Jean-Paul Rosaye

TRADITION AND MEANING: ONE INTERPRETATION OF RENAISSANCE THEOLOGY IN THE TWENTIETH CENTURY

The present article intends to look at Renaissance theology from the point of view of the historian of ideas. As a matter of fact, we cannot but adopt the perspective of the twentieth century whenever we deal with Renaissance theology: how far is our interpretation objective is a problem partaking not only in the nature of historical events but also in our ideological preferences. I shall therefore try to define first of all the general conditions in which history can be deemed objective before introducing T.S. Eliot's famous interpretation of Renaissance Anglicanism. How he came to be interested in it and how he evaluated it is not extremely revealing as far as Christian thought is concerned, it also expresses some tendencies in the history of ideas, literary, philosophical, and theological, in the twentieth century.

History is not a scientific sermon forsaking human experience and substituting facts or events to it: it integrates the most active and sometimes even the most prolix aspect of human experience, the act of interpretation. Whenever we deal with objectivity in history, we have to remove from our thoughts and practice the romantic idea that there is a truth hidden somewhere and that a thorough research, setting up a strategy of unveiling the mystery and using the most exact sciences, can come to a discovery of truth. The paradox lies in the fact that objectivity is rather reached when subjectivity is taken into account, and when it is pursued for its own sake. As the meaning of historical facts is forever changing, historical objectivity comprehends, in theory at least, the totality of the rectifications, or the reinterpretations, of the past: it proscribes one simple and unique interpretation and it includes human experience in the whole process. As history is represented in a series of narratives, of interpretations articulating a meaning, every event or period of history is dependant on a plot mediating the materialisation of a strictly human experience of time. And since meaning is implied in a narrative, the various metamorphoses of meaning are displayed in the successive re-interpretations of history between the moment of the occurrence up to present day. History can therefore be conceived as dependant on the metamorphoses of this meaning through traditions and revolutions. Our subject matter being here the English Reformation, or on a broader scale, Renaissance theology, how can the historian, and more specifically the historian of ideas, deal with it? The status of the events (the separation from Rome, the advent of Protestantism and the situation during the Renaissance in the British Isles) is sometimes put into question as being not fundamental by some thinkers (John Henry Newman for example, given the religious controversies during the Victorian period). Yet, the status of the interpretation of what happened at that time proved historically successful as Britain gradually emerged and owed her specificity thanks to the Anglican Schism. Therefore, whenever we try to constitute an interpretation which would, in our eyes, be totally founded, we have to remind ourselves that whatever a Catholic or a Protestant interpretation would agree on is of very little importance and also that a community of synchronic interpretations on a event or a period would not do because they would tend to coincide, eventually; a diachronic display of interpretations is largely to be preferred. Moreover, to have a whole set of different interpretations of the same period or event would perhaps not be worth dealing with, if ever it were desired: we have to take care of interpretations which retain a certain value, and which throw light on the substantial importance of past events at certain periods of time. As the Christian religion is fundamentaly based on teleological presuppositions, time has a great role to play in discriminating and in focusing on some elements of the doctrine more than others. As the main object of study in the present article is the theology of the via media of Renaissance Anglicans, I have therefore chosen to focus on one interpretation given in the twentieth century and emanating from a distinguished man of letters, to assess the evolution of the translation / interpretation of the Word into words through history. As Newman suggested, the Christian religion is apt to assert its attractive power through stages of psychological development in the mind of the believer and stages of historical development in the world. The idea of development implies that time favours and also selects the elements of doctrine which are the most to the purpose and the only ones to be taken into account for a true appraisal and assessment of sound theology. If the theologians of the Elizabethan and Caroline via media tried to define a perspective to advocate the Real Presence and to maintain the notion of a fundamental mystery alluding to the meaning of mankind and the quest that has to be pursued, the sanction of history and the innumerable interpretations that have reinstituted the debate up to the present day are necessarily and inexorably led towards a purer and purer understanding of their basic tenets. The diffusion of the doctrine set up by Elizabethan and Caroline Anglicans through the centuries developed ideas which are still upheld, venerated, and sometimes even adapted to our modern world: these are therefore to be considered the most central to the truth they tried to establish. To measure how the teaching of these theologians has developed allows us not only to highlight some of their initial ideas but also to show where their originality is really conspicuous. Every twentieth century perspective on the essentials of Renaissance Anglican theology is sure to capture a fundamental dimension of its thought in its development. My own way of looking at the event of Anglicanism today shall follow Thomas Stearns Eliot's special view on it, not only because he was awarded the Nobel Prize and the Order of Merit but also because he incarnated the tendencies of his age: that is the first half of our century. Eliot's perspective is not only interesting for reasons attached to the quality of his views (as a poet he was particularly receptive to the floating ideas or the certitudes of our modern world), it is also attached to conditions of thought which were bound to develop in a direction akin to principles held in the Anglican view of the world.

