# A Bounded Measure for Estimating the Benefit of Visualization (Part I): Theoretical Discourse and Conceptual Evaluation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Identifying a shortcoming of using the KL-divergence in the information-theoretic measure proposed by Chen and Golan [1] and evidencing the shortcoming using practical examples (Parts I and II);
- Presenting a theoretical discourse to justify the use of a bounded measure for finite alphabets (Part I);
- Proposing a new bounded divergence measure, while studying existing bounded divergence measures (Part I);
- Analyzing nine candidate measures using seven criteria reflecting desirable conceptual or mathematical properties, and narrowing the nine candidate measures to six measures (Part I);
- Conducting several case studies for collecting instances for evaluating the remaining six candidate measures (Part II);
- Demonstrating the uses of the cost–benefit measurement to estimate the benefit of visualization in practical scenarios and the human knowledge used in the visualization processes (Part II);
- Discovering a new conceptual criterion that a divergence measure is a summation of the entropic values of its components, which is useful in analyzing and visualizing empirical data (Part II);
- Offering a recommendation to revise the information-theoretic measure proposed by Chen and Golan [1] based on multi-criteria decision analysis (Parts I and II).

## 2. Related Work

## 3. Overview and Motivation

## 4. Mathematical Notations and Problem Statement

#### 4.1. Mathematical Notation

**Note**: In this paper, to simplify the notations in different contexts, for an information-theoretic measure, we use an alphabet $\mathbb{Z}$ and its PMF P interchangeably, e.g., $\mathcal{H}\left(P\right(\mathbb{Z}\left)\right)=\mathcal{H}\left(P\right)=\mathcal{H}\left(\mathbb{Z}\right)$. An arXiv report [43] provides a short introduction to the cost–benefit analysis and the relevant mathematical background of information theory, which some readers may find helpful.

#### 4.2. Problem Statement

## 5. Bounded Measures for Potential Distortion (PD)

#### 5.1. A Conceptual Proof of Boundedness

#### 5.2. Existing Candidates of Bounded Measures

#### 5.3. New Candidates of Bounded Measures

## 6. Conceptual Evaluation of Bounded Measures

#### 6.1. Criterion 1: Is It a Bounded Measure?

#### 6.2. Criterion 2: How Many PMFs Does It Have as Dependent Variables

#### 6.3. Criterion 3: Is It an Entropic Measure?

#### 6.4. Criterion 4: Is It a Distance Measure?

- identity: $d(x,y)=0\iff x=y$,
- symmetry: $d(x,y)=d(y,z)$,
- triangle inequality: $d(x,y)\le d(x,z)+d(z,y)$,
- non-negativity: $d(x,y)\ge 0$.

#### 6.5. Criterion 5: Is It Intuitive or Easy to Understand?

#### 6.6. Criterion 6: Visual Analysis of Curve Shapes in the Range of $(0,1)$

#### 6.7. Criterion 7: Visual Analysis of Curve Shapes in a Range near Zero, i.e., $[{0.1}^{10},0.1]$

## 7. Discussions and Conclusions

- It is not intuitive to interpret a set of values that would indicate that the amount of distortion in viewing a visualization that features some information loss, could be much more than the total amount of information contained in the visualization.
- It is difficult to specify some simple visualization phenomena. For example, before a viewer observes a variable x using visualization, the viewer incorrectly assumes that the variable is a constant (e.g., $x\equiv 10$, and probability $p\left(10\right)=1$). The KL-divergence cannot measure the potential distortion of this phenomenon of bias because this is a singularity condition, unless one changes $p\left(10\right)$ by subtracting a small value $0<\u03f5<1$.
- If one tries to restrict the KL-divergence to return values within a bounded range, e.g., determined by the maximum entropy of the visualization space or the underlying data space, one could potentially lose a non-trivial portion of the probability range (e.g., $13\%$ in the case of a binary alphabet).

