original paper, Lee et al. compared SparkClouds to three other trend visualizations, and found that participants could read SparkClouds comparatively quickly across several tasks, and preferred them over other representations. Nguyen et al. [START_REF] Nguyen | Visualizing tags with spatiotemporal references[END_REF] follow a similar approach, and propose several tag designs that use the background of a word to convey temporal information, while placing tags on a map to further convey geospatial relationships. Beck et al. [START_REF] Beck | Word-sized eye tracking visualizations[END_REF] propose word-sized eye tracking visualizations ( ) which are meant to be used alongside think-aloud transcripts in a tabular view. The authors re-analyzed data from a previous small user study using the new graphics and show how their observations could be reported using word-sized eye tracking visualizations inline.

Finally, our work relates to the body of research on data glyphs which are data-driven visual marks that "make use of different visual channels to encode multiple data dimensions" [START_REF] Fuchs | A systematic review of experimental studies on data glyphs[END_REF]. While not always the case, glyphs are often small-scale and used in smallmultiple display settings. Several overview papers and chapters have been published on data glyphs [START_REF] Fuchs | A systematic review of experimental studies on data glyphs[END_REF], [START_REF] Borgo | Glyph-based visualization: Foundations, design guidelines, techniques and applications[END_REF], [START_REF] Chung | Glyph-based multi-field visualization[END_REF], [START_REF] Ropinski | Survey of Glyph-based Visualization Techniques for Spatial Multivariate Medical Data[END_REF], [START_REF] Ward | A taxonomy of glyph placement strategies for multidimensional data visualization[END_REF], [START_REF] Ward | Multivariate data glyphs: Principles and practice[END_REF] that focus on different characteristics of their design space. The work of Fuchs et al. [START_REF] Fuchs | A systematic review of experimental studies on data glyphs[END_REF] is informative for our investigation, as it summarizes user studies on data glyphs, and points out the many open questions regarding the performance of representations drawn at small sizes-that similarly apply to word-scale graphics.

Word-Scale Graphics

Almost all of the previous work, including our own, has focused on data-driven graphics. In fact, even the study reported here began with this assumption. However, small pictogrammatic, iconic, and symbolic graphics have been used alongside text for centuries. Galilei's drawings of Jupiter and its moons (

[21]) are one famous example. Emoticons ( ) are a more recent example of iconic representations that are incredibly common in computer mediated communication. Dresner and Herring describe emoticons' rich set of functions within written communication that goes well beyond the expression of emotion [START_REF] Dresner | Functions of the nonverbal in CMC: Emoticons and illocutionary force[END_REF]. Children's books are another place where iconic graphics are sometimes used to replace words in a sentence to help beginning readers as in "Benny's Big Bubble" by J. O'Connor [START_REF] O'connor | Benny's Big Bubble[END_REF]. Designers have also used symbols or icons as word-scale graphics to tag words in text to provide meta-information. Verjat [START_REF] Verjat | Semantic word-scale graphics[END_REF], for example, uses a human profile icon to make names stand out on his online CV. Given the frequent use of non-data-driven small graphics in text documents, we introduce the term word-scale graphics throughout this paper. The term refers to the broader category of graphics that includes both data-driven and non-data-driven graphics.

Previous work on word-scale graphics has focused mainly on the design of these graphics. However, there is little understanding related to how, when, and why such graphics can or should be used. Our main focus in this article is thus the authoring of word-scale graphics, and in a lesser state, their integration in

STUDY: DESIGNING WORD-SCALE GRAPHICS

To explore this diversity of possible uses and designs of word-scale graphics, we conducted a qualitative study in which we asked nine professional graphic designers to graphically annotate existing documents. We chose Wikipedia articles as the documents to annotate, because they provide a realistic scenario and environment for the integration of word-scale graphics. We designed the study expressly to elicit new and interesting possible designs for wordscale graphics, and to explore how designers might use them to enhance existing documents and engage readers.

Participants

We recruited nine designers (4 male, 5 female) via word-of-mouth. All designers had formal training in visual communication and storytelling and were actively employed in a variety of related fields. In addition, we ensured that participants were expert users of Adobe Illustrator (mean = 8.5 years using Illustrator, SD = 2.60), with professional experience through their work or field of study (mean = 6.6 years as a practitioner, SD = 3.97). The expertise in Illustrator was necessary to allow flexible digital manipulation and generation of text and graphics. Fig. 2 gives details on the participants' occupations and age ranges. All were graduates or did a post-master at a prominent French design school.

We compensated participants with a 20 e Amazon gift certificate.

Task

We provided participants with a Wikipedia article in Adobe Illustrator format. Participants had to read the article carefully, and envision how small visualizations embedded in the text could provide a richer reading experience. We asked them to create small word-scale graphics, and embed them into the document using Illustrator. Participants repeated this task for three articles in total.

Dataset

We selected subsets of three English language Wikipedia articles for participants to annotate. Fig. 3 shows a high-level overview of the curated articles-the full documents can be found in the supplemental material. We chose these articles for their high density of various data types and diversity of sub-topics. Taken together, the articles contained examples of the seven data types highlighted Fig. 3: Thumbnails of the three Wikipedia articles. From left to right: "Regions of France", "Europe", and "Human Body".

by Shneiderman [START_REF] Shneiderman | The eyes have it: A task by data type taxonomy for information visualizations[END_REF], including 1D, 2D, 3D, multi-dimensional (MD), temporal, hierarchical (tree), and network data. We used the shortest article on the "Regions of France" as the starting article. It consisted of three short paragraphs describing the administrative regions of France and the history of their creation. At the bottom of the article, a large table provided basic information about the regions, including flags, capitals, and current presidents. To the right, an infobox included a schematic map of France and additional facts, such as the populations of the largest and smallest regions. This article had only a limited set of data types: 1D, 2D, temporal, and MD.

The second text was an article on "Europe". It contained five general paragraphs about the continent, followed by separate sections on Europe's history, economy, languages, climate, biodiversity, and the creation of the European Union. Each of these sections included one image or data table. At the beginning of the article, an infobox provided data on various indicators (area, population, etc.) and a map. In total, the "Europe" article contained 1D, 2D, MD, temporal, hierarchical, and network data.

The third article, the "Human Body", contained three introductory paragraphs followed by sections about the body's structure, its physical composition, and its anatomical systems. The article also included a large table at the bottom, which provided more detailed information about individual anatomical systems. Each section had at least one associated image. Unlike the other articles, this article contained 3D data (including diagrams of the human body). It also contained 1D, MD, hierarchical, network data.

For each article we created an Adobe Illustrator file. To improve performance and make it easier for participants to manipulate the document, we grouped the characters in each line of text together as a single rasterized object. As a result, the smallest movable units in each document were lines of text, rather than words or characters. This allowed designers to concentrate on designing the graphics rather than manipulating and rearranging the text.

Instead of the "Human Body" article, our very first participant (P0) used a Wikipedia article on the city of "Zurich" which contained no 3D or network data. Despite this change, we chose to include P0's data in our analysis as it did not significantly differ in quality or creativity from the other data we collected.

Study Procedure

One experimenter was present throughout each design session. The study started with a questionnaire, in which we asked about participants' age and experience. Next, the experimenter gave a short presentation on word-scale visualizations, showed past examples of word-scale visualizations, and explained the study procedure. After this presentation, participants started the study with the article "Regions of France" ("Zurich" for P0). For 15 minutes, we allowed participants to familiarize themselves with the text of the article, generate initial ideas, and start to instantiate them with Illustrator. In order to reduce the overall study time, we allowed participants to approximate the actual data, rather than trying to retrieve and reproduce exact values in their designs.

After a short break, participants then started with one of the two larger remaining Wikipedia articles. We counter-balanced the order of the larger articles between participants to mitigate fatigue effects. We gave participants as much time as they wanted to familiarize themselves with each larger article. For the next 20 minutes, participants created and integrated their graphics into the article using Illustrator. We instructed participants to insert graphics into the interline space of the main text, but not inside of tables, images, captions, infoboxes, or the navigation bar. After 20 minutes of design work, we gave participants the option to spend 5-10 additional minutes finishing their designs. All participants asked for at least 5 more minutes. We then allowed participants another short break before moving on to the second article, following the same procedure.

At the end of the study we debriefed each participant, and asked them to explain each of their designs, describing their intended message and where the data originated. We also discussed the rationale behind our study, and asked for any additional thoughts on the subject. Finally, we asked participants to describe how future authoring tools might make it easier for them to create and integrate graphics and text. On average, participants took roughly 2.5 hours to complete the entire study.

