Romy Sauvayre
email: romysauvayreromy.sauvayre@uca.fr

Misreferencing Practice of Scientists: Inside Researchers' Sociological and Bibliometric Profiles

Keywords: Questionable research practice, reference error, citation analysis, bibliographical databases, research ethics, peer-review system, publication pressure

Scientists, as human beings, sometimes make mistakes. The aim of this case study was to examine the misreferencing practice of scientists. Citations of two documents about the neuroimaging of a dead salmon were collected. A total of 173 references were extracted from Google Scholar and analyzed by both qualitative and quantitative methods. The findings reveal a reference error rate of 93.1%, involving 419 authors. Sociological and bibliometric data about the authors were collected and revealed that referencing errors can be committed by all types of researchers. Further, a major referencing error was discovered in 22.5% of the citing documents, involving 121 authors, that is, the citation of a poster as if it were an article published in a prestigious journal. This major error was committed by the more prolific and recognized researchers. The higher the first author's number of citations, number of publications or h-index, the more major reference errors (citing a "ghost" paper) were committed. Nevertheless, the ScienceDirect database bears some responsibility, as it indexed abstracts of the Organization for Human Brain Mapping meeting as journal articles that had never been published in the prestigious NeuroImage journal. Finally, this case study raises epistemological concerns about research and its dissemination.

Introduction

Reference accuracy reflects, on the one hand, the precision, ethics, and credibility of scientists [START_REF] Wilks | ʺReference Accuracy Among Research Articles Published in Research on Social Work Practice[END_REF][START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF]) and, on the other hand, their knowledge of the literature [START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF][START_REF] Gilbert | ʺReferencing as Persuasion[END_REF]. According to the literature [START_REF] Wilks | ʺReference Accuracy Among Research Articles Published in Research on Social Work Practice[END_REF][START_REF] Fenton | ʺThe Accuracy of Citation and Quotation in Otolaryngology/Head and Neck Surgery Journals[END_REF][START_REF] George | ʺReference Accuracy in the Dermatological Literature[END_REF][START_REF] Evans | ʺQuotational and Reference Accuracy in Surgical Journals: A Continuing Peer-Review Problem[END_REF], reference errors should also involve the responsibility and credibility [START_REF] Fenton | ʺThe Accuracy of Citation and Quotation in Otolaryngology/Head and Neck Surgery Journals[END_REF] of both editors and reviewers. Even though reference accuracy is an integral part of scientific activity, this research practice, namely, committing reference errors, occurs regularly and may lead to misinformation [START_REF] Granter | ʺCalciphylaxis and the Persistence of Medical Misinformation in the Era of Google[END_REF]. Reference accuracy entails a lack of citation errors ("inaccuracies in the components of the cited reference" (Montenegro et al. 2021, 14) or the quotation errors ("inconsistency between the author's assertion and the content of its supporting source" (Montenegro et al. 2021, 14).

In studies on reference accuracy, the purpose is often to identify and count reference errors (i.e. citation errors). Such errors have been investigated in numerous research areas [START_REF] Spivey | ʺReference List Accuracy in Social Work Journals[END_REF][START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF]): medical sciences (biomedicine, biomedical informatics, orthopedics, general surgery, otolaryngology, emergency medicine, nursing, burns, anatomy, public health, and veterinary medicine); entomology; psychology (counseling psychology, and experimental psychology); social work; library and information science; business and economics; and clinical chemistry.1 Thus, the bibliographical material used is most often published in peer-reviewed journals [START_REF] Speck | ʺEffectiveness of a Reference Accuracy Strategy for Peer-Reviewed Journal Articles[END_REF] specializing in specific research areas [START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF][START_REF] Fenton | ʺThe Accuracy of Citation and Quotation in Otolaryngology/Head and Neck Surgery Journals[END_REF][START_REF] George | ʺReference Accuracy in the Dermatological Literature[END_REF][START_REF] Montenegro | ʺHow Accurate Is the Neurosurgery Literature? A Review of References[END_REF]. Rarely, the reference accuracy of a single highly cited journal article is examined [START_REF] Liang | ʺScientists' Referencing (Mis)Behavior Revealed by the Dissemination Network of Referencing Errors[END_REF]. By contrast, books and book chapters are often excluded from reference accuracy analyses. Several errors in the following parts of the citing reference are usually counted: the author (omitted name, incorrect name order, and misspelled name and initials), title, publication year, journal, volume, issue, and page numbers (e.g., [START_REF] Wilks | ʺReference Accuracy Among Research Articles Published in Research on Social Work Practice[END_REF][START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF][START_REF] Spivey | ʺReference List Accuracy in Social Work Journals[END_REF])). To summarize, the error rates reported in journal articles published between 1975 and 2017 [START_REF] Wilks | ʺReference Accuracy Among Research Articles Published in Research on Social Work Practice[END_REF][START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF][START_REF] Spivey | ʺReference List Accuracy in Social Work Journals[END_REF]) are between 11% and 54%. The error rate may vary depending on the considered indicators and area of research. Therefore, data cannot be compared, all things being equal, and caution is when interpreting them. Nevertheless, these studies show that authors often incorrectly reference cited authors (45%), titles (25%), page numbers (16%), journals (6%), volumes and issues (6%), and publication years (4%). In addition, these studies suggest that inaccuracies may be introduced by manual transcription [START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF] and can be reduced by using bibliographical software and databases [START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF]). However, these databases are not fully accurate. Indeed, the error rates reported in the literature are between 0.05% [START_REF] Haddaway | ʺThe Role of Google Scholar in Evidence Reviews and Its Applicability to Grey Literature Searching[END_REF]) and 6.6% [START_REF] Franceschini | ʺEmpirical analysis and classification of database errors in Scopus and Web of Science[END_REF] for Web of Science (WoS); between 2.0% [START_REF] Moed | ʺA new methodology for comparing Google Scholar and Scopus[END_REF] and 12.4% [START_REF] Valderrama-Zurián | ʺA systematic analysis of duplicate records in Scopus[END_REF] for Scopus; between 2.9% [START_REF] Haddaway | ʺThe Role of Google Scholar in Evidence Reviews and Its Applicability to Grey Literature Searching[END_REF] and 53.5% [START_REF] Granter | ʺCalciphylaxis and the Persistence of Medical Misinformation in the Era of Google[END_REF] for Google Scholar (GS); 1.25% for PubMed [START_REF] Valderrama-Zurián | ʺA systematic analysis of duplicate records in Scopus[END_REF]; and 1.1% for PsycINFO (García-Pérez 2010).

Context and methods

The aim of this case study was to examine the misreferencing practice of scientists who cited a notorious, unusual work about the neuroimaging of a dead salmon [START_REF] Bennett | ʺNeural Correlates of Interspecies Perspective Taking in the Post-Mortem Atlantic Salmon: An Argument for Multiple Comparisons Correction[END_REF][START_REF] García-Pérez | ʺAccuracy and Completeness of Publication and Citation Records in the Web of Science, PsycINFO, and Google Scholar: A Case Study for the Computation of h Indices in Psychology[END_REF]. Originally, Craig Bennett and colleagues wanted to calibrate a scanner in preparation for psychological experiments using functional magnetic resonance imaging (fMRI) at Dartmouth College [START_REF] Bennett | ʺThe Story Behind the Atlantic Salmon[END_REF]. They used various objects to do this: a pumpkin, a Cornish game hen, and an Atlantic salmon. The data were recorded and stored at that time. Three years later, data analysis was conducted, revealing what appeared to be brain activity in this dead salmon. In June 2009, the data from the dead salmon experiment were presented as a poster at the 15th Congress of the Organization for Human Brain Mapping (OHBM poster) [START_REF] Bennett | ʺNeural Correlates of Interspecies Perspective Taking in the Post-Mortem Atlantic Salmon: An Argument for Multiple Comparisons Correction[END_REF]. Thereafter, in 2010, the data were published in a peer-reviewed journal article, namely, in the Journal of Serendipitous and Unexpected Results2 (JSUR article) [START_REF] Bennett | ʺNeural Correlates of Interspecies Perspective Taking in the Post-Mortem Atlantic Salmon: An Argument For Proper Multiple Comparisons Correction[END_REF]. Last, Bennett and colleagues received an Ig® Nobel Prize3 in 2012 for their dead salmon case study (OHBM poster and JSUR article). This neuroscience prize was awarded "for demonstrating that brain researchers, by using complicated instruments and simple statistics, can see meaningful brain activity anywhere -even in a dead salmon" (ʺWinners of the Ig® Nobel Prizeʺ s. d.). Since 2009, this result has been used to illustrate the false-positive issue (type I error) that may occur in fMRI analysis and thus to alert the scientific community [START_REF] Margulies | ʺThe Salmon of Doubt. Six Months of Methodological Controversy within Social Neuroscience[END_REF]. Furthermore, this study became one of the few studies of neuroimaging methods to be cited more than a hundred times4 and to receive worldwide media coverage.

