Computational Comparison and Visualization of Viruses in the Perspective of Clinical Information
Identifieur interne : 000272 ( Pmc/Corpus ); précédent : 000271; suivant : 000273Computational Comparison and Visualization of Viruses in the Perspective of Clinical Information
Auteurs : Ant Nio M. Lopes ; J. A. Tenreiro Machado ; Alexandra M. GalhanoSource :
- Interdisciplinary Sciences, Computational Life Sciences [ 1913-2751 ] ; 2017.
Abstract
This paper addresses the visualization of complex information using multidimensional scaling (MDS). MDS is a technique adopted for processing data with multiple features scattered in high-dimensional spaces. For illustrating the proposed techniques, the case of viral diseases is considered. The study evaluates the characteristics of 21 viruses in the perspective of clinical information. Several new schemes are proposed for improving the visualization of the MDS charts. The results follow standard clinical practice, proving that the method represents a valuable tool to study a large number of viruses.
Url:
DOI: 10.1007/s12539-017-0229-4
PubMed: 28391493
PubMed Central: 7090701
Links to Exploration step
PMC:7090701Le document en format XML
<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Computational Comparison and Visualization of Viruses in the Perspective of Clinical Information</title><author><name sortKey="Lopes, Ant Nio M" sort="Lopes, Ant Nio M" uniqKey="Lopes A" first="Ant Nio M." last="Lopes">Ant Nio M. Lopes</name><affiliation><nlm:aff id="Aff1"><institution-wrap><institution-id institution-id-type="ISNI">0000 0001 1503 7226</institution-id><institution-id institution-id-type="GRID">grid.5808.5</institution-id><institution>UISPA - LAETA/INEGI, Faculty of Engineering,</institution><institution>University of Porto,</institution></institution-wrap>Rua Dr. Roberto Frias, 4200-465 Porto, Portugal</nlm:aff></affiliation></author><author><name sortKey="Machado, J A Tenreiro" sort="Machado, J A Tenreiro" uniqKey="Machado J" first="J. A. Tenreiro" last="Machado">J. A. Tenreiro Machado</name><affiliation><nlm:aff id="Aff2">Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, R. Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal</nlm:aff></affiliation></author><author><name sortKey="Galhano, Alexandra M" sort="Galhano, Alexandra M" uniqKey="Galhano A" first="Alexandra M." last="Galhano">Alexandra M. Galhano</name><affiliation><nlm:aff id="Aff2">Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, R. Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal</nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">28391493</idno><idno type="pmc">7090701</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7090701</idno><idno type="RBID">PMC:7090701</idno><idno type="doi">10.1007/s12539-017-0229-4</idno><date when="2017">2017</date><idno type="wicri:Area/Pmc/Corpus">000272</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000272</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Computational Comparison and Visualization of Viruses in the Perspective of Clinical Information</title><author><name sortKey="Lopes, Ant Nio M" sort="Lopes, Ant Nio M" uniqKey="Lopes A" first="Ant Nio M." last="Lopes">Ant Nio M. Lopes</name><affiliation><nlm:aff id="Aff1"><institution-wrap><institution-id institution-id-type="ISNI">0000 0001 1503 7226</institution-id><institution-id institution-id-type="GRID">grid.5808.5</institution-id><institution>UISPA - LAETA/INEGI, Faculty of Engineering,</institution><institution>University of Porto,</institution></institution-wrap>Rua Dr. Roberto Frias, 4200-465 Porto, Portugal</nlm:aff></affiliation></author><author><name sortKey="Machado, J A Tenreiro" sort="Machado, J A Tenreiro" uniqKey="Machado J" first="J. A. Tenreiro" last="Machado">J. A. Tenreiro Machado</name><affiliation><nlm:aff id="Aff2">Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, R. Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal</nlm:aff></affiliation></author><author><name sortKey="Galhano, Alexandra M" sort="Galhano, Alexandra M" uniqKey="Galhano A" first="Alexandra M." last="Galhano">Alexandra M. Galhano</name><affiliation><nlm:aff id="Aff2">Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, R. Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal</nlm:aff></affiliation></author></analytic><series><title level="j">Interdisciplinary Sciences, Computational Life Sciences</title><idno type="ISSN">1913-2751</idno><idno type="eISSN">1867-1462</idno><imprint><date when="2017">2017</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p id="Par1">This paper addresses the visualization of complex information using multidimensional scaling (MDS). MDS is a technique adopted for processing data with multiple features scattered in high-dimensional spaces. For illustrating the proposed techniques, the case of viral diseases is considered. The study evaluates the characteristics of 21 viruses in the perspective of clinical information. Several new schemes are proposed for improving the visualization of the MDS charts. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">Interdiscip Sci</journal-id><journal-id journal-id-type="iso-abbrev">Interdiscip Sci</journal-id><journal-title-group><journal-title>Interdisciplinary Sciences, Computational Life Sciences</journal-title></journal-title-group><issn pub-type="ppub">1913-2751</issn><issn pub-type="epub">1867-1462</issn><publisher><publisher-name>Springer Berlin Heidelberg</publisher-name><publisher-loc>Berlin/Heidelberg</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">28391493</article-id><article-id pub-id-type="pmc">7090701</article-id><article-id pub-id-type="publisher-id">229</article-id><article-id pub-id-type="doi">10.1007/s12539-017-0229-4</article-id><article-categories><subj-group subj-group-type="heading"><subject>Original Research Article</subject></subj-group></article-categories><title-group><article-title>Computational Comparison and Visualization of Viruses in the Perspective of Clinical Information</article-title></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name><surname>Lopes</surname><given-names>António M.</given-names></name><address><email>aml@fe.up.pt</email></address><xref ref-type="aff" rid="Aff1">1</xref></contrib><contrib contrib-type="author"><name><surname>Machado</surname><given-names>J. A. Tenreiro</given-names></name><xref ref-type="aff" rid="Aff2">2</xref></contrib><contrib contrib-type="author"><name><surname>Galhano</surname><given-names>Alexandra M.</given-names></name><xref ref-type="aff" rid="Aff2">2</xref></contrib><aff id="Aff1"><label>1</label><institution-wrap><institution-id institution-id-type="ISNI">0000 0001 1503 7226</institution-id><institution-id institution-id-type="GRID">grid.5808.5</institution-id><institution>UISPA - LAETA/INEGI, Faculty of Engineering,</institution><institution>University of Porto,</institution></institution-wrap>Rua Dr. Roberto Frias, 4200-465 Porto, Portugal</aff><aff id="Aff2"><label>2</label>Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, R. Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal</aff></contrib-group><pub-date pub-type="epub"><day>8</day><month>4</month><year>2017</year></pub-date><pub-date pub-type="ppub"><year>2019</year></pub-date><volume>11</volume><issue>1</issue><fpage>86</fpage><lpage>94</lpage><history><date date-type="received"><day>1</day><month>11</month><year>2016</year></date><date date-type="rev-recd"><day>17</day><month>3</month><year>2017</year></date><date date-type="accepted"><day>25</day><month>3</month><year>2017</year></date></history><permissions><copyright-statement>© Springer-Verlag 2017</copyright-statement><license><license-p>This article is made available via the PMC Open Access Subset for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic.</license-p></license></permissions><abstract id="Abs1"><p id="Par1">This paper addresses the visualization of complex information using multidimensional scaling (MDS). MDS is a technique adopted for processing data with multiple features scattered in high-dimensional spaces. For illustrating the proposed techniques, the case of viral diseases is considered. The study evaluates the characteristics of 21 viruses in the perspective of clinical information. Several new schemes are proposed for improving the visualization of the MDS charts. The results follow standard clinical practice, proving that the method represents a valuable tool to study a large number of viruses.</p></abstract><kwd-group xml:lang="en"><title>Keywords</title><kwd>Multidimensional scaling</kwd><kwd>Computational visualization</kwd><kwd>Complex data</kwd><kwd>Viruses diseases</kwd></kwd-group><custom-meta-group><custom-meta><meta-name>issue-copyright-statement</meta-name><meta-value>© International Association of Scientists in the Interdisciplinary Areas 2019</meta-value></custom-meta></custom-meta-group></article-meta></front><body><sec id="Sec1"><title>Introduction</title><p id="Par2">Presently, reliable and assertive data about many real-world phenomena are available for computer processing. One example consists of clinical information about viral diseases. Viruses infections are an important cause of mortality and morbidity. More than 2000 viruses were identified and many can infect humans, or animals [<xref ref-type="bibr" rid="CR1">1</xref>]. In general, viral diseases have very diverse characteristics and complexity, and computational methods for data mining and feature extraction are relevant strategies to adopt. As usually occurs with real-world data, information is scattered, and exhibits multiple characteristics with distinct levels of relevance. Therefore, it is important to explore reliable algorithms for highlighting the main details, and to take advantage of modern computational resources to visualize the relations embedded within the data.</p><p id="Par3">Herein, we adopt the multidimensional scaling (MDS) technique to compare the relationships among several viruses responsible for human diseases. New schemes for improving the visualization of the MDS charts are proposed. In what concerns selection of the “objects” under study, most are based on their impact on people and visibility in communication media (e.g., subtype H5N1 of Influenza A virus, Ebola, Chikungunya and Zika), others due to historical reasons (e.g., Rabies, Poliomyelitis, and Smallpox), and some because of their incidence and prevalence in humans (e.g., Influenza, Rhinovirus, and Norovirus). The viruses are compared by means of their characteristics and the symptoms of the diseases that they may cause in humans.