Lost in Magnitudes: Exploring the Design Space for Visualizing Data with Large Value Ranges
Katerina Batziakoudi Florent Cabric Stéphanie Rey Jean-Daniel Fekete















    < g r a p h i c s >

 Our design space encompasses four dimensions: MARKS (green), DATA (blue), VISUAL CHANNELS (red), and TASKS (yellow). The image illustrates an example of using our design space as an interactive table, where a mark is selected, and visual channels are assigned to data attributes. Grey cells are invalid, according to the visualization literature. After checking for integrity constraints, a visualization is generated to perform the tasks. Note the ⊕ signs (top left); enable the assignment of the scale to positions.
   
   


Introduction

Adaptive Mixed-Scale Feature Fusion Network for Blind AI-Generated Image Quality Assessment
    Tianwei Zhou, Songbai Tan, Wei Zhou, Yu Luo, Yuan-Gen Wang, and Guanghui Yue
    April 29, 2024
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In this article, we explore the design space for the static visualization of datasets with attributes encompassing several orders of magnitude. Such data, hereafter referred to as Order of Magnitude Values (), include values that span a broad range of exponents when presented in scientific notation <cit.>.
Our goal is to achieve effective static visualization of .
An illustrative example is the budget allocations of the French government (<ref>), which range from tens of millions of Euros (10^7) to hundreds of billions of Euros (10^11), thereby covering five orders of magnitude or five distinct exponents. 

  


are integral to various domains of daily life, including but not limited to financial analysis, pandemic tracking, demographic studies, environmental monitoring, and social media metrics. For instance, the “WDC Web Table Corpus 2015” dataset <cit.>, which contains 90M tables extracted from the Web in 2015,

includes 25,175 tables with . 

The effective static visualization of is essential for providing a clear and informative overview accessible to a broad audience. This audience ranges from the general public, who may seek insights into their country's taxation system, to scientists and professionals looking to analyze and understand complex datasets. We argue that should be treated as a specific kind of quantitative data type in visualization, and thus, visualization systems should support these values effectively.


Common visualization techniques, such as linear and logarithmic bar charts, prove ineffective,

as illustrated in <ref>. In the example of the French budget, using a linear scale (left) makes it impossible to read and compare budget categories with smaller magnitudes, resulting in information being entirely lost. 
While a logarithmic scale (right) can partially mitigate this issue by ensuring that all budget values are visible, it is not well understood by the general public <cit.> and may even lead to misconceptions among scientists <cit.>. Beyond legibility issues, logarithmic scales also hinder efficient quantitative comparisons <cit.>, such as the fundamental tasks of estimating differences or ratios <cit.>, between values within the same or neighboring magnitudes.

In recent studies in visualization <cit.>, authors split the into mantissa and exponent parts, inspired by scientific notation,
and visualize them as the combination of two distinct parts. 
For example, 16,000 is represented as  where 1.6 is the mantissa and 4 the exponent of the value.
Thus, one quantitative attribute is represented by two new quantitative attributes that are strongly dependent, thereby introducing the challenge of visually maintaining their numerical relationship in a cohesive way.

The existing literature <cit.> supports the idea that the mantissa/exponent separation could enhance the visualization of , facilitating reading and comparison tasks. According to their studies, performed by non-specialists, these visualizations are effective and understandable. Nevertheless, at the date of this article, the number of visualization designs exploring this approach remains limited to six. 
We aim to discover whether other visualizations are effective and elucidate possible rationales behind their effectiveness.

Following the Grammar of Graphics (GoG) rules <cit.>, 
the number of combinations of marks and visual channels that could integrate both a mantissa and exponent for a specific dataset is very large. It is the Cartesian product of different data attributes (including the mantissa and exponent) to visual channels (position, length, color intensity), using different marks (points, lines, areas). Given the dimensions of our design space (<ref>), this number reaches 6,240 combinations. However, not every combination is likely to be effective; only a few may prove visually effective for performing specific tasks. 

To discern these effective combinations, we draw upon insights from the literature on visualization effectiveness and graphical perception. We apply these insights as integrity constraints, following a similar approach to the one used from visualization recommendation systems <cit.>. Through the application of these constraints, we refined the design space to 336 viable combinations (168 factoring the symmetrical ones). Conducting controlled evaluations for all these combinations, however, would be prohibitively expensive. Instead, our qualitative study systematically explores these combinations to provide a more holistic understanding of the factors influencing the effectiveness of visualizations <cit.>. Building on this exploration and qualitative evaluation, we introduce design guidelines and recommendations for static visualizations of .

















In summary, we make the following contributions in this paper:


  * We present a design space, 
for the static visualization of by decomposing them into mantissa and exponent parts.

  * From our design space, we generate several (25) effective visualizations for —some of them being novel—, and other combinations that could be effective 
as discussed in <ref>.

  * We refine the definition of a scale, already alluded to in previous work <cit.>, that leads to effective visualizations for ; we call it .


  * We introduce guidelines and recommendations for designing effective visualizations. 



Additionally, we introduce our open-source tool, OMVis (Fig. <ref>), that enables systematic exploration and facilitates the future empirical evaluation of the generated visualizations by saving and exporting them in Observable Plot <cit.> (called Plot in the remaining).





§ BACKGROUND

In this section, we first discuss the definition and numerical relationship between the mantissa and exponent. Then, we present related work in data visualization with varying orders of magnitude and visualization effectiveness.




 §.§ Definition and Terminology

We borrow our definitions and terminology from the https://en.wikipedia.org/wiki/Scientific_notationscientific notation. 
A numerical value v is represented as a pair (m, e) with v=m × 10^e, where m ∈ℝ∖{0} is called mantissa (or significand) and e ∈ℕ is called exponent. In scientific normalized notation, e is chosen so that |m| ∈ [1,10[. The mantissa defines the significant digits of a value inside an order of magnitude and is represented with a given precision, the number of digits p ∈ℕ^+, that may depend on v's intrinsic precision and on the desired display precision.

We call e (using the scientific normalized notation) the exponent or order of magnitude of a number, also known as its level of magnitude.