Usually, a neuroimaging study has a higher chance of being cited if it is published in a journal indexed by Journal Citation Reports (JCR), which provides impact factors. However, the dead salmon study was disseminated via a poster (OHBM poster) and a journal that existed only to publish a single article (JSUR article). Moreover, an indexing anomaly appears in the ScienceDirect and GS databases: the OHBM poster is referenced as an article published in a prestigious neuroimaging journal (NeuroImage). Without careful checking, this error is not easy to identify. The aim is, in the present study, to understand how citing authors deal with major difficulties in finding information to reference the dead salmon study accurately and thus highlight their reference practices, their rigor and their ethical values. First, a comprehensive analysis was performed to define the types of errors committed by researchers5 ; a broad category of document types was considered (journal articles, books, books chapter, and conference proceedings). Second, a specific analysis was conducted among publications in which an article that does not exist has been cited; that is, those in which a "ghost paper citation error" has been committed. A sociological profile of the citing authors was constructed, and the rationale for their particular misreferencing practice was analyzed. Third, four hypotheses were proposed to understand and explain this particular misreferencing practice involving a wellknown bibliographical database (ScienceDirect) and how these authors' referencing errors and this database anomaly were possible.

Consequently, the present study aimed to examine publications that cited the OHBM poster or JSUR article (based on the OHBM poster), namely, the "Dead Salmon fMRI References" (DSfR).

These citation data were extracted from GS via the Publish or Perish (PoP) software on October 6, 2017 (for more methodological details, please see the supplemental material document). A total of 281 references were examined. After a manual data cleaning process, 173 unique citing publications were included in the analysis. Thereafter, the following supplementary data were manually collected and analyzed to explore the sociological and bibliometric profile of citing authors, mainly with nonparametric statistical tests: 1) Concerning the citing documents: type of DSfR cited, number of authors, publication year, publication type, journal impact factor from Journal Citation Reports (JCR); 2) Concerning the first authors of these citing documents: gender, academic position, research area, field of study or department, institutional affiliation, institutional Shanghai ranking, country, number of citations in WoS, number of publications in WoS, and h-index in WoS. Lastly, the accuracy of DSfR cited in each document reference list was examined, and citation errors were categorized and counted.

Results

Reference errors were categorized into two types: 1) errors usually found in the reference accuracy literature (see above) and 2) a new major error never presented before.

Usual reference errors

The reference errors found in the present study were categorized into 10 elements: title, year, authors, journal or meeting name, volume or issue number, page numbers, "poster" mention, state, city, and month.

A total of 631 reference errors were counted (Table 1), with a median of 3 errors per citing document.6 Furthermore, 161 citing documents (93.1%) involving 419 authors contained at least one error. Only 12 publications (6.9%) were accurate. This error rate is so unusually high that an additional data check was conducted to ensure the precision of the results. The most common error is incorrect volume or issue numbers (62.1% of citing documents)7 .

1. Author error. The OHBM poster reference was mainly found to be affected by this type of error (70.0%), and the second author, Abigail Baird, was particularly affected. Although Baird is the coauthor of both DSfR, she was often omitted. In fact, neither the poster session program, which recorded this OHBM poster reference, nor the ScienceDirect database, which still indexes this poster program, makes mention of this coauthor. Thus, the omission of this author may have contributed to the improper referencing. Last, two other types of author errors rarely occurred: inversion of the first and last names of the fourth author, and replacement of the second, third, and fourth authors by "et al."

2. Year error. The JSUR article reference was mainly found to be affected by this type of error (89.7%). Note that it is not easy to find this JSUR publication year (i.e., 2010) because it did not appear in the full text of the article downloadable from the first author's blog (http://prefrontal.org/files/papers/Bennett-Salmon-2010.pdf).

Moreover, bibliographical databases have not indexed this article, and the journal and its website no longer exist. Consequently, authors made incorrect choices: from 2005 (-5 years) to 2013 (+3 years). Furthermore, among the miscited years, 2011 was the most frequent.

3. Title error. Authors mainly omitted part of the DSfR title (74.2%) or added an irrelevant part (19.4%). This error seems to have been caused by a strong similarity between the two DSfR titles; indeed, only a single word ("proper") in the second part of the title differs. In addition, the JSUR article title was misreferenced more often (75.0%) than the OHBM title.

4. Journal or meeting name error. The OHBM poster reference was mainly found to be affected by this type of error (67.8%). These names were incorrect, incomplete or missing, and they also contained some capitalization and abbreviation errors.

5.

Volume or issue error. The JSUR article reference was mainly found to be affected by this type of error (62.0%). The most common error was the omission of either the volume or the issue number. Note that this number is the same for both (i.e., "1") because the journal published a unique article in a unique volume. In other cases, the journal issue was replaced by an irrelevant digital object identifier (DOI) or by an irrelevant page number (i.e., "125").

6. Page error. The OHBM poster reference was found to be almost universally affected by this type of error (90.0%). The accurate page number was replaced by that of the OHBM program (i.e., "S125"), which contained the schedules of the poster presentations. In other cases, a double page was referenced (i.e., "S125-125"). In addition, in a single case, another part of the program was referenced (i.e., "S39-S41").

7. Poster error. The OHBM poster reference was found to be almost universally affected by this type of error (97.1%). The word "poster" was often missing and was, in rare instances, replaced by the word "paper." 8. State error; 9. City error; 10. Month error. These parts of the reference were missing mainly in the OHBM-citing documents (96.9%, 98.3%, and 97.2%, respectively). Rarely, the following replacements were detected: July instead of June and Chicago or Santa Barbara instead of San Francisco.

In addition, there was a significant difference in the number of errors between the two types of DSfR cited. The chances of reference errors were 9 times higher in documents citing the OHBM poster than in those citing the JSUR article (odds ratio = 8.899, 95% confidence interval = 1.12-70.56, N = 173). Furthermore, considering both the number of errors and the number of publications that contain at least one error, a significant difference was found between documents citing the OHBM poster reference and those citing the JSUR article reference (U = 537.5, p < 0.001, N = 173; U = 3201, p = 0.006, N = 173, respectively).

After the presentation of these 10 reference error types, it may be interesting to explore the sociological and bibliometric profiles of the citing authors. Therefore, it is reasonable to suppose-or expect-that the more experienced or cited the researcher is, the more accurate his or her referencing will be. However, the data revealed no significant difference or correlation between the number of errors and several sociological and bibliometric indicators, such as academic position, university ranking, journal impact factor, first author's h-index, and first author's number of citations8 . In other words, all types of researchers are likely to commit at least one reference error.

A major reference error: the "ghost paper citation error"

The data on these 10 error types demonstrate that researchers committed numerous reference errors, but these results are very typical, as previous studies demonstrate. However, one of these error types reveals more about research practice and may explain the significant difference in error rate between the two DSfR cited. In fact, the data contain an error never reported before in the reference accuracy literature, that is, the replacement of the OHBM poster reference with a reference to an irrelevant article reportedly published in a prestigious neuroscience journal. This type of major and questionable error was found in 39 publications (22.5%) involving 121 authors. How is that possible? The first-the Human Brain Mapping ghost paper (N = 1)-was a journal name error probably caused by its strong similarity to the meeting name, that is, "Organization for Human Brain Mapping." The second-the NeuroImage ghost paper (N = 38)-is indexed by the ScienceDirect database (produced by Elsevier since 1997). This database has also provided a full webpage for the NeuroImage ghost paper since August 28, 2009, which has been visited more than one thousand times (https://www.sciencedirect.com/science/article/pii/S1053811909712029?via%3Dihub). This ScienceDirect webpage contains the OHBM poster title, three of the four authors, a detailed summary, a DOI (10.1016/S1053-8119(09)71202-9), a copyright, an "Article Metrics" section, and two downloadable documents available in PDF format to authorized users only. In fact, these documents are an extract of the OHBM meeting program (pages S39-S41 and page S125) that were published in a supplemental issue of NeuroImage. In place of articles, these PDFs provide the dates, times, places, titles, and author names of the poster sessions, including the poster of Bennett and colleagues. However, the following reference, which resembles a journal article, can then be recorded from ScienceDirect: However, a simple action would have avoided this major citation error. Thus, opening and reading one of the two downloadable documents would prevent this ghost paper citation error because of its content.

In addition, because ScienceDirect indexes the NeuroImage ghost paper using the same structure applied for journal articles 9 , GS includes this false reference in its database. More importantly, the results obtained from GS through a title search (with or without a Bennett mention) show that only the reference to the NeuroImage ghost paper is available, unlike the OHBM poster 10 . Therefore, the only visible and available reference in both ScienceDirect and GS is the false reference. Nevertheless, GS provides for free the full text of the OHBM poster (https://www.psychology.mcmaster.ca/bennett/psy710/readings/BennettDeadSalmon.pdf). However, again, the opening and reading of this PDF would reveal to authors that this document cannot be a NeuroImage journal article, but is instead a meeting program. Therefore, it is unlikely that the authors had read the OHBM poster before citing the NeuroImage ghost paper. This surprising fact raises questions about the types of researchers who committed this ghost paper citation error. For this purpose, the collected variables described below were analyzed to define the sociological characteristics of those scientists (highest frequencies per variable are enclosed in brackets below)11 . Thus, the citing document was found to be an article (59.0%) published in a journal with a median impact factor (IF) (73.9%) of 3.8. The 20 journals that published this citing document included the prestigious Nature Reviews Neuroscience (1st neuroscience journal, 2017 JCR, IF = 32.6) and, unexpectedly, NeuroImage itself. For instance, NeuroImage published, in 2015 and 2016, two articles that referenced the NeuroImage ghost paper. Furthermore, the citing publication was written by a median of two coauthors who committed a median of 8 reference errors on the DSfR cited. The first author was most often a man (73.7%) affiliated (100.0%) with a top 100-ranked university (41.0%). This university was most frequently American (48.7%), such as the Massachusetts Institute of Technology, Stanford University, or Northwestern University. The first author was a junior researcher (e.g., Assistant Professor) (58.3%) in social sciences (56.3%), mainly in psychology (34.2%). He or she had accumulated a median of 280 citations (WoS), 27 publications (WoS), and an h-index (WoS) of 8.012 . Finally, correlations between these indicators of scientific productivity and recognition and the type of citation error were statistically significant (Fig. 1). In other words, the higher the first author's number of citations (rs = 0.199, p = 0.011), publications (rs = 0.226, p = 0.004) or hindex (rs = 0.238, p = 0.002) were, the more major reference errors (a ghost paper citation error) were committed. C) First author's hindex (WoS) by the type of citation error (rs = 0.238, p = 0.002). The following type of citation was analyzed as an ordinal classification: 1) no citation error; 2) usual citation error, that is, the main types of reference inaccuracies described in the literature (see above the 10 types of errors categorized in the present study); and 3) ghost paper citation error, that is, as defined above, referencing an article that does not exist. * p < 0.05, ** p < 0.01.