</p><p id="Par4">The MDS can lead to a new perspective in the study of human pathologies. MDS is a statistical technique for analyzing similarities in information that generates geometric representations for complex objects [<xref ref-type="bibr" rid="CR2">2</xref>]. MDS appeared in the context of behavioral sciences, for understanding judgments of individuals about features in a set of objects [<xref ref-type="bibr" rid="CR3">3</xref>, <xref ref-type="bibr" rid="CR4">4</xref>]. Presently, the MDS is used in real-world data, such as biological taxonomy [<xref ref-type="bibr" rid="CR5">5</xref>], finance [<xref ref-type="bibr" rid="CR6">6</xref>], marketing [<xref ref-type="bibr" rid="CR7">7</xref>], sociology [<xref ref-type="bibr" rid="CR8">8</xref>], physics [<xref ref-type="bibr" rid="CR9">9</xref>], geophysics [<xref ref-type="bibr" rid="CR10">10</xref>–<xref ref-type="bibr" rid="CR12">12</xref>], communication networks [<xref ref-type="bibr" rid="CR13">13</xref>], biology and biomedicine [<xref ref-type="bibr" rid="CR14">14</xref>], among others [<xref ref-type="bibr" rid="CR15">15</xref>].</p><p id="Par5">The paper is organized as follows. Section <xref rid="Sec2" ref-type="sec">2</xref> introduces the MDS technique. Section <xref rid="Sec3" ref-type="sec">3</xref> studies and compares data characterizing the clinical effects of 21 viruses. Finally, Sect. <xref rid="Sec10" ref-type="sec">4</xref> draws the conclusions.</p></sec><sec id="Sec2"><title>Multidimensional Scaling</title><p id="Par6">We consider <italic>s</italic> objects defined in a <italic>m</italic>-dim space, <inline-formula id="IEq1"><alternatives><tex-math id="M1">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}</tex-math><mml:math id="M2"><mml:mi mathvariant="script">M</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq1.gif"></inline-graphic></alternatives></inline-formula>, and a proximity measure, <inline-formula id="IEq2"><alternatives><tex-math id="M3">\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M4"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq2.gif"></inline-graphic></alternatives></inline-formula>, between objects <italic>i</italic> and <italic>j</italic>. The first step consists of calculating <inline-formula id="IEq3"><alternatives><tex-math id="M5">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {C}=[\delta _{ij}]$$\end{document}</tex-math><mml:math id="M6"><mml:mrow><mml:mi mathvariant="bold">C</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq3.gif"></inline-graphic></alternatives></inline-formula> (<inline-formula id="IEq4"><alternatives><tex-math id="M7">\documentclass[12pt]{minimal}
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\begin{document}$$\dim(\mathbf {C})=s \times s$$\end{document}</tex-math><mml:math id="M8"><mml:mrow><mml:mo>dim</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold">C</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>s</mml:mi><mml:mo>×</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq4.gif"></inline-graphic></alternatives></inline-formula>), of item-to-item dissimilarities. The MDS produces a configuration <inline-formula id="IEq5"><alternatives><tex-math id="M9">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {X}$$\end{document}</tex-math><mml:math id="M10"><mml:mi mathvariant="bold">X</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq5.gif"></inline-graphic></alternatives></inline-formula> (<inline-formula id="IEq6"><alternatives><tex-math id="M11">\documentclass[12pt]{minimal}
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\begin{document}$$\dim(\mathbf {X})=s \times q$$\end{document}</tex-math><mml:math id="M12"><mml:mrow><mml:mo>dim</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold">X</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>s</mml:mi><mml:mo>×</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq6.gif"></inline-graphic></alternatives></inline-formula>), where the dimension <inline-formula id="IEq7"><alternatives><tex-math id="M13">\documentclass[12pt]{minimal}
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\begin{document}$$q < m$$\end{document}</tex-math><mml:math id="M14"><mml:mrow><mml:mi>q</mml:mi><mml:mo><</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq7.gif"></inline-graphic></alternatives></inline-formula> is chosen by the user. Thus, <inline-formula id="IEq8"><alternatives><tex-math id="M15">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {X}$$\end{document}</tex-math><mml:math id="M16"><mml:mi mathvariant="bold">X</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq8.gif"></inline-graphic></alternatives></inline-formula> attempts to replicate in a low-dimensional space, <inline-formula id="IEq9"><alternatives><tex-math id="M17">\documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$$\mathcal {Q}$$\end{document}</tex-math><mml:math id="M18"><mml:mi mathvariant="script">Q</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq9.gif"></inline-graphic></alternatives></inline-formula>, the proximities between the <italic>s</italic> elements in <inline-formula id="IEq10"><alternatives><tex-math id="M19">\documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$$\mathcal {M}$$\end{document}</tex-math><mml:math id="M20"><mml:mi mathvariant="script">M</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq10.gif"></inline-graphic></alternatives></inline-formula>. In general, the MDS unveils the embedded data patterns, being different from other techniques [<xref ref-type="bibr" rid="CR16">16</xref>, <xref ref-type="bibr" rid="CR17">17</xref>], not only because it requires no a priori assumptions for each dimension, but also due to its good convergence [<xref ref-type="bibr" rid="CR18">18</xref>, <xref ref-type="bibr" rid="CR19">19</xref>].</p><p id="Par7">To arrive to configuration <inline-formula id="IEq11"><alternatives><tex-math id="M21">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {X}$$\end{document}</tex-math><mml:math id="M22"><mml:mi mathvariant="bold">X</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq11.gif"></inline-graphic></alternatives></inline-formula>, MDS evaluates different alternative values to minimize some fitness function, such as [<xref ref-type="bibr" rid="CR20">20</xref>] the raw stress, <inline-formula id="IEq12"><alternatives><tex-math id="M23">\documentclass[12pt]{minimal}
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\begin{document}$$\sigma ^2$$\end{document}</tex-math><mml:math id="M24"><mml:msup><mml:mi mathvariant="italic">σ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq12.gif"></inline-graphic></alternatives></inline-formula>:<disp-formula id="Equ1"><label>1</label><alternatives><tex-math id="M25">\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sigma ^2=\displaystyle \sum _{i=2}^s \sum _{j=1}^{i-1} z_{ij}\left( \delta _{ij}-d_{ij}\right) ^2, \end{aligned}$$\end{document}</tex-math><mml:math id="M26" display="block"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:msup><mml:mi mathvariant="italic">σ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mi>s</mml:mi></mml:munderover><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mfenced close=")" open="("><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mstyle></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12539_2017_229_Article_Equ1.gif" position="anchor"></graphic></alternatives></disp-formula>where <inline-formula id="IEq13"><alternatives><tex-math id="M27">\documentclass[12pt]{minimal}
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\begin{document}$$z_{ij}>0$$\end{document}</tex-math><mml:math id="M28"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq13.gif"></inline-graphic></alternatives></inline-formula> is a weight and <inline-formula id="IEq14"><alternatives><tex-math id="M29">\documentclass[12pt]{minimal}
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\begin{document}$$d_{ij}$$\end{document}</tex-math><mml:math id="M30"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq14.gif"></inline-graphic></alternatives></inline-formula> measures the dissimilarities among the items <italic>i</italic> and <italic>j</italic> in the embedding space <inline-formula id="IEq15"><alternatives><tex-math id="M31">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Q}$$\end{document}</tex-math><mml:math id="M32"><mml:mi mathvariant="script">Q</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq15.gif"></inline-graphic></alternatives></inline-formula>. Therefore, a distance measure is often adopted for implementing <inline-formula id="IEq16"><alternatives><tex-math id="M33">\documentclass[12pt]{minimal}
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\begin{document}$$d_{ij}$$\end{document}</tex-math><mml:math id="M34"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq16.gif"></inline-graphic></alternatives></inline-formula> [<xref ref-type="bibr" rid="CR21">21</xref>].</p><p id="Par8">Besides (<xref rid="Equ1" ref-type="">1</xref>), there are several stress measures [<xref ref-type="bibr" rid="CR22">22</xref>], namely, the normalized raw stress, the Kruskal’s stress-1 and stress-2, and the <italic>S</italic> stress.</p><p id="Par9">To assess the quality of the MDS solutions, it is used the Shepard diagram that represents the pairs <inline-formula id="IEq17"><alternatives><tex-math id="M35">\documentclass[12pt]{minimal}
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\begin{document}$$(d_{ij}, \, \delta _{ij})$$\end{document}</tex-math><mml:math id="M36"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq17.gif"></inline-graphic></alternatives></inline-formula>. The Shepard diagram displays the outliers and residuals resulting from the MDS. A narrow scatter following the 45<inline-formula id="IEq18"><alternatives><tex-math id="M37">\documentclass[12pt]{minimal}
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\begin{document}$$^{\circ }$$\end{document}</tex-math><mml:math id="M38"><mml:msup><mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:msup></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq18.gif"></inline-graphic></alternatives></inline-formula> line corresponds to a good fit between <inline-formula id="IEq19"><alternatives><tex-math id="M39">\documentclass[12pt]{minimal}
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\begin{document}$$d_{ij}$$\end{document}</tex-math><mml:math id="M40"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq19.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq20"><alternatives><tex-math id="M41">\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M42"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq20.gif"></inline-graphic></alternatives></inline-formula>.</p><p id="Par10">Another test to the MDS quality is the stress plot that represents <inline-formula id="IEq21"><alternatives><tex-math id="M43">\documentclass[12pt]{minimal}