In accordance with Höhn et al. <cit.>, a dataset contains orders of magnitude values () (or a large value range) when one of its numerical attributes contains values that span four orders of magnitude or more.
Note that the order of magnitude of 0 is undefined. 







    

    

 §.§ Visualization of Large Value Ranges
 
    In the context of data visualization, visually representing for the general public poses significant challenges. While linear scales are widely understood, they fall short of effectively depicting large value ranges, rendering only the larger values legibly. If the screen has a width and height in the 10^3 pixels, it cannot use position to distinguish values that are more than three orders of magnitude apart.
    
    This distinctiveness issue <cit.> hinders fundamental analytical tasks, including value estimation, and comparison (ratio and difference) between different ranges. Although dual scale charts <cit.> and innovative designs like the Du Bois Wrapped Bar Chart <cit.> offer improvements for analyzing two disparate ranges, their effectiveness diminishes with  <cit.>, as they require the accurate representations of multiple (more than four) data ranges in the same view. 
    
    Logarithmic scales, capable of showcasing diverse ranges, facilitate qualitative comparisons of , like finding extrema. Nonetheless, they pose difficulties for quantitative tasks, such as estimating differences or ratios between values of similar or close magnitudes <cit.>. 
    Despite their prevalence in scientific contexts, they may lead to  misconceptions <cit.> and are less understandable for the general public, challenging perception and interpretation <cit.>. Landy et al. <cit.>, conducted a study focusing on in the range of 10^5 to 10^13 and found that adults rarely follow a purely logarithmic interpretation of the scale
    when estimating magnitudes or performing quantitative tasks. About half of the study participants estimated numbers linearly across this spectrum, while nearly all others followed a segmented linear model. In this model, magnitudes are uniformly distributed as in logarithmic scales, but numbers within these magnitudes are linearly interpolated. This behavior reflects our verbal description of large numbers, using units as categories (millions, billions) for magnitude and digits from 1 to 10 for precision <cit.>. Normalized scientific notation also embodies this separation, with the exponent indicating the order of magnitude and the mantissa providing higher precision within the order of magnitude.

    Over the last decade, researchers have introduced visualizations that separate into their exponent and mantissa components. The effectiveness of these visualizations was evaluated based on tasks such as value retrieval, sorting, ratio estimation, difference estimation, and trend identification. 
    Hlawatsch et al. <cit.> introduced the Scale-Stack Bar Chart (SSBSSB) that visualizes large value ranges by stacking values at multiple scales, beginning from zero and mapping numbers linearly. Following a segmented linear model approach, this visualization demonstrates its effectiveness over traditional linear and logarithmic bar charts for quantitative comparisons. Borgo et al. <cit.> proposed visualizing exponents and mantissa, using two separate marks, as overlapping bars on a linear scale within a classic bar chart, termed Order of Magnitude Markers (OMMOMM). Their approach showcased high accuracy for value retrieval and comparison tasks. Höhn et al. <cit.> introduced the Width-Scale Bar Charts (WSBWSB), which uses position Y for the mantissa and a redundant combination of color intensity and width for the exponent. They compared with in a controlled experiment, showing that is more effective for value retrieval and for sort, ratio, and trend tasks. Additionally, Braun et al. <cit.> illustrated the use of color, specifically their Order of Magnitude Color (OMCOMC) scheme, to aid in perceiving changes in orders of magnitude by mapping exponents to hue and mantissa to a sequential color scale. Lastly, Braun et al. <cit.>, focused on visualizing large-scale time-series data, introducing the Order of Magnitude Horizon Graph () and the Order of Magnitude Line chart (OMLOML). OMHOMH modifies the standard horizon graph <cit.> by using position Y for mantissa and color intensity for the exponents, while combines an approach similar to the segmented linear model that uses for position and the color scheme from . Unlike , where values are shown per magnitude in stacked rows, combines both exponent and mantissa in the position Y, introducing a new visualization scale termed “linear scaling within orders of magnitude” <cit.>. 
    We clarify their scale and call it  in <ref>. 
    We summarise the existing literature in <ref>. 

    

    The studies highlighted above have conclusively demonstrated the advantages of visualizations that separate into their mantissa and exponent components. This separation enhances analytical tasks' accuracy, confidence, and efficiency by offering a more effective representation of large numerical ranges. However, these studies have explored only a subset of the potential combinations of marks and visual channels for encoding the mantissa and exponent (<ref>). Recognizing this gap, our objective is to conduct a systematic exploration of these combinations along with different attribute types: nominal, ordinal, time, and quantitative. By doing so, we aim to identify and discuss which visualizations are most effective for , proposing guidelines for their representation and enhancing the comprehension of vast numerical data.


    
    

 §.§ Visualization Effectiveness

    The concept of visual channel effectiveness in visualization has significantly evolved since Bertin first introduced it <cit.>. Building on Bertin's foundation, Cleveland and McGill <cit.> conducted experiments to evaluate the ranking of visual channels for quantitative data, with a specific focus on the accuracy of sorting and comparison tasks. Mackinlay <cit.> expanded this ranking to include both nominal and ordered data, thus broadening our understanding of visual channel effectiveness. Several subsequent experiments <cit.>, have empirically validated and refined the proposed effectiveness rankings of visual channels, revealing additional factors that could influence them.
    
    Zeng and Battle <cit.>, through a systematic review of the literature on channel effectiveness, pointed out that position encodings are the most accurate channels across all data types. The accuracy
    
    of other visual channels (area, length, or color intensity) varies, influenced by factors like discriminability — the ability to perceive distinct values for each attribute <cit.>—, and separability—the independence of channels from one another <cit.>. Discriminability is affected by several factors, including the number of objects <cit.>, the type and size of a mark <cit.>, the background color <cit.>, visual attention limitations <cit.>, and the interference effects with other channels <cit.>.
    
    Interference effects with other channels lead to separability issues. Separability allows for focused attention on one visual channel without interference from another <cit.>. Separable visual channels, such as position and color, show minimal cross-channel effects, causing no effectiveness issues when assigned to different attributes of a visualization. In contrast, when the encoding of one channel affects the perception of another, the channels are called integral <cit.>. Integral visual channel pairs include color hue and color intensity <cit.>, height and width <cit.>, shape and size <cit.>, shape and color intensity <cit.>, size and color intensity <cit.>, and size and position <cit.>. The redundant combination of integral visual channels for the same attribute facilitates decoding. However, encoding two unrelated attributes with two integral channels can lead to interference, making it difficult to identify the attributes and affecting value perception <cit.>.