The results showed that a prolific or recognized researcher was more likely to misreference a DSfR and especially to commit a ghost paper citation error. Therefore, all affiliated researchers may be prone to committing a ghost paper citation error, but prolific and recognized researchers are more likely to commit such an error than others.

Discussion

One may think that referencing errors are a neglected aspect of questionable scientific activity, but they constitute a unique objective and measurable indicator of the rigor in practice and ethical values of scientists. In sum, citation practice provides valuable information about researchers. However, some reference errors have different consequences: some may seem inconsequential, such as using the wrong month for a conference meeting or making capitalization errors in journal names. Other reference errors cause citing authors to have difficulty accessing the documents, such as citing an article instead of a poster, citing the wrong journal name or the wrong date. This present case study, because of its unusual nature, questions an epistemological problem involving, in particular, the dissemination of knowledge.

The results presented reveal that 93.1% of citing documents contained at least one reference error. This error rate is substantially higher than in previous findings, where the error rate lies between 11% and 54% [START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF][START_REF] Spivey | ʺReference List Accuracy in Social Work Journals[END_REF][START_REF] Wilks | ʺReference Accuracy Among Research Articles Published in Research on Social Work Practice[END_REF]. This difference can be explained by 1) the similarity between the titles of the OHBM poster and JSUR article (only one word was added in the latter title); 2) the fact that the JSUR article was published in a journal with a very short life because it published only one article before disappearing; 3) the indexing anomaly of the OHBM poster in both ScienceDirect and GS; 4) the ghost paper citation error detected in 22.5% of citing documents, in which the OHBM poster reference was replaced with a paper that does not exist.

The last of these four causes-the ghost paper citation error-is the most striking and a new reference error detected in this study involving 39 publications, 20 academic journals, 12 books, 34 institutions or universities and 123 coauthors. Last, although numerous thorough studies have been conducted in this field, none of them have reported the citation of references that do not exist [START_REF] Wilks | ʺReference Accuracy Among Research Articles Published in Research on Social Work Practice[END_REF][START_REF] Davies | ʺReference Accuracy in Library and Information Science Journals[END_REF][START_REF] Spivey | ʺReference List Accuracy in Social Work Journals[END_REF]). How can one explain why this major error appeared, was published and was disseminated in the peer-review system? Four hypothetical explanations are discussed: 1) confidence in bibliographical databases; 2) ethical failures on the part of researchers who do not read or verify their cited references; 3) the absence of a control policy; and 4) publication pressure. Note that these four "explanatory hypotheses" produced by "abduction" (Peirce 1998, CP 5.171) 13 are intended to bring a reflexive point of view to the epistemological problems raised by the present study about science practice and the dissemination of knowledge, that must be confronted with data beyond that in the present case study. 1) Confidence in bibliographical databases. Two trusted sources-ScienceDirect and GSindex the NeuroImage ghost paper. The first, ScienceDirect, provides "peer-reviewed scholarly literature" (ʺWhat is ScienceDirectʺ s. d.), including "articles from over 3,800 journals and more than 37,000 book titles" (ʺWhat is ScienceDirectʺ s. d.). Note that ScienceDirect makes no mention that it is providing poster abstracts, in addition to journal articles and books, in its database. As mentioned above, the complete ScienceDirect webpage describes the NeuroImage ghost paper with standard elements that also contributed to strengthening the reliability of those data. In fact, OHBM poster abstracts are presented with the same webpage structure as are NeuroImage journal articles. Therefore, researchers had no reason to doubt the ScienceDirect bibliographical information, which is expected-and claimed-to provide the full text of journal articles. The second, GS, indexes the ScienceDirect webpage and, as a result, the NeuroImage ghost paper. However, GS has relegated the OHBM poster reference to the "version" subcategory. In the GS search results, the accurate reference (OHBM poster) was replaced by that for the ghost paper. Therefore, only the two following references could be obtained from GS: the JSUR article (163 citations) and the NeuroImage ghost paper (132 citations). Nevertheless, again, researchers had no reason to doubt the GS bibliographical information. Consequently, ScienceDirect should remove the NeuroImage ghost paper indexation and the 2,000 other OHBM posters indexed each year from its website or should at least change the presentation of these communication poster abstracts to prevent them from being confused with journal articles.

2) The ethical failure of researchers. Part of ethical research practice requires that researchers read their sources before citing them and subsequently verify their references. However, several studies have suggested that researchers do not do so [START_REF] Evans | ʺQuotational and Reference Accuracy in Surgical Journals: A Continuing Peer-Review Problem[END_REF][START_REF] Liang | ʺScientists' Referencing (Mis)Behavior Revealed by the Dissemination Network of Referencing Errors[END_REF][START_REF] Simkin | ʺRead Before You Cite!ʺ[END_REF][START_REF] Mogull | ʺAccuracy of Cited "Facts" in Medical Research Articles: A Review of Study Methodology and Recalculation of Quotation Error Rate[END_REF]. Though the present case study cannot illustrate the reading practice of citing authors, the probability of unethical conduct is high enough to hypothesize that several authors did not read the OHBM poster before citing the NeuroImage ghost paper or did not complete the necessary checks to avoid this type of major error. Furthermore, according to [START_REF] Mogull | ʺAccuracy of Cited "Facts" in Medical Research Articles: A Review of Study Methodology and Recalculation of Quotation Error Rate[END_REF], the major errors counted "might suggest that authors are not reading" or "suggest a deliberate attempt by authors to mislead readers". If authors had read even one of the documents attached to the NeuroImage ghost paper reference, then they would have discovered either an OHBM meeting program (downloadable from ScienceDirect) or the OHBM poster (downloadable from several coauthors' websites, including GS). Consequently, authors who cited the NeuroImage ghost paper probably did not read it. In addition, the considerable reference error rate (93.1%) found in the present study also suggests that researchers did not sufficiently verify their reference list.

3) The absence of a control policy. Concerning reference inaccuracy, the literature [START_REF] Wilks | ʺReference Accuracy Among Research Articles Published in Research on Social Work Practice[END_REF][START_REF] Fenton | ʺThe Accuracy of Citation and Quotation in Otolaryngology/Head and Neck Surgery Journals[END_REF][START_REF] George | ʺReference Accuracy in the Dermatological Literature[END_REF] has suggested for the last 25 years [START_REF] Evans | ʺQuotational and Reference Accuracy in Surgical Journals: A Continuing Peer-Review Problem[END_REF]) that responsibility should be shared by authors, reviewers, and editors. According to Franceschini and colleagues (2015), errors persist despite improvements. Indeed, the present study showed that 39 documents (22.5% of the citing documents) were published despite citing a ghost paper. Among these documents, 23 manuscript drafts were examined by reviewers and editors, who failed to detect the ghost paper reference. Furthermore, NeuroImage itself published two articles that cited the NeuroImage ghost paper. Nevertheless, it is important to mention that editors rely on the ethicality of their authors, and they do not require reviewers to examine the accuracy of manuscript reference lists. Neither editors nor reviewers could be reasonably expected to have identified this ghost paper citation error. However, the current academic publication system, which is based in part on the ethical practices of researchers, is imperfect, as indicated by the publication and diffusion of citations of retracted articles (Madlock-Brown and Eichmann 2015; Bar-Ilan and Halevi 2017; Wray and Andersen 2018) and citations of the present ghost paper. Furthermore, in the absence of any control policy for references, the peer review process tacitly validated the ghost paper reference and then increased readers' confidence in this erroneous citation. Subsequently, the ghost paper may have been cited by these readers as well causing them to unintentionally, contribute to the diffusion of the ghost paper citation error and hence to the vicious cycle of misreferencing. Indeed, as Figure 2 shows, the ghost paper citation error has been committed each year since 2010 and may be perpetuated in the next few years as well. 4) Publication pressure. The literature [START_REF] Fenton | ʺThe Accuracy of Citation and Quotation in Otolaryngology/Head and Neck Surgery Journals[END_REF][START_REF] Begley | ʺRaise standards for preclinical cancer research[END_REF][START_REF] Hicks | ʺBibliometrics: The Leiden Manifesto for research metrics[END_REF] suggests that the professional survival of researchers-in terms of tenure, academic advancement, and funding-may depend on citation analysis. Consequently, it is tempting for researchers to publish their findings as quickly as possible but at the expense of quality or rigor. Some researchers may therefore avoid the considerably time-consuming task of reading their citing sources and verifying their reference list. The present study showed that nearly all the researchers who cited the dead salmon study seemed to make this choice. In contrast, few of the researchers (6.9%) properly referenced DSfR, and they were less prolific and less cited than those who did not. However, this case study does not permit us to generalize this hypothesis. Nevertheless, the data show a significant correlative link between the number of reference errors and the academic productivity of the citing authors (number of publications).