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\begin{document}$$\sigma ^2$$\end{document}</tex-math><mml:math id="M44"><mml:msup><mml:mi mathvariant="italic">σ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq21.gif"></inline-graphic></alternatives></inline-formula> versus <italic>q</italic>. The curve <inline-formula id="IEq22"><alternatives><tex-math id="M45">\documentclass[12pt]{minimal}
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\begin{document}$$\sigma ^2(q)$$\end{document}</tex-math><mml:math id="M46"><mml:mrow><mml:msup><mml:mi mathvariant="italic">σ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq22.gif"></inline-graphic></alternatives></inline-formula> is monotonic decreasing and the user chooses <italic>q</italic> as a compromise between reducing <inline-formula id="IEq23"><alternatives><tex-math id="M47">\documentclass[12pt]{minimal}
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\begin{document}$$\sigma ^2$$\end{document}</tex-math><mml:math id="M48"><mml:msup><mml:mi mathvariant="italic">σ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq23.gif"></inline-graphic></alternatives></inline-formula> and having small values of <italic>q</italic>.</p><p id="Par11">The MDS interpretation focuses on the emerging clusters and considers the distances between points in the produced chart. Therefore, the user can rotate, shift, or zoom the chart, while the distances remain invariant. Usually, <inline-formula id="IEq24"><alternatives><tex-math id="M49">\documentclass[12pt]{minimal}
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\begin{document}$$q=2$$\end{document}</tex-math><mml:math id="M50"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq24.gif"></inline-graphic></alternatives></inline-formula>, or <inline-formula id="IEq25"><alternatives><tex-math id="M51">\documentclass[12pt]{minimal}
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\begin{document}$$q=3$$\end{document}</tex-math><mml:math id="M52"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq25.gif"></inline-graphic></alternatives></inline-formula>, is adopted, since they allow a direct graphical representation.</p></sec><sec id="Sec3"><title>Data Analysis and Visualization</title><p id="Par12">We analyze data for <inline-formula id="IEq26"><alternatives><tex-math id="M53">\documentclass[12pt]{minimal}
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\begin{document}$$s = 21$$\end{document}</tex-math><mml:math id="M54"><mml:mrow><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn>21</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq26.gif"></inline-graphic></alternatives></inline-formula> viruses responsible for infectious diseases. These are <inline-formula id="IEq27"><alternatives><tex-math id="M55">\documentclass[12pt]{minimal}
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\begin{document}$$\{$$\end{document}</tex-math><mml:math id="M56"><mml:mo stretchy="false">{</mml:mo></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq27.gif"></inline-graphic></alternatives></inline-formula>Bird Flu, Chicken Pox, Chikungunya, Dengue Fever, Ebola, Hepatitis B, HIV, Marburg disease, Measles, MERS, Mumps, Norovirus, Polio, Rabies, Rhinovirus, Rotavirus, Rubella, SARS, Seasonal Flu, Smallpox, and Zika virus infection<inline-formula id="IEq28"><alternatives><tex-math id="M57">\documentclass[12pt]{minimal}
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\begin{document}$$\}$$\end{document}</tex-math><mml:math id="M58"><mml:mo stretchy="false">}</mml:mo></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq28.gif"></inline-graphic></alternatives></inline-formula>, with acronyms <inline-formula id="IEq29"><alternatives><tex-math id="M59">\documentclass[12pt]{minimal}
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\begin{document}$$\{$$\end{document}</tex-math><mml:math id="M60"><mml:mo stretchy="false">{</mml:mo></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq29.gif"></inline-graphic></alternatives></inline-formula>BFlu, CPox, Chi, Den, Ebo, HepB, HIV, Mar, Mea, MERS, Mum, Nor, Pol, Rab, Rhi, Rot, Rub, SARS, SFlu, Sma, and ZIKV<inline-formula id="IEq30"><alternatives><tex-math id="M61">\documentclass[12pt]{minimal}
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\begin{document}$$\}$$\end{document}</tex-math><mml:math id="M62"><mml:mo stretchy="false">}</mml:mo></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq30.gif"></inline-graphic></alternatives></inline-formula>.</p><p id="Par13">For the <italic>i</italic>th virus, <inline-formula id="IEq31"><alternatives><tex-math id="M63">\documentclass[12pt]{minimal}
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\begin{document}$$i = 1,\ldots , \, s$$\end{document}</tex-math><mml:math id="M64"><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mi>s</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq31.gif"></inline-graphic></alternatives></inline-formula>, we associate <inline-formula id="IEq32"><alternatives><tex-math id="M65">\documentclass[12pt]{minimal}
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\begin{document}$$m=5$$\end{document}</tex-math><mml:math id="M66"><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq32.gif"></inline-graphic></alternatives></inline-formula> quantitative attributes, namely, (i) the fatality rate, (ii) the average basic reproductive number, (iii) the average serial interval, (iv) the incubation period, and (v) the virus survival time outside a host. Table <xref rid="Tab1" ref-type="table">1</xref> lists the data, where the numerical values correspond to the matrix <inline-formula id="IEq33"><alternatives><tex-math id="M67">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {\tilde{U}} = [\tilde{u}_{ik}]$$\end{document}</tex-math><mml:math id="M68"><mml:mrow><mml:mover accent="true"><mml:mi mathvariant="bold">U</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>u</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover><mml:mrow><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq33.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq34"><alternatives><tex-math id="M69">\documentclass[12pt]{minimal}
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\begin{document}$$i = 1,\ldots , \, s$$\end{document}</tex-math><mml:math id="M70"><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mi>s</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq34.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq35"><alternatives><tex-math id="M71">\documentclass[12pt]{minimal}
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\begin{document}$$k = 1,\ldots , \, m$$\end{document}</tex-math><mml:math id="M72"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mi>m</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq35.gif"></inline-graphic></alternatives></inline-formula>.<table-wrap id="Tab1"><label>Table 1</label><caption><p>Attributes of the viruses</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="left" rowspan="3"><italic>i</italic></th><th align="left" rowspan="3">Virus</th><th align="left" rowspan="3">Acronym</th><th align="left" colspan="5"><italic>k</italic></th></tr><tr><th align="left">Fatality rate (%)</th><th align="left">Average basic reproductive number</th><th align="left">Average serial interval (days)</th><th align="left">Incubation period (days)</th><th align="left">Survival outside host (days)</th></tr><tr><th align="left">1</th><th align="left">2</th><th align="left">3</th><th align="left">4</th><th align="left">5</th></tr></thead><tbody><tr><td align="left">1</td><td align="left">Bird Flu</td><td align="left">BFlu</td><td char="." align="char">59.00</td><td char="." align="char">2.00</td><td align="left">3.00</td><td char="." align="char">3.0</td><td char="." align="char">30.0</td></tr><tr><td align="left">2</td><td align="left">Chicken Pox</td><td align="left">CPox</td><td char="." align="char">0.00</td><td char="." align="char">7.50</td><td align="left">14.00</td><td char="." align="char">14.0</td><td char="." align="char">2.0</td></tr><tr><td align="left">3</td><td align="left">Chikungunya</td><td align="left">Chi</td><td char="." align="char">0.40</td><td char="." align="char">4.00</td><td align="left">23.00</td><td char="." align="char">2.5</td><td char="." align="char">–</td></tr><tr><td align="left">4</td><td align="left">Dengue Fever</td><td align="left">Den</td><td char="." align="char">5.00</td><td char="." align="char">3.00</td><td align="left">16.00</td><td char="." align="char">7.0</td><td char="." align="char">63.0</td></tr><tr><td align="left">5</td><td align="left">Ebola</td><td align="left">Ebo</td><td char="." align="char">75.00</td><td char="." align="char">2.50</td><td align="left">15.30</td><td char="." align="char">11.4</td><td char="." align="char">50.0</td></tr><tr><td align="left">6</td><td align="left">Hepatitis B</td><td align="left">HepB</td><td char="." align="char">0.75</td><td char="." align="char">6.00</td><td align="left">25.00</td><td char="." align="char">75.0</td><td char="." align="char">28.0</td></tr><tr><td align="left">7</td><td align="left">HIV</td><td align="left">HIV</td><td char="." align="char">2.10</td><td char="." align="char">3.50</td><td align="left">–</td><td char="." align="char">60.0</td><td char="." align="char">42.0</td></tr><tr><td align="left">8</td><td align="left">Marburg virus disease</td><td align="left">Mar</td><td char="." align="char">25.00</td><td char="." align="char">1.60</td><td align="left">9.00</td><td char="." align="char">6.0</td><td char="." align="char">21.0</td></tr><tr><td align="left">9</td><td align="left">Measles</td><td align="left">Mea</td><td char="." align="char">0.20</td><td char="." align="char">15.00</td><td align="left">11.70</td><td char="." align="char">11.0</td><td char="." align="char">0.1</td></tr><tr><td align="left">10</td><td align="left">MERS</td><td align="left">MERS</td><td char="." align="char">27.00</td><td char="." align="char">0.50</td><td align="left">7.60</td><td char="." align="char">5.0</td><td char="." align="char">3.0</td></tr><tr><td align="left">11</td><td align="left">Mumps</td><td align="left">Mum</td><td char="." align="char">0.01</td><td char="." align="char">5.50</td><td align="left">18.00</td><td char="." align="char">17.0</td><td char="." align="char">0.3</td></tr><tr><td align="left">12</td><td align="left">Norovirus</td><td align="left">Nor</td><td char="." align="char">0.08</td><td char="." align="char">3.70</td><td align="left">1.86</td><td char="." align="char">1.5</td><td