    Studies also suggest that the effectiveness of visualizations should not only be evaluated in relation to the used channels alone but also in terms of the tasks <cit.>. McColeman et al. <cit.> highlight that the effectiveness of visual channels is task-dependent, asserting that data characteristics might significantly impact task performance more than the choice of visual channel. Hollands and Spence <cit.> demonstrate that the original ranking may not sufficiently explain average performance in “discrimination” tasks, where participants identify the larger of two quantities. Kim et al. <cit.> observed that different combinations of visual channels perform differently for comparison and summary tasks depending on a dataset's cardinality (number of items). Similarly, Yuan et al. <cit.> discovered that the ranking does not apply to tasks involving comparisons of multiple values, such as average across data points. 
    
    
    In our exploration of the design space, we consider a broad spectrum of factors that could influence effectiveness. This exploration begins with channel effectiveness in relation to the accuracy, discriminability, and separability of the channels used. Alongside these theoretical considerations, we also address task-related factors, such as the subjective effort and confidence required to perform various tasks with visualizations. Our goal is to engage in a nuanced discussion about the effectiveness of visualizations for and to propose guidelines informed by critical observations that affect it.
    







§ DESIGN SPACE

    In this section, we define the specific visualization design space addressed in our paper. This design space is informed and constrained by 1) the principles of the Semiology of graphics <cit.> and the implementations of the Grammar of Graphics <cit.>, particularly the Plot <cit.> library, which is used to generate the visualizations in our study, 2) studies on graphical perception <cit.> and visualization effectiveness <cit.>, and 3) the literature about visualization of large value ranges <cit.>. We start by establishing the boundaries of our design space, proceed to describe its dimensions—how each visualization within this space is described— then we define the scale, and conclude by outlining the constraints implemented to refine the design space, with the goal of improving visualization effectiveness.
    
    
    

 §.§ Boundaries

    Our design space's boundaries are shaped by the objectives of our research, insights gained from our review of literature on the visualization of , and the choice of the visualization library used (Plot) . Firstly, we concentrate on tabular datasets with a single attribute with positive exponents, excluding types such as networks, fields, and geometries.  Given that Plot primarily uses the Cartesian coordinate system, our study is focused on visualizations within this system, ensuring a uniform layout and excluding polar or parallel layouts. Aimed at enhancing the effectiveness of visualizations in overview, we limit our scope to static visualization designs and exclude interactive elements. Moreover, our design space does not entail data transformations; these are handled during dataset creation. Our analysis explicitly targets visualizations that employ a single mark 
    
    , and consequently we are removing  <cit.> from our design space. Additionally, we omit logarithmic scales, a decision detailed in the background section. Contrary to other studies, we do not include sort and trend tasks because McColeman et al. <cit.> argue that these tasks, like many others in visualization, rely on the most fundamental “two-value ratio judgment task”, already included in our tasks. Their quantitative evaluation may reveal significant differences; however, within the context of our design space, we evaluate the tasks qualitatively.

    

 §.§ Dimensions

    After setting the design space boundaries, we define the dimensions that specify each visualization. Plot <cit.> describes a single visualization as the combination of data types, marks, visual channels, facets, scales, transforms, and projections. Considering the boundaries established, we describe each observed visualization design by a combination of the following elements: data, marks, and visual channels, considering faceting as part of visual channels.  
    <ref> showcases the dimensions of our design space.
    
    

  
Data
    In this context, “data” pertains to the type and range of attributes. We consider as a specific type of data attribute that is separated into two quantitative attributes: mantissa and exponent, as defined in the background section. We explore visualizations that display the mantissa, the exponent, and an additional attribute type (we call ) among nominal, ordinal, time, and quantitative. The range of these additional data attributes, and therefore the cardinality of the datasets chosen for our exploration, was seven, small enough to remain discriminable with all the visual channels <cit.>. <ref> displays the DATA (blue) part, as rows on the top left of the table.






    

  
Marks
    Marks are defined based on their spatial dimension requirements: points represent zero-dimensional (0D) marks, lines correspond to one-dimensional (1D) marks, and areas are considered two-dimensional (2D) marks <cit.>. Within the framework of the , as exemplified by Plot, we represent points either with the Dot or with the Rule- mark—when the visual channel of Length is chosen—, for lines, we use the Bar mark for nominal and ordinal data, and the Rect mark for time, and last for areas we use the Area mark. <ref> displays the MARKS (green) part, as rows on the left of the table.



    

  
Visual channels
    Visual channels (VC) are ways to control the appearance of marks <cit.>. In our design space, we selected channels based on insights from the literature on visualization effectiveness <cit.> and constraints from Plot specifications. We focus on the following nine visual channels: Position X (), Position Y (), , Column (), , , Color Intensity (), Color Hue (), and . Length refers to both width and height, depending on the orientation of the line and the choice of the mark. and are channels derived from the faceting operation, as done by Moritz et al. <cit.>. <ref> displays the VISUAL CHANNELS (red), as columns on the top of the table.


    

  
Description of Visualizations
    Considering the dimensions above, we describe a visualization that belongs to our design space as a list of four assignments forming a visualization configuration:
 Mark, Exponent ↦VC, Mantissa ↦VC, Attribute-2 ↦VC.
    
    <ref> displays the combination on the bottom of the generated visualization as:
     

    

  
Tasks
    The type of task performed using a visualization affects its overall effectiveness <cit.>. Therefore, as effective visualizations are the desired outcome of our study, we consider tasks an additional dimension of our design space <cit.>. Considering previous studies in and the boundaries we set, we evaluate task effectiveness in relation to the following tasks: value identification, difference estimation, and ratio estimation. 
    <ref> displays the TASKS (yellow) part on top of the generated visualization.