The true meaning of Christianity for him lay in the Incarnation, which was a historical fact and a principle for him, to be found nowhere better explained and interpreted in the English language but during the Elizabethan period. What is paradoxical is that T.S. Eliot advocated a return to the values of the Renaissance and made of the Elizabethan via media the model theology to be copied and followed today if we wanted to rejuvenate the world, after having ascribed the decadence of our modern age (see The Waste Land ), to a dissociation of thought starting precisely in the Renaissance period.

His evaluation of the Anglican theology of the via media proves enlightening in so far as he situated it in time and out of time, as a moment and a process operating the transition from the Middle Ages to the Modern Age and reassuming the true meaning of the Christian religion despite a gradual and inexorable 'Disintegration of the Intellect' 1 . Eliot's reappraisal and refreshment of Hooker, Andrewes and Bramhall enunciated his views on the fundamental importance of the via media and of the Incarnation for Anglicans: he made use of these theologians to oppose their world of light and mediation to the secularized world of darkness and loneliness. If the Meaning was for him the Incarnation, the only tradition he wanted to proclaim was the Christian Tradition symbolized by the Elizabethan via media. But all of this needs some clarification.

T.S. Eliot was attracted to Anglicanism through Lancelot Andrewes, and his own slant on Renaissance Anglicanism bears the importance of this encounter. In an article on Lancelot Andrewes written shortly before his conversion to Anglo-Catholicism, Eliot exposed the major reasons justifying his interest in Renaissance theology. The cultural atmosphere of the Elizabethan period represented a kind of ideal to him: the Church of England, the via media, were the very spirit of Anglicanism, as they amounted to a sublime intellectual, political and religious achievement. If Eliot went on praising the English Church of the period, comparing it even to the Catholic and Apostolic Church of Dante's time, he also gave the background motives for the picture offered:

A Church is to be judged by its intellectual fruits, by its influence on the sensibility of the most sensitive and on the intellect of the most intelligent, and it must be made real to the eye by monuments of artistic merit 2 .

The most important feature of Anglicanism for him (and here again he showed his perspective as a man of letters) is to be found in the prose of Hooker and Andrewes:

The intellectual achievement and the prose of Hooker and Andrewes came to complete the structure of the English Church as the philosophy of the thirteenth century crowns the Catholic Church (…) If the Church of Elizabeth is worthy of the age of Shakespeare and Jonson, that is because of the works of Hooker and Andrewes 3 . Three things then are combined to make the Church of England the intellectual and aesthetic equivalent of the Roman Catholic Church: the breadth of culture, the prose qualities and the impersonality of thought of her exponents:

Andrewes's emotion is purely contemplative; it is not personal, it is wholly evoked by the object of contemplation, to which it is adequate; his emotions wholly contained in and explained by its objects 4 .

To understand this rather mystifying sentence, one should refer to the Clark Lectures Eliot had given in Cambridge in 1926 (one year before his conversion) which dealt with the question of metaphysical poetry and more largely with the reasons why European culture had known a decadence since the sixteenth century. After the 'dissociation of sensibility' theory he had developed in an article in 1921 [START_REF]The Metaphysical Poets[END_REF] , Eliot wanted to write a series of books going much further into the study of this trend and dealing with what he called then 'the Disintegration of the Intellect'. In metaphysical poetry, Eliot wrote,

(…) the thought is so to speak fused into poetry at a very high temperature…to draw within the orbit of feeling and sense what hed existed only in thought. It creates a unity of feeling (…) 6