Thus, the present study pursues further questions: 1) in a "publish or perish" context, do researchers adopt questionable reference practices rather than rigorous ones because prolific researchers are more highly valued? 2) As a consequence, is publication pressure-caused by the current research evaluation process-partially responsible for the increase in researchers' misreferencing practice? Finally, an epistemological consequence may be highlighted with reference to the ghost paper citation error, which is leading readers to assign higher epistemological value to the dead salmon study and increasing its dissemination. The more prestigious the journal (the higher the impact factor) in which an article is published, the more the article will tend to be cited [START_REF] Callaham | Journal Prestige, Publication Bias, and Other Characteristics Associated With Citation of Published Studies in Peer-Reviewed Journals[END_REF]. The dead salmon case study appears to have been published in a prestigious journal after having been evaluated by experts in the neuroimaging field. Considering that, as [START_REF] Gilbert | ʺReferencing as Persuasion[END_REF] reports, a citation gives scientific support to and implicitly presents the results as "valid", misreferencing may lead to the dissemination and validation of a case study that had been rejected by prestigious neuroimaging journals. The present study may contribute to research in the citation analysis field [START_REF] Nicolaisen | ʺCitation Analysis[END_REF].

Conclusion

In conclusion, the present case study showed that nearly all the researchers examined misreferenced work based on fMRI data obtained from a dead salmon. These researchers included junior and senior researchers who were highly and less highly cited, affiliated with an average-ranked or well-ranked university, and publishing papers in unranked and prestigious academic journals. However, a major reference error, namely, referencing a poster as if it were an article published in a prestigious journal, was committed by the more prolific and cited researchers. Therefore, scientific excellence does not seem to prevent reference errors. Finally, this case invites us to epistemologically question the practice of research and its dissemination based on an activity that seems innocuous (referencing) and that nevertheless speaks volumes.

  Two journals are affected by this error: Human Brain Mapping and NeuroImage. In other words, neither Human Brain Mapping nor NeuroImage published any journal article written by Bennett and colleagues and entitled "Neural correlates of interspecies perspective taking in the post-mortem Atlantic Salmon". The coauthors of the OHBM poster do not include any supposed Human Brain Mapping or NeuroImage journal article about the dead salmon neuroimaging work in their CV publication lists. Those two references refer to journal articles that do not exist. In other words, citing authors committed a "ghost paper citation error."

"

  Bennett C. M., M. B. Miller, G. L. Wolford, ʺNeural correlates of interspecies perspective taking in the post-mortem Atlantic Salmon: an argument for multiple comparisons correction.ʺ NeuroImage 47: S125 (2009). doi:10.1016/S1053-8119(09)71202-9."

Figure 1 .

 1 Figure 1. Reference errors as a function of the first author's bibliometric indicators. (A) First author's number of citations (WoS) by the type of citation error (rs = 0.199, p = 0.011). (B) First author's number of publications (WoS) by the type of citation error (rs = 0.226, p = 0.004). (C) First author's hindex (WoS) by the type of citation error (rs = 0.238, p = 0.002).The following type of citation was analyzed as an ordinal classification: 1) no citation error; 2) usual citation error, that is, the main types of reference inaccuracies described in the literature (see above the 10 types of errors categorized in the present study); and 3) ghost paper citation error, that is, as defined above, referencing an article that does not exist. * p < 0.05, ** p < 0.01.

Figure 2 .

 2 Figure 2. Number of citing documents by publication year (between 2009 and 2017) and by type of citation error (including a cumulative frequency curve).

Note that studies report results that are not strictly comparable because of the use of different criteria for identifying reference errors and calculating error rates.

Note that this journal had a short editing life: it published only this article. This journal probably existed, but it is currently not possible to determine this with certainty. The founders of the journal did not respond to our requests for comment, but the authors (Bennett and colleagues) list this publication in their CVs.

The Ig Nobel Prizes are "the wayward son of the more righteous Nobels"[START_REF] Pilcher | ʺLaughter in the Lab[END_REF]). The ceremony has been organized by the Annals of Improbable Research magazine at Harvard University since 1991: "The Ig Nobel Prizes honor achievements that make people LAUGH, then THINK. The prizes are intended to celebrate the unusual, honor the imaginative -and spur people's interest in science, medicine, and technology" (ʺAbout The Ig® Nobel Prizesʺ s. d.). Each year, 10 laureates are selected from among 9,000 candidates (ʺIg Nobel Nominationsʺ s. d.). This unusual prize is "Increasingly covered by the media"(Gingras and Vécrin 2010, 67), thus greatly popularizing the research of the laureates.

Only 4.12% (53/1287) of journal articles published on reliability and fMRI have at least 100 citations (source: WoS All databases; timespan: 2009-2019).

For more detail about the reference errors considered, see the supplemental material, section "1.3 Identification of reference accuracy inconsistencies".

The median was used because this measure of central tendency is more appropriate than the mean in skewed distributions affected by extreme values.

See complete results in the supplemental material, tableS2.

See the complete results in the supplemental material, tableS10.

See the supplemental material, figure S2.

See the supplemental material, figureS3.

Note that the complete aggregated data are available in the supplemental material, TableS3(cf. the column entitled "Documents with ghost paper citation errors").

See the comparative descriptive statistics in the supplemental material in table S3 and the complete list of publications publishing documents that cite at least one DSfR in tableS9.

"Abduction is the process of forming an explanatory hypothesis. It is the only logical operation which introduces any new idea[...]. Its only justification is that from its suggestion deduction can draw a prediction which can be tested by induction, and that, if we are ever to learn anything or to understand phenomena at all, it must be by abduction that this is to be brought about"(Peirce 1998, CP 5.171).

Funding

This study was conducted without funding.

Data Availability

All relevant data are present in the manuscript and supplemental material. However, restrictions apply to the availability of the bibliometric data, which is used under license from Thomson Reuters. Readers can contact Thomson Reuters at the following URL: http://thomsonreuters.com/en/products-services/scholarly-scientific-research/scholarly-searchand-discovery/web-of-science.html.

Conflict of Interest

The author declares no competing interests.

Notes on contributor

Romy Sauvayre (PhD, HDR) is Maîtresse de conférences (Associate Professor) in sociology at the engineering school Polytech Clermont (Clermont Auvergne INP, Clermont Auvergne University, France). Her current work focuses on various fields such as epistemology, contemporary history of science, ethics of belief, belief change, sociology of diffusion, sociological methods.

ORCID

Romy Sauvayre https://orcid.org/0000-0003-0806-6234

LinkedIn

Romy Sauvayre https://www.linkedin.com/in/romysauvayre/


Gabriel Jarry
email: gabriel.jarry@recherche.enac.fr

Almoctar Hassoumi
email: almoctar.hassoumi@enac.fr

Daniel Delahaye
email: daniel.delahaye@enac.fr

Christophe Hurter
email: christophe.hurter@enac.fr

Interactive trajectory modification and generation with FPCA

Keywords: Geographic/geospatial visualization, Data aggregation, Data cleaning, Data clustering, Data transformation and representation, Data editing, Manipulation and deformation, Multidimensional data, Geometry-based techniques

Moving object analysis is a constantly growing field with numerous concrete applications in terms of traffic understanding, prediction and simulation. While many algorithms and analytic processes exist, there are still areas of investigation with novel trajectory analysis methods. As such, the geometric information analyses data with respect to its statistical distribution along extracted dimensions. This opens new ways of gaining a better understanding of large and complex trajectory data sets while providing flexible data manipulations. In this paper, we report our investigations with the development of an interactive methodology based on the geometric information analytic process where users can analyze trajectories sets, cluster and deform them maintaining the actual statistical properties of the investigated trajectories. As a contribution, this paper shows how Functional Data Analysis can provide novel support for trajectory analyses taking into account the statistical properties of the investigated clusters. We also provide recommendations for efficient usage of the process, considering trajectory registration, initial clustering, trajectory deformation and generation. These recommendations are illustrated with actual examples validated by a domain expert of air traffic flow analysis.

Introduction

Our society has entered a data-driven era, in which not only enormous amounts of data are being generated every day, but also growing expectations are placed on their analysis [START_REF] Thomas | A visual analytics agenda[END_REF]. Trajectory data (i.e. flows of cars, airplanes or people) are collected every day and analyzing these massive and complex data sets is essential to making new discoveries and creating benefits for people. Processing such data is a challenging task due to their intrinsic, time-dependent nature. While machine learning heralds a solution to address the issues of big data and efficient knowledge extraction, alternatives do exist where humans play a central role with the usage of interactive visualization systems [2].