    

  
Scales According to the GoG, a visualization scale is a function that transforms attribute values into visual channel values, such as position, length, or color intensity. In our design space, scales are handled by Plot and are determined automatically by the attribute type. Band scales are utilized for nominal attributes, ordinal scales for ordinal attributes, and linear scales for quantitative attributes. 
    To simplify interpretation and eliminate the need for calculations, we implement a tick transformation representing the exponent value as 10^e. For instance, if the exponent is 3, the tick label displays 1,000 (10^3). Given that scales are selected based on the data attribute type, they are not discussed within our design space, with the exception of (see below).

    

 §.§ Scale

    As mentioned in the background, studies in cognitive psychology <cit.> suggest that people perceive following a segmented linear model, with magnitudes distributed uniformly as in logarithmic scales and numbers within these magnitudes being linearly interpolated. We believe the scale used by Braun et al. <cit.>, and inspired by the positional encoding of  <cit.>, reflects this segmented linear model well and represents an effective way to visualize . 
    
    We call this scale and refine its definition, aiming to facilitate its experimental evaluation and wider adoption in visualizations. According to the figures of Braun et al. <cit.>, we infer that their scale is s(v) = e(v) + (m(v)-1)/ 9, a piecewise linear scale. With this scale, the units are the exponent values e, and, to fill up the [0,1[ interval between two exponents, it maps the range of the mantissa m ∈ [1,10[ to the domain [0,1[ by rescaling it: (m-1)/9.
    
    When used for position, the axis units represent the values of the exponents, and the subunits correspond to the mantissa values. Labeling/ticks for this scale uses the exponent for the major labels/ticks, and the mantissa's values for the minor ticks. Braun et al. <cit.> do not show subunit labels, but we realized that showing minor ticks and labels, even just one at m=5, expresses well the linearity of the scale within orders of magnitude. In our design space, this scale is expressed by assigning both exponent and mantissa to the same positional channel.

    

 §.§ Constraints

    To ensure the effectiveness and expressiveness of the visualizations in our design space, we apply constraints related to the boundaries we have set, the expressiveness of marks and visual channels, and the interference of different channels. A visualization is considered viable only if all constraints are satisfied. Our approach is inspired by the literature on recommendation systems for visualization <cit.>, where integrity constraints are used to encode expressiveness and restrictions to the dimensions of a visualization specification. 
    
    

  
Constraints based on layout 
    As we limit our design space to the Cartesian system, we include only combinations that contain at least one position channel (or ). 

    

  
Constraints for the expressiveness of channels 
    To ensure the expressiveness of each assignment <cit.>, we use and only for nominal data, for ordinal and quantitative data, and and only for quantitative data. For the and channels, we assign only the exponent, as mantissa is a quantitative attribute with continuous range, and 
    
    has a one-to-one relationship with the , leading to visualizations similar to those using a positional channel. An exception to this constraint is our decision to retain visualizations that include both row and column for the exponent and as a categorical attribute, as they result in novel visualizations.
    
    

  
Constraints based on mark type 
    for each type of mark, we impose constraints in accordance with its defined dimensions. The channels and can be used only for point marks, and the channel is not feasible for area marks. We also apply constraints in relation to the dataset. We use area marks only for time data, and we exclude lines for quantitative data.

    

  
Constraints based on interference with identity channels 
    channels like and serve identification purposes. Combining with within the same mark generates a new color. Similarly, merging with can alter the shape (transforming a circle into an ellipse). These combinations modify the identity of the primary channel, and therefore, we do not consider them valid.
   



    
    


§ EXPLORATION

Our exploration has been carried out using our tool, OMVis. We explored four datasets, generated all the viable visualizations, and explored them visually on our wall-size display to identify key properties, related to the visual channels, that made them effective. The four coauthors then performed the three tasks defined in our design space and subjectively assessed the visualizations in terms of suitability, accuracy, confidence, and effort. Following this, we convened to reach a consensus on the coding, resulting in a set of effective visualizations, rationales, and design guidelines.

    

 §.§ Order of Magnitudes Tool
 

    To facilitate the exploration of the design space, we implemented a web-based open-source tool, OMVis (<ref>), written in JavaScript. Our tool includes two main components: 1) a menu for the interactive exploration of all possible combinations and 2) the resulting visualization. Within the menu, users can upload or select a dataset from our database. They then choose an to separate into its mantissa and exponent parts, followed by the selection of additional attributes (nominal, ordinal, time, or quantitative). Subsequently, users decide on a mark—point, line, or area, and assign visual channels to the mantissa, exponent, and selected attributes. We apply the constraints by disabling the corresponding choices; for example, quantitative visual channels such as are not applicable to nominal attributes. A visualization is generated based on these selections.  We use Plot to create the visualizations. The user can export the produced visualization either as an image or as JavaScript code. More details are provided in the supplemental material.

    

 §.§ Datasets

    We generated four datasets for our exploration, one for each data type combined with the attribute.
    We used the “Trending YouTube Video Statistics” dataset from Kaggle <cit.>. This dataset comprises daily metrics (Views, Likes, Comments) for the top trending YouTube videos across categories (Music, Entertainment, Gaming) in nine Countries. We selected this dataset as it contains (Views) and has diverse attributes, including nominal (video Categories), ordinal (7 Likert scale Sentiments), time (Days), and quantitative (Likes, Comments). For simplification, we created four smaller datasets (cardinality = 7), each featuring Views () and another attribute: nominal, ordinal, time, and quantitative, used to aggregate the source dataset using a “group by” query.  
    
    The nominal dataset includes total Views (ranging from 10^4 to 10^11) across all countries by 7 video categories. For the ordinal dataset, we created 7 categories of sentiment analysis based on the comments, from “Very Positive” to “Very Negative”, and assigned “Total Views” ranging from 10^5 to 10^9. The time dataset comprises total views (from 10^3 to 10^7) over 7 days for the “Nonprofits and Activism” category. For the quantitative attribute, we determined the likes-to-comments ratio for the “Music” category, selecting 7 videos with total views spanning six exponents (from 10^4 to 10^9). We employed Python Pandas for data cleaning and dataset creation. The original dataset and the Jupyter Notebook for generating the three datasets used in our exploration are available in the supplemental material. 
    