This unity of feeling is achieved when thought is in tune with feeling, but this is only possible during periods when sensibility is enlarged 7 , periods which are called 'metaphysical periods', when people are given 'the gift of incarnation' in their writings, when sense is 'elevated for a moment to regions ordinarily attainable only to abstract thought' 8 . Andrewes is one outstanding example of this unity of feeling and Eliot associated him to a tradition of thought he endeavoured to link to the spirit of the Middle Ages, which he thought was the key period of history, when Christianity was united and universality expressed through a common language, Latin. Namely, this tradition goes back to Hugh and Richard of Saint Victor (De Gratia Contemplationis ) who suceeded in writing a prose consisting in 'an intellectual preparation for spiritual contemplation' and not in 'a spiritual haschisch, a drugging of the emotions' 9 , the latter elements being infused in the writings to be found after the Elizabethan period, when this tradition was dissociated. If the trend of this metaphysical thought is to be found in Hugh and Richard of Saint Victor, Thomas Aquinas and Dante, this is a tradition of ontological thought conveying the gift of incarnation, present in Andrewes and reigning supreme during the Elizabethan period.

In fact, T.S. Eliot felt that the shift from ontology to psychology was the most meaningful movement at the beginning of the seventeenth century, and that its practical fulfilment was performed through the Counter-Reformation movement, inspired by Jesuitism, and also through Protestantism. The problem for Eliot was not between Catholicism and Protestantism but between the passage from ontological thought where philosophy was not separated from theology, to psychological thought present both in Protestantism and Jesuitism:

(…) the unique character of this century (the seventeenth century) … [is that] it experienced the acute crisis of the transition from the old to the new Europe (to the Europe, we might say, of 1914) 10 .

The greatest difference between Andrewes and John Donne then lay in this fundamental opposition accounting for the gradual dissociation up to our present age. Andrewes is part of a tradition which goes back to the Greek, through the Church of the Middle Ages, whereas there is something impure in Donne, owing to the influence of Jesuitism on him.

'That Jesuitism is a phenomenon typically of the Renaissance is a fact upon which I insist' 11 . Placing Jesuitism outside the classical tradition borne by Greek and Roman traditions, outside the tradition of meditation and contemplation of the Middle Ages, Eliot opposed the military spirit of the Jesuits fighting heresy to the contemplative minds with a view to showing that the spirit of controversy made everybody a theologian in the seventeenth century, exacerbated the theological controversies and entailed a destruction of religion:

Theology which is bent on political controversy, theology at bay, extinguishes the light of pure ideas, the Greek disinterestedness of mind, which the Middle Ages had revived; but it does not extinguish religious sentiment…Religion and theology, abandonning the pursuit of metaphysical truth, develop in the seventeenth century in the direction of psychology 12 .

The tradition of thought based on a search for divine contemplation thanks to reason, on a quest for beatitude through prose and narrative, having Aristotle, Aquinas, Dante and Andrewes as its proponents is strongly opposed to the tradition of thought based on life in the flux of immediate experience, separated from reason. Moreover, the diversion of enquiry and thought entailed an alteration of the human mind so as to adapt it to the new categories of thought; the ideas were drawn not from the objects but from the ideas of these objects (Descartes's sixth meditation) hence the development of individualism, of self-consciousness and of epistemology.