Set-Up and Materials

We conducted the study in a small office with tables and chairs for up to three people. We provided blank sheets of paper for notetaking or quick idea sketches. Each participant brought their own laptop with Adobe Illustrator already installed. We asked them to do the study on their own laptop so that they would have access to their own customized Illustrator configuration. We provided participants with a USB flash drive with the three curated Wikipedia articles, as well as a PDF copy of the slides we used to explain the study and task. All participants were conversant in English, and used English when including text in their word-scale graphics, but completed the interviews and demographic questionnaire in French.

Analysis Procedure

During our analysis, we first extracted all word-scale graphics produced by our participants. For each graphic, we took a snapshot containing the entire illustration, along with any words or sentences associated with it. Two of the authors then printed and clustered these snapshots using an open-coding approach, then refined the clusters through several rounds of discussion with all four authors. This process helped us identify a number of dimensions along which to compare and categorize word-scale graphics.

RESULTS

We collected exactly 200 different word-scale graphics-a mean of 22 (SD = 10) per participant (see Fig. 4). In order to better understand the broader space of possible word-scale designs or potential uses for them, we coded all graphics according to: 1) whether or not they encoded data, 2) what types of visual representation they used, 3) how they were connected to the text, 4) how they supported comparison and provided context, 5) what interaction they suggested, and 6) the designer's intent.

Our main focus was to describe the diversity of designs, with a particular attention to the designers' editorial and rhetorical choices.

Data vs. Non-Data-Driven Encodings

We first examined whether the graphics in our collection represented quantitative data or more conceptual ideas. We coded graphics that used one or more visual variables (position, length, hue, value, . . . ) to encode data as data-driven. We refer to the 79.5% (159/200) of word-scale graphics that were data-driven as word-scale visualizations. The remaining 20.5% generally represented broad concepts, ideas, or messages not backed by abstract, quantitative data (see Fig. 5).

Data Provenance

Having determined which word-scale graphics were data-driven, we examined the provenance of the data they used. We found that data could come from one or multiple sources. Word-scale visualizations with just one data source took their data either from the associated sentence (49.7%; 79/159), or from somewhere else in the article (1.9%; 3/159). Graphics with multiple sources either included data from the associated sentence + elsewhere in the article (11.3%; 18/159), or from the associated sentence + outside the article (37.1%; 59/159). While in theory word-scale visualizations can also visualize data exclusively from outside the article, we found no examples of this in our collection.

Types of Data

We then coded the types of data used in our word-scale visualizations. While we designed the three Wikipedia articles to contain all of Shneiderman's data types [START_REF] Shneiderman | The eyes have it: A task by data type taxonomy for information visualizations[END_REF], our coding deviated from his set to make more fine-grained distinctions. From Shneiderman's set we kept locations, temporal data and relations, but added proportions & cumulative quantities, absolute quantities, rankings, and change, variation, or trends (see Fig. 6).

Proportions, cumulative quantities: , , . Graphics in this category-such as stacked bars or pie charts-put , . Rankings were present in 15 (9.4%) designs. These showed ordered sets, in which typically a focus object was highlighted.

Relational data:

. A single design (0.1%) included relational data depicting connections between different systems in the human body.

Design of Word-Scale Representations

We coded the types of visual representations used in our word-scale graphics, and noted whether or not designers followed any specific process in making their designs.

Types of Visual Representation

We found examples of quantitative abstract visualizations, geographic or schematic maps, icons and symbols, and illustrations (see Fig. 7). Quantitative abstract visualizations:

, . The 70.5% (141/200) of word-scale graphics in this category

France is divided into 18 administrative regions (French: région, [ʁe.ʒjɔ]), 13 of which are in Metropolitan France and five of which are overseas regions. [1] The mainland regions and Corsica are each further subdivided into departments, ranging in number from 2 to 13 per region for the Fig. 8: Example of an implementation of a reusable graphical system. The square stands for a region unit and the circle for a department. A bar chart and a trend visualization are assembled from these units. used visual variable(s) like length, position, or color to encode quantities-which were typically numbers drawn from the text. Geographic and schematic maps:

, . 8% (16/200) of the word-scale graphics were geographic and/or schematic maps. We coded these separately from quantitative abstract visualizations, as it was often unclear whether they actually encoded quantitative data (beyond the geo-coordinates of regional boundaries). For example, N is a schematic representation of the topological surroundings of the city of Zurich-from left to right: a mountain, a lake, and a forest. Icons and symbols:

, , . Of the word-scale graphics, 18% (36/200) did not encode any quantitative data at all. Instead, they replaced a single concept or word in the text. Some icons and symbols directly resembled the object they referred to (such as a standing for cranial bones), while others described an entity-these needed to be learned to be understood (for example, standing for France). Icons and symbols were also sometimes used as a part of quantitative abstract visualizations. Here, they had a secondary function to help relate the shown quantities to known ones [START_REF] Chevalier | Using concrete scales: a practical framework for effective visual depiction of complex measures[END_REF], or to give contextual cues. For example in the standard heights of men and woman are related to a "known" height of a cow (according to designer who created the graphic). In a second example , the graphic shows a ranking with a human figure on the left signaling that the graphic shows population data. Illustrations: , <(portrait de mitterrand) PS . This group of 7 (3.5%) word-scale graphics included more complex diagrams and sketches. Often, these illustrations tried to convey a message. For example, the following small graphic , shows two human skeletons with one sitting in a wheelchair. The designer wanted to convey the message that people with disabilities are often not considered in articles on the human body.

Reusable Graphical Systems

Several designers used a coherent visual theme for their wordscale graphics. They largely remained consistent in terms of coloring schemes and connection between entity and graphic (see Sect. 4.3.2). One designer (P0) went further, proposing a visual language based on the assembly of small data-driven units (see Fig. 8). This system consisted in introducing tokens (token grammar), and assembling them (assembly model) to express various quantities-similarly to the construction of unit-based visualizations discussed by Huron et al. [START_REF] Huron | Constructive visualization[END_REF].

Connections Between Text and Graphics

In our previous work [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF], we defined the notion of an entitya single word or a word sequence associated with a word-scale graphic. However, this definition did not specify in detail how entities and their corresponding graphics can be connected. As part of our study, we wanted to analyze these text-to-graphic connections in more detail and build a better understanding of how designers chose and communicated these relationships.

Size of the Associated Text

Designers often associated graphics with one-or multi-word entities. In the study results, we identified 139 entities, with lengths ranging from a single number or word up to the size of a sentence, paragraph, or document subsection (mean = 4.0 words per entity, SD = 4.9). One-word (39%; 54/139) and twoword (25.9%; 36/139) entities were the most common. However, the relationships between entities and graphics was not always oneto-one. Although 42.5% (85/200) of graphics were connected to a single entity, 9.5% (19/200) were connected to multiple distinct entities at different locations. We also found a number of wordscale graphics (48%; 96/200) that were not clearly associated with any specific subset of the text.

Type of Connection

The designers in our study linked 52% (104/200) of the graphics they created to one or more entities using a variety of different graphical techniques (see Fig. 9): Font / hyperlink color: In 49% (51/104) of the cases, a font color associated with a word served as the graphical connection between word-scale graphic and text. In 19% (20/104) of the cases, the designer altered the color of the text to create the mapping-often using color present in the graphic itself. In 30% (31/104) of the cases, designers used the existing highlighted Wikipedia links in the article as entities, sometimes even reusing the link color (dark blue) in the graphic to emphasize the connection.

Frame: The second most common technique (35%; 36/104) used a frame around associated words, and sometimes added a reference link between frame and graphic. Underlining: In 11% (11/104) of the cases, designers used a line drawn under an associated word to connect text to graphic. Background color: In 7% (7/104) of the cases, designers created a visual connection by changing the background color of the text. Another 48% (96/200) of our word-scale graphics were not linked to text by graphical techniques, but rather using spatial proximity. In most cases, designers either or center-aligned

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The European Union, a political entity composed of 28 European states, comprise largest single economic area in the world. 18 EU countries share the euro as a com currency. Five European rank in the top ten of the world's largest nationa economies in GDP (PPP).