- Rømer—0 degree: freezing brine, 7.5 degree: the freezing point of water, 60 degree: the boiling point of water;
- Fahrenheit (original)—0 degree: the freezing point of brine (a high-concentration solution of salt in water), 32 degree: ice water, 96 degree: average human body temperature;
- Fahrenheit (present)—32 degree: the freezing point of water, 212 degree: the boiling point of water;
- Réaumur—0 degree: the freezing point of water, 80 degree: the boiling point of water;
- Delisle—0 degree: the boiling point of water, $-1$ degree; the contraction of the mercury in hundred-thousandths.
- Celsius* (original)—0 degree: the boiling point of water, 100 degree: the freezing point of water;
- Celsius (1743–1954)—0 degree: the freezing point of water, 100 degree: the boiling point of water;
- Celsius (1954–2019)—redefined based on absolute zero and the triple point of VSMOW (specially prepared water);
- Celsius (2019–now)—redefined based on the Boltzmann constant.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AC | Alphabet Compression |

JS | Jenson–Shannon |

KL | Kullback–Leibler |

MCDA | Multi-Criteria Decision Analysis |

MIP | Maximum Intensity Projection |

PD | Potential Distortion |

PMF | Probability Mass Function |

## Appendix A. Conceptual Boundedness of 𝓗_{CE}(P,Q) and 𝓓_{KL}

**Definition**

**A1.**

- if $Q=\{\frac{1}{2},\frac{1}{2}\}$, codeword 0 for ${z}_{1}$ and codeword 1 for ${z}_{2}$;
- if $Q=\{\frac{3}{4},\frac{1}{4}\}$, codeword 0 for ${z}_{1}$ and codeword 10 for ${z}_{2}$;
- ⋯
- if $Q=\{\frac{63}{64},\frac{1}{64}\}$, codeword 0 for ${z}_{1}$ and codeword 111111 for ${z}_{2}$;
- ⋯

**Lemma**

**A1.**

**Theorem**

**A1.**

**Corollary**

**A1.**

**Further Discussion:**The code created using Huffman encoding is also considered to be optimal for source coding (i.e., assuming without the need for error correction and detection). A formal proof can be found in [55].