char="." align="char">24.0</td></tr><tr><td align="left">13</td><td align="left">Polio</td><td align="left">Pol</td><td char="." align="char">22.00</td><td char="." align="char">6.00</td><td align="left">–</td><td char="." align="char">13.0</td><td char="." align="char">160.0</td></tr><tr><td align="left">14</td><td align="left">Rabies</td><td align="left">Rab</td><td char="." align="char">0.00</td><td char="." align="char">1.60</td><td align="left">–</td><td char="." align="char">40.0</td><td char="." align="char">6.0</td></tr><tr><td align="left">15</td><td align="left">Rhinovirus</td><td align="left">Rhi</td><td char="." align="char">0.00</td><td char="." align="char">3.70</td><td align="left">7.50</td><td char="." align="char">3.0</td><td char="." align="char">1.0</td></tr><tr><td align="left">16</td><td align="left">Rotavirus</td><td align="left">Rot</td><td char="." align="char">0.00</td><td char="." align="char">3.50</td><td align="left">7.00</td><td char="." align="char">1.5</td><td char="." align="char">60.0</td></tr><tr><td align="left">17</td><td align="left">Rubella</td><td align="left">Rub</td><td char="." align="char">0.00</td><td char="." align="char">6.50</td><td align="left">18.30</td><td char="." align="char">17.7</td><td char="." align="char">0.9</td></tr><tr><td align="left">18</td><td align="left">SARS</td><td align="left">SARS</td><td char="." align="char">11.00</td><td char="." align="char">3.50</td><td align="left">10.00</td><td char="." align="char">8.0</td><td char="." align="char">9.0</td></tr><tr><td align="left">19</td><td align="left">Seasonal Flu</td><td align="left">SFlu</td><td char="." align="char">0.01</td><td char="." align="char">1.30</td><td align="left">3.30</td><td char="." align="char">2.0</td><td char="." align="char">2.0</td></tr><tr><td align="left">20</td><td align="left">Smallpox</td><td align="left">Sma</td><td char="." align="char">15.00</td><td char="." align="char">6.00</td><td align="left">17.70</td><td char="." align="char">14.0</td><td char="." align="char">1.5</td></tr><tr><td align="left">21</td><td align="left">Zika virus disease</td><td align="left">ZIKV</td><td char="." align="char">0.3</td><td char="." align="char">0.5</td><td align="left">13.0</td><td char="." align="char">6.0</td><td char="." align="char">180.0</td></tr></tbody></table></table-wrap></p><p id="Par14">For constructing Table <xref rid="Tab1" ref-type="table">1</xref>, data were obtained from several distinct sources: Influenza A virus, subtype H5N1 (or “Bird Flu”) [<xref ref-type="bibr" rid="CR23">23</xref>–<xref ref-type="bibr" rid="CR25">25</xref>]; Chicken Pox (varicella-zoster infection) [<xref ref-type="bibr" rid="CR26">26</xref>–<xref ref-type="bibr" rid="CR28">28</xref>]; Chikungunya [<xref ref-type="bibr" rid="CR29">29</xref>–<xref ref-type="bibr" rid="CR31">31</xref>]; Dengue Fever [<xref ref-type="bibr" rid="CR32">32</xref>, <xref ref-type="bibr" rid="CR33">33</xref>]; Ebola [<xref ref-type="bibr" rid="CR34">34</xref>–<xref ref-type="bibr" rid="CR36">36</xref>]; Hepatitis B [<xref ref-type="bibr" rid="CR37">37</xref>–<xref ref-type="bibr" rid="CR39">39</xref>]; Human Immunodeficiency Virus (HIV) [<xref ref-type="bibr" rid="CR40">40</xref>–<xref ref-type="bibr" rid="CR42">42</xref>]; Marburg hemorrhagic fever [<xref ref-type="bibr" rid="CR36">36</xref>, <xref ref-type="bibr" rid="CR43">43</xref>]; Measles [<xref ref-type="bibr" rid="CR44">44</xref>–<xref ref-type="bibr" rid="CR47">47</xref>]; Middle East Respiratory Syndrome (MERS) [<xref ref-type="bibr" rid="CR48">48</xref>–<xref ref-type="bibr" rid="CR50">50</xref>]; Mumps [<xref ref-type="bibr" rid="CR51">51</xref>, <xref ref-type="bibr" rid="CR52">52</xref>]; Norovirus [<xref ref-type="bibr" rid="CR53">53</xref>, <xref ref-type="bibr" rid="CR54">54</xref>]; Poliomyelitis [<xref ref-type="bibr" rid="CR55">55</xref>–<xref ref-type="bibr" rid="CR57">57</xref>]; Rabies [<xref ref-type="bibr" rid="CR58">58</xref>–<xref ref-type="bibr" rid="CR60">60</xref>]; Rhinovirus [<xref ref-type="bibr" rid="CR61">61</xref>–<xref ref-type="bibr" rid="CR63">63</xref>]; Rotavirus [<xref ref-type="bibr" rid="CR64">64</xref>–<xref ref-type="bibr" rid="CR67">67</xref>]; Rubella [<xref ref-type="bibr" rid="CR46">46</xref>, <xref ref-type="bibr" rid="CR68">68</xref>]; Severe Acute Respiratory Syndrome (SARS) [<xref ref-type="bibr" rid="CR49">49</xref>, <xref ref-type="bibr" rid="CR69">69</xref>]; Seasonal flu [<xref ref-type="bibr" rid="CR25">25</xref>, <xref ref-type="bibr" rid="CR70">70</xref>, <xref ref-type="bibr" rid="CR71">71</xref>]; Smallpox [<xref ref-type="bibr" rid="CR72">72</xref>, <xref ref-type="bibr" rid="CR73">73</xref>]; Zika virus disease [<xref ref-type="bibr" rid="CR74">74</xref>, <xref ref-type="bibr" rid="CR75">75</xref>].</p><sec id="Sec4"><title>MDS Analysis using the Arc-cosine Distance</title><p id="Par15">Previous to applying MDS, the data are “normalized” to avoid saturation effects of the numerical values. Therefore, the elements of each column of matrix <inline-formula id="IEq36"><alternatives><tex-math id="M73">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {\tilde{U}}$$\end{document}</tex-math><mml:math id="M74"><mml:mover accent="true"><mml:mi mathvariant="bold">U</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq36.gif"></inline-graphic></alternatives></inline-formula> are converted to the interval [0, 1], producing the data matrix <inline-formula id="IEq37"><alternatives><tex-math id="M75">\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {U}}$$\end{document}</tex-math><mml:math id="M76"><mml:mi mathvariant="bold">U</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq37.gif"></inline-graphic></alternatives></inline-formula>. The vectors of features for item-to-item comparison correspond to the lines of <inline-formula id="IEq38"><alternatives><tex-math id="M77">\documentclass[12pt]{minimal}
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\begin{document}$${\mathbf {U}}$$\end{document}</tex-math><mml:math id="M78"><mml:mi mathvariant="bold">U</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq38.gif"></inline-graphic></alternatives></inline-formula> and will be denoted by <inline-formula id="IEq39"><alternatives><tex-math id="M79">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {u}_i$$\end{document}</tex-math><mml:math id="M80"><mml:msub><mml:mi mathvariant="bold">u</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq39.gif"></inline-graphic></alternatives></inline-formula>.</p><p id="Par16">Various distance measures were tested for constructing the matrix <inline-formula id="IEq40"><alternatives><tex-math id="M81">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {C}$$\end{document}</tex-math><mml:math id="M82"><mml:mi mathvariant="bold">C</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq40.gif"></inline-graphic></alternatives></inline-formula>. Here, we present results for the arc-cosine distance, <inline-formula id="IEq41"><alternatives><tex-math id="M83">\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M84"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq41.gif"></inline-graphic></alternatives></inline-formula>, since it leads to charts that are easy to interpret. Other distances are possible and have also been used in distinct applications [<xref ref-type="bibr" rid="CR6">6</xref>, <xref ref-type="bibr" rid="CR12">12</xref>], but several numerical tests confirmed that the arc-cosine leads to reliable results. Therefore, for items <italic>i</italic> and <italic>j</italic><inline-formula id="IEq42"><alternatives><tex-math id="M85">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(i, \, j = 1,\ldots , \, s)$$\end{document}</tex-math><mml:math id="M86"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mi>s</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq42.gif"></inline-graphic></alternatives></inline-formula>, we have<disp-formula id="Equ2"><label>2</label><alternatives><tex-math id="M87">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
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\begin{document}$$\begin{aligned} \delta _{ij}=\arccos \left( \frac{\displaystyle \sum _{k=1}^{m} \alpha _k^2u_{ik} u_{jk}}{\sqrt{\displaystyle \sum _{k=1}^{m} \alpha _k^2 u_{ik}^2 \cdot \displaystyle \sum _{k=1}^{m} \alpha _k^2 u_{jk}^2}}\right) , \end{aligned}$$\end{document}</tex-math><mml:math id="M88" display="block"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>arccos</mml:mo><mml:mfenced close=")" open="("><mml:mfrac><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>m</mml:mi></mml:munderover><mml:msubsup><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mstyle><mml:msqrt><mml:mstyle displaystyle="true" scriptlevel="0"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>m</mml:mi></mml:munderover><mml:msubsup><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msubsup><mml:mi>u</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>·</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>m</mml:mi></mml:munderover><mml:msubsup><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msubsup><mml:mi>u</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:mstyle></mml:msqrt></mml:mfrac></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><graphic xlink:href="12539_2017_229_Article_Equ2.gif" position="anchor"></graphic></alternatives></disp-formula>where <inline-formula id="IEq43"><alternatives><tex-math id="M89">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha _k>0$$\end{document}</tex-math><mml:math id="M90"><mml:mrow><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq43.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq44"><alternatives><tex-math id="M91">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$k=1,\ldots , \, m$$\end{document}</tex-math><mml:math id="M92"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mi>m</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq44.gif"></inline-graphic></alternatives></inline-formula>, represent weights specified by the user. Given expression (<xref rid="Equ2" ref-type="">2</xref>), the matrix <inline-formula id="IEq45"><alternatives><tex-math id="M93">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
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\begin{document}$$\mathbf {C} = [\delta _{ij}]$$\end{document}</tex-math><mml:math id="M94"><mml:mrow><mml:mi mathvariant="bold">C</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq45.gif"></inline-graphic></alternatives></inline-formula> can be computed for feeding the MDS.</p><p id="Par17">Figure <xref rid="Fig1" ref-type="fig">1</xref> represents the 2D and 3D charts (<inline-formula id="IEq46"><alternatives><tex-math id="M95">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