Fig. 10: The word-scale graphic has been positioned to the left of the paragraph, showing the proportion of countries using the Euro [START_REF] Ropinski | Survey of Glyph-based Visualization Techniques for Spatial Multivariate Medical Data[END_REF] versus the number of countries actually in Europe [START_REF] Huron | Constructive visualization[END_REF]

Main article: Economy of Europe

As a continent the economy of Europe is currently the largest on Earth an Fig. 11: The word-scale graphic has been added to the title of the paragraph. It is a horizontal bar chart comparing the assets under management in Europe and the North America the word-scale graphics with a related word or phrase. In some cases, designers associated graphics with whole paragraphs or subsections. For example, in Fig. 10, P4 visually associated the word-scale graphic with a complete paragraph by placing the chart immediately to the left of the entire text block. Similarly, in Fig. 11, the same designer placed a word-scale graphic next to a section heading, effectively associating the graphic with the entire subsequent section. Two designers (P4, P8) added few (P4), or almost all (P8) of their designs below the entity. Several of these designs showed long vertical bars (for example in a stacked bar chart) that served as an underline in terms of its connection to the associated entity. In total, 14.5% (29/200) of the designs were positioned below the entity instead of above, but used the same general connection types as discussed above.

Legends and Captions

The design of small-scale visualizations allows little space to add captions or labels to explain a visualization. Interestingly, the designers in our study addressed this challenge in a variety of ways, summarized in Fig. 12.

The top graphic in Fig. 12, for example, shows how word-scale graphics themselves can serve as part of a legend to a larger graphic. Here, small icons added to the individual element names (carbon, calcium, phosphorus) label the elements of the stacked bar chart above. In cases such as this one, the larger graphic was

Type of Legend or Caption Examples

Word-scale graphics as legends including carbon, calcium and phosphorus.

Entities as legends including carbon, calcium and phosphorus

Entities as captions

the human body is Fig. 12: Several ways in which graphics and entities serve as legends or captions. Atlantic currents tempering winters and enabling warm summers on much of the continent, even on latitudes that have severe climates in North America and Asia. Further from the Atlantic, seasonal differences increase, but the mildness of the climate remains.

Europe, in particular ancient Greece, is the birthplace of Western culture. [7][8]9] The fall of the Western Roman Empire, during the migration period, marked the end of ancient history and the beginning of an era known as the "middle ages". The Renaissance humanism, exploration, art, and science led the "old continent", and eventually the rest of the world, to the modern era. From this period onwards, Europe played a predominant role in global affairs. Between the 16th and 20th centuries, European nations controlled at typically connected to multiple entities, which in turn had small associated iconic representations placed in close proximity to them. For example, the designer who introduced a token-based grammar for the design of her graphics (Sect. 4.2.2) used this mechanism to introduce the tokens of her grammar, placing individual instances of each token near the first occurrance of the corresponding term.

The second example in Fig. 12 shows the same text illustrated by another designer. The designer colored the element names to serve as the legend to the associated pie chart. In general, graphics with associated entities as legends included multi-entity associations, in which the entities were colored according to the elements of the graphic they were associated with.

These first two techniques use entity text to label parts of wordscale graphics. The last example on the bottom, instead, uses an entity as a caption to describe the overall content of the graphic, without giving any details about the individual stacked bars.

While Fig. 12 summarizes three common uses of legends and captions, designers also explored a range of other variations. In a few cases, designers even created compound captions and legends like the one shown in Fig. 13a. Here, the designer introduced a multi-line entity with nested caption and legend elements that help readers interpret the larger bar chart.

Finally, one designer also created an explicit legend in the margin explaining the types of graphics introduced in her article, including representations showing geographic, temporal, and population data.

Supporting Comparison and Providing Context

Designers also used a variety of strategies to support comparison between word-scale graphics in the document: Shared reference structure: Multiple designers used shared reference structures, including reference lines and grids which spanned multiple graphics. In one example (Fig. 14), P1 added a grid connecting two word-scale graphics on the same line such that the viewers could more easily compare them. In cases where graphics were further apart in the document, some designers created duplicate reference structures to preserve comparability. In Fig. 15, for example, P3 repeated the same reference grid and icons in two word-scale graphics, highlighting small differences between heights more clearly. Traces: Another static method we saw (P1, P4, P7) for comparing word-scale graphics used a graphical overlay or trace of one graphic in another (Fig. 16).

can vary from a high 75% in a newborn infant to a lower 45% in an obese p Sequencing: Several designers (P3, P5, P7) created sequences of word-scale graphics in which each graphic built upon the previous ones. For example, in Fig. 17, P7 placed a small timeline over each date in the text. Each successive timeline also included desaturated versions of all of the previous dates, helping to place the current date in context with the earlier ones. Other examples of sequencing included sets of maps which built upon one another, each containing elements of the previous one.

Interaction

We did not ask the designers to consider interaction when creating their word-scale graphics. However, during the post-study debrief several participants proposed ways of augmenting their designs with interaction: Connecting word-scale graphics: P6 suggested adding interactive brushing and linking to the graphics in Fig. 18-allowing readers to interactively explore the relationship between hormonal changes and the development of fat and muscle tissue. Fig. 17: Sequencing of timeline word-scale graphics.

shape (and female body shape) is influenced by the distribution of muscle and fat tissue and is also effected by various hormones.

react to the mouse cursor Fig. 18: Three connected word-scale graphics. Moving the cursor in one graphic results in highlighted values in the others (we added the red dots and the cursor to explain the interaction) land area. Of Europe's approximately 50 countries, Russia is by far the largest a and population, taking up 40% of the continent (although the country has both Europe and Asia), while Vatican City is the smallest. Europe is the third-Russie Fig. 19: Using interaction to overlay the word-scale graphic for one country on top of the graphic for another. (Sizes are approximateas estimated by the participant. We added the cursor and line for illustration.)

Comparing word-scale graphics: P3 proposed several techniques for bringing multiple word-scale graphics together to facilitate comparison. By dragging one word-scale graphic onto another (as in Fig. 19) a reader could overlay the two graphics on top of one other. Similarly, dragging one graphic next to another could facilitate side-by-side comparison. Both approaches would allow for dynamic comparison without the use of static support structures like those seen in the preceding section. Revealing word-scale graphics: P2 suggested using interaction to show word-scale graphics on demand-displaying graphics only in the paragraph that a reader is currently attending to. All word-scale graphics outside of the paragraph could either be hidden, desaturated, or scaled down to reduce visual clutter. Data interactions: Interaction could also be used to trigger filtering or highlighting on the graphics themselves. For example, P2 proposed using interaction with labels on the right side of the word-scale graphic in Fig. 20 to transition between alternative views. In this case, hovering over the labels for "population" and "size" would toggle which of the two data sets the chart displayed.

Designer's Goals

During their debrief sessions, designers described a diverse range of goals that guided the ways they designed and integrated word-scale graphics with the documents. Fig. 20: Word-scale graphic showing the contrast between population and surface area for the EU, Russia, and Vatican City.

using graphics to illustrate concepts (3 out of 9 designers), make quantities explicit (6 out of 9 designers), provide scales (3 out of 9 designers), or put information into context (4 out of 9 designers). One designer also suggested using word-scale graphics to provide more detailed information about the entity as a way to facilitate understanding.

Summarizing the content: Six designers expressed the use of word-scale graphics to help summarizing the content of a sentence. Designers discussed using word-scale graphics to provide a preview of the content or to expose the most important information from the text, providing readers with a clearer take-home message. Emphasis: Two designers mentioned using word-scale graphics to emphasize and draw attention to content they felt was particularly important. We were surprised that only two designers mentioned this explicitly, as the presence of word-scale graphics in text already provides a type of intrinsic emphasis effect [START_REF] Hall | Formalizing emphasis in information visualization[END_REF] where the process of mapping content to a graphic increases its visual prominence. In addition, many of the text-to-graphic connections (font color, frames, underlining, background color) give further extrinsic emphasis [START_REF] Hall | Formalizing emphasis in information visualization[END_REF] to the associated entities. Providing alternative representations: Two designers also explicitly stated that they used word-scale graphics to provide alternative representations as a supplement to the written text. They emphasized that introducing a redundant encoding of the information in the text could help make the text both more understandable and more memorable [START_REF] Borkin | Beyond memorability: Visualization recognition and recall[END_REF]. For example one of the designer suggested to add a visual representation of a cell-as known from Biology text books-to the entity "cell " in order to make it more memorable. Providing contrasting viewpoints: Interestingly, two designers also described creating graphics to express a contrasting viewpoint that differed from what was written in the text. Here, designers mentioned wanting to encourage readers to think more deeply about the text and consider alternative interpretations of it. For example, one designer proposed to add a map of the world showing where Europe was perpetrating slavery in the 16th century to provide a contrasting viewpoint on how Europe "controlled" the world at that time. Especially, pointing out that the word choice "controlled " seems too neutral.