## References

- Chen, M.; Golan, A. What May Visualization Processes Optimize? IEEE Trans. Vis. Comput. Graph.
**2016**, 22, 2619–2632. [Google Scholar] [CrossRef] [Green Version] - Kullback, S.; Leibler, R.A. On information and sufficiency. Ann. Math. Stat.
**1951**, 22, 79–86. [Google Scholar] [CrossRef] - Ishizaka, A.; Nemery, P. Multi-Criteria Decision Analysis: Methods and Software; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Lin, J. Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theory
**1991**, 37, 145–151. [Google Scholar] [CrossRef] [Green Version] - Chen, M.; Abdul-Rahman, A.; Silver, D.; Sbert, M. A Bounded Measure for Estimating the Benefit of Visualization (Part II): Case Studies and Empirical Evaluation. (earlier version: arXiv:2103.02502). arXiv, 2022; under review. [Google Scholar]
- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Cover, T.M.; Thomas, J.A. Elements of Information Theory; John Wiley & Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Chen, M.; Feixas, M.; Viola, I.; Bardera, A.; Shen, H.W.; Sbert, M. Information Theory Tools for Visualization; A K Peters: Natick, MA, USA, 2016. [Google Scholar]
- Feixas, M.; del Acebo, E.; Bekaert, P.; Sbert, M. An Information Theory Framework for the Analysis of Scene Complexity. Comput. Graph. Forum
**1999**, 18, 95–106. [Google Scholar] [CrossRef] - Rigau, J.; Feixas, M.; Sbert, M. Shape Complexity Based on Mutual Information. In Proceedings of the IEEE Shape Modeling and Applications, Cambridge, MA, USA, 13–17 June 2005. [Google Scholar]
- Gumhold, S. Maximum entropy light source placement. In Proceedings of the IEEE Visualization, Boston, MA, USA, 27 October–1 November 2002; pp. 275–282. [Google Scholar]
- Vázquez, P.P.; Feixas, M.; Sbert, M.; Heidrich, W. Automatic View Selection Using Viewpoint Entropy and its Application to Image-Based Modelling. Comput. Graph. Forum
**2004**, 22, 689–700. [Google Scholar] [CrossRef] - Feixas, M.; Sbert, M.; González, F. A unified information-theoretic framework for viewpoint selection and mesh saliency. ACM Trans. Appl. Percept.
**2009**, 6, 1–23. [Google Scholar] [CrossRef] - Ng, C.U.; Martin, G. Automatic selection of attributes by importance in relevance feedback visualisation. In Proceedings of the Information Visualisation, London, UK, 14–16 July 2004; pp. 588–595. [Google Scholar]
- Bordoloi, U.; Shen, H.W. View selection for volume rendering. In Proceedings of the IEEE Visualization, Minneapolis, MN, USA, 23–28 October 2005; pp. 487–494. [Google Scholar]
- Takahashi, S.; Takeshima, Y. A Feature-Driven Approach to Locating Optimal Viewpoints for Volume Visualization. In Proceedings of the IEEE Visualization, Minneapolis, MN, USA, 23–28 October 2005; pp. 495–502. [Google Scholar]
- Wang, C.; Shen, H.W. LOD Map—A Visual Interface for Navigating Multiresolution Volume Visualization. IEEE Trans. Vis. Comput. Graph.
**2005**, 12, 1029–1036. [Google Scholar] [CrossRef] [Green Version] - Viola, I.; Feixas, M.; Sbert, M.; Gröller, M.E. Importance-Driven Focus of Attention. IEEE Trans. Vis. Comput. Graph.
**2006**, 12, 933–940. [Google Scholar] [CrossRef] [Green Version] - Jänicke, H.; Wiebel, A.; Scheuermann, G.; Kollmann, W. Multifield Visualization Using Local Statistical Complexity. IEEE Trans. Vis. Comput. Graph.
**2007**, 13, 1384–1391. [Google Scholar] - Jänicke, H.; Scheuermann, G. Visual Analysis of Flow Features Using Information Theory. IEEE Comput. Graph. Appl.
**2010**, 30, 40–49. [Google Scholar] [CrossRef] [PubMed] - Wang, C.; Yu, H.; Ma, K.L. Importance-Driven Time-Varying Data Visualization. IEEE Trans. Vis. Comput. Graph.
**2008**, 14, 1547–1554. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bruckner, S.; Möller, T. Isosurface similarity maps. Comput. Graph. Forum
**2010**, 29, 773–782. [Google Scholar] [CrossRef] [Green Version] - Ruiz, M.; Bardera, A.; Boada, I.; Viola, I.; Feixas, M.; Sbert, M. Automatic transfer functions based on informational divergence. IEEE Trans. Vis. Comput. Graph.
**2011**, 17, 1932–1941. [Google Scholar] [CrossRef] [Green Version] - Bramon, R.; Ruiz, M.; Bardera, A.; Boada, I.; Feixas, M.; Sbert, M. Information Theory-Based Automatic Multimodal Transfer Function Design. IEEE J. Biomed. Health Inform.
**2013**, 17, 870–880. [Google Scholar] [CrossRef] - Bramon, R.; Boada, I.; Bardera, A.; Rodríguez, Q.; Feixas, M.; Puig, J.; Sbert, M. Multimodal Data Fusion based on Mutual Information. IEEE Trans. Vis. Comput. Graph.
**2012**, 18, 1574–1587. [Google Scholar] [CrossRef] [PubMed] - Wei, T.H.; Lee, T.Y.; Shen, H.W. Evaluating Isosurfaces with Level-set-based Information Maps. Comput. Graph. Forum
**2013**, 32, 1–10. [Google Scholar] [CrossRef] - Bramon, R.; Ruiz, M.; Bardera, A.; Boada, I.; Feixas, M.; Sbert, M. An Information-Theoretic Observation Channel for Volume Visualization. Comput. Graph. Forum
**2013**, 32, 411–420. [Google Scholar] [CrossRef] - Biswas, A.; Dutta, S.; Shen, H.W.; Woodring, J. An Information-Aware Framework for Exploring Multivariate Data Sets. IEEE Trans. Vis. Comput. Graph.