In this regard, this paper investigates a novel analytic method for trajectory processing using information geometry [3]. While general trajectory analysis relies on distance and time algorithms, information geometry uses differential geometry and probability theory [START_REF] Amari | Differential-geometrical methods in statistics, 1st edn[END_REF]. Such analytic tools capture the intrinsic statistical properties of the investigated trajectories. Previous work [START_REF] Hurter | Functional decomposition for bundled simplification of trail sets[END_REF] showed its potential to support visual simplification and visual flow modeling. Geometry information deserves further investigation which goes beyond its usage for visualization purposes.

Considering trajectory input data as a set of ℝ d curves, the standard multivariate statistical representation of a set of curves would be a set of d-dimensional samples [START_REF] Mark | Statistics in spectroscopy[END_REF][START_REF] Prado | Time series: modeling, computation, and inference[END_REF][START_REF] Winkler | Interpreting economic and social data: a foundation of descriptive statistics, ser, mathematics and statistics[END_REF]. However, this representation may not capture all relevant curve characteristics-e.g. its shape or smoothness. Functional data analysis [START_REF] Ramsay | Functional data analysis ser[END_REF] enables a better representation of multivariate data functions like curves. A curve is then modeled as a point in an infinite-dimensional space, usually the L 2 space of square-integrable functions [START_REF] Ramsay | Functional data analysis ser[END_REF]. Geometry information can then be used to obtain a finite representation of the data by means of Functional Principal Component 1 3

Analysis (FPCA). This tool captures the data variability around the mean curve while estimating the Karhunen-loève expansion [START_REF] Ramsay | Functional data analysis ser[END_REF]. In other words, FPCA yields a finite basis describing the main variability modes contained in the data. Learning the distribution of the data on this basis enables two powerful applications: the generation of new samples with the same behavior, and the creation of samples with a user-deformed mean consistent with the collected data.

Traditional methods for trajectory generation are generally based on a series of space-time points generated by aircraft model data such as Based of Aircraft Data (BADA) [START_REF] Nuic | User manual for the base of aircraft data (bada) revision 3.10[END_REF]. Such techniques have shown their efficiency for trajectory simulation [START_REF] Nuic | Bada: An advanced aircraft performance model for present and future ATM systems[END_REF], characterization, and analysis [START_REF] Yanto | Aircraft fuel burn performance study: a data-enhanced modeling approach[END_REF], and have led to the development of fast-time air traffic modeling, and simulation software. However, they are not optimal for group of trajectory analysis since they usually apply a per trajectory analysis. Such limitation relies on the shortcoming of these tools to capture the intrinsic statistical properties of trajectory sets. In the context of noise contour simulation for new approach procedures, it is important to ensure that the trajectory flow statistical distribution is capture by the model to produce reliable results. This motivates the use of Functional Data Analysis with FPCA, which addresses such limitation and enables generation, while keeping a consistent flow distribution.

This paper applies geometry information for analytic purposes and proposes an analytic pipeline to support trajectory processing. This pipeline handles trajectory clustering, data cleaning, flow simplification, flow generation and flow transformation. This methodology was built with the help of air traffic experts to ensure the accuracy of the processed information. This paper's contributions rely on the analytic pipeline and its guidelines to leverage trajectory analysis with geometry information tools. The article is structured as follows: Sect. 1 presents related works on existing trajectory processing algorithms. Section 2 lays the mathematical foundations for trajectory analysis limited to flow understanding and management which is one of the core task for an air traffic controller to insure smooth and safe traffic. The following section gives the basis of the information geometry. Next, we detail the pipeline followed by its use cases. Next, we discuss this paper with an extract recommendation for good usage of the tools. Finally, we conclude the paper with possible work extension.

Related work

There is abundant literature concerning the analysis of moving object trajectories. Even if it is a well-explored topic, it remains a popular area of research where geometry information has barely been used [START_REF] Hurter | Functional decomposition for bundled simplification of trail sets[END_REF]. This paper fills this gap with a thorough usage of geometry information-based algorithms for trail set analysis and deformation. This section presents the main research challenges encountered in the fields of trajectory analysis, trajectory deformation and interactive exploration of trajectories.

Functional modeling

When manipulating objects that have a functional nature or are raised from a functional model, it is advised to preserve this model using appropriate tools. Functional Data Analysis (FDA), [START_REF] Ramsay | Functional data analysis ser[END_REF] is a tool that aims to precisely preserve the functional nature of data by expanding it into an appropriate finite functional basis. The input object is transformed into a coefficient vector, which can then be used in a multivariate framework. This enables the use of traditional multivariate statistics but with the insurance of keeping the functional behavior of the underlying objects. The choice of the basis is important since an appropriate basis choice can better capture some data features such as smoothness. Nevertheless, the main applications of FDA are real-valued functions such as spectrometric data [START_REF] Saeys | Potential applications of functional data analysis in chemometrics[END_REF] or weather data [START_REF] Jacques | Functional data clustering: a survey[END_REF]. Nevertheless, there are few applications on vector-valued functions, such as the 2D or 3D curves considered in [START_REF] Olive | Quantitative assessments of runway excursion precursors using mode s data 06[END_REF], and [START_REF] Jarry | Aircraft atypical approach detection using functional principal component analysis[END_REF][START_REF] Jarry | Aircraft atypical approach detection using functional principal component analysis[END_REF] or trail set brushing [START_REF] Hassoumi | Interactive shape based brushing technique for trail sets[END_REF].

Trajectory clustering, simplification and generation

Trajectory analysis often relies on clustering algorithms.

Clustering can be performed on the geographical space [START_REF] Adrienko | Spatial generalization and aggregation of massive movement data[END_REF] with density maps [START_REF] Scheepens | Interactive visualization of multivariate trajectory data with density maps[END_REF], with pattern similarities [START_REF] Giannotti | Trajectory pattern mining[END_REF] or with time clustering [START_REF] Kalnis | On discovering moving clusters in spatio-temporal data[END_REF]. It is also possible to define distances between trajectories to enable clustering [START_REF] Delahaye | Trajectory mathematical distance applied to airspace major flows extraction[END_REF], or to use dimensional reduction processes [START_REF] Olive | Quantitative assessments of runway excursion precursors using mode s data 06[END_REF]. Using the Functional Principal Component Analysis for trajectory clustering has barely been investigated yet, which makes this study a precursor in the area. New methods such as Generative Adversarial Networks allow to generate trajectories. A recent publication [START_REF] Jarry | On the use of generative adversarial networks for aircraft trajectory generation and atypical approach detection[END_REF] proposes the use of GANs for the generation of aircraft trajectories and the detection of atypical approaches.

Trail-set simplification and manipulation

Introduced by Holten [START_REF] Holten | Hierarchical edge bundles: visualization of adjacency relations in hierarchical data[END_REF], visual simplification helps to remove clutter in dense graphs or trail-set visualization such as road traffic or air traffic visualization. Thanks to this interactive visual aggregation, several goals can be achieved supporting select, navigate, filter and arrange tasks [START_REF] Brehmer | A multi-level typology of abstract visualization tasks[END_REF][START_REF] Lhuillier | State of the art in edge and trail bundling techniques[END_REF]. [START_REF] Lhuillier | State of the art in edge and trail bundling techniques[END_REF] provides a review of existing edge bundling techniques and details the existing algorithms and their usages. A more recent work shows how to use FPCA tools to support visual 1 3 simplification [START_REF] Hurter | Functional decomposition for bundled simplification of trail sets[END_REF] but this work only operates on the visual perspective and no analytic processing is provided.

Other trajectory manipulation can be carried out, some based on the user's interaction [START_REF] Hurter | MoleView: an attribute and structure-based semantic lens for large element-based plots[END_REF], with variant layout deformation such as fisheye and bring & go techniques [START_REF] Tominski | Fisheye tree views and lenses for graph visualization[END_REF][START_REF] Lambert | Living flows: Enhanced exploration of edge-bundled graphs based on GPU-intensive edge rendering[END_REF] and edge plucking [START_REF] Wong | Supporting interactive graph exploration using edge plucking[END_REF]. The fisheye deformation is illustrated in Fig. 1 on air traffic trajectories over France. Such techniques are refined in more recent papers [START_REF] Riche | Exploring the design space of interactive link curvature in network diagrams[END_REF][START_REF] Luo | Ambiguity-free edge-bundling for interactive graph visualization[END_REF]. Transmogrification also interactively transforms graphs or trail-sets with users' input [START_REF] Brosz | Transmogrification: Causal manipulation of visualizations[END_REF], but again such transformation only operated on the visualization side and no further data processing can be performed. The work presented in the following sections is inspired by these techniques to perform trail-set deformation while allowing its analytic processing.

Trajectory exploration tools

Exploring, analyzing and visualizing temporal data such as trajectories has a long history. Time series analysis [START_REF] Cryer | Time series analysis[END_REF] helps the extraction of relevant information. Frameworks [START_REF] Andrienko | A conceptual framework and taxonomy of techniques for analyzing movement[END_REF] are available to gain a better understanding of such complex time-varying data sets thanks to aggregation techniques [START_REF] Andrienko | A general framework for using aggregation in visual exploration of movement data[END_REF]. A recent visualization framework has been provided to structure efficient temporal data representations [START_REF] Bach | A descriptive framework for temporal data visualizations based on generalized space-time cubes[END_REF].