T.S. Eliot's aesthetic and intellectual version of Elizabethan Anglicanism therefore strips the via media of its religious meaning and turns it into a metaphysical principle. His article on Bishop Bramhall 13 provides an other dimension as well: that of a conception of civilisation and culture which is greatly endangered today. In this article, Eliot went on explaining how the spirit of the Anglicans (and in that respect, Bramhall was presented as very close to Hooker) was different from a trend debilitating the true nature of man. Thomas Hobbes's writings are held responsible for this new tradition, as they are seen as the adumbration of the physical science of atoms, of materialistic determinism, and of totalitarianism. Through the controversy which opposed Bramhall to Hobbes, Eliot wanted to stress the idea that to follow the way of the via media in the twentieth century, presented as a 'period of debility', was becoming an impossible task. To understand Eliot's view, it is necessary to see in what context he started to be interested in Anglicanism. It was mainly because of a certain disillusion with the modern age and its values that he decided to revert to a sphere of thought which would be displaying everything but 'values', that is mere appearances excluding any possibility to find reality in the world. Eliot was looking for a way of life in which thought is not separated from feeling: his attitude is therefore not the result of an epistemological choice even if his explorations towards reality, or an absolute, closely followed the path of the philosopher or the artist 14 . For him, Anglicanism articulated belief with the wisdom accumulated in centuries, associated a tradition to the meaning it vehiculated: a meaning not subject to constant ratiocination but leading to constant contemplation. The end of The Waste Land had asserted such a search for a unifying principle: it rang like an appeal for the restoration of tradition, so as to set the lands in order [START_REF]Shall I at least set my lands in order?[END_REF] . The criticism of decadence had led eliot to look for a renaissance of style and order and to support the view of an ideal to follow which would be close to classicism, that is to say to an authoritarian and superior conception of art and values in general. To his mind, if the modern age was the age of slipshod thought, of the disintegrated intellect, if, in order to renew with this metaphysicality which pertained to the greatest ages of mankind it became necessary to rejuvenate the tradition issued from the Antiquity, it was then essential to come back to the Elizabethan period, when thought and feeling were fused. The Elizabethan Age actualized for Eliot the fabulous and marvellous significance of a combination of feeling, thought, reason and beauty.

In Lancelot Andrewes, Eliot pointed to the idea that to revert to via media Anglicanism would represent, as the prose of Andrewes clearly shows, the possibility to reach the Absolute through words, through discourse: the quest for the Word, which is one feature of his poetics after his conversion, combined with the continuation of the meaning attributed to a tradition already perpetuated by andrewes, and which took the Incarnation to be the key principle. The tradition linking Eliot to Andrewes and to Christianity at large is a common study and a common glorification of the Incarnation, felt as being the central Meaning not only of their own existence, but also of European civilisation. One poem is emblematic of the importance of Andrewes in that respect. 'Journey of the Magi', an Ariel Poem celebrating the birth of Christ, actually started with a verbatim quotation from one of Andrewes's sermons on the Incarnation of Christ, thus reiterating and adapting to modern times Andrewes's most essential preoccupation, centred on the mystery of the Incarnation. Eliot noted in his article, how often Andrewes had spoken on the Incarnation, and most of Eliot's later writings indicate the very clear relationship he drew between the historical Incarnation and its metaphoric possibilities:

What poetry should do in the theatre is a kind of humble shadow analogy of the Incarnation, whereby the human is taken up into the divine (… ) [START_REF]The Aims of Poetic Drama[END_REF] . The creation of a sensuous embodiment [in poetry] is the making the Word Flesh (…) 17 . The Christian thinker finds himself inexorably committed to the dogma of the Incarnation (…) 18 . I spoke at one point of the culture of a people as an incarnation of its religion; and while I am aware of the temerity of employing such an exalted term, I cannot think of any other term which would convey so well the intention to avoid relation on the one hand and identification on the other 19 .

The Incarnation helped Eliot discover what kind of Christianity he could accept. Through Andrewes, who had drawn Eliot to Anglicanism, the Elizabethan period was given the status of an age which had achieved the fusion of thought and feeling, an analogy and therefore a mataphoric incarnation according to him. The Incarnation is, so to speak, the medium through which the Elizabethan via media is given its real meaning.

T.S. Eliot's views of the Incarnation was the essential element inspiring the the dynamics of his work following his conversion to Anglo-Catholicism. Through the notion of Incarnation, Eliot tried to articulate a sense, a Meaning giving impetus to his own personal quest for the Word and to an attempt at showing the supernatural vocation of mankind. In his Anglican play Murder in the Cathedral, the Meaning to which Thomas Beckett refered in his sermon (the pattern of action and suffering 20 ) had the value of a model repeating, as one of the secondary kairoi, the essential Kairos of Christ's passion. But Eliot's kenotic version of the Incarnation 21 also shifted to an existential conception in so far as it gave a significance to everyday life through a necessary Imitatio Christi. There lies perhaps the essential unity which linked Eliot to Andrewes: Eliot did not only appreciate Andrewes's prose but also adopted the doctrinal content of Andrewes's liturgical conception of time, where the past, the present and the future are mysteriously combined, where everyday experience integrates and recapitulates the true sense of the Incarnation [START_REF]See the beginning of Burnt Norton: 'Time present and time past / Are both perhaps present in time future / And time future present in time past. / If all time is eternally present / All time is unredeemable[END_REF] .