**2013**, 19, 2683–2692. [Google Scholar] [CrossRef] - Chen, M.; Jänicke, H. An Information-theoretic Framework for Visualization. IEEE Trans. Vis. Comput. Graph.
**2010**, 16, 1206–1215. [Google Scholar] [CrossRef] - Chen, M.; Walton, S.; Berger, K.; Thiyagalingam, J.; Duffy, B.; Fang, H.; Holloway, C.; Trefethen, A.E. Visual multiplexing. Comput. Graph. Forum
**2014**, 33, 241–250. [Google Scholar] [CrossRef] - Purchase, H.C.; Andrienko, N.; Jankun-Kelly, T.J.; Ward, M. Theoretical Foundations of Information Visualization. In Information Visualization: Human-Centered Issues and Perspectives; LNCS 4950; Springer: Berlin, Germany, 2008; pp. 46–64. [Google Scholar]
- Xu, L.; Lee, T.Y.; Shen, H.W. An information-theoretic framework for flow visualization. IEEE Trans. Vis. Comput. Graph.
**2010**, 16, 1216–1224. [Google Scholar] [PubMed] - Wang, C.; Shen, H.W. Information Theory in Scientific Visualization. Entropy
**2011**, 13, 254–273. [Google Scholar] [CrossRef] [Green Version] - Tam, G.K.L.; Kothari, V.; Chen, M. An analysis of machine- and human-analytics in classification. IEEE Trans. Vis. Comput. Graph.
**2017**, 23, 71–80. [Google Scholar] [CrossRef] [Green Version] - Kijmongkolchai, N.; Abdul-Rahman, A.; Chen, M. Empirically measuring soft knowledge in visualization. Comput. Graph. Forum
**2017**, 36, 73–85. [Google Scholar] [CrossRef] - Chen, M.; Gaither, K.; John, N.W.; McCann, B. cost–benefit analysis of visualization in virtual environments. IEEE Trans. Vis. Comput. Graph.
**2019**, 25, 32–42. [Google Scholar] [CrossRef] [Green Version] - Chen, M.; Ebert, D.S. An ontological framework for supporting the design and evaluation of visual analytics systems. Comput. Graph. Forum
**2019**, 38, 131–144. [Google Scholar] [CrossRef] - Streeb, D.; El-Assady, M.; Keim, D.; Chen, M. Why visualize? Untangling a large network of arguments. IEEE Trans. Vis. Comput. Graph.
**2019**, 27, 2220–2236. [Google Scholar] [CrossRef] - Viola, I.; Chen, M.; Isenberg, T. Visual Abstraction. In Foundations of Data Visualization; Springer: Berlin, Germany, 2020. [Google Scholar]
- Tennekes, M.; Chen, M. Design Space of Origin-Destination Data Visualization. Computer Graphics Forum
**2021**, 40, 323–334. [Google Scholar] [CrossRef] - Chen, M.; Grinstein, G.; Johnson, C.R.; Kennedy, J.; Tory, M. Pathways for Theoretical Advances in Visualization. IEEE Comput. Graph. Appl.
**2017**, 37, 103–112. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, M. Cost–benefit Analysis of Data Intelligence—Its Broader Interpretations. In Advances in Info-Metrics: Information and Information Processing across Disciplines; Oxford University Press: Oxford, UK, 2020. [Google Scholar]
- Chen, M. A Short Introduction to Information-Theoretic cost–benefit Analysis. arXiv
**2021**, arXiv:2103.15113. [Google Scholar] - Chen, M.; Sbert, M. On the Upper Bound of the Kullback–Leibler Divergence and Cross Entropy. arXiv
**2019**, arXiv:1911.08334. [Google Scholar] - Moser, S.M. A Student’s Guide to Coding and Information Theory; Cambridge University Press: Cambridge, MA, USA, 2012. [Google Scholar]
- Endres, D.M.; Schindelin, J.E. A new metric for probability distributions. IEEE Trans. Inf. Theory
**2003**, 49, 1858–1860. [Google Scholar] [CrossRef] [Green Version] - Ôsterreicher, F.; Vajda, I. A new class of metric divergences on probability spaces and its statistical applications. Ann. Inst. Stat. Math.
**2003**, 55, 639–653. [Google Scholar] [CrossRef] - Liese, F.; Vajda, I. On divergences and informations in statistics and information theory. IEEE Trans. Inf. Theory
**2006**, 52, 4394–4412. [Google Scholar] [CrossRef] - Van Erven, T.; Harremos, P. Rényi Divergence and Kullback–Leibler Divergence. IEEE Trans. Inf. Theory
**2014**, 60, 3797–3820. [Google Scholar] [CrossRef] [Green Version] - Klein, H.A. The Science of Measurement: A Historical Survey; Dover Publications: Mineola, NY, USA, 2012. [Google Scholar]
- Pedhazur, E.J.; Schmelkin, L.P. Measurement, Design, and Analysis: An Integrated Approach; Lawrence Erlbaum Associates: Mahwah, NJ, USA, 1991. [Google Scholar]
- Haseli, G.; Sheikh, R.; Sana, S.S. Base-criterion on multi-criteria decision-making method and its applications. Int. J. Manag. Sci. Eng. Manag.
**2020**, 15, 79–88. [Google Scholar] [CrossRef] - Chen, M.; Sbert, M. Is the Chen-Sbert Divergence a Metric? arXiv
**2021**, arXiv:2101.06103. [Google Scholar] - Newton, I. Scala graduum caloris. Philos. Trans.
**1701**, 22, 824–829. [Google Scholar] - Golin, M.J. Lecture 17: Huffman Coding. Available online: http://home.cse.ust.hk/faculty/golin/COMP271Sp03/Notes/MyL17.pdf (accessed on 15 March 2020).