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\begin{document}$$q=2$$\end{document}</tex-math><mml:math id="M96"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq46.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq47"><alternatives><tex-math id="M97">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q=3$$\end{document}</tex-math><mml:math id="M98"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq47.gif"></inline-graphic></alternatives></inline-formula>) resulting from the MDS using the weights <inline-formula id="IEq48"><alternatives><tex-math id="M99">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha _k = \{5,\,2,\,1,\,1,\,1\}$$\end{document}</tex-math><mml:math id="M100"><mml:mrow><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq48.gif"></inline-graphic></alternatives></inline-formula>, where the points represent viruses. The relationships between the items are inferred from the coordinates of the points. Objects that are similar (dissimilar) appear closer (farther) to each other in space.</p><p id="Par18">With alternative distances, we capture different characteristics of the phenomena that yield distinct plots, but in general lead to identical conclusions. A “good” distance is the one that produces a MDS reflecting the real-world phenomenon in a direct and clear visualization.<fig id="Fig1"><label>Fig. 1</label><caption><p>MDS charts resulting from the arc-cosine distance <inline-formula id="IEq49"><alternatives><tex-math id="M101">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M102"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq49.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq50"><alternatives><tex-math id="M103">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q=2$$\end{document}</tex-math><mml:math id="M104"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq50.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq51"><alternatives><tex-math id="M105">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q=3$$\end{document}</tex-math><mml:math id="M106"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq51.gif"></inline-graphic></alternatives></inline-formula></p></caption><graphic xlink:href="12539_2017_229_Fig1_HTML" id="MO3"></graphic></fig></p><p id="Par19">Figure <xref rid="Fig2" ref-type="fig">2</xref> depicts the Shepard diagram for <inline-formula id="IEq52"><alternatives><tex-math id="M107">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
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\begin{document}$$q=1,\ldots , \, 5$$\end{document}</tex-math><mml:math id="M108"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>5</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq52.gif"></inline-graphic></alternatives></inline-formula> and the stress plot. The Shepard diagram depicts a good scatter of points around the 45<inline-formula id="IEq53"><alternatives><tex-math id="M109">\documentclass[12pt]{minimal}
\usepackage{amsmath}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$^{\circ }$$\end{document}</tex-math><mml:math id="M110"><mml:msup><mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:msup></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq53.gif"></inline-graphic></alternatives></inline-formula> line for <inline-formula id="IEq54"><alternatives><tex-math id="M111">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q \ge 3$$\end{document}</tex-math><mml:math id="M112"><mml:mrow><mml:mi>q</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq54.gif"></inline-graphic></alternatives></inline-formula>, demonstrating a good fit between the distances and the dissimilarities. The curvature of the stress plot is often adopted for deciding the value of <italic>q</italic>. In this case, we observe that <inline-formula id="IEq55"><alternatives><tex-math id="M113">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
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\begin{document}$$q =2$$\end{document}</tex-math><mml:math id="M114"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq55.gif"></inline-graphic></alternatives></inline-formula> is insufficient, <inline-formula id="IEq56"><alternatives><tex-math id="M115">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q=3$$\end{document}</tex-math><mml:math id="M116"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq56.gif"></inline-graphic></alternatives></inline-formula> seems to be a good choice, and <inline-formula id="IEq57"><alternatives><tex-math id="M117">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q> 3$$\end{document}</tex-math><mml:math id="M118"><mml:mrow><mml:mi>q</mml:mi><mml:mo>></mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq57.gif"></inline-graphic></alternatives></inline-formula> leads to limited improvements. However, if we adopt <inline-formula id="IEq58"><alternatives><tex-math id="M119">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q=3$$\end{document}</tex-math><mml:math id="M120"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq58.gif"></inline-graphic></alternatives></inline-formula> the question remains of visualizing efficiently the MDS information, since for 3D representations, we often have to zoom, shift, and rotate the MDS graph to perceive assertively the real location of the objects in space. This question will further discussed in Sect. <xref rid="Sec8" ref-type="sec">3.3.2</xref>.<fig id="Fig2"><label>Fig. 2</label><caption><p>Quality of the MDS solution for the arc-cosine distance <inline-formula id="IEq59"><alternatives><tex-math id="M121">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
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\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M122"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq59.gif"></inline-graphic></alternatives></inline-formula> assessed by the Shepard diagram for <inline-formula id="IEq60"><alternatives><tex-math id="M123">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q=1,\ldots , \, 5$$\end{document}</tex-math><mml:math id="M124"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>5</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq60.gif"></inline-graphic></alternatives></inline-formula> and the stress plot</p></caption><graphic xlink:href="12539_2017_229_Fig2_HTML" id="MO4"></graphic></fig></p><p id="Par20">Before continuing, two numerical aspects need to be clarified: the weights <inline-formula id="IEq61"><alternatives><tex-math id="M125">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
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\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha _k$$\end{document}</tex-math><mml:math id="M126"><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq61.gif"></inline-graphic></alternatives></inline-formula> used and the missing data in Table <xref rid="Tab1" ref-type="table">1</xref>. The weights were tuned for highlighting the importance of the features recognized as being more harmful from the medical point of view: first, the fatality rate and, second, the average basic reproductive number. However, the question remains on how to choose <inline-formula id="IEq62"><alternatives><tex-math id="M127">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
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\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha _k$$\end{document}</tex-math><mml:math id="M128"><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq62.gif"></inline-graphic></alternatives></inline-formula>. In this perspective, we performed several experiments varying the weights. Figure <xref rid="Fig3" ref-type="fig">3</xref> depicts the results obtained with four distinct sets of values, namely, <inline-formula id="IEq63"><alternatives><tex-math id="M129">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
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\usepackage{amsbsy}
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\begin{document}$$\alpha _k = \{1,\,1,\,1,\,1,\,1\},$$\end{document}</tex-math><mml:math id="M130"><mml:mrow><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq63.gif"></inline-graphic></alternatives></inline-formula><inline-formula id="IEq64"><alternatives><tex-math id="M131">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\alpha _k = \{2.5,\,1.5,\,1,\,1,\,1\}$$\end{document}</tex-math><mml:math id="M132"><mml:mrow><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>2.5</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1.5</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq64.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq65"><alternatives><tex-math id="M133">\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
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\begin{document}$$\alpha _k = \{5,\,2,\,1,\,1,\,1\}$$\end{document}</tex-math><mml:math id="M134"><mml:mrow><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq65.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq66"><alternatives><tex-math id="M135">\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _k = \{7.5,\,2.5,\,1,\,1,\,1\}$$\end{document}</tex-math><mml:math id="M136"><mml:mrow><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mn>7.5</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>2.5</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>1</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq66.gif"></inline-graphic></alternatives></inline-formula>. For each set <inline-formula id="IEq67"><alternatives><tex-math id="M137">\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _k$$\end{document}</tex-math><mml:math id="M138"><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq67.gif"></inline-graphic></alternatives></inline-formula>, we generate one MDS chart, and afterwards, the charts are combined using Procrustes analysis [<xref ref-type="bibr" rid="CR76">76</xref>]. Procrustes involves the operations of translation, reflection, orthogonal rotation, and scaling, to best conform the points in a given matrix under modification in relation with the points of a reference matrix.</p><p id="Par21">In our case, we (i) choose the first chart for initial reference, (ii) use Procrustes to superimpose the next MDS chart into the current reference, (iii) make the current set of superimposed charts the new reference, and (iv) continue to step (ii) until all charts have been conformed. The results obtained reveal identical patterns, meaning that the method is robust to distinct values of <inline-formula id="IEq68"><alternatives><tex-math id="M139">\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _k$$\end{document}</tex-math><mml:math id="M140"><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq68.gif"></inline-graphic></alternatives></inline-formula>.<fig id="Fig3"><label>Fig. 3</label><caption><p>MDS global chart for <inline-formula id="IEq69"><alternatives><tex-math id="M141">\documentclass[12pt]{minimal}