DISCUSSION: USING WORD-SCALE GRAPHICS

Having developed a fine grained understanding of the different design dimensions of word-scale graphics, we here propose a higher-level definition of use-cases for word-scale graphics. We intend these to serve as inspiration and starting points for authors and designers who wish to integrate word-scale graphics into text documents. We believe this higher-level approach is generally more accessible for generating new designs and applications, in contrast to the low-level characteristics from Sect. 4, which are more suitable for describing existing examples. Furthermore, we reflect upon the related work in light of our study results, and propose an extension to the notion of associated entity. Moreover, we discuss challenges that arose during the study.

Applications of Word-Scale Graphics

We characterize three different ways in which designers can apply word-scale graphics to alter the rhetorical structure of a document. Specifically, word-scale graphics can support the content, extend the content, or go beyond the content.

Supporting the Content

From an editorial perspective, one of the simplest ways of applying a word-scale graphic is using it to support its associated entities. In these cases, word-scale graphics display data (left image) that comes directly from the entity text itself (shown above using a green arrow), or use non-data-driven symbolic representations (right image) that visually emphasize or reiterate the content of the text. Thus, when creating word-scale graphics that support the content, editorial choices are relatively restricted. The editor or designer needs to choose which entities to visualize or illustrate, how to design them, and how to place and connect them to the associated text. However, they do not need to locate or prepare additional data. For example, an editor could choose a number in the text as an entity and represent that value using a chart. Alternatively, an author could add a symbolic representation such as a flag to represent the name of a country already included in the text. Although they restate the existing content, these graphics can still provide useful emphasis-facilitating better understanding of numbers or concepts [START_REF] Borkin | Beyond memorability: Visualization recognition and recall[END_REF], providing summaries, or highlighting alternative versions of the entities (see Sect. 4.6).

Extending the Content

Word-scale graphics can also be used to provide contextual data that extends the ideas put forward by their associated entities. These word-scale graphics extend the content with compatible information (shown above using blue arrows), and do so without purposefully changing the message of the sentence, or proposing novel interpretations. Yet, the creation of word-scale graphics that extend content is somewhat more complicated than for the previous group-it involves finding data from elsewhere in the text or from outside the document. It can also involve searching for or creating non-datadriven graphics that provide additional information. For example, an author could extend a timeline connected to the date of a WWII battle by adding outside data such as the start and end dates of the war. Graphics that extend the content can help provide context or comparisons for data already contained in the text, or provide additional material about specific entities.

Going Beyond the Content

The previous two applications of word-scale graphics generally complement and reinforce the content of the original document. However, authors and editors can also use word-scale graphics that go beyond the content, including contrasting ideas and messages (shown above using red arrows) which may significantly broaden, annotate, or even subvert the original meaning. As a simple example, an editor can choose to add illustrations or symbolic representations that express an opinion, make a statement, or show emotion. For example, by adding a "thumbs up" icon next to an entity expressing an action, the editor can express agreement. Similarly, by including a word-scale graphic showing external data that contradicts the existing narrative, an editor or designer can suggest alternative, contrasting viewpoint (see Sect. 4.6). While these contrasting viewpoints may originate with the original author, they can also be added in response to the article at a later point.

Defining "Word-Scale"

As we alluded to in Sect. 2, prior work has offered several competing terms and definitions for describing small graphics and visualizations embedded in text. Here, we revisit these descriptions in light of our study results.

Tufte's definition of sparklines refers to them as "datawords" and describes them as "word-sized" [START_REF] Tufte | Beautiful Evidence[END_REF]. While many of Tufte's examples-such as his baseball win/loss charts-are much longer than typical words, most are still essentially word-height. In our study and elsewhere, we have observed many word-scale graphics that go well beyond this size limit. As such, our initial definition of word-scale visualizations [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF]-which allows graphics "as small as a single letter to as large as a whole sentence or paragraph"-better captures the variety of graphics we saw in the study.

Parnow and Dörk's work on micro visualizations [START_REF] Parnow | Micro visualizations: Data-driven typography and graphical text enhancement[END_REF], [START_REF] Parnow | Micro visualizations: How can micro visualisations enhance text comprehension, memorability, and exploitation?[END_REF] provides another definition for small embedded visualizations. Rather than focusing explicitly on size, their definition instead stresses simplicity-emphasizing that such visualizations contain few data dimensions or points. While we found a number of simple data-driven graphics that do fit this definition, we also found more data-intense graphics that would not be covered. As such, our definition of word-scale visualizations better captures the graphics we saw, as it does not explicitly restrict the number of data dimension or data points.

However, while our definition of word-scale visualizations seemed to be a good term to capture the data-driven graphics we saw in the study, it does not encompass the full range of non-datadriven graphics designers produced. Yet, our study clearly suggests that non-data-driven graphics at these scales can express a variety of design intents, and may be just as useful a communication tool as their data-driven counterparts. Our higher-level designation "wordscale graphics" captures both data-driven and non-data-driven variants, and serves as a unifying term for characterizing a wide range of small graphics that can be embedded in text.

What is an Entity?

In our previous work, we had assumed that each visualization would explicitly be associated or linked with a single entity (see [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF]), i. e. a single word or a sequence of words in the text. In this previous work, we had discussed how proximity-or placement of a wordscale visualization in close context to an entity-would establish this connection. However, participants did not always follow this association we had envisioned, which led us to re-evaluate our initial definition. For example, in addition to using proximity to establish the connection between a word-scale graphic and the text, they also linked multiple disconnected pieces of text to one word-scale graphic (see Sect. 4.3). Designers also created wordscale graphics without any explicit or implicit entity. In some cases, designers added word-scale graphics to the beginning of a paragraph, or after a title. They also added word-scale graphics vertically along a paragraph-effectively associating the graphics with the entire block of text.

A particularly interesting finding of our study showed that entities also served a function that went beyond the mere provision of metadata we had envisioned. Instead, they served a reference function, and were used as parts of legends.

As a result, we propose an updated definition of what an associated entity is. An entity is still a concrete piece of text that can have associated metadata, however, it also serves as a way to link the word-scale graphic to the text, and can be part of the word-scale graphic's legend, or of its caption. Word-scale graphics can have one entity, multiple entities, or no entities associated to them. If there is an entity, it is marked using for example font color or a frame. In cases where the word-scale graphic has no marked entities, it means the graphic is associated with the sentence or the paragraph, depending on its positioning.

Finally, it may be productive to more deeply consider the relationship between entities and word-scale graphics using the semiotic concepts of referent, signifier, and signified [31], [START_REF] Nöth | Handbook of semiotics[END_REF], [START_REF] De Saussure | Course in general linguistics[END_REF]. These semantic connections are particularly interesting, as entities and word-scale graphics sometimes represent the same concept, but may also introduce indirection or serve as an alternative representation of an underlying idea.

Challenges when Creating Word-Scale Graphics

One of the main challenges our participants had was to find suitable pieces of text they could complement with word-scale graphics. The process of deciding what should and could be represented usually happened at the very beginning of the process of integrating wordscale graphics. The designers also expressed concerns that the associated entities in the text had to be below a certain length to be representable or connectable to a word-scale graphic. We, therefore, updated our definition of an entity. A word-scale graphic can have one or multiple entities, or none, and proximity can be a way to link graphic and text. We argue that this updated definition provides more freedom to support the linking of word-scale graphics to the text. Entities can also serve multiple purposes-including acting as a label or caption for the graphic-a fact that was not recognized in previous work. Moreover, this definition explicitly acknowledges that word-scale graphics do not necessarily need an entity if it is clear what the word-scale graphic represents, and the connection to the text is clear.

In addition, and linked to this thematic of defining associated text, our participants had difficulties finding suitable places to position their word-scale graphics. In our instructions, we restricted the positioning of word-scale graphics to the interline space. However, designers still hesitated at times, and sought alternative placement options-for example at the end of a title or along a paragraph (see Sect. 4.3). One participant also told us that she would have liked to rewrite the sentences to make it easier for her to integrate word-scale graphics. She found the sentences were too convoluted, which led her to find it difficult to find suitable positions for her word-scale graphics. Moreover, some of the information or data that the designers wanted to integrate into the word-scale graphics was spread over multiple lines. This made it more difficult to find good places to position graphics. Sometimes, designers also started to create a graphic first, and then had a difficult time finding suitable text to associate with the word-scale graphic. As Sect. 4.3 and Sect. 4.4 show, there are multiple ways to link entities to a word-scale graphic. Future authoring tools for word-scale graphics should take these possibilities into account and support different placement options and linking possibilities.

However, simple authoring tools could make the process of creating and integrating word-scale graphics considerably easier by streamlining or even automating portions of the creation and placement process.