Many interactive tools and systems for trail-set exploration and manipulation exist. Selection boxes help to filter objects of interest [START_REF] Hochheiser | Interactive exploration of time series data[END_REF] [START_REF] Buono | Interactive pattern search in time series[END_REF], particle systems help to understand flow directions [START_REF] Scheepens | Visualization, selection, and analysis of traffic flows[END_REF]. More recently, image based techniques [START_REF] Hurter | Image-based visualization: Interactive multidimensional data exploration[END_REF] have been applied for trajectory analysis [START_REF] Hurter | Interactive image-based information visualization for aircraft trajectory analysis[END_REF]. Boolean operation can be performed to combine selections of trajectories [START_REF] Hurter | Fromdady: Spreading aircraft trajectories across views to support iterative queries[END_REF] on a 2D screen or in virtual reality [START_REF] Hurter | Fiberclay: Sculpting three dimensional trajectories to reveal structural insights[END_REF]. Overall, no previous system used the FPCA tools in a unified framework for trajectory analysis and this paper provides the first of the kind.

Mathematical foundations

This section provides the mathematical foundations to understand the Functional Data Analysis process. Section 2.1 underlines the representation of discrete trajectories in a function space. Section 2.2 explains how principal curves help to represent a set of trajectories. Section 3.1

shows how to generate a new distribution using these principal coefficients. Finally, Sect. 4 illustrates how to modify curves thanks to controlled deformations.

Curve functional modeling

Functional Data Analysis considers curves as objects in an infinite dimensional space. This enables certain curve behaviours such as their shape or smoothness to be taken into account. To retrieve the functional model from discrete data, curves must be reconstructed in a dedicated functional space. It is mandatory that curves have two continuous first derivatives and thus belong to the L 2 space of square-integrable functions. Before applying functional decomposition, curves must belong to this space 2 , so called Sobolev [START_REF] Adams | Sobolev spaces[END_REF]. 2 is composed of functions f that meets the following criteria:

To obtain a functional representation of the discrete curves, the choice of a cubic spline kernel K is made since it has smoothing properties and respects the condition of being in the Sobolev Space. A set of curves P = { i } is then rep- resented by a matrix A where each row a i represents i in terms of spline coefficients. The reproducing kernel theory assumes a decomposition of a function i as a series of a reproducing kernel (in our case a third degree polynomial) taken at a defined number of timestamp t j called centers. The number of spline coefficients a i,j of a i corresponds to the number of center plus two:

Functional principal component analysis modeling

Let C = {𝛾 1 , … , 𝛾 N } ⊂ P be a set of N curves. The Func- tional Principal Component Analysis (FPCA) process consists in modeling C with its mean curve γ and the variance around it. A classic hypothesis is that C comes from an underlying hidden stochastic process Γ ∶ Ω × [0, 1] , where Ω is the probability space of all possible outcomes and [0, 1] the time interval. The empirical covariance estimator Ĥ ena- bles the capturing of the variability of C around its mean using the Karhunen-Loève expansion [START_REF] Stark | Probability, random processes, and estimation theory for engineers[END_REF] with ∈ Ω and t ∈ [0, 1] :

(1)

𝕎 2 ={f ∈ C 1 ([0, 1], ℝ), f � abs. cont., ∫ 1 0 f (x) 2 + f �� (x) 2 dx < +∞} (2) i (t) = a i,0 + a i,1 t + j=n ∑ j=0 a i,j+2 K(t, t j )
Fig. 1 Fisheye deformation applied to air traffic trajectories over France where b j are real-valued random variables called principal component scores. j are the (vector-valued) eigenfunctions of the covariance operator with eigenvalues j . For the discrete implementation of such functional decomposition see [START_REF] Hurter | Functional decomposition for bundled simplification of trail sets[END_REF]. With this model and knowing the mean curve γ and the principal component functions j , a group of curves can be described and reconstructed (Inverse FPCA) with the matrix of the principal component score b j of each curve. Usually, a finite vector (fixed dimension d) of b j scores is selected such that the explained variance is more than a defined percentile.

To sum up, each trajectory can be represented through the FPCA process by the Mean plus the sum of the Principal Component Functions weighted by the Principal Component Score. The Inverse FPCA (IFPCA) process consists in reconstructing the trajectory from the Principal Component Scores knowing the Mean and the Principal Component Functions.

Tools

This section is divided into two parts. Section 3.1 explains the curve generation process, and Sect. 3.2 presents the clustering task.

Curve generation

In [START_REF] Hurter | Functional decomposition for bundled simplification of trail sets[END_REF], Hurter et al. generated curves with a random selection of principal coefficient scores with a centered independent simple Gaussian distribution hypothesis. Usually, coefficients are not simply Gaussian. Consequently, curves generated with this model do not present realistic behavior. Two alternatives are proposed in the following.

Neighborhood generation

To ensure the generated curves are sufficiently realistic, a neighborhood generation was developed in this study. The curve regeneration process needs to take into account an important number of principal coefficients (typically more than 60% of the total principal components). The process is as follows: First, each principal coefficient of the dimension variance is computed using a Gaussian centered model. Then, a curve is randomly selected and its principal component scores are kept. Finally, a new score is randomly generated in the neighborhood of the selected sample. The range of the neighborhood is defined with the variance among each dimension. In addition, the user is able to tune a coefficient between 0 and 1 that is multiplied to the range of the

(3) Γ( , t) = + +∞ ∑ j=1 b j ( ) j (t)
neighborhood. This coefficient enables the user to modify the similarity between the original and the generated trajectory. The following Algorithm 1 illustrates an implementation of the neighborhood algorithm.

Multivariate Gaussian mixture model generation

An alternative to the neighborhood generation model consists in applying a multivariate Gaussian Mixture model, i.e. an Expectation-Maximization (EM) algorithm [START_REF] Mclachlan | The EM algorithm and extensions[END_REF], on the principal component scores that concentrate more than a user-defined percentage of the explained variance. This process does not assume the independence of the principal component scores and enables a richer representation with a Gaussian Mixture instead of a simple Gaussian Distribution. With this generation, it is usually more difficult or even impossible to properly estimate the distribution for a large number of components. This is the well known problem referred to as the curse of dimensionality [START_REF] Hinneburg | Optimal grid-clustering: Towards breaking the curse of dimensionality in high-dimensional clustering[END_REF]. In high dimensional space, the volume of space increases rapidly and samples are usually isolated. The choice was made in this study to estimate only the distribution of the first components that explain most of the variance with the dependence hypothesis. The last components, which mostly correspond to the noise, are then assumed to be independent.

Clustering

Clustering for regeneration

Clustering is a very important initial step before applying the FPCA process. To be efficient, FPCA must operate on clusters with representative mean curves. A two step clustering process was derived for this study. A cutting down clustering, which aims at reducing large data-sets, is applied in a first step. For example, one can use a k-means clustering (or other simple literature algorithms) on arrival or departure trajectory locations. For the study of aircraft landing trajectories, the initial clustering is here done on the destination runways. The second step is to apply a refinement clustering based on the FPCA decomposition score. Displaying first coefficient dimension, the user is able to apply another clustering algorithm to group together similar trajectories.

The choice of the Expectation-Maximization algorithm [START_REF] Mclachlan | The EM algorithm and extensions[END_REF] is made here but other algorithms such as k-mean [START_REF] Bickel | Springer series in statistics[END_REF], or Hierarchial Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) [START_REF] Campello | Density-based clustering based on hierarchical density estimates[END_REF] are also applicable. The choice of the algorithm and/or the number of clusters should be guided by the visualization of the FPCA score and by expert knowledge of the investigated data-set. In addition, the user is able to select the number of dimensions of the principal component score to use for the clustering and visualize the clustering result on the trajectory to decide which clustering method produces the most representative clusters (i.e. mean curve dissimilarity).

Clustering for Classification

The distribution of the principal component score can be used to cluster data. Indeed, the finite dimension representation enables the computation of distance. Besides, the euclidean norm of the principal component score is equal to the L2-norm in the Sobolev Space [START_REF] Ramsay | Functional data analysis ser[END_REF]. In a situation where the behavior of the group of trajectories to classify is known, this knowledge can be used to define a classification process using unsupervised learning techniques. First, trajectories are decomposed using the FPCA process. Then, the HDBSCAN [START_REF] Campello | Density-based clustering based on hierarchical density estimates[END_REF] clustering algorithm is applied to all the trajectories principal component scores. Since the FPCA process clusters together similar data, it means that similar trajectories will be grouped together. HDBSCAN is really highly efficient in determining density-based clusters with irregular shapes, i.e. clusters that are generated from the same distribution with no assumptions on the type of distribution. In addition, the HDBSCAN algorithm gives the probability of being in a cluster. Knowing the behavior of the group to be detected, it is possible to identify to which cluster it corresponds. Finally, the user defines a probability value above which the trajectory is attributed to a cluster. This enables the user to choose the characteristics of their classification algorithm in terms of accuracy or specificity.

Data cleaning

To improve generation efficiency, the user must remove outliers that are not considered as representative members of a class of trajectory. This data cleaning process is also an important step for clustering refinement. Since curves are already clustered, it is possible to define the probability that a sample is in a cluster. This probability is usually defined using the distance between a sample and a representative sample of the cluster. Therefore, outliers can be considered as samples whose probability of being in a cluster is lower than a threshold value. The user is able to select this likelihood threshold and therefore select outliers they wish to remove. It is also possible to define an interaction to perform outlier cleaning by visualizing the principal coefficient scores and selecting the samples to remove. This interaction is similar to a brushing interaction but is carried out in the principal component score space.

In Fig. 2, the process of outlier detection is illustrated. Each ellipse represents a likelihood value level, which is user defined.