It can also be remarked that his idea of a decadence (disintegration, diversion, dissociation, and so on) was just an instance of a general attitude starting towards the middle of the nineteenth century and to which the history of ideas attributes Ernst Cassirer's notion of Kulturkritik. As the word implies, it consists in a critical analysis of Western Culture and qualifies a philosophical and literary genre often giving expression to the Weltschmerz paradigm. This Kulturkritik generally articulated forms illustrating the sense of displacement and opened out on spirituality.

At the crossroads of kenoticism and existentialism and as a representative of the Kulturkritik movement, T.S. Eliot's interpretation of Renaissance Anglicanism is therefore both personal and collective. Personal because in a world in crisis, his life was in crisis, because to become a convert meant for him to live a sacramental life and to be incorporated into the supernatural life, finding out the reasons and the excuse for it in conditions which would favour Anglo-Catholicism. Collective also because his desire for reality in religion had turned him into an Anglican playwright and had made him aware of the fact that his interpretation of Anglicanism could also be favoured by a collective interpretation. Mainly because society as a whole was at stake for him, but also because the necessity for a via media corresponded to his idea of a redemption of the Church of England through a fight against liberalism and secularization. Speaking of 'the vital importance of the reunion of Christendom' [START_REF]Catholicism and International Order[END_REF] , he remarked that the choice was between the formation of a new Christian culture, and the acceptance of a Pagan one, that the Christian world-order would be victorious and that the ideal Christian society could only be realized through the Church of England. The renaissance in life he advocated for himself also implied a renaissance of the falling world in which he lived. Christians versus non-Christians: mankind has to be rejuvenated because it is in a Waste Land, because civilisation has to be rebuilt: there lay the importance of a Renaissance for T.S. Eliot.


Cristhyne León

Guy Carrault

Patrick Pladys

Alain Beuchée

Early Detection

Keywords: Heart rate variability, visibility graph, late onset sepsis, machine learning, premature infants, predictive monitoring

I. INTRODUCTION

Late onset sepsis (LOS) is one of the main causes of morbidity and mortality in neonates [START_REF] Liu | Global, regional, and national causes of child mortality: an updated systematic analysis for 2010 with time trends since 2000[END_REF]. Very preterm infants are especially vulnerable, with 10-25% of them presenting at least one episode of late onset sepsis, which in turn increases the length of hospital stay, morbidity, and mortality [START_REF] Stoll | Late-Onset Sepsis in Very Low Birth Weight Neonates: The Experience of the NICHD Neonatal Research Network[END_REF]. Some studies have found that prompt diagnosis and administration of antibiotics can significantly reduce mortality ( [START_REF] Kumar | Duration of hypotension before initiation of effective antimicrobial therapy is the critical determinant of survival in human septic shock[END_REF], [START_REF] Nguyen | Implementation of a bundle of quality indicators for the early management of severe sepsis and septic shock is associated with decreased mortality[END_REF]). However, indiscriminate administration of antibiotics is also counterproductive, as it can further increase the level of antimicrobial resistance [START_REF] Kuppala | Prolonged Initial Empirical Antibiotic Treatment is Associated with Adverse Outcomes in Premature Infants[END_REF], which is already considered a threat to global public health by the World Health Organization [START_REF] Organization | Antimicrobial resistance[END_REF]. In addition, antibiotics themselves can cause harmful side effects to the patients ( [START_REF] Alverdy | Collapse of the Microbiome, Emergence of the Pathobiome, and the Immunopathology of Sepsis[END_REF], [START_REF] Singh | Side effects of antibiotics during bacterial infection: Mitochondria, the main target in host cell[END_REF]). Therefore, prompt and accurate diagnosis, leading to adequate use of antibiotics is the key to decrease sepsis-related morbidity and mortality, while C. León, G. Carrault, P. Pladys, and A. Beuchée are with Univ Rennes, Inserm, LTSI -UMR 1099, F-35000 Rennes, France also protecting patients from unnecessary antibiotic treatment. However, blood cultures and other laboratory tests used to diagnose sepsis are invasive, take time, and present variations in their predictive value, especially in the early phases of infection [START_REF] Malik | Beyond the Complete Blood Cell Count and C-Reactive Protein[END_REF]. Changes in heart rate variability (HRV) have been associated with neonatal sepsis [START_REF] Fairchild | Heart Rate Characteristics: Physiomarkers for Detection of Late-Onset Neonatal Sepsis[END_REF].