Based on observations from our study, as well as our own experience in designing word-scale graphics, we identify four major decision points at which new tools could support the process of creating and integrating a graphic. In order to add a word-scale graphic to a document, designers or authors must specify the design of the graphic, as well as any entities to which it corresponds, the data (if any) it displays, and its placement relative to the text. Our study suggests a number of important takeaways for the design of each of these operations:

Designing the Word-Scale Graphic: An authoring tool needs to offer some mechanism for selecting existing word-scale graphic designs, or for creating new ones. As part of this process, an author may need to make an explicit choice between data-driven and non-data-driven graphics-since datadriven graphics may require additional design choices. In our study, a large portion of the designs used familiar chart types, and nearly a third (31%) of the visualizations were unit-based. This suggests that support for generating simple, scalable visualization designs based on basic geometric shapes could support a large number of the data-driven design alternatives that we observed.

In contrast, non-data-driven designs in our study were quite diverse. Allowing authors to insert appropriately-sized icons, images, and cliparts from existing libraries may cover a considerable number of cases. However, more fully-fledged image editing tools would be necessary to create many of the designs we observed. Selecting an Entity: While a word-scale graphic does not always need to be associated with text, authors in our study chose to connect the majority of their graphics to one or more entities. This suggests that it can be useful for tools to provide a mechanism for associating a word-scale graphic with one or more blocks of text. These associations can make it easier to position the graphic, and can also provide context for interpreting it. Selecting Data: Our study also suggests that, in many cases, the data needed to populate a design is actually contained in the associated entity (see Sect. 4.1.1). This means that data selection and entity selection can often be treated as a single operation. However, the ability to link data from outside an entity or from outside a document can support a richer and more nuanced use of word-scale graphics (see Sect. 5.1). For non-data-driven word-scale graphics the data selection component can be bypassed. Placing the Word-Scale Graphic: While selecting an entity specifies a relationship between that text and the associated graphic, it does not necessarily determine the graphic's placement relative to the text. In previous work [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF] we discussed options for placing word-scale graphics in text documents, and explored the trade-offs associated with placing graphics immediately above an entity, in-line with the entity, or overlapping the graphic and text. Authoring systems could build on this work, showing possible placement options for each graphic and analyzing how they will impact the visibility of the graphic and the flow of the text.

More generally, tools for authoring word-scale graphics in text could take different forms depending on the design expertise and investment of the authors. For writers with little interest in customizing designs, mechanisms for including simple word-scale graphics could be integrated into existing text entry interfaces. For example, an interface could allow authors to create wordscale graphics using markup embedded directly in the document text (akin to entering equations in L A T E X), or via in-editor search interfaces, like those used to select and insert emoji in most modern operating systems. Such tools could make a number of simplifying assumptions, including supporting a limited set of chart types, and reduced control over their placement. Libraries like Peity [START_REF] Pickles | Peity[END_REF], which generates simple word-scale SVG charts based on markup included in HTML documents represent a first step towards this sort of minimalist authoring.

Alternatively, the process of adding very basic word-scale graphics to documents could potentially be automated entirely. For example, a system could generate and place graphics automatically next to known entities like dates or stock symbols. In this case, decisions about the design, data, and placement might be made in advance by a developer, then applied to new pages with little or no user interaction.

However, more expressive, unique, customized, or interactive designs like those described by designers in our study likely call for interfaces that provide greater control. These might include the ability to create new visual forms using image editing or coding tools, as well as greater control over the relationship between the text and graphics.

CONCLUSION, LIMITATIONS, & FUTURE WORK

We presented the results of an exploratory study on the use of word-scale graphics in text documents. The study resulted in a rich collection of different types of graphic designs and also helped us to identify specific design challenges unique to word-scale graphics. Unlike larger visualizations, designers creating word-scale graphics must consider the immediate connections between visualizations and text, and also find alternatives to common visual elements like axes, legends, and captions that work given reduced space. Designing interactions also presents a challenge, since graphics may be dispersed, small, and only loosely associated with one another. At the same time, our results also illustrate a variety of ways in which word-scale graphics can enrich text documents and also suggest implications for the design of future authoring tools.

Our overall goal was to improve our understanding of how, when, and why one might use word-scale graphics in text documents. Yet, the results of our study and our discussion need to be interpreted in light of several specific study design decisions. In this paper, we focused on the design of word-scale graphics and not on their placement. To give participants enough study time to focus on design, we only allowed them to place graphics between lines of text. This made placement easier, since designers did not have to shift individual words or reflow text and could instead focus on creating interesting new designs. To increase opportunities for creative designs, we allowed participants to freely adjust the amount of inter-line space, allowing them to explore graphics with a greater variety of different heights, widths, and positions relative to the text.

Participants were not able to edit the text itself, and thus did not have full control over the phrasing, language, or organization of the content they were building on. In the post-session interview, participants had a chance to express their ideas and concerns about the placement of their word-scale graphics. Only one of our participants mentioned that she felt her design had been restricted by the fixed text-noting that she would have liked to rewrite sentences to better fit her word-scale graphics and use graphics inline as a legend. Thus, while 8/9 participants did not report design restrictions due to the fixed placement, it is likely that our results do not show the full spectrum of creativity that the participants could have applied to word-scale graphic placement.

Two further factors influencing the reported designs are the choice of text and the restricted study time. Wikipedia articles are usually much more descriptive and data-driven than articles, novels, poems, or other forms of text. Participants also did not have time to make multiple iterations over their designs, as we asked them to maximize the diversity of their designs rather than produce a few refined ones. Therefore, the designs should be seen as initial ideas, which would require iteration and further development before their effectiveness could be tested.

Overall, we believe this work serves as a strong starting point for considering how word-scale graphics might complement a variety of types of text documents. Building on the work presented here, we plan to study several aspects of word-scale graphic integration more closely. In particular, we plan to focus on interfaces that will make it easier to design and integrate wordscale graphics into documents, both during and after authoring the text. Ultimately, we hope that these kinds of tools can help lower the threshold for authoring and including word-scale graphicsmaking integrating text and other word-scale representations as fluid as writing text is today.

  both in terms of weightThe vast majority of cellsEuropeEurope is the world's second smallest continent North America's $27.carbon, calcium and phosphorus.

Fig. 1 :

 1 Fig. 1: Examples of word-scale graphics collected in our study.

Fig. 2 :

 2 Fig. 2: Participant details

Fig. 6 :Fig. 7 :.

 67 Fig. 6: Levels of types of data, their counts and examples.

Fig. 9 :

 9 Fig.9: Types of explanatory structures for word-scale graphics.

Fig. 13 :

 13 Fig. 13: A nested legend and caption explains the top bar chart in (a). The right image (b) shows an example of an explicit legend created and placed in the margin by a designer.

Fig. 14 :Fig. 15 :

 1415 Fig. 14: Connecting multiple graphics with a support grid.

Fig. 16 :

 16 Fig. 16: Overlaying a trace of one graphic over another.

  Europe ( i/ˈjʊərəp/ or /ˈjɜːrəp/[START_REF] Victor | Explorable explanation[END_REF]) is a continent that comprises the westernmost part of Eurasia. Europe is bordered by the Arctic Ocean to the north, the Atlantic Ocean to the west, and the Mediterranean Sea to the south. To the east and southeast, Europe is generally considered as separated from Asia by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways of the Turkish Straits.[START_REF]Use sparklines to show data trends[END_REF] Yet the borders of Europe-a concept dating back to classical antiquity-are arbitrary; as the primarily physiographic term "continent" also incorporates cultural and political elements.Europe is the world's second-smallest continent by surface area, covering about 10,180,000 square kilometres (3,930,000 sq mi) or 2% of the Earth's surface and about 6.8% of its land area. Of Europe's approximately 50 countries, Russia is by far the largest by both area and population, taking up 40% of the continent (although the country has territory in both Europe and Asia) while Vatican City is the smallest. Europe is the thirdmost populous continent after Asia and Africa, with a population of 739-743 million or about 11% of the world's population.[START_REF]Use sparklines to show data trends[END_REF] Europe has a climate heavily affected by warm

		Geographic elements
	The average adult body
		Temporal elements
	contains between 5 and 5½	Population elements
	litres of blood and
	approximately 10 litres of
	interstitial fluid.
	(a) Nested example of
	legends and a caption.

  Facilitating understanding:The main goal all nine designers cited was facilitating understanding. Concretely, they mentioned

	EU Russia Vatican City	population	POPULATION surface area	legend (population, size)
			SIZE SIZE

by both area and population, taking up 40% of the continent (although the country has Of Europe's approximately 50 countries Russia is by far the largest territory in both Europe and Asia), while Vatican City is the smallest.