Visual simplification (Edge bundling)

The edge bundling is performed thanks to the Inverse FPCA process. This reconstruction method uses all principal component functions. However, one can choose to reconstruct the trajectories with only a certain number of principal components. In [START_REF] Hurter | Functional decomposition for bundled simplification of trail sets[END_REF], Hurter et al. suggested removing the component by a percentage of the explained variance.

Usually, 99% of the variance is explained by a small number of components, less than 10% of the total number of principal components. A bundling coefficient between 0 and 1 was defined as representing the percentage of explained variance kept: A value of 0 means that only the mean curve is kept, while a value of 1 means that the original trajectory is fully reconstructed. For computational purposes, the transition between 0.99 and 1 is achieved by adding 90% of the principal components. In terms of total data set variance this is negligible. However, individually this transition may induce very large modifications to the behavior of certain trajectories.

To avoid this type of visual artifacts, the current study favors a piece-wise linear mapping for the bundling coefficient. For example, from 0 to 0.90, add the number of components equivalent to 0 to 99% of the variance. For the last 0.9 to 1 add the remaining components linearly. Figure 3 shows the bundling result with a set of landing trajectory and increasing explained variance (0%, 25%, 50%, 75%, 100%).

Curve shape modification

While simple trajectory deformations can be performed with Cartesian dimensions, it become more complex with additional data dimensions such as altitude. Furthermore, deformation becomes cumbersome when it has to be applied to many trajectories. FPCA can help solely with the deformation of the cluster mean curve and its principal components to modify every trajectory of the investigated cluster. Hurter et al. [START_REF] Hurter | Functional decomposition for bundled simplification of trail sets[END_REF] only modified the mean curve to perform trajectory modifications, which leads to many visual artifacts. Indeed, the mean curve modification is not sufficient, the principal components also have to be modified to correctly model the temporal behavior which was embedded in the undistorted original FPCA model. Modifying the trajectory behavior implies being sure that the principal components, and therefore their underlying variation on the mean curve behaviour, are applied at the right timestamp. Modifying the mean curve without insuring that the role of the principal components was not modified, resulted in most cases, with aberrant curve behaviors.

In Fig. 4 good and poor usage of curve modifications are illustrated. This shows aircraft vertical profiles (altitude function of the distance) modifications, where the landing procedure was increased in altitude (1000ft higher for noise sustainability issues). This use case will be further detailed in Sect. 6.2.1. The bottom Fig. 4 shows the result of the solely mean curve modification. In this case, the principal components are no longer aligned with the mean curve and result in an unrealistic trajectory with artifacts around the level-off flight (red circle in Fig. 4). The top Fig. 4 The mean curve modification and its principal component modifications are not an easy task. First, a curve registration is needed to align curve landmarks. In our case, trajectories are aligned by distance (along the curve) with respect to the runway threshold considering that the time stamp 0 is the last point over the runway threshold. Then, the key idea is to apply the same temporal modifications to both the mean and the principal components functions. This enables a consistent modification which produces reliable results without artefacts.

Curves translation: The translation operator of a curve is defined as v the translation vector for any t ∈ [0, 1] as : This is the sole operator that can be applied only to the mean curve since it does not affect landmark time position.

Curves 2D rotation: The 2D rotation of a curve = ( x , y ) at time t 1 with angle for any t ∈ [0, 1] as is defined as:

Temporal compression and dilatation: Temporal compression/dilatation operator of a curve between t 1 and t 2 with the compression coefficient ∈ ℝ + with t CD = t 2 -(1-)⋅t 1 , the temporal compression/dilatation is defined for any t ∈ [0, 1] as :

(4)

Translation (v)(t) = (t) + v (5) Rot 𝛾 (t 1 , 𝜃) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝛾(t), if t < t 1 𝛾(t 1 ) + � cos(𝜃) -sin(𝜃) sin(𝜃) cos(𝜃) � (𝛾(t) -𝛾(t 1 )) else (6)
TCD 𝛾 (t 1 , t 2 , 𝛼)(t) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝛾(t), if t < t 1 𝛾(𝛼 ⋅ t + (1 -𝛼) ⋅ t 1 ), if t 1 ≤ t < t CD 𝛾(t -(t CD -t 2 )), if t ≥ t CD

Curve temporal cut or extension:

The cutting or extending operator of a curve at t 1 with width t ∈ [-t 1 , 1t 1 ] , for any t ∈ [0, 1t ] is defined as :

In addition, this operator modifies the definition interval of the curve. A good use consists in applying the TCD (Eqn. 6) operator between 0 and 1t with compression coefficient = 1t . Smoothness: The three last operators (Eqn. 7, Eqn. 6, Eqn. 5, only insure the continuity but do not ensure the smoothness of the obtained curve. Nevertheless, it may be restored for an operational or visual purpose using an additional filtering algorithm (Laplacian filtering or other).

Pipeline

The Fig. 5 shows the pipeline with a trajectory data-set as input data. The first step performs an initial clustering to reduce the data-set size into clusters with similar trajectories. This initial clustering is data-set dependent. For instance, with aircraft trajectories, it can be performed on departure or arrival airport. The FPCA process can then be applied to each cluster to compute the mean curve, the principal component functions and the principal component scores. ( 7) Then, a clustering refinement step can be computed based on the principal component scores as previously explained. User input is needed at this clustering step: with the suggested EM algorithm, the user has to define the number of clusters and the number of principal components to use.

CE 𝛾 (t 1 , 𝛿 t )(t) = 𝛾(t), if t < t 1 𝛾(t + 𝛿 t ) + 𝛾(t 1 ) -𝛾(t 1 + 𝛿 t ), elif t < 1 -𝛿 t
The following step shows different possible trajectory processing. The Inverse FPCA produces size varying trajectory with respect to their shape and statistical properties. Two different types of trajectory generation are available 3.1

Use cases

This section is divided into two parts. Each part illustrates a specific feature of the pipeline (Fig. 5). The first part illustrates the clustering process, the second one the trajectory modification operator through concrete examples.

Clustering and classification

Clustering refinement

In this section, the clustering method based on the FPCA is illustrated. To show the flexibility of our method, it is applied to a different data-set. We use the trace of 150352 brain fibers extracted from high fractional anisotropy areas in a 128 × 128 × 51 DTI volume (dataset from Everts et al. [52]). The initial clustering with a 5-means on the departure arrival points was applied. Then a refinement clustering of the FPCA decomposition of each cluster using HDBSCAN algorithm was applied. On Fig. 6 we represented on the left image the initial clustering, then we selected one cluster to which a second FCPA process we applied. Finally, on the right, 4 clusters were extracted. Clustering refinement is an important task before further FPCA usage to insure an optimal mean curve and principal component extraction.

Landing sequence classification

In this section we will apply the unsupervised classification process defined in Sect. 3.2.2 to the identification of landing procedure at Bordeaux Merignac airport (one of the major airports in France). When landing on runway 05, aircraft follow four kinds of trajectory (RNAV,Visual-RNAV, VOR+DME arc, other radar vectoring). The RNAV approaches, are GNSS paths. They are very characteristic since they follow a path from defined way-points (geographical points on a map). It also means that this type of approach will be very similar in the FPCA space. We recorded 2597 suitable landing sequences (one record every 4 seconds, 995963 points in total) during a three month period in 2018 (summer time). We fixed the probability threshold of each cluster in order to minimize the false positive samples. Indeed, we know the behaviour of these approach trajectories, so we fixed the threshold such that all the detected trajectories correspond to this behavior.

Figure 7 shows the result of the clustering algorithm where classes of landing sequence are clearly separated within four clusters. Figure 7 bottom shows the distribution of the first four principal components with the identified cluster. Figure 8 illustrates the influence of each principal components on the mean curve. As an explanation, we observe that influence of PC1 in the 2D trajectories path is mainly the position of the base leg. Therefore, it explains the spatial disparity between blue vs green and purple PC1 values in Fig. 7.

Curve modification and generation

Interception altitude modification

After The Grenelle de l'environnement 2018 annual meeting to discuss sustainability issues, landing procedures at Charles de Gaulle airports were raised by 1000ft (around Fig. 6 This figure illustrates the clustering process applied to a brain fibers data-set. On the left side are illustrated the fibers after an initial clustering using 5-means clustering algorithm on the origin-departure pair of points. From this clustering, the FPCA process is applied to each cluster. Then, the HDBSCAN algorithm is applied to the principal component score in order to extract subclusters. On the right side are illustrated four of the extracted clusters and the full two step clustering result 1 3 300m) to reduce noise emission. In this regard, we worked in collaboration with the Environmental Office of the French Civil Aviation Authority.

In this section, we report the simulation results where we processed traffic before the rise and modified them with the Grenelle 300 meter rise. We then computed the resulting noise emission and compared it with the actual trajectories after the rise. This comparison provides a good assessment of the accuracy of the trajectory generation and modification pipeline.

The process is the following. First, curves were registered by distance considering the starting point at the runway threshold. Second, the FPCA decomposition and a clustering refinement of the trajectories with the EM algorithm [START_REF] Mclachlan | The EM algorithm and extensions[END_REF] were applied.

Then, for each cluster, curves were modified to follow the 300 meter pull-up. Two timestamps are inferred: t 1 , the timestamp of the mean curve at 1000ft (300m), t 2 the times- tamp of the mean curve at 2000ft (600m); and the difference t = t 1t 2 is computed. Finally, the curve extension opera- tor CE (t 2 , t ) is applied.