    

 §.§ Generation of Visualizations

        
        
    
        
        We systematically generated all visualizations by first calculating every combination of the nine visual channels (n=9) in our design space with our data attributes (k=3) {exponent, mantissa, }, where   represents a nominal, ordinal, time, or quantitative attribute. For our exploration, we considered only one-to-one assignments, where one data attribute is encoded with one visual channel. This constitutes a permutation problem since the order of assignment matters (which visual channel is assigned to which data attribute). The formula for permutations of n items taken k at a time is given by:  P(n, k) = n!/(n-k)! so, in our case, P(9,3)=504 visualizations. 

        An exception to the one-to-one rule involves the visual channels of and , where both the mantissa and exponent can be assigned to the same visual channel, resulting in the scale denoted as . The total number of combinations for the  scale for both and is 2 × P(8,1) = 16 visualizations.
    
        This method was uniformly applied across all mark types {point, line, area} and data attribute types for {nominal, ordinal, time, quantitative}, resulting in the total number of combinations before imposing any constraints being equal to:
        
    3 × 4 × (504 + 16) = 6,240


        The result after the application of constraints mentioned in the previous section was 336 viable combinations. We used our tool to generate all the viable visualizations and exported them as images. For details about the creation of the combinations and the applications of constraints, see the program  in the supplemental material.









 §.§ Systematic Exploration of Visualizations

Discussing the effectiveness of visualization is challenging and multi-faceted. We chose to explore the design space with a qualitative approach to collect observations about possible rationales for effectiveness <cit.>. We explore our design space in two stages: 1) Exploration of channel effectiveness and 2) Exploration of task effectiveness, gathering insights for design guidelines during the two stages. 



  §.§.§ Channel Effectiveness

The first stage of our exploration consisted of two three-hour meetings between three co-authors in which each visualization was displayed on a wall-sized display to facilitate both discussion and comparison (<ref>). During this phase, we coded the channel assigned to the mantissa, exponent, and , of the 336 visualizations according to three factors: Accuracy, Discriminability, and Separability with other channels. 
We first present results for each channel and then discuss the relationship between the factors.



  
Accuracy We identified 48 (14.3%) visualizations conveying high accuracy for all three of the exponent, mantissa, and . We classified 168 visualizations as accurate for two attributes (48 for exponent and mantissa, 84 for exponent and , and 36 for and mantissa), enabling accurate retrieval of just one attribute's value (48 for exponent, 24 for mantissa and 48 for ).



  
Discriminability We classified 48 visualizations that convey discriminability of exponent, mantissa, and (14.3%). We counted 168 visualizations allowing discriminability for only two attributes (48 for exponent and mantissa, 84 for exponent and , and 36 for and mantissa), and 120 that discriminate efficiently only one attribute (48 for exponent, 24 for mantissa and 48 for ).



  
Separability We classified 224 (66.7%) that enable separability among the three pairs of data attributes ({exponent and mantissa},{exponent and },{and mantissa}). The remaining 112 visualizations (33.3%) enable separability between two pairs of attributes (20 for {exponent and mantissa} and {mantissa and }, 40 for {exponent and mantissa} and {exponent and }, 52 for {and mantissa} and {exponent and }).



  
Cross-analysis 

By analyzing the visualizations, we noticed a correlation among the three factors. Visualizations with separability issues between two attributes (48 visualizations for exponent and mantissa, 20 for exponent and , and 40 for mantissa and ) were deemed non-discriminable for those respective attributes. This classification indicates that the interference effect between two channels led to compromised discriminability. Furthermore, all visualizations classified with low discriminability for an attribute exhibited low accuracy for the same attribute, resulting in identical counts for our accuracy and discriminability analysis. Consequently, in our discussion, we treat accuracy and discriminability as a single factor influenced by the assignment of channels and the interference effects between them. 


  By analyzing accuracy and discriminability, we noticed that each visualization that provide or reduce accuracy for one visual channel also provide or reduce discriminability for the same channel. In other words, the visualizations that enable accuracy for three (resp. two and one) components also enable discriminability for the same three (resp. two and one) components. The 48 visualizations that enable discriminability and accuracy for the exponent, the mantissa and the attribute also enable separability between these three components. Otherwise, some visualizations (96) that allows separability between all pairs of visual channels suffer from accuracy and discriminability problem for one component (36 for exponent, 80 mantissa and 48 for attribute)	or two components (4 for exponent and attribute and 8 accuracy and mantissa). The three remaining categories are visualizations that do not provide good discriminability and accuracy for two components and consequently leads to a low separability between them (48 visualizations for exponent and mantissa, 20 for exponent and attribute and 40 for mantissa and attribute).   



        
        
        
        
        
        
        

        
        

We also observed that all symmetrical visualizations shared the same set of strengths/weaknesses. Therefore, we removed the duplicated symmetrical visualizations, reducing our set of visualizations by half, from 336 to 168. We considered symmetrical combinations of both position and faceting, preferring to retain and , over and  <cit.>. Coding the visualizations informed us about good practices in visualizing . However, a second phase regarding task effectiveness was deemed necessary to assess visualization effectiveness in relation to the required tasks <cit.>.


            




  §.§.§ Task Effectiveness

Visualization effectiveness based on tasks is primarily evaluated with controlled user studies. However, performing this evaluation with the 168 viable visualizations in our design space would be intractable. Instead, we decided to explore visualization effectiveness based on tasks with a qualitative-coding approach. Each visualization was attributed to two coders selected between the four coauthors of this paper. After the first coding phase, three coders entered a collaborating coding phase where all the encodings were discussed and reviewed. The coders had to answer a set of 11 questions regarding three tasks: value retrieval, difference estimation, and ratio estimation. 
We subjectively evaluated the three tasks that belong to our design space in terms of suitability, accuracy, confidence, and effort.



  
Value retrieval With many representations, retrieving a value is a two-step process: retrieving the exponent first, then the mantissa, to compute or visually perceive the whole value. Retrieving the exponent is a critical task as it can lead to an “order-of-magnitude” error (the retrived value can be ten times higher or smaller than the actual value). Therefore, we binary-coded whether the exponent retrieval was feasible without any potential confusion. As the mantissa defines the significant digits of a value inside an order of magnitude, the effect of an error in its estimation is narrower. In addition, the mantissa can be a number with an infinite number of decimals, making its exact retrieval impossible, so we focused on estimating the mantissa between two ranges: ± 0.5 and ± 1.  