IMPLICATIONS FOR AUTHORING TOOLSUnlike our study participants, it is clear that not all editors or authors are also competent designers or visualization specialists with the expertise to create word-scale graphics. Currently, the amount of time necessary to manually create and integrate wordscale graphics into a document is high-with individual graphics taking significantly more time to author than raw text.

ACKNOWLEDGMENTS

This work is sponsored by the French Research Organization, project grant ANR-11-JS02-003 and supported by the Collaborative European Digital Archive Infrastructure project CENDARI (cendari.eu).

Petra Isenberg is a research scientist at Inria, Saclay, France in the AVIZ project team. Prior to joining Inria, Petra received her PhD from the University of Calgary. Her main research areas are information visualization and visual analytics with a focus on collaborative work scenarios, interaction, and evaluation. She is interested in exploring how people can most effectively work together when analyzing large and complex data sets on novel display technology such as small touch-screens, wall displays, or tabletops.


Pascal Goffin
email: pascal.goffin@inria.fr

Wesley Willett

Petra Isenberg
email: petra.isenberg@inria.fr

Sharing Information from Personal Digital Notes using Word-Scale Visualizations

Keywords: Information visualization, personal visualization, digital note-taking, awareness, collaboration, word-scale visualizations

We describe how small visualizations embedded in text (word-scale visualizations) can be used to share information from and in personal notes. From our previous research, we learned that people see many opportunities for sharing personal notes, for example among a small social group. Yet, people reported that they were hesitant to share raw notes due to the notes' often disorganized structure, haphazard writing style, or due to the fact that notes may contain a number of unrelated or irrelevant pieces of information. In this paper, we discuss how word-scale visualizations can be used in a collaborative personal visualization setting-to show abstracted information from a shared set of notes in the context of personal notetaking. In particular, we discuss potential kinds of data about notes that can be shared and motivate why sharing them may be helpful. Additionally, we provide two examples that illustrate the challenges and implications of using word-scale visualizations to share data in notes. The first example describes how notes in a shared notebook can be combined with private notes. The second example shows how data from public sources can be mixed with private comments to support sharing between notes on a common topic.

INTRODUCTION

Digital notebooks can store a variety of different types of personal information, including recipes, health related information, results of web searches, shopping lists, or personal diaries. Digital notes are, thus, a unique and common type of personal data. They offer numerous opportunities for personal analysis-for example, examining the evolution of a person's culinary tastes or retracing a person's daily activity.

While note-taking is typically a solitary activity, for certain situations and tasks sharing the content of notes is important. Examples include a family planning their summer vacation or the members of a book club reading a book and preparing themselves for the next meeting. Many note-taking tools support content sharing. However, note-takers typically refrain from sharing their notes because they are concerned about how their notes and note-taking style will be perceived by others [START_REF] Willett | Understanding digital notetaking practice for visualization[END_REF]. As a result note-authors clean, reformat, and even rewrite their notes before sharing or simply refrain from sharing their notes at all. Yet, they seem to be comfortable with sharing some information about the notes' content-including which terms in the note are also used by their collaborators [START_REF] Willett | Understanding digital notetaking practice for visualization[END_REF].

Sharing information and ideas among acquaintances and family members can be a perfect way to spark discussions, prepare events, and increase awareness of other people's thoughts. For example when planning a summer vacation, several family members may use their personal digital notebooks to collect information about potential destinations, what activities or points of interest are available at these locations, or which friends live close by and could be visited. Later, the family must collaborate and consolidate these notes in order to settle on a single vacation plan.

Here, we explore the use of word-scale visualizations-small data graphics that display information associated with specific words or word combinations in text-to enrich personal notes with information from collaborators' notes. In the vacation planning scenario above, small visualizations embedded in a note about possible vacation spots could show which other family members have listed similar properties, locations, or countries and, thus, aid in the process of finding a joint destination.

In this paper, we discuss how word-scale visualizations can be used to show descriptive metadata from collaborators' notes. We specifically focus on sharing only metadata extracted from notes as a simple channel for communicating shared ideas and understanding collaborators' activities. By focusing on metadata we also free collaborators from the burden of feeling that they need to reformat and clean their notes before sharing.

Here, we explore and describe which types of information originating in notes could be shared between friends and family members. We discuss potential problems that can arise when sharing personal information and then elaborate on how word-scale visualizations could help to support sharing relevant pieces of information between members of a small social group. Finally, we illustrate our discussions with two examples.

RELATED WORK

In recent years, digital note-taking tools have become more and more available both inside and outside the workplace. Yet, general information on the characteristics of personal digital notes is relatively sparse. We base most of our knowledge on digital notes on our own previous study [START_REF] Willett | Understanding digital notetaking practice for visualization[END_REF] of digital note-taking practices and their implications for visualization. We found that while notes that are digital can theoretically be easily shared and distributed, people typically felt the need to edit notes prior to sharing. Hence, we discussed how visualization can be a good way to abstract notes and prepare them for sharing. In that previous work, we discussed visualizations that can create awareness of overlapping concepts and phrases in collaborators' digital notes. Here, we discuss in particular how small-scale visualizations can integrate metadata from friends' and family's notes in-context of one's own notes.

Chuah and Roth [START_REF] Chuah | Visualizing common ground[END_REF] previously discussed how visualizations can provide common ground during collaboration. One example of the use of visualization to provide common ground was introduced by Brennan et al. [START_REF] Brennan | Toward a multi-analyst, collaborative framework for visual analytics[END_REF]. In their work, private views of a graph visualization in a distributed work scenario are merged to show shared information and common ground. Keel [5] discusses a similar idea, using computational agents to identify information in private views that should be relayed to a larger group of collaborators. Similarly, we use word-scale visualizations to provide common ground for small social groups.

Other researchers have used visualization to create awareness about collaborators' search activity. Awareness of other people's data and actions is an important prerequisite for common ground formation and several tools have attempted to provide mechanisms for facilitating awareness. For example, in Cambiera [START_REF] Isenberg | Collaborative brushing and linking for colocated visual analytics of document collections[END_REF] Isenberg et al. use small visualizations (e. g. bars) as Scented Widgets [START_REF] Willett | Scented widgets: Improving navigation cues with embedded visualizations[END_REF] to show the overlap between the documents two participants had searched for and used to provide awareness of the pair's reading histories. The authors termed this concept of explicitly visualizing collaborators' activities "collaborative brushing and linking." Similarly, Mayar and Tory [START_REF] Mahyar | Supporting communication and coordination in collaborative sensemaking[END_REF] use partial merging to provide subtle awareness of other people's work. They term this concept of merging "linked common work" and implement it in their tool CLIP. The tool includes node-link graphs in which nodes have color coded overlapping segments to represent common and different information among collaborators. This is similar to our approach, which uses word-scale visualizations to provide subtle awareness of information that is shared between collaborators' notes. Finally, Morris and Horovitz's SearchTogether [START_REF] Morris | SearchTogether: An interface for collaborative web search[END_REF] creates awareness of web search histories by showing what collaborators have searched for and by explicitly sharing information via recommendations. In contrast to these previous three tools, we use small visualizations embedded directly within the text of digital notes rather than stand-alone visualization tools.

WORD-SCALE VISUALIZATIONS AS AN APPROACH TO PRESENT SHARED INFORMATION

Here, we discuss how word-scale visualizations embedded in notes can support shared awareness among a small social group. We begin with a short definition of word-scale visualizations before discussing the types of data that can be shared.

Why word-scale visualizations?

Word-scale visualizations are small data graphics that are associated with words or sets of words (which we refer to as "entities") and integrated alongside them in the text. Word-scale visualizations are similar to sparklines [START_REF] Tufte | Beautiful Evidence[END_REF], in that they are "small, intense, simple, word-sized graphic with typographic resolution", but can use a wider variety of visual encodings and sizes [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF]. Small word-scale visualizations can be the size of a single letter, while larger ones can approach the size of a sentence or paragraph. Word-scale visualizations are small and can easily embed additional information in text without being obtrusive. As a result, we can use word-scale visualizations to add additional information in the context of a note-taker's original notes. This can help notetakers better understand the overlap between their notes and their collaborators', without requiring a separate interface.

Sharing metadata and snippets

In our previous study of digital note-taking [START_REF] Willett | Understanding digital notetaking practice for visualization[END_REF], we found that notetakers were generally open to sharing their notes but were worried that unedited notes would overwhelm collaborators with unimportant information. In order to facilitate sharing notes, we therefore propose using word-scale visualizations that show abstracted metadata and snippets from notes. Using only metadata and brief snippets alleviates collaborators from the burden of having to edit their notes and, thus, simplifies sharing.