Diagnosis relying on heart rate and HRV have the advantage of being non-invasive and readily and continuously available in the context of neonatal intensive care unit (NICU). HRV analysis typically relies on three different categories of features: time-domain, frequency-domain, and non-linear measurements. Previous studies have shown that machine learning algorithms (MLAs) using HRV, based on these features as input, can be useful in early detection of sepsis both in infants and adults ([11]- [START_REF] Chen | Heart rate variability measures as predictors of in-hospital mortality in ED patients with sepsis[END_REF]). More recently, network-based time series analysis has also been applied to HRV ( [START_REF] Marwan | Recurrence based complex network analysis of cardiovascular variability data to predict pre-eclampsia[END_REF], [START_REF] Sun | Analyzing spatial characters of the ecg signal via complex network method[END_REF]), and visibility graph features have shown to be of interest for the diagnosis of different conditions known to alter HRV characteristics ( [START_REF] Shao | Network analysis of human heartbeat dynamics[END_REF], [START_REF] Li | Detection and prediction of the onset of human ventricular fibrillation: An approach based on complex network theory[END_REF]).

Studies have found that some of the features derived from the visibility graph analysis of the HRV time series have a weak correlation to the traditional features, both in adults [START_REF] Madl | Network analysis of heart beat intervals using horizontal visibility graphs[END_REF] and infants [START_REF] Nguyen Phuc Thu | Improving methodology in heart rate variability analysis for the premature infants: Impact of the time length[END_REF], which suggest that these features might add complementary information to HRV analysis. One previous study used network analysis of heart rate and blood pressure as input features, alongside multiscale entropy features and clinical measurements from the patients' electronic medical record, for a machine learning algorithm (MLA) that successfully predicted sepsis in adults, achieving an AUROC of 80% on the test population; an improvement of 7% over the AUROC obtained by their model trained on only the multiscale entropy features and clinical measurements [START_REF] Shashikumar | Multiscale Network Representation of Physiological Time Series for Early Prediction of Sepsis[END_REF].

In this study we aimed to test the diagnostic value of HRV analysis integrating new visibility graph indexes when used in MLAs, in combination with the traditional HRV features, to discriminate between septic and non-septic infants in a selected population of premature infants.

In the following sections, we describe the database used for the study, and the chain of treatment from acquisition of the ECG to computation of the HRV features. We describe the preprocessing of the data and generation of different variants of the feature set, and the MLAs employed to predict sepsis in our population. Finally, we present an evaluation of the machine learning models used on the different variants of the feature set, and then present the results for two sample cases as examples of the differences in HRV between septic and non-septic infants, and the predictions made by the best performing MLA. In the last section we discuss these results and compare them with other results reported in the literature.

II. MATERIALS AND METHODS

A. Population

The data used in this study is part of the database of the Digi-NewB cohort (NCT02863978, EU GA n • 689260). The cohort prospectively included preterm infants born before 30 weeks of gestation, hospitalized in the NICU of six university hospitals in the western region of France (University Hospitals of Rennes, Angers, Nantes, Brest, Poitiers, and Tours) in 2017-2019. The collection of the data was carried out after approval by the ethics committee (CPP Ouest 6-598) and informed parental consent. All the patients with available data having received more than five days of antibiotics, beginning at least 72 hours after birth, were included in the LOS group. The control group consisted of infants who did not receive antibiotics after the first three days of life. For this study, we used data coming from 24 infants who developed LOS, and 25 control infants.