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\begin{document}$$q=2$$\end{document}</tex-math><mml:math id="M142"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq69.gif"></inline-graphic></alternatives></inline-formula> and the arc-cosine distance <inline-formula id="IEq70"><alternatives><tex-math id="M143">\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M144"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq70.gif"></inline-graphic></alternatives></inline-formula>, obtained by Procrustes with four different sets of weights <inline-formula id="IEq71"><alternatives><tex-math id="M145">\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _k$$\end{document}</tex-math><mml:math id="M146"><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq71.gif"></inline-graphic></alternatives></inline-formula></p></caption><graphic xlink:href="12539_2017_229_Fig3_HTML" id="MO5"></graphic></fig></p><p id="Par22">In Fig. <xref rid="Fig1" ref-type="fig">1</xref>, the unknown data, denoted by ‘-’ in Table <xref rid="Tab1" ref-type="table">1</xref>, are considered zero. Therefore, these values do not contribute to the distance used for comparing items. Moreover, as the missing data occur only in four values of the less weighted features, their influence is not as significant as for the rest of the information. In addition, as will be shown in Sect. <xref rid="Sec5" ref-type="sec">3.2</xref>, the results reveal small sensitivity to possible noise in the data, which includes the uncertainty in the unknown values that were set to zero. Nonetheless, a different criterion for dealing with that problem could be adopted. Experiments with the missing data replaced not only by zero, but also by the minimum, average, and maximum values in the third and fifth columns of Table <xref rid="Tab1" ref-type="table">1</xref> led to results qualitatively similar, as depicted in Fig. <xref rid="Fig4" ref-type="fig">4</xref>, revealing the effectiveness of the criterion adopted.<fig id="Fig4"><label>Fig. 4</label><caption><p>MDS global chart for <inline-formula id="IEq72"><alternatives><tex-math id="M147">\documentclass[12pt]{minimal}
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\begin{document}$$q=2$$\end{document}</tex-math><mml:math id="M148"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq72.gif"></inline-graphic></alternatives></inline-formula> and arc-cosine distance <inline-formula id="IEq73"><alternatives><tex-math id="M149">\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M150"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq73.gif"></inline-graphic></alternatives></inline-formula>, obtained by Procrustes with missing data replaced by zero, minimum, average, and maximum values of the third and fifth features</p></caption><graphic xlink:href="12539_2017_229_Fig4_HTML" id="MO6"></graphic></fig></p></sec><sec id="Sec5"><title>Sensitivity Analysis</title><p id="Par23">The 21 viruses were compared in the perspective of quantitative features. However, the data diverge slightly, depending on factors such as the time of the study or the operational conditions, namely, environmental conditions, geographic region, development level, or medical assistance. Therefore, we analyze here the sensitivity results with respect to the input data.</p><p id="Par24">We start by adding random noise to the quantitative features, <inline-formula id="IEq74"><alternatives><tex-math id="M151">\documentclass[12pt]{minimal}
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\begin{document}$$k=1,\ldots , \, 5$$\end{document}</tex-math><mml:math id="M152"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mn>5</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq74.gif"></inline-graphic></alternatives></inline-formula>, with amplitude <inline-formula id="IEq75"><alternatives><tex-math id="M153">\documentclass[12pt]{minimal}
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\begin{document}$$\pm 10\%$$\end{document}</tex-math><mml:math id="M154"><mml:mrow><mml:mo>±</mml:mo><mml:mn>10</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq75.gif"></inline-graphic></alternatives></inline-formula> of the values in Table <xref rid="Tab1" ref-type="table">1</xref>. Moreover, any negative values are avoided by limiting numbers to zero. A sample of 50 experiments, each yielding one MDS chart, is performed and the charts are combined using the Procrustes scheme.</p><p id="Par25">Figure <xref rid="Fig5" ref-type="fig">5</xref> illustrates the MDS chart for <inline-formula id="IEq76"><alternatives><tex-math id="M155">\documentclass[12pt]{minimal}
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\begin{document}$$q=2$$\end{document}</tex-math><mml:math id="M156"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq76.gif"></inline-graphic></alternatives></inline-formula> produced by the Procrustes algorithm. We verify that the method has low sensitivity to variations in the quantitative features, since the location of the points reveals minor variations.<fig id="Fig5"><label>Fig. 5</label><caption><p>MDS global chart for <inline-formula id="IEq77"><alternatives><tex-math id="M157">\documentclass[12pt]{minimal}
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\begin{document}$$q=2$$\end{document}</tex-math><mml:math id="M158"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq77.gif"></inline-graphic></alternatives></inline-formula> and arc-cosine distance, <inline-formula id="IEq78"><alternatives><tex-math id="M159">\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M160"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq78.gif"></inline-graphic></alternatives></inline-formula>, generated by Procrustes with random variations added to the values of the five features</p></caption><graphic xlink:href="12539_2017_229_Fig5_HTML" id="MO7"></graphic></fig></p></sec><sec id="Sec6"><title>Data Clustering and Visualization</title><p id="Par26">The MDS interpretation focuses on the distances between points in the produced charts. For identifying clusters, we can adopt some kind of ad hoc strategy based on the direct visualization of the MDS plots, or we can implement an algorithm for obtaining an automatic clustering. In addition, the configuration, <inline-formula id="IEq79"><alternatives><tex-math id="M161">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {X}$$\end{document}</tex-math><mml:math id="M162"><mml:mi mathvariant="bold">X</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq79.gif"></inline-graphic></alternatives></inline-formula>, produced by the MDS tries to replicate, in the low-dimensional space, <inline-formula id="IEq80"><alternatives><tex-math id="M163">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Q}$$\end{document}</tex-math><mml:math id="M164"><mml:mi mathvariant="script">Q</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq80.gif"></inline-graphic></alternatives></inline-formula>, the original proximities between pairwise elements. For <inline-formula id="IEq81"><alternatives><tex-math id="M165">\documentclass[12pt]{minimal}
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\begin{document}$$q=2$$\end{document}</tex-math><mml:math id="M166"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq81.gif"></inline-graphic></alternatives></inline-formula>, this leads to a direct visualization, but neglects the information described in the higher dimensional components of <inline-formula id="IEq82"><alternatives><tex-math id="M167">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {X}$$\end{document}</tex-math><mml:math id="M168"><mml:mi mathvariant="bold">X</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq82.gif"></inline-graphic></alternatives></inline-formula>. In this line of thought, in the next subsections, we introduce the non-hierarchical clustering algorithm <italic>K</italic>-means for automatic cluster identification and we propose a technique for an improved visualization of MDS information in the 2D space by embedding information of the extra dimensions.</p><sec id="Sec7"><title>The <italic>K</italic>-Means Clustering</title><p id="Par27">Clustering is a technique that groups objects similar to each other in some sense. The <italic>K</italic>-means is a non-hierarchical clustering algorithm [<xref ref-type="bibr" rid="CR77">77</xref>] that starts with a set of <italic>s</italic> objects, where each one is represented by a point in a <italic>q</italic>-dim space, and a certain number of clusters, <italic>K</italic>, specified in advance. The <italic>K</italic>-means groups the <italic>s</italic> objects into <inline-formula id="IEq83"><alternatives><tex-math id="M169">\documentclass[12pt]{minimal}
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\begin{document}$$K \le s$$\end{document}</tex-math><mml:math id="M170"><mml:mrow><mml:mi>K</mml:mi><mml:mo>≤</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq83.gif"></inline-graphic></alternatives></inline-formula> clusters, to minimize the sum of distances between the points and the centers of their clusters. The <italic>K</italic>-means produces a solution where objects in a cluster are close to each other and far from objects in other clusters.</p><p id="Par28">An important issue in <italic>K</italic>-means is to specify <italic>K</italic>, since the notion of “good clustering” is subjective. Nevertheless, we can adopt different measures for assessing the quality of the solution, such as the Calinski-Harabasz, Davies-Bouldin, and silhouette [<xref ref-type="bibr" rid="CR78">78</xref>].</p><p id="Par29">Here, we consider the silhouette, <italic>S</italic>, to assess if an object lies “adequately” within its cluster. The silhouette varies in the interval <inline-formula id="IEq84"><alternatives><tex-math id="M171">\documentclass[12pt]{minimal}
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\begin{document}$$S \in [\!-1,1]$$\end{document}</tex-math><mml:math id="M172"><mml:mrow><mml:mi>S</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mspace width="-0.166667em"></mml:mspace><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq84.gif"></inline-graphic></alternatives></inline-formula>, so that values close to <inline-formula id="IEq85"><alternatives><tex-math id="M173">\documentclass[12pt]{minimal}