The generated trajectories are then studied through Eurocontrol IMPACT software that uses Base of Aircraft Data (BADA) models [START_REF] Nuic | User manual for the base of aircraft data (bada) revision 3.10[END_REF] and standard profiles, and Aircraft Noise and Performance (ANP) data base, which includes Noise Power Distance (NPD) tables to computed noise maps from estimated thrust. This is a compliant procedure with International Civil Aviation Organization (ICAO) DOC 29 Vol. 2 [START_REF] Icao | Ecac.ceac doc 29[END_REF] regarding noise contour assessment for ANSPs. Figure 9 illustrates, on top, the noise level for the real traffic (after the altitude rise), and at the bottom, the noise generated from the pipeline with the modified traffic. The noise indicator is the NA62 indicator which is computed over one day of traffic. This indicator is mainly used by the environmental office. It corresponds to the number of aircraft emitting noise above 62dB during the period. The area for 5 to 20 events above this threshold is represented here.

The result shows that this noise computation is close to the actual recorded noise with main identical parts even if a few differences in terms of areas can be observed. The simulation covers 92% of the area because the real noise map is slightly more extended on the left side. This is due to the fact that in the real context, approaches tend to have a longer level-off flight before starting the final descent. Nevertheless, this shows that the pipeline can produce valuable simulated trajectories and can be used for realistic flow simulation.

New departure flow investigation

Before building a new aircraft departure flow, it is valuable that it is simulated and its noise impact assessed. This study's pipeline assists in this matter. To test such a possibility, we considered a novel departure flow at Nantes-Atlantic Airport (one of the major airports in France) using an existing flow at Bordeaux-Merignac Airport as a reference model. Such flow duplication is not straightforward since the original departure flow (i.e. Bordeaux) has to be modified to follow mandatory way points at the destination airport (i.e. Nantes).

As a result, Fig. 10 represents at the top the original flow of trajectories at Bordeaux and at the bottom the generated and distorted departure flow with the pipeline after its modification to fit Nantes airport landing procedure. To make this transformation several steps are needed. First, we apply a translation whose vector is the difference between the coordinates of the two runway thresholds. Second, the angle 0 between the two runway headings is computed and a rotation Rot(0, 0 ) is applied to align the final approach paths. Third, knowing the distances d 1 between the runway and the Final Approach Fix EOOO1, d 2 between EOOO1 and way- point FB003, and d 3 between FB003 and waypoint BOBRI, corresponding timestamps ( t 1 , t 2 , t 3 ) from mean curve are inferred. We first consider the angle [THR, EO001, FB003] and compute the angle difference 1 with the corresponding angle on the mean curve after the first rotation [ (0), (t1), (t2)) ]. Then, we apply the rotation Rot(t 1 , 1 ) of angle 1 at timestamp t 1 . Finally, we consider the angle [EO001, FB003, BOBRI] and compute the differences 2 with the corresponding angle on the the mean curve after the second rotation [ (t1), (t2), (t3) ] and apply the rotation Rot(t 2 , 2 ) of angle 2 at timestamp t 2 .

User interactions

In this version of the software, the interactions have been implemented in a simple way. To modify the interception altitude, the user chooses the curve extension mode, he enters the reference altitude (1000ft) and the deviation (+1000ft), the software infers the time delta and applies the curve extension operator. For the modification of the departure flow, the translation and rotation from one airport At the bottom, the modified trajectories at Nantes Airport. In addition, the expected procedure following different aeronautical way points is shown in red to another are automatically computed from the runway coordinates. Then, the user chooses the rotation mode to modify the trajectories. For each of the successive rotations, he enters the coordinates of the four points to rotate: the three points that make the original angle, and the final point after rotation. The software infers the angle and applies the rotation to the intermediate point.

Potential improvements can be made to enhance the process by making it even more interactive. We can imagine a drag-and-drop interaction to directly modify the trajectories and as an aid to the user, the software could automatically give information about the modification of the rotation angle in degrees or the modification of the altitude in feet.

Discussion and FPCA good practice

This section discusses the paper outcomes (clustering, trajectory distortion, trajectory generation) and provides a summary of FPCA good practice.

The objective of clustering refinement is to compute cluster with consistent mean curve. As previously explained, the FPCA process is efficient when groups of curves have similar shapes to correctly capture their variability around a mean curve. In addition to the clustering refinement, it is also important to remove outliers which may impact the mean curve and potentially induce weak results in the next FPCA processing steps.

The system, derived from the pipeline, contains a set of specific tools for trajectory modifications. Modifying the mean curve without keeping the principal components aligned will generate artifacts. Users need to visually assess the modification results and fine tune the regenerated curves. Currently, the methodology has predefined modification presets, but in our future work, the user will be given the ability to choose the modifications and interact on the mean and on the principal components to directly see the effect on the regenerated curves.

For the generation process, the user has to select which kind of generation process they wish to apply and define how many curves to generate. The size of the original data-set can be adjusted while keeping a consistent distribution or extending the data-set in number to simulate traffic growth or decrease. The choices in the generation process and in the parameters are guided by the visualizations of the principal component score and the desired proximity in shape to the original trajectories. Besides, the user can also adapt the generation process with the visualization of the reconstructed trajectory and ensure that the generated curves have the shape expected.

As a summary, recommendations for efficient usage of FPCA tools for trajectory analysis are provided here:

• Trajectory registration: This initial step is mandatory to efficiently capture the variance around the mean curve of the considered clusters, • Initial clustering: An initial clustering is mandatory to have a meaningful mean trajectory, • Trajectory deformation: trajectory deformation only operates with the mean curve deformation associated with the principal component function modifications, • Trajectory generation: many possible methods exist to increase or decrease the number of trajectories. We proposed three methods taking into account the global, local and neighbor variance.

Conclusion

In this paper, we propose a new approach to analyze trajectories from a functional decomposition perspective for the underlying data-set. Thus, we developed a functionally based pipeline to support the following trajectory processing: clustering, trajectory deformation and trajectory generation. Thanks to the pipeline, trajectories can be clustered taking into account trajectory curvature and their variability around the mean curve. This provides another clustering tool which mainly considers trajectory shapes as a grouping parameter.

Through concrete examples, an aircraft path deformation and the corresponding noise computation, we show that the pipeline can produce reliable solutions for trajectory simulation. The results seem to be relevant regarding operational metrics. Rather than processing every trajectory to deform it and make it compliant with new air traffic flow constraints, the pipeline enables the deformation of a single mean curve to produce an equivalent result. Furthermore, this pipeline is flexible, since the user can also increase or decrease the number of trajectories while keeping a coherent distribution around the mean curve. While we show quantitative and accurate results with this pipeline, many improvements can be considered. Firstly, FPCA tools need some fine tuning and the underlying parameters require some prior knowledge in statistical tool manipulations. Secondly, the pipeline provides trajectory deformations applied to the mean curve and the principal component of the considered cluster. We currently provide simple transformation like rotation, stretching and bending. Some additional work is needed to make this transformation applicable to any kind of trajectory.

Fig. 2

 2 Fig. 2 Illustration of the data cleaning or outlier removal process. The blue colored dots represent the different likelihood levels. The user is able to select a likelihood level outside of which a sample is considered as an outlier

Fig. 3

 3 Fig.3This figure illustrates the Edge Bundling (i.e. trail visual simplification) process for landing trajectories at Charles de Gaulle Airport. On the left side, the bundling coefficient is chosen so that it explains 100% of the variance, i.e. the original curves are rep-

Fig. 4

 4 Fig.[START_REF] Amari | Differential-geometrical methods in statistics, 1st edn[END_REF] This figure illustrates examples of good and poor usage of curve modifications for aircraft approach trajectory altitude profile. At the top, both the mean and the principal components were modified. At the bottom, only the mean was modified. The behavior observed while modifying only the mean curve presents artifacts. Level-off flight was expected, but the trajectories present descent phases. These behaviors are not nominal and underline that the process was not executed properly

Fig. 5

 5 Fig. 5 Illustration of our pipeline. On the left, the input curve dataset passes through an initial clustering step. After this, trajectory registration based on landmarks is applied. Then, the FPCA process is computed for the first time for each cluster. This FPCA process gives two pieces/elements of information: the principal component scores (top figure) and the mean curve with the principal component functions (bottom figure). The principal component scores are used in two situations: First for clustering refinement and data cleaning, second

Fig. 7 Fig. 8

 78 Fig. 7 This figure illustrates the clustering task of the landing trajectory at Bordeaux airport. On the top image is illustrated the result of the classification of the Bordeaux airport approaches from the 2017 data records. At the bottom, the first four principal component score distributions are represented

Fig. 9

 9 Fig.9At the top, the noise NA62 indicator map of the real aircraft traffic (aircraft noise above 62dB). At the bottom, the same indicator map for the simulated traffic obtained with the pipeline. The modification consists in raising by 1000ft (300m) the level-off flight of landing aircraft before landing. This modification was applied following the Grenelle de l'environnement for noise reduction purposes

Fig. 10

 10 Fig. 10 This figure illustrates the departure procedure deformation. On top, the original trajectories at Bordeaux Airport are represented. At the bottom, the modified trajectories at Nantes Airport. In addition, the expected procedure following different aeronautical way points is shown in red

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