The clinical characteristics of the preterm infants studied is presented in Table I. The results are presented as either median and interquartile range (IQR) or as the number of cases and corresponding percentage. Comparisons between the two populations were performed using Mann-Whitney U test or Chi-squared.

B. Proposed approach

The general approach we propose is described in Figure 1. In general terms, we acquired the ECG data from both septic and non-septic patients. This data was processed to detect the R peak and thus obtain the RR time series, which is then segmented in periods of 30 minutes. The HRV features are then extracted from each of these periods.

Afterwards there was a labeling process for each neonate, in which the first hours of measurements were used as a calibration period, to which the measurements from the remaining hours are compared. Then the hours before the beginning of antibiotic therapy, in the case of septic infants, were labeled as infected. For the control population, we randomly selected a time between the third and tenth day of life (corresponding to the time of late onset sepsis diagnosis in the LOS group), and the hours before it were labeled as no-sepsis.

Different methods were then used to select the features to be passed as input to four different MLAs. We used the leaveone-out cross-validation (LOOC) method, leaving one infant out in each iteration, so each infant was at some point the test patient, while the rest formed the training set. We used 8-fold cross validation, grouped by patients, in the training set to optimize the hyperparemeters for each MLA.

The following sections will explain in greater detail each of the steps of our proposed approach.

C. Signal Processing

The ECG from the infants were obtained with a sampling rate of 500Hz. The RR intervals were detected using a modified version of the Pan and Tompkins algorithm, with filter coefficients specifically adapted for newborns, as proposed in [START_REF] Doyen | Robust, real-time generic detector based on a multi-feature probabilistic method[END_REF]. Afterwards, a sliding window of 30 minutes, with no overlap, was applied to calculate the RR series and from it all the HRV parameters that will be described in section II-D.

D. Extraction and Analysis of HRV Parameters a) Time-Domain Measurements:

The time domain parameters calculated for this study were the mean of the RR intervals (meanRR), the standard deviation (sdRR), the root mean square of successive RR intervals (RMSSD), the maximum and minimum value for the RR intervals in the time series (maxRR and minRR, respectively) [START_REF] Shaffer | An Overview of Heart Rate Variability Metrics and Norms[END_REF], skewness, kurtosis [START_REF] Bolanos | Comparison of heart rate variability signal features derived from electrocardiography and photoplethysmography in healthy individuals[END_REF], and AC and DC, which characterize the acceleration and deceleration capacity, respectively, of the heart rate [START_REF] Bauer | Deceleration capacity of heart rate as a predictor of mortality after myocardial infarction: cohort study[END_REF].

b) Frequency-Domain Measurements: In the frequency domain we calculated the low frequency power (LF), with limits 0.02-0.2Hz, the high frequency power (HF), with limits 0.2-2Hz. We also calculated these features in normalized units (LFnu and HFnu, respectively), and the LF/HF ratio [START_REF] Shaffer | An Overview of Heart Rate Variability Metrics and Norms[END_REF].

c) Non-linear Measurements: The non-linear parameters include the sample entropy (SampEn) and approximate entropy (ApEn), which estimates the level of regularity and predictability of the signal; the coefficients α1 and α2, obtained from the detrended fluctuation analysis of the time series, and which represent, respectively, the short-range and long-range fractal correlations of the signal; and the parameters SD1 and SD2, derived from the Poincaré plot, and which reflect the short and long term variability, respectively [START_REF] Shaffer | An Overview of Heart Rate Variability Metrics and Norms[END_REF]. d) Visibility Graph Indexes: The visibility graph (VG) is a network-based time series analysis; it converts the series into a graph that inherits several of its properties, by transforming every data point of the time series into a node. For this we used the visibility graph criterion proposed by Lacasa et al. in [START_REF] Lacasa | The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion[END_REF]. The horizontal visibility graph (HVG) is a subset of the VG, which we calculated using the HVG criterion proposed by Luque et al. in [START_REF] Luque | Horizontal visibility graphs: Exact results for random time series[END_REF].