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\begin{document}$$\{-1,0,1\}$$\end{document}</tex-math><mml:math id="M174"><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq85.gif"></inline-graphic></alternatives></inline-formula> correspond to <inline-formula id="IEq86"><alternatives><tex-math id="M175">\documentclass[12pt]{minimal}
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\begin{document}$$\{incorrect, neutral, correct\}$$\end{document}</tex-math><mml:math id="M176"><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>u</mml:mi><mml:mi>t</mml:mi><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq86.gif"></inline-graphic></alternatives></inline-formula> object assignments.</p><p id="Par30">Knowing the coordinates of the <inline-formula id="IEq87"><alternatives><tex-math id="M177">\documentclass[12pt]{minimal}
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\begin{document}$$s=21$$\end{document}</tex-math><mml:math id="M178"><mml:mrow><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn>21</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq87.gif"></inline-graphic></alternatives></inline-formula> objects produced by the MDS in the <inline-formula id="IEq88"><alternatives><tex-math id="M179">\documentclass[12pt]{minimal}
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\begin{document}$$q=3$$\end{document}</tex-math><mml:math id="M180"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq88.gif"></inline-graphic></alternatives></inline-formula> dim space, we assess the quality of the clusters in the interval <inline-formula id="IEq89"><alternatives><tex-math id="M181">\documentclass[12pt]{minimal}
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\begin{document}$$K \in [2,6]$$\end{document}</tex-math><mml:math id="M182"><mml:mrow><mml:mi>K</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq89.gif"></inline-graphic></alternatives></inline-formula>. Figure <xref rid="Fig6" ref-type="fig">6</xref> depicts the corresponding silhouettes and the mean value for each cluster (blue marks). The optimum value is obtained <inline-formula id="IEq90"><alternatives><tex-math id="M183">\documentclass[12pt]{minimal}
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\begin{document}$$K=4,$$\end{document}</tex-math><mml:math id="M184"><mml:mrow><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq90.gif"></inline-graphic></alternatives></inline-formula> corresponding to the maximum silhouette mean value <inline-formula id="IEq91"><alternatives><tex-math id="M185">\documentclass[12pt]{minimal}
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\begin{document}$$S_M = 0.77$$\end{document}</tex-math><mml:math id="M186"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>M</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.77</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq91.gif"></inline-graphic></alternatives></inline-formula>.<fig id="Fig6"><label>Fig. 6</label><caption><p>Silhouettes assessing the quality of the clustering for <inline-formula id="IEq92"><alternatives><tex-math id="M187">\documentclass[12pt]{minimal}
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\begin{document}$$K \in [2,6]$$\end{document}</tex-math><mml:math id="M188"><mml:mrow><mml:mi>K</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq92.gif"></inline-graphic></alternatives></inline-formula>, the arc-cosine distance <inline-formula id="IEq93"><alternatives><tex-math id="M189">\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M190"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq93.gif"></inline-graphic></alternatives></inline-formula>, and <inline-formula id="IEq94"><alternatives><tex-math id="M191">\documentclass[12pt]{minimal}
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\begin{document}$$q=3$$\end{document}</tex-math><mml:math id="M192"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq94.gif"></inline-graphic></alternatives></inline-formula>. The blue marks depict the mean silhouette value for each cluster</p></caption><graphic xlink:href="12539_2017_229_Fig6_HTML" id="MO8"></graphic></fig></p><p id="Par31">For <inline-formula id="IEq95"><alternatives><tex-math id="M193">\documentclass[12pt]{minimal}
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\begin{document}$$K=4$$\end{document}</tex-math><mml:math id="M194"><mml:mrow><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq95.gif"></inline-graphic></alternatives></inline-formula>, the clusters are <inline-formula id="IEq96"><alternatives><tex-math id="M195">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A} = \{$$\end{document}</tex-math><mml:math id="M196"><mml:mrow><mml:mi mathvariant="script">A</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">{</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq96.gif"></inline-graphic></alternatives></inline-formula>CPox, Mea, Mum, Nor, Rhi, Rot, Rub, SFlu<inline-formula id="IEq97"><alternatives><tex-math id="M197">\documentclass[12pt]{minimal}
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\begin{document}$$\}$$\end{document}</tex-math><mml:math id="M198"><mml:mo stretchy="false">}</mml:mo></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq97.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq98"><alternatives><tex-math id="M199">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {B} = \{$$\end{document}</tex-math><mml:math id="M200"><mml:mrow><mml:mi mathvariant="script">B</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">{</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq98.gif"></inline-graphic></alternatives></inline-formula>HepB, HIV, Rab<inline-formula id="IEq99"><alternatives><tex-math id="M201">\documentclass[12pt]{minimal}
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\begin{document}$$\}$$\end{document}</tex-math><mml:math id="M202"><mml:mo stretchy="false">}</mml:mo></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq99.gif"></inline-graphic></alternatives></inline-formula>, <inline-formula id="IEq100"><alternatives><tex-math id="M203">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {C} = \{$$\end{document}</tex-math><mml:math id="M204"><mml:mrow><mml:mi mathvariant="script">C</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">{</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq100.gif"></inline-graphic></alternatives></inline-formula>BFlu, Ebo, Mar, MERS, Pol, SARS, Sma<inline-formula id="IEq101"><alternatives><tex-math id="M205">\documentclass[12pt]{minimal}
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\begin{document}$$\}$$\end{document}</tex-math><mml:math id="M206"><mml:mo stretchy="false">}</mml:mo></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq101.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq102"><alternatives><tex-math id="M207">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {D} = \{$$\end{document}</tex-math><mml:math id="M208"><mml:mrow><mml:mi mathvariant="script">D</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">{</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq102.gif"></inline-graphic></alternatives></inline-formula>Chi, Den, ZIKV<inline-formula id="IEq103"><alternatives><tex-math id="M209">\documentclass[12pt]{minimal}
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\begin{document}$$\}$$\end{document}</tex-math><mml:math id="M210"><mml:mo stretchy="false">}</mml:mo></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq103.gif"></inline-graphic></alternatives></inline-formula>. These clusters are further discussed in the next subsection.</p></sec><sec id="Sec8"><title>Improved Visualization in 2D Space</title><p id="Par32">The geometrical shape of the chart produced by MDS varies significantly with the distance measure adopted for quantifying the distances between items. However, this characteristic does not precludes that we use the MDS chart taking full advantage of all its properties. Consequently, we may interpret the collection of points as “samples” of an abstract locus corresponding to the projection of the <italic>m</italic> initial dimensions into a lower dimensional (abstract) space.</p><p id="Par33">We adopt a scheme that allows for a direct visualization of the MDS, while including information up to <inline-formula id="IEq104"><alternatives><tex-math id="M211">\documentclass[12pt]{minimal}
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\begin{document}$$q=3.$$\end{document}</tex-math><mml:math id="M212"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq104.gif"></inline-graphic></alternatives></inline-formula> Therefore, we approximate the dimension <inline-formula id="IEq105"><alternatives><tex-math id="M213">\documentclass[12pt]{minimal}
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\begin{document}$$x_3$$\end{document}</tex-math><mml:math id="M214"><mml:msub><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq105.gif"></inline-graphic></alternatives></inline-formula> of <inline-formula id="IEq106"><alternatives><tex-math id="M215">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {X}$$\end{document}</tex-math><mml:math id="M216"><mml:mi mathvariant="bold">X</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq106.gif"></inline-graphic></alternatives></inline-formula> with a contour generated by means of a linear radial basis function interpolation [<xref ref-type="bibr" rid="CR79">79</xref>]. Moreover, we improve the identification of patterns by superimposing a tree in the MDS chart. The nodes of the tree are the <inline-formula id="IEq107"><alternatives><tex-math id="M217">\documentclass[12pt]{minimal}
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\begin{document}$$s=21$$\end{document}</tex-math><mml:math id="M218"><mml:mrow><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn>21</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq107.gif"></inline-graphic></alternatives></inline-formula> points representing items (viruses). In a first phase, we connect the group of points that are closer in the MDS chart producing the sets, <inline-formula id="IEq108"><alternatives><tex-math id="M219">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}$$\end{document}</tex-math><mml:math id="M220"><mml:mi mathvariant="script">P</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq108.gif"></inline-graphic></alternatives></inline-formula>, of interconnected points (nodes). In a second phase, the sets, <inline-formula id="IEq109"><alternatives><tex-math id="M221">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}$$\end{document}</tex-math><mml:math id="M222"><mml:mi mathvariant="script">P</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq109.gif"></inline-graphic></alternatives></inline-formula>, are compared through the distances between their constitutive nodes. The distance can be calculated taking into account any number <inline-formula id="IEq110"><alternatives><tex-math id="M223">\documentclass[12pt]{minimal}