  
Difference estimation Estimating the difference between two is a contrasting task that depends on the magnitude level to which the two values belong. Following Braun et al. <cit.>, we differentiate three kinds of difference estimation tasks when two values belong to: the same order of magnitude, neighboring orders of magnitude, or distant orders of magnitude.  



  
Ratio estimation Similarly to difference estimation, estimating the ratio depends on the order of magnitude of the values. However, estimating the ratio between neighboring or distant orders of magnitude results in the same task. Therefore, we only evaluated ratio estimation for values that belong to either the same order of magnitude or distant orders of magnitude.
       

    

 §.§ Results

  
     From the 168 visualizations that we qualitatively evaluated in relation to their effectiveness to represent , 32 were identified as suitable. Within this subset, 25 allowed task performance without effort, and 7 required some effort. Of the 136 visualizations not classified within the effective group, 55 were deemed to necessitate a medium effort, and 81 were found to either demand considerable effort for the tasks or were outright impractical. We classify the combinations into four groups: 
   
        
  Effortless and Effective (25/168): Visualizations that exhibit pre-attentive task performance for both value estimation and quantitative tasks, demonstrating high effectiveness for the visualization of . <ref> shows three examples of these visualizations.
        
  Suitable with Effort (7/168): Visualizations that are generally effective for value estimation but demand additional effort for quantitative comparison tasks (<ref>).
        
  Medium Effort and Low Confidence (55/168): Visualizations that require a moderate amount of effort and result in low confidence in their outcome (<ref>).
        
  High Effort or Impractical (81/168): Visualizations that are either too challenging to use effectively for the intended tasks or are outright infeasible for visualizing (<ref>).
    
    The 25 visualizations we classified as “Effortless and Effective” are described using eight combinations of visual channels across different marks and attribute types, as shown in <ref>. Through our analysis of these 25 visualizations and their comparison with those requiring more effort, we identified common patterns that are both necessary and sufficient for the effective visualization of . We present these patterns as design guidelines and recommendations, which are detailed in the following section. The 168 images used in our evaluation are available at <https://www.omvs-designspace.com/> and in the supplemental material.

 





§ DISCUSSION AND RECOMMENDATIONS

    In this section, we discuss our results about the channel and task effectiveness. We introduce guidelines for the visualization of , suggest recommendations, and present rationales about how the channel effectiveness affected the perceived effort and confidence for the tasks. We also assess the impact of mark choice and data type on visualization effectiveness, underscoring the challenges and considerations for each. Furthermore, we explore redundant encoding and the risk for interference between channels. With these guidelines and rationales in mind, we revisit existing literature on visualizations. Finally, we outline our study's limitations.
    

   

 §.§ Design Guidelines

    
    We identified four requirements that we present as design guidelines: Accuracy for Magnitude, Detail Inside Magnitudes, Continuity Between Magnitudes, and Parsimony in Channels.

            

    

  §.§.§ AcM — Accuracy for Magnitude

    
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            The exponent should be encoded with a highly accurate and discriminable channel to prevent order-of-magnitude errors. Errors in the estimation of the exponent could lead to large mistakes in value estimation and comparisons. Therefore, we consider this guideline the most critical for the effective visualization of . 
            This guideline applies to all visualizations we consider suitable, including in the “Effortless and Effective” (25/25) and “Suitable with Effort” (7/7) categories. It is the main differentiating factor between “Medium Effort and Low Confidence” (51/51) and “High Effort or Impractical” (27/81). Non-compliance with this guideline results in visualizations that require high effort to accomplish tasks, potentially leading to an increased error rate (Example in <ref>).
            
            Recommendations: To comply with this guideline, we recommend favoring the use of highly accurate channels for the exponent, such as , , , and , while avoiding low discriminability channels as exclusive channels for the exponent, such as , , and , especially in cases where the dataset contains a range of five or more exponents <cit.>.
             

    

  §.§.§ DeM — Detail Inside Magnitudes

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        The mantissa should be encoded with a highly accurate and discriminable channel to facilitate the estimation of difference and ratio between values of the same magnitude.  
        This guideline applies to all visualizations classified as suitable for : “Effortless and Effective” (25/25) and “Suitable with Effort” (7/7). Visualizations violating this guideline fall into the categories “Medium Effort and Low Confidence” (7/55) and “High Effort or Impractical” (42/81). Non-compliance with this guideline decreases the accuracy and confidence in value retrieval. Furthermore, it leads to visualizations that require significant effort to perform quantitative comparisons for values of the same magnitude and values of neighboring magnitudes. Specifically, employing the channel or not aligned or orthogonal to the exponent for the mantissa necessitates frequent legend consultation to identify values. This complexity intensifies with the use of , severely complicating the estimation of ratios between two (Example in <ref>).

        Recommendations: To comply with this guideline, we recommend favoring the use of or for encoding the mantissa. When this is not possible due to the violation of other guidelines (CoM), we advise using for point marks,  requiring the drawn lines to be aligned, as alignment facilitates comparison. However, we do not advocate for using and , as these channels are not discriminable within the defined range of the mantissa <cit.>. 

    

  §.§.§ CoM — Continuity Between Magnitudes

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         The channel used for the mantissa should express a transition from one exponent to the next, reflecting the mathematical relationship between the two attributes. CoM ensures a coherent and understandable progression across the entire range of values and facilitates the quantitative comparison of values that belong to neighboring exponents. 
         Compliance with this guideline is the primary differentiating factor between the visualizations in the “Effortless and Effective” (25/25) category and those in the “Suitable with Effort” (1/7) category. It is also notably low in the “Medium Effort and Low Confidence” (4/55) and “High Effort or Impractical” (5/81) categories. While CoM is not necessary to perform the tasks, the absence of it leads to higher efforts to estimate differences between OMVs in neighboring magnitudes (Example in <ref>). 
    