We specifically discuss descriptive metadata that is created when a note is saved or edited and snippets extracted from the notes themselves.

Technical metadata

Where the note was created: Sometimes digital notebooks store GPS locations when notes are created on mobile devices with location services turned on. Seeing where notes were written could, for example, allow you to see if a collaborator has visited and taken notes on a potential vacation destination. While location data can be particularly sensitive, it is also easy to abstract (e.g. reporting only the city, country, or even continent) and obfuscate [START_REF] Krumm | A survey of computational location privacy[END_REF].

Who authored or edited a note: In personal notebooks, notes are typically first authored and later edited by the same person. When sharing data from notes, the author/editor information might be crucial in order to judge the potential usefulness of information given existing background information in a social group. This information could also make it easier to contact the author directly and request the full content of a note.

When was a note created or edited: This information can be an indicator of how recently someone has engaged him or herself with the topic of a note. In addition, this information could give additional information on the note type. In our previous work [START_REF] Willett | Understanding digital notetaking practice for visualization[END_REF] we found that certain types of notes, such as project logs or running lists were edited relatively frequently while notes containing information scraps or references and copied content were rarely if ever edited by note-takers. Thus, data about note creation and editing patterns-coupled with other metadata-can provide a powerful tool for deducing the importance of a note.

Other metadata: There is a wealth of other technical metadata available for notes that can be the basis of a shared visualization. For example, the length of a note may hint at how much information a note contains. The note content type describes if the note has been authored by the note-taker or if it only contains clippings from public sources (what Willett et al. [START_REF] Willett | Understanding digital notetaking practice for visualization[END_REF] call "references and copied content") or if it is a mix of authored and clipped content. These clippings can be text or image clippings from websites such as Wikipedia or any other website. The tags or notebooks used to file the note might also be useful when identifying related notes.

Extracted keywords

In order to avoid sharing the raw text from notes but still provide cues on the semantic content of notes, automatically extracted key terms from notes could serve as a data source. These terms can be keyword labels applied to the note by its author or named entities like locations, dates, and surnames that are extracted directly from the text.

User-authored keywords or entities can succinctly describe the topic or ideas of a note without disclosing much unrelated information or revealing a note's disorganized structure. Entities can be found in the text through entity recognition and text processing. They can also be manually tagged by the original note-taker. For visualization purposes, entities or keywords can be used to cluster notes that are of interest to a specific topic. For example for planning a trip to New York City, notes containing entities such as "New York City", "Manhattan restaurants", or "MoMA" are relevant and may provide inspirations of what to visit and where to eat.

In addition to the mere presence of a specific entity, a diverse set of statistics can be gathered around entities. One of them is the number of occurrences of an entity in a note. Others include cooccurrence patterns with other entities or entity types and semantic relationships between entities extracted from knowledge structures such as WordNet. The number of occurrences, for example, can be a good indicator of what to expect from a note in terms of information relevant to a certain topic.

EXAMPLES

We discuss two information-sharing scenarios that illustrate how word-scale visualizations embedded in notes can show some of the data mentioned in the previous section and help support information sharing in a small social group. Additionally, we provide sketches of potential word-scale visualizations.

Shared recipes with Roommates

In this scenario, four flatmates live in a shared apartment. They like to cook together in the evenings and on weekends, and enjoy creat- This shared notebook provides a mechanism for sharing specific recipes, as well as for planning future meals, identifying interesting recipe pairings, and selecting sets of recipes that share ingredients. However, if the shared folder contains a large number of recipes it may be difficult to get a sense of which recipes others in the group are interested in or which have overlapping ingredients.

Sketch

Figure 1 shows a note containing a recipe for "Zitronencake" which has been shared by one of the flatmates. A small bar chart next to each ingredient highlights how many other recipes in the shared folder contain this ingredient as well as which roommates added them. Each of the bars corresponds to a single roommate and indicates how many recipes with that ingredient he or she has contributed. Hovering over a bar opens a small window that provides links to the three recipes with that ingredient which that roommate has most recently read or added. These visualizations make it easier for the flatmates to navigate the large collection of shared notes and identify interesting recipes added by their peers that might pair well together or which use the same ingredients.

Book Club

Our second scenario-a book club with about ten membersdiffers in that no notes are shared directly. Instead, each member of the club keeps their own personal notes about the current book and those notes can be annotated to provide peripheral indicators of other members' activity. Each member's individual notes on the book usually contain direct quotes from the work as well as observations about the characters, places, events, and ideas mentioned in the book, written in their own words. For example, a member may take notes discussing why they think a character behaves in one way or another or describing interesting situations that appear in the book. He or she may also discuss relationships between characters or describe feelings and ideas that occurred when reading a memorable passage.

Sketch

Figure 2 shows how a member's personal notes about the book could be augmented to provide more information about which as- pects of the book were interesting to their fellow members. Specifically, the augmented note highlights the shared use of quotes from the book as well as how often other members used the names of important characters, places, and ideas. The quotes can be seen as data from a public source-the book-and can be presented in more detail, whereas the comments about the characters and places might be seen as more private. As a result, the note can be augmented to highlight shared mentions of those entities-letting the note-taker know which other members have something to say about these points, without directly revealing their personal thoughts on the topic. This information about shared quotes and entities can then be used to steer the conversations when the club meets in person.

In Figure 2, the colored circle in the upper right corner of the note indicates its owner-Mrs. Green. The bar charts over each character's name depicts how many times that character was mentioned in each of the other members' notes. Interaction can be used to show more details. By hovering over a bar, the note-taker can reveal the name of the book club member and the number of occurrences of the entity. Additionally, hovering highlights all bars belonging to Mrs. Green in the paragraph. Any characters that are mentioned in the same paragraph are visually connected via a line. The strength of this relationship-number of co-occurrences of the characters in the note's paragraphs-is also shown on the line.

Quotes drawn from the book are marked with a large quotation mark, indicating that this complete snippet of public material is present in multiple members' notes. The bar on the left of the big quotation mark indicates which other members included the quote as well as how many times they included it (for example, Mrs. Green included the second quote three separate times, when discussing different aspects of the book). The note-taker can also click on the bar to see the names of the characters associated with the quote.

CHALLENGES AND IMPLICATIONS

Using word-scale visualizations to share metadata and entities drawn from personal notes requires careful consideration of the content of the visualizations and openness of the collaborators. Moreover, designers must create visualizations that help to abstract note content but also do not unintentionally remove or create links between pieces of information.

Visualization

The design space for word-scale visualizations is very large. However, an interesting implication for word-scale visualizations in the context of notes is that notes are often edited. As a result, even though embedding visualizations in-line can minimize the degree to which the text is disrupted [START_REF] Goffin | Exploring the placement and design of word-scale visualizations[END_REF], it may break note-takers' expectations about the behavior of common commands like cut, copy, paste, or backspace. For example, what happens if the cursor is to the right of a word-scale visualization? Will it be okay to delete using backspace? As a result, the best position for the word-scale visualizations may depend on whether or not the note is likely to be edited. Consider the word-scale visualizations we placed in-line in the sketch for the recipe scenario. Since ingredients are typically given as itemized lists with plenty of empty white space to the right of each item, and recipes are less often edited than other notes [START_REF] Willett | Understanding digital notetaking practice for visualization[END_REF] this placement is less likely to cause problems.

Privacy

The degree of privacy attached to information in a note has a wide spectrum. On one end is the raw note, that when shared with associated metadata discloses everything. The other extreme is a setting in which nothing at all is shared and all notes are kept private. In between these two endpoints we have many possibilities. Certainly there are tradeoffs between sharing and providing awareness and privacy. Word-scale visualizations can be a way to deal with this tradeoff. By the sharing metadata and entities of a note there is a level of flexibility available. Not all metadata, entities, or types of entities have to be shared if not wanted. Entities can be chosen through entity recognition and text processing. For digital notebooks, however, this means that an additional infrastructure has to be available to provide the possibility to regulate which entities and metadata are shared and to whom.

Sharing useful data

Word-scale visualizations provide a degree of abstraction from the original note. Yet, when they are based on metadata such as entities or keywords, word-scale visualizations might provide less information than originally contained in the note. For example, relationships can disappear or be created by inferring that two characters from a book appear in the same paragraph. In fact, the two characters might appear in unrelated sentences or even in quotes from separate books, but the abstraction of the paragraph may seem to imply a relationship. The challenge of choosing the right information to share through word-scale visualizations can be complex. This has implications for the design of the word-scale visualizations. For example, a word-scale visualization that groups entities based on co-occurrence at the sentence level could help avert incorrect assumptions about the relationship between entities, but runs a greater risk of revealing the raw content.