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\begin{document}$$p</tex-math> <mml:math id="M224"><mml:mrow><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq110.gif"></inline-graphic></alternatives></inline-formula> of <inline-formula id="IEq111"><alternatives><tex-math id="M225">\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {X}$$\end{document}</tex-math><mml:math id="M226"><mml:mi mathvariant="bold">X</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq111.gif"></inline-graphic></alternatives></inline-formula> components. A connection is established in the <italic>q</italic>-dim chart, only between the two closest nodes (i.e., <inline-formula id="IEq112"><alternatives><tex-math id="M227">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}_i$$\end{document}</tex-math><mml:math id="M228"><mml:msub><mml:mi mathvariant="script">P</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq112.gif"></inline-graphic></alternatives></inline-formula> and <inline-formula id="IEq113"><alternatives><tex-math id="M229">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}_j$$\end{document}</tex-math><mml:math id="M230"><mml:msub><mml:mi mathvariant="script">P</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq113.gif"></inline-graphic></alternatives></inline-formula>). This calculation generates a second level of interconnection, and the scheme is repeated iteratively until there is a continuous route between all points. Therefore, the interpretation of the MDS chart is based not only in the relative position of the points, but also in the structure interconnecting them.</p><p id="Par34">Figure <xref rid="Fig7" ref-type="fig">7</xref> depicts a projection of the MDS chart for <inline-formula id="IEq114"><alternatives><tex-math id="M231">\documentclass[12pt]{minimal}
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\begin{document}$$q=2,$$\end{document}</tex-math><mml:math id="M232"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq114.gif"></inline-graphic></alternatives></inline-formula> the contour that approximates the dimension <inline-formula id="IEq115"><alternatives><tex-math id="M233">\documentclass[12pt]{minimal}
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\begin{document}$$x_3$$\end{document}</tex-math><mml:math id="M234"><mml:msub><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq115.gif"></inline-graphic></alternatives></inline-formula>, and the superimposed interconnections generated by calculating the distances between objects with <inline-formula id="IEq116"><alternatives><tex-math id="M235">\documentclass[12pt]{minimal}
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\begin{document}$$p=5$$\end{document}</tex-math><mml:math id="M236"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq116.gif"></inline-graphic></alternatives></inline-formula>. We observe easily the four clusters identified in the previous subsection. Moreover, we verify that the proposed methodology leads to a clear visualization and produces a richer chart of the objects.<fig id="Fig7"><label>Fig. 7</label><caption><p>MDS chart for <inline-formula id="IEq117"><alternatives><tex-math id="M237">\documentclass[12pt]{minimal}
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\begin{document}$$q=2$$\end{document}</tex-math><mml:math id="M238"><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq117.gif"></inline-graphic></alternatives></inline-formula> and the arc-cosine distance <inline-formula id="IEq118"><alternatives><tex-math id="M239">\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{ij}$$\end{document}</tex-math><mml:math id="M240"><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq118.gif"></inline-graphic></alternatives></inline-formula>. The contour represents the dimension <inline-formula id="IEq119"><alternatives><tex-math id="M241">\documentclass[12pt]{minimal}
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\begin{document}$$x_3$$\end{document}</tex-math><mml:math id="M242"><mml:msub><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq119.gif"></inline-graphic></alternatives></inline-formula> and the superimposed tree allows for an easier identification of patterns</p></caption><graphic xlink:href="12539_2017_229_Fig7_HTML" id="MO9"></graphic></fig></p><p id="Par35">In synthesis, besides the observation based on the relative distances in 2D space, we now verify that the ZIKV has a relevant position along the <inline-formula id="IEq120"><alternatives><tex-math id="M243">\documentclass[12pt]{minimal}
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\begin{document}$$x_3$$\end{document}</tex-math><mml:math id="M244"><mml:msub><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq120.gif"></inline-graphic></alternatives></inline-formula> dimension, somehow strengthening the characteristics revealed by the Chikungunya and Dengue.</p></sec><sec id="Sec9"><title>Discussion of the Results</title><p id="Par36">The clusters <inline-formula id="IEq121"><alternatives><tex-math id="M245">\documentclass[12pt]{minimal}
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\begin{document}$$\{\mathcal {A}, \, \mathcal {B}, \, \mathcal {C}, \, \mathcal {D}\}$$\end{document}</tex-math><mml:math id="M246"><mml:mrow><mml:mo stretchy="false">{</mml:mo><mml:mi mathvariant="script">A</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mi mathvariant="script">B</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mi mathvariant="script">C</mml:mi><mml:mo>,</mml:mo><mml:mspace width="0.166667em"></mml:mspace><mml:mi mathvariant="script">D</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq121.gif"></inline-graphic></alternatives></inline-formula> do not follow an epidemiological line of thought, but may be of medical value, since they reflect characteristics measured by health care practice. In cluster, <inline-formula id="IEq122"><alternatives><tex-math id="M247">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document}</tex-math><mml:math id="M248"><mml:mi mathvariant="script">A</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq122.gif"></inline-graphic></alternatives></inline-formula> are included viruses of Risk Group 2 that in general do not cause serious illness nor life threatening.</p><p id="Par37">In cluster <inline-formula id="IEq123"><alternatives><tex-math id="M249">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {B}$$\end{document}</tex-math><mml:math id="M250"><mml:mi mathvariant="script">B</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq123.gif"></inline-graphic></alternatives></inline-formula>, we find the <italic>Lentivirus</italic> that is responsible for HIV and acquired immunodeficiency syndrome (AIDS), a Risk Group 3 agent. We find also the Hepatitis B and the Rabies virus, a <italic>Lyssavirus</italic> genus and <italic>Rhabdoviridae</italic> family virus, of Risk Group 2.</p><p id="Par38">In cluster <inline-formula id="IEq124"><alternatives><tex-math id="M251">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {C}$$\end{document}</tex-math><mml:math id="M252"><mml:mi mathvariant="script">C</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq124.gif"></inline-graphic></alternatives></inline-formula>, we can consider two subclusters. The first subcluster includes the Ebola and Marburg viruses that belong to the Risk Group 4. In addition, in this subcluster, the agents responsible for MERS and Bird flu are classified as Risk Group 3. The second subcluster includes viruses of different Risk Groups, namely, the Polio virus and the SARS–associated <italic>coronavirus</italic>, belonging to Risk Groups 2 and 3, respectively. Smallpox is also present [<xref ref-type="bibr" rid="CR80">80</xref>].</p><p id="Par39">Cluster <inline-formula id="IEq125"><alternatives><tex-math id="M253">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {D}$$\end{document}</tex-math><mml:math id="M254"><mml:mi mathvariant="script">D</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq125.gif"></inline-graphic></alternatives></inline-formula> includes Chikungunya, considered a Risk Group 3 pathogen. Also included in <inline-formula id="IEq126"><alternatives><tex-math id="M255">\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {D}$$\end{document}</tex-math><mml:math id="M256"><mml:mi mathvariant="script">D</mml:mi></mml:math><inline-graphic xlink:href="12539_2017_229_Article_IEq126.gif"></inline-graphic></alternatives></inline-formula> are the Dengue fever virus, a Risk Group 2 <italic>arbovirus</italic> pathogenan, and ZIKV, recognized as being similar to Chikungunya and Dengue viruses.</p><p id="Par40">In conclusion, we verified that the MDS provides a powerful computational visualization technique of viruses data and the obtained charts may be of medical interest in the scope of present and future viral outbreaks.</p></sec></sec></sec><sec id="Sec10"><title>Conclusions</title><p id="Par41">This paper discussed the computational analysis of real-world data describing viruses main quantitative characteristics. By encompassing complex scattered data, researchers have to choose between comparing all aspects and detecting the main properties. This problem represents a challenge since some information (or its absence) may lead to incomplete or eventually to incorrect conclusions. Therefore, complex information calls for computational and visualization tools capable of revealing the most relevant issues. The MDS technique was adopted, leading to substantive results that follow present-day scientific knowledge.</p></sec></body><back><notes notes-type="ethics"><title>Compliance with ethical standards</title><notes notes-type="COI-statement"><title>Conflict of interest</title><p>The authors declare that there is no conflict of interest regarding the publication of this paper.</p></notes></notes><ref-list id="Bib1"><title>References</title><ref id="CR1"><label>1.</label><element-citation publication-type="book"><person-group person-group-type="author"><name><surname>Murray</surname><given-names>PR</given-names></name><name><surname>Rosenthal</surname><given-names>KS</given-names></name><name><surname>Pfaller</surname><given-names>MA</given-names></name></person-group><source>Medical microbiology</source><year>2013</year><publisher-loc>Philadelphia</publisher-loc><publisher-name>Elsevier Sounders</publisher-name></element-citation></ref><ref id="CR2"><label>2.</label><element-citation 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