        Recommendations: To comply with this guideline, we recommend encoding the mantissa with when either (scale) or is used for the exponent, and, symmetrically, using when either or is employed for the exponent. CoM can also be achieved by placing the exponent in a positional channel and using or for points, provided the marks are drawn in the same direction as the exponents, start from the same position, and reach their maximum range at the position of the next exponent. Given the boundaries we have established for our design space, no other combination can achieve CoM. 
        
    
    

  §.§.§ PaC — Parsimony in Channels

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     Borrowing the term from philosophy, we define PaC as the principle of using the fewest visual channels required to effectively visualize the exponent and mantissa. This approach serves two primary purposes: firstly, it minimizes interference effects between channels; secondly, it conserves visual capacity, allowing for the efficient integration of additional data attributes into the visualization. In our design space, we consider visualizations that adhere to PaC as those that employ the same visual channel for both the exponent and mantissa, specifically and resulting in the scale. Additionally, visualizations that use or for the exponent and or for the mantissa, respectively, are also considered compliant, as they create a similar visual effect to using a single channel for both attributes. 
    
    This guideline is applicable to the majority of the “Effortless and Effective” category (21/25). It exhibits low compliance in the subsequent categories: “Suitable with Effort” (1/7), “Medium Effort and Low Confidence” (2/55), and “High Effort or Impractical” (7/81). While PaC is not essential for performing tasks, we believe that its absence could lead to ineffective visualizations due to unnecessary complexity (Example in <ref>). 
    
    Recommendations: Similarly to CoM, to comply with this guideline, we recommend using the scale, or encoding the mantissa with when is used for the exponent, and symmetrically, using when is employed for the exponent. Given the boundaries of our design space, no other combination can achieve PaC.

    

 §.§ Effectiveness in Relation to Mark and Data

    
    

    

    

  
Effectiveness per mark The choice of mark significantly influences the effectiveness of visualizing . Our findings indicate that different mark types, even with identical combinations of visual channels, exhibit varied levels of effectiveness. The point mark, creating the least interference, facilitates the encoding of multiple channels, yielding more expressive visualizations. We noted that using the scale, while highly effective for points, demanded greater effort for ratio tasks when lines were employed (<ref>). However, this difficulty was not observed when using for the exponent and line for the mark. A possible reason for this difference is that, while bars are interpreted in a linear manner, the scale that results from both exponent and mantissa in is not linear, causing confusion in the visual comparison of the two lines.  Conversely, areas behave differently; faceting (exponent encoded with ) led to decreased performance. Other studies, such as Braun et al. <cit.>, address this by overlaying facets to create a horizon chart. Moreover, area marks demand more design considerations to circumvent common pitfalls. Employing redundant encoding and additional marks, like point marks to highlight values, could enhance their effectiveness. 

    

  
Effectiveness in relation to data Different types for introduce various constraints that limit the effective combinations for visualizing . Nominal data might interfere with the mantissa and exponent if 
    encoded using or , while the attributes are represented with or . Ordinal data could interfere with an attribute when is utilized for its encoding, and is allocated to either the mantissa or exponent. Time data, 
    limited in expressiveness, do not significantly interfere with . Lastly, quantitative data is the most challenging, as their integration with appears feasible only when a positional channel is employed. 
    Constraints or considerations could also be introduced in relation to the . For instance, when there are exponent values with gaps (sparse), the use of or obscures these gaps, 
    increasing the effort required for comparing values and potentially leading to errors. Yet, when data cardinality is high, positional channels are recommended for all attribute types, underscoring the need for PaC for the .
    
    

 §.§ Redundant Encoding

    For our qualitative analysis, we focused on combinations where each attribute is encoded using a single visual channel. This approach was chosen because allowing for more flexibility in encoding would lead to an unmanageable large number of combinations. However, employing redundant encoding can enhance total discrimination and information transmission <cit.>, potentially improving the accuracy of the mantissa and exponent and thereby enhancing the overall viability and effectiveness of a visualization design (AcM, DeM). To support this, our tool (<ref>) enables users to interactively explore the entire spectrum of possible redundant encoding configurations available, considering the boundaries and the dimensions of our design space.
    
    Achieving more effective visualizations through redundant encoding, however, demands a meticulous approach to mitigate any interference between the employed channels. This is particularly crucial because, for redundancy to benefit a visualization, the selected combination of visual channels must exhibit a correlation <cit.>. This principle applies, for example, when both width and height or and are assigned to the same attribute. Nevertheless, this strategy comes with its caveats. Performance can significantly suffer when there is interference between channels assigned to different attributes, leading to confusion and misinterpretation of the data. For example, when a line is encoded with both height () and width () for two different attributes, people cannot separate the two dimensions when estimating the visualized magnitude <cit.>. Therefore, it becomes imperative to apply separability rules when considering an attribute alongside an additional attribute.
    
    When implementing redundant encoding, it is crucial to maintain the inherent numerical relationship between the mantissa and exponent. Treating the exponent and mantissa as two completely separable attributes can lead to continuity issues, complicating task execution and potentially fostering incorrect perceptions of the derived from their combination. For instance, assigning to the mantissa and to the exponent could lead to confusion regarding which channel represents the magnitude, thereby increasing the likelihood of errors. Such a setup might obscure the integral link between the mantissa and exponent. Consequently, we recommend against assigning , , and to 
    the mantissa or exponent if one of the mentioned channels is already 
    used by the other attribute. Nevertheless, we advocate for selecting redundant encoding strategies that favor the exponent to enhance its accuracy (AcM). This approach helps ensure a coherent and effective representation of the , facilitating easier comprehension and minimizing the risk of misinterpretation.    

    
    
    

   
    

 §.§ Revisiting Existing Visualizations

        In this section, we revisit the existing literature and critically discuss the proposed visualizations in light of our design space dimensions, exploration results, and proposed guidelines.
        

        

  
SSB falls within the “Effortless and Effective” category of our design space. Specifically, it uses a line mark, for the exponent, for the mantissa, and for a nominal attribute, aligning with the guidelines we recommend for efficiently visualizing . However, our visualization design for the same combination diverges in a key aspect. We adhere to the scientific normalized notation, thus encoding the mantissa in within the range [1,10[. In contrast, 's approach initiates the range within each row from 0. This leads to the depiction of all previous magnitudes within the [0,1[ range for each row. 's authors contend that this approach facilitates the independent interpretation of each row, suggesting that visualizations might be misleading if the axis does not start at 0. Nonetheless, we argue that this viewpoint contradicts the continuity guideline (CoM), potentially explaining why subsequent studies utilizing reported that participants found comparisons challenging <cit.> and identified that the majority of errors in occurred with values in neighboring orders of magnitude <cit.>. This hypothesis deserves further research.
        