**Figure 1.**Visual encoding typically features many-to-one mapping from data to visual representations, hence information loss. For example, (

**a**) in volume visualization, the color of each pixel results from a complex process of combining a sequence of voxel values, and (

**b**) in metro maps, different geographical paths are often represented using indistinguishable line segments. The significant amount of information loss in volume visualization and metro maps suggests that viewers not only can abide the information loss but also benefit from it. Measuring such benefits can lead to new advancements of visualization, in theory and practice.

**Figure 2.**Each process in a data intelligence workflow can be characterized using three abstract measures: alphabet compression (AC), potential distortion (PD), and cost. They can be used to reason about the shortcomings in a workflow and identify possible solutions in abstraction [37]. For example, increasing data filtering in visualization (AC) may reduce the cost of ${P}_{i}$ and ${P}_{i+1}$, especially when human knowledge can reduce perceptual errors (PD).

**Figure 3.**A volume dataset was rendered using the maximum intensity projection (MIP) method, which causes curved surfaces of arteries to appear rather flat. Posing a question about a “flat area” in the image can be used to tease out a viewer’s knowledge that is useful in a visualization process.

**Figure 4.**In this 2D illustration of a simplified scenario of volume visualization, three sequences of voxels are rendered using the MIP method. The volume on the right features a curved surface defined by those brightest voxels. By projecting the maximum voxel values to the pixels in the middle, the curvature information of the surface is lost. A viewer needs to determine if the surface in the volume is curved or flat, for which the viewer’s knowledge is critical.

**Figure 5.**The different measurements of the divergence of two PMFs, $P=\{{p}_{1},1-{p}_{1}\}$ and $Q=\{{q}_{1},1-{q}_{1}\}$. The x-axis shows ${p}_{1}$, varying from 0 to 1, while we set ${q}_{1}=(1-\alpha ){p}_{1}+\alpha (1-{p}_{1}),\alpha \in [0,1]$. When $\alpha =1$, Q is most divergent away from P. The curve $0.3{\mathcal{D}}_{\mathrm{KL}}(\alpha =0.5)$ is shown in a dashed black line, and is used as a benchmark for observing the corresponding curves (in orange) produced by the candidate measures in (

**c**–

**i**).

**Figure 6.**A visual comparison of the candidate measures in a range near zero. Similar to Figure 5, $P=\{{p}_{1},1-{p}_{1}\}$ and $Q=\{{q}_{1},1-{q}_{1}\}$, but only the curve $\alpha =1$ is shown, i.e., ${q}_{1}=1-{p}_{1}$. The line segments of ${\mathcal{D}}_{\mathrm{KL}}$ and $0.3{\mathcal{D}}_{\mathrm{KL}}$ in the range $[0,{0.1}^{10}]$ do not represent the actual curves. The ranges $[0,{0.1}^{10}]$ and $[0.1,0.5]$ are only for references to the nearby contexts as they do not use the same logarithmic scale as in $[{0.1}^{10},0.1]$.

**Figure 7.**Some of the major temperature scales considered by scientists in the history. It took four decades from Isaac Newton’s instance-based proposal to arrive at the most-commonly used Celsius scale. It took another century to discover absolute zero as the lower bound.

**Table 1.**Imaginary scenarios where probability data is collected for estimating knowledge related to alphabet $\mathbb{A}=\{\mathit{curved},\phantom{\rule{0.166667em}{0ex}}\mathit{flat}\}$. The ground truth (G.T.) PMFs are defined with $\u03f5=0.01$ and $0.0001$ respectively. The potential distortion (as “→ value”) is computed using the KL-divergence.

Scenario 1 | Scenario 2 | |
---|---|---|

$Q\left({\mathbb{A}}_{\mathrm{G}.\mathrm{T}.}\right)$: | $\{0.99,0.01\}$ | $\{0.9999,0.0001\}$ |

$P\left({\mathbb{A}}_{\mathrm{MIP}}\right)$: | $\{0.01,0.99\}\to 6.50$ | $\{0.0001,0.9999\}\to 13.28$ |

$P\left({\mathbb{A}}_{\mathrm{doctors}}\right)$: | $\{0.99,0.01\}\to 0.00$ | $\{0.99,0.01\}\to 0.05$ |

$P\left({\mathbb{A}}_{\mathrm{patients}}\right)$: | $\{0.7,0.3\}\to 1.12$ | $\{0.7,0.3\}\to 3.11$ |

**Table 2.**Imaginary scenarios for estimating knowledge related to alphabet $\mathbb{B}=\{\mathit{wrinkles}-\mathit{and}-\mathit{bumps},$$\mathit{smooth}\}$. The ground truth (G.T.) PMFs are defined with $\u03f5=0.1$ and $0.001$ respectively. The potential distortion (as “→ value”) is computed using the KL-divergence.