CONCLUSION

Word-scale visualizations provide a lightweight way of sharing important pieces of information from personal notes, and can support awareness and decision making without requiring family, friends and collaborator to reveal their raw notes. Personal digital notes are a treasure trove for personal analytics. This work represents a first attempt at helping make family, friends and collaborators more aware of one another while keeping sharing light-weight, avoiding the need for note editing prior to sharing, and placing shared data in-context to already existing personal information.

Figure 1 :

 1 Figure 1: This sketch is an excerpt of a recipe from the flatmates' shared recipe notebook. Each ingredient is accompanied by a wordscale visualization depicting the number of recipes with that ingredient shared by other members of the group.

2 Figure 2 :

 22 Figure 2: This sketch is an excerpt of one of the book club member's note about the current book. The bar chart describes the number of occurrences of the character in each book club member's current book note.

ACKNOWLEDGEMENTS

This work is sponsored by the French Research Organization, project grant ANR-11-JS02-003 and supported by the Collaborative European Digital Archive Infrastructure project CENDARI (cendari.eu).


Thomas Le Mézo
email: thomas.lemezo@ensta-bretagne.fr

Luc Jaulin

Benoît Zerr

An Interval Approach to Solve an Initial Value Problem

This paper proposes an original guaranteed interval-based method to solve an Initial Value Problem (IVP) for ordinary differential equations (ODE). Our method uses the geometrical properties of the vector field given by the ODE and a state space discretization to compute an enclosure of the trajectories set that verifies the IVP problem.

INTRODUCTION

Guaranteed integration is an elementary component for the study of dynamical systems and has been an important subject of research the last decades [START_REF] Berz | Computational differentiation: techniques, applications, and tools[END_REF]. It is used from guaranteed system simulation [START_REF] Konečnỳ | Enclosing the behavior of a hybrid system up to and beyond a zeno point[END_REF] to the computation of viability kernel [START_REF] Lhommeau | [END_REF].

In this paper, we propose a method to solve guaranteed integration problems which is orthogonal to literature as we will not use the Picard operator nor any interval counterpart of integration method (Euler [START_REF] Moore | Methods and applications of interval analysis[END_REF], Runge-Kutta [START_REF] Dit Sandretto | [END_REF] or Taylor [6]). The first section formalizes the problem; then the notion of pathbubble and new contractors are introduced; a method to solve the problem is presented; and a final section shows how the algorithm can be used on two test-cases.

FORMALIZATION OF THE PROBLEM

Let consider a Box Initial Value Problem (BIVP) S defined by the following equation:

S :              ẋ(t) = f(x(t)), x(t 0 ) = x 0 , x 0 ∈ [x 0 ], t 0 ∈ [t 0 ]. (1)
where x(t) ∈ R n is the state vector and f : R n → R n is the evolution function of S and where [t 0 ] is an interval and [x 0 ] is a box. We can note that both here time and initial condition are uncertain.

Problem We want to find the set S of points solution of (1) such that:

S = {(t, x) | ∃t 0 ∈ [t 0 ], ∃x 0 ∈ [x 0 ] | x = Φ(t -t 0 , x 0 )}, ( 2
)
where Φ is the flow map of ẋ = f(x).

Hypothesis We know a search box [r] of R n such that for all x(t) solution of the BIVP, ∀t, x(t) ∈ [r].

PATHBUBBLE AND PROPAGATION CONTRACTORS

We introduce now the pathbubble object which is composed of three set : a bubble X which is a compact of R n , an input set I ⊂ ∂X and an output set O ⊂ ∂X where ∂X is the boundary of X. A pathbubble will be noted (X, I, O) and is a mathematical objects that will be used to support paths. Fig. 1 illustrates a pathbubble with its input set (in red) and output set (in blue). Dashed orange paths are incompatible with this pathbubble because they enter or leave the pathbubble in a wrong sense whereas the plain orange path.

Let Q be a paving of R n . We can define two contractors that will be applied to the pathbubble (X, I, O), where X ∈ Q, in order to contract I and O.

Continuity propagation contractor This contractor has the property to match the input set (I) of a pathbubble to the output sets (O) of its neighbor pathbubbles.

I X = Y∈N(X) O Y ∩ ∂X (3)
where S X is the set (input or output) associated with the pathbubble (X, I, O) and N(X) are the neighbor pathbubbles of X i.e. N(X) = {Y ∈ Q|X ∩ Y ∅ or singleton}. Fig. 2 shows an example of a pathbubble continuity contraction.

Flow contractor This contractor propagate the input set I of a pathbubble to its output set according to the geometrical direction given by the cone of consistent trajectories, f(X) which can be computed using interval arithmetics, and do also the backward operation (output to input). The corresponding contraction is:

O = (I + {y = β • f(x), x ∈ X, β ≥ 0}) ∩ ∂X. (4)
Fig. 3 shows the principle of a pathbubble flow contraction.

METHOD

In this section, we will provide an algorithm to solve our BIVP.

FIGURE 2 .

 2 FIGURE 2. Illustration of the principle of continuity propagation contractor of a pathbubbles with its neighbors. Left, before the contraction, Right after the contraction. The plain orange path is a valid whereas the dashed path is incompatible with the border conditions. The corresponding contraction is O 1 := (I 2 ∪ I 3 ∪ . . .) ∩ O 1 and I 1 := (O 2 ∪ O 3 ∪ . . .) ∩ I 1 .

FIGURE 4 .

 4 FIGURE 4. Results of the algorithm in the state space. The yellow shape corresponds to the outer approximation of valid trajectories and gray cones correspond to the vector field in each pathbubblees. The initial condition is the red box and the search set r ∈ [0, 5] × [-2, 2].

FIGURE 5 .

 5 FIGURE 5. Right: The shape of the car on the hill problem. Left:Results of the algorithm in the state space. The search set is r ∈ [-2, 13] × [-6, 6], the color code is the same as Fig. 4. The Initial condition is on the left part of the figure

CONCLUSION

In this paper, we have proposed a new approach to compute an enclosure set of paths that verify a Box Initial Valued Problem for Ordinary Differential Equation. The main idea of our method is to follow the geometrical directions given by the vector field of the ODE. Whereas existing approaches which need a discretization of time, our method is only based on a discretization of the space state. The principle of our method has been illustrated on the guaranteed integration of the car on the hill problem and on a sinusoidal function.

Algorithm We propose the following algorithm to compute an enclosure of I:

1. Cover R n with a paving Q. 2. For each X of Q, initialize the pathbubble with I = ∂X and O = ∂X except for paving that intersects the initial condition which will be initialized with I = ∂X and O = ∅. 3. For each pathbubble, apply the continuity contractor to match neighbor pathbubble output sets (O) to the pathbubble input set (I). 4. For each pathbubble, apply the flow propagation contractor. 5. Go to step 3 until a fix point is reached i.e. output and input sets of any pathbubble stay identical by applying contractors. 6. Bisect all boxes of Q associated with a non-empty pathbubble (i.e. I and O ∅).

Go to Step 3.

We can notice that the main idea of the method is that following vector field directions is sufficient to compute an outer approximation set. By introducing the time variable as a component of the state vector, we can generalize the approach: in the following test-case section, we will use this point of view.

TEST-CASES

In this section, we will illustrate our algorithm on two test-cases. We used a single core i5-2520M CPU@2.50GHz processor for all the computation. To solve our BIVP, we will use here a pathbubble where X is a box which can be represented on a computer.

Simple function integration

The first problem we will consider is to solve the following system ẋ =sin x + e, The algorithm was stopped after 13 iterations which corresponds to less than 1 second of computation. At t = 5.0, the size of the x interval is inferior to [-0.126, 0.126] as shown on Fig. 4.

Car on the hill problem

Consider the car on the hill problem [7] given by the following equation:

which corresponds to the evolution of a car on a hill (see Fig. 5). We started the car at the initial condition x 0 ∈ [-1, 1] × [-1, 1] and stopped the algorithm after 18 iterations which takes less than 11 seconds. Fig. 5 shows the result of the algorithm in the state space. Due to uncertainty of the initial condition, we can see that there is two possibilities for the car when it arrives at the last third hill: to continue ahead and exit the figure or falls back into the valley toward an equilibrium. We can see, on Fig. 5, that our method is able to separate the two possible trajectories of the car.