        

  
WSB uses line for marks, redundant encoding for the exponent, combining (width) and , and for the mantissa.  This setup corresponds to two distinct combinations in our design space due to the redundant encoding for the exponent. Both combinations are categorized as “High Effort or Impractical” for two main reasons: 1) the channels used for the exponent, and , are low in accuracy (AcM); 2) as mentioned in the background, there is an interference effect between both the height of the line (defined by for the mantissa) and width (defined by for the exponent), and between size (defined by for the mantissa) and for the exponent. However, the fact that they have a very small range for exponents (4) and the choice to use redundant encoding diminish the accuracy limitations for estimating the exponent. Nevertheless, the visualization does not follow the guidelines of CoM, which could lead to difficulties in performing quantitative comparisons, a fact verified by their empirical study. Lastly, the use of many channels to achieve accuracy for the exponent limits the potential for effectively visualizing an additional attribute (PaC).
        


        

  
OMC is a colormap for , combining for the exponent and for the mantissa. It is applied to a point mark alongside two quantitative attributes in and , demonstrating the flexibility to be combined with additional attributes (PaC). In our design space, we exclude combinations of both and within the same mark, in alignment with literature on the interference caused by integral channels <cit.>. Nonetheless, we recognize that since the exponent and mantissa originate from the separation of a single value, encoding them with integral channels might be justifiable if such a combination accurately represents their mathematical relationship, with the mantissa indicating a transition from one exponent to the next (CoM). , however, does not provide continuity between magnitudes, and as they state, their approach “is not feasible for comparing values that are close to the borders of the exponents” <cit.>.
        
        
        


        

  
OML integrates the positional encodings of with the colormap of . Thus, it employs an area mark, , and for the exponent, and for the mantissa, and for a time-series attribute. The positional encoding for the exponent and mantissa ensures high accuracy (AcM, DeM) and continuity (CoM). However, we contend that the decision to use for redundant encoding, which contravenes CoM, diminishes the visualization's perceived continuity. They align with this argument as they also indicate that the slower response times and increased cognitive load might result from the redundant encoding by position and color. Finally, they conclude that “the separation of mantissa and exponent needs to be discrete, as the continuous separation in the case of does not seem efficient.” Nevertheless, we argue that continuous separation could be efficient, provided it adheres to the principle of CoM. This hypothesis warrants further investigation.
         


        

  
OMH modifies the standard horizon graph, using area mark, for exponent, for mantissa and for the time attribute. Employing for the exponent contradicts the AcM principle 
        but could be effective when the exponent range is lower than five. The authors also note that “the choice of colors could be improved to facilitate easier distinction between the bands,” indicating potential issues with discriminability. Utilizing for the mantissa aids in quantitative comparisons within the same magnitude (DeM), yet the lack of CoM complicates comparisons between neighboring exponents.

    

    
    
    
    
    

 §.§ Limitations

    The main limitations of our design space arise from the specific boundaries we established. 
    We focused 
    on static visualizations of for overviews, thereby not exploring the potential benefits of interactive techniques. Our study is confined to tabular data and visualizations within the Cartesian coordinate system, suggesting that other data types and layouts could offer additional insights. We also did not include negative mantissa and exponents in our analysis, although we speculate their behavior would be similar to that of positive ones. Furthermore, we explore only visualizations that include one mark for both mantissa and exponent, excluding the from our design space. Moreover, our exploration centers on three tasks we deem fundamental for . However, further tasks, such as correlation, could be explored. 

     A general limitation of design spaces, including the one we explore in this paper, is that it is not possible to encompass all the factors that have an impact on design. Our focus was on perception and we did not consider additional factors that might influence the overall effectiveness of visualization design, such as attention <cit.>, visualization literacy <cit.>, visual aesthetics <cit.>, and the background and preferences of the user <cit.>. 
     
     

       





§ CONCLUSIONS AND FUTURE WORK

In this article,

we presented the design space of visualization decomposing values into mantissa and exponent parts, presented a list of effective visualizations for ,
define the scale, 

and introduced guidelines and recommendations for designing effective visualizations.
Our design space is descriptive, 
generative since it allows presenting new designs, 
and evaluative since it provides criteria to eliminate ineffective combinations, select viable ones, and improve their designs.
Yet, it cannot quantitatively compare the effectiveness of different viable combinations.


We also introduced OMVis, an open-source web tool that enables the systematic interactive exploration of . 
We hope our study will motivate visualization system designers to 

define operators to easily separate attributes into mantissa and exponent parts, and define the scale to support them with positional visual channels. Our study addressed several questions regarding effective visualization but raised new ones, and was not able to answer all the questions. Controlled experiments are still needed to compare the effectiveness of several visualizations, using our set of tasks, as well as more tasks for specific applications and attribute types. Our design space is a subset of the whole visualization design space, and might also be useful for networks, fields, maps, and more specialized visualizations and application domains.







The authors wish to thank 
Ludovic David for his support in the development and bulk generation of images,
Olivier Gladin for his support with the wall-size display, 
Lorenzo Ciccione,
and Emanuele Santos for their useful feedback and thoughtful comments. 
This work has been partially supported by Berger-Levrault.






§ SUPPLEMENTAL MATERIALS


All supplemental materials are available on OSF at <https://osf.io/msvbe/?view_only=061aa4a15028449b91304339df89d40c>, released under a CC BY 4.0 license. In particular, they include (1) the dataset we used in our study, (2) the combinations.js file that we used to apply the constraints and generate the viable combinations, (3) a pdf file giving more details for the implementation of our tool and showcasing examples of its use, (4) a website with the 168 visualizations that are included in our design space, also accessible via:  <https://www.omvs-designspace.com/>, and (5) csv files with the results of our qualitative evaluation.






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