Scenario 3 | Scenario 4 | |
---|---|---|

$Q\left({\mathbb{B}}_{\mathrm{G}.\mathrm{T}.}\right)$: | $\{0.9,0.1\}$ | $\{0.999,0.001\}$ |

$P\left({\mathbb{B}}_{\mathrm{MIP}}\right)$: | $\{0.1,0.9\}\to 2.54$ | $\{0.001,0.999\}\to 9.94$ |

$P\left({\mathbb{B}}_{\mathrm{doctors}}\right)$: | $\{0.8,0.2\}\to 0.06$ | $\{0.8,0.2\}\to 1.27$ |

$P\left({\mathbb{B}}_{\mathrm{patients}}\right)$: | $\{0.1,0.9\}\to 2.54$ | $\{0.1,0.9\}\to 8.50$ |

**Table 3.**A summary of multi-criteria decision analysis in the first part of this paper. Each measure is scored against a conceptual criterion using an integer in [0, 5] with 5 being the best. The symbol ▸ indicates an interim conclusion after considering one or a few criteria. In the second part of the paper [5], we will discuss another five criteria.

Criteria | Importance | $0.3{\mathcal{D}}_{\mathbf{KL}}$ | ${\mathcal{D}}_{\mathbf{JS}}$ | $\sqrt{{\mathcal{D}}_{\mathbf{JS}}}$ | $\mathcal{H}\left(\mathit{P}\right|\mathit{Q})$ | ${\mathcal{D}}_{\mathbf{new}}^{\mathit{k}=1}$ | ${\mathcal{D}}_{\mathbf{new}}^{\mathit{k}=2}$ | ${\mathcal{D}}_{\mathbf{ncm}}^{\mathit{k}=1}$ | ${\mathcal{D}}_{\mathbf{ncm}}^{\mathit{k}=2}$ | ${\mathit{D}}_{\mathbf{M}}^{\mathit{k}=2}$ | ${\mathit{D}}_{\mathbf{M}}^{\mathit{k}=200}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

1. Boundedness | critical | 0 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 3 | 3 |

▸$0.3{\mathcal{D}}_{\mathrm{KL}}$ is eliminated but used below only for comparison. The other scores are carried forward. | |||||||||||

2. Number of PMFs | important | 5 | 5 | 5 | 2 | 5 | 5 | 5 | 5 | 5 | 5 |

3. Entropic measures | important | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 1 | 1 |

4. Distance metric | helpful | 2 | 3 | 5 | 2 | 4 | 3 | 2 | 2 | 5 | 5 |

5. Easy to understand | helpful | 4 | 4 | 3 | 4 | 4 | 3 | 4 | 3 | 5 | 4 |

6. Curve shapes (Figure 5) | helpful | 5 | 5 | 3 | 1 | 2 | 4 | 2 | 4 | 2 | 2 |

7. Curve shapes (Figure 6) | helpful | 5 | 3 | 4 | 1 | 3 | 5 | 3 | 5 | 2 | 3 |

▸Eliminate$\mathcal{H}\left(P\right|Q)$, ${\mathcal{D}}_{\mathrm{M}}^{2}$, ${\mathcal{D}}_{\mathrm{M}}^{200}$based on criteria 1–7 | sum: | 30 | 30 | 20 | 28 | 30 | 26 | 29 | 23 | 23 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, M.; Sbert, M.
A Bounded Measure for Estimating the Benefit of Visualization (Part I): Theoretical Discourse and Conceptual Evaluation. *Entropy* **2022**, *24*, 228.
https://doi.org/10.3390/e24020228

**AMA Style**

Chen M, Sbert M.
A Bounded Measure for Estimating the Benefit of Visualization (Part I): Theoretical Discourse and Conceptual Evaluation. *Entropy*. 2022; 24(2):228.
https://doi.org/10.3390/e24020228

**Chicago/Turabian Style**

Chen, Min, and Mateu Sbert.
2022. "A Bounded Measure for Estimating the Benefit of Visualization (Part I): Theoretical Discourse and Conceptual Evaluation" *Entropy* 24, no. 2: 228.
https://doi.org/10.3390/e24020228