Several indexes were computed to characterize each graph:

• The mean degree (MD_VG and MD_HVG, respectively) of all the nodes in the graph, where the degree of a node is defined as the number of connected edges of the node. The mean degree is a measure of the complexity of the network [START_REF] Da | Characterization of complex networks: A survey of measurements[END_REF]. • The cluster coefficient (C_VG and C_HVG, respectively) is the average of the local cluster coefficient of all the nodes in the graph, where the coefficient of a node is a measure of how much its neighbors are also connected to each other, and is defined as the ratio between all triangles involving that node, and the number of connected triples centered on that node [START_REF] Da | Characterization of complex networks: A survey of measurements[END_REF]. • The transitivity (Tr_VG and Tr_HVG, respectively) is a global version of the cluster coefficient, and is obtained as the ratio between the number of triangles in the graph, and the number of connected triples [START_REF] Da | Characterization of complex networks: A survey of measurements[END_REF]. • The assortativity (r_VG and r_HVG), which is a correlation between the degrees of all nodes on two opposite ends of an edge, with a graph being assortative if the connected nodes have comparable degree (r > 0), and disassortative otherwise (r < 0) [START_REF] Newman | Assortative Mixing in Networks[END_REF]. The details of how the graphs were constructed and how the indexes were calculated are presented in appendix A.

E. Data Analysis and Machine Learning

To prepare the data for analysis and machine learning, we first excluded all the 30-minute segments with a maxRR greater than one second, or a minRR of less than 0.19 seconds. Afterwards, for the infected infants we selected all the remaining segments prior to the time of administration of antibiotics (t 0 ); for the infants in the control group we selected a moment at random between the third and tenth day of recording as t 0 (due to the fact that for our LOS group the median value for infection onset is eight days after birth), and selected all the segments prior to that moment. Then, for each infant we calculated the median value of each parameter over a calibration period; we tried three different lengths of calibration period: (i) the first 24 hours of recording, (ii) the first 48 hours of recording, and (iii) the first 72 hours of recording. For the rest of the 30-minute segments of each infant, we subtracted from each feature the median value of that feature obtained over the calibration period. In this way we obtained the calibrated features (∆features). Thus, we generated three variants of the feature set, one for each different length of the calibration period. Additional variants of the feature set were obtained by changing the time we considered as the learning window (t i ): we considered all cases starting from t i = -72 hours (that is, 72 hours before t 0 ) [START_REF] Mithal | Vital signs analysis algorithm detects inflammatory response in premature infants with late onset sepsis and necrotizing enterocolitis[END_REF], until t i = -6 hours, with six hours increments. For the infected group we labeled the entire duration of the learning window as infected, and for the control group as not infected. Finally, for each variant of the feature set we considered two cases: one including the visibility graph features, and one were they were excluded.

For the statistical analysis and machine learning process we used the LOOC method, leaving one patient out in each iteration. For each variant of the feature set, we used the Mann-Whitney U test to compare the LOS and control population in the training set on each HRV feature, and thus retain only the features that yielded a p-value under 0.1 (MW). Then, we performed principal component analysis (PCA) on the training data with all the features, as well as PCA with only the features with p-value < 0.1 (PCA_MW); in both cases we retained the components for 95% of the variance of the feature set. Thus, we created two different sets of features based on PCA. We created a third set with only the features for which p-value < 0.1, which were standardized (S_MW) before being passed to the MLAs.

We used each of these sets of features to train four different machine learning algorithms: k-nearest neighbors (KNN), logistic regression (LogR), random forest (RandF), and support vector machine (SVM). We used grid search with a per subject 8-fold cross validation split in the training set to find the best hyperparameters specific to each algorithm to maximize recall for each MLA. Finally, each algorithm trained with the best hyperparameters was tested on the patient left out. This process was repeated until every infant in the database had been used as the test subject (the patient left out). The probability curve thus obtained for every patient was then smoothed by calculating for each point in time (equivalent to a 30-minute segment), the probability of infection as the median value between the current predicted probability, and the probability of the two previous segments.

F. Evaluation Method

The evaluation of the performance of each MLA has been done in terms of its area under the receiver operating characteristics curve (AUROC). Our main analysis focused on the performance on the time window comprising the six hours before t 0 . However, we were also interested in evaluating the performance during earlier periods of time to determine how early the infection could be detected by the MLAs tested. For this purpose, we evaluated the AUROC on a sliding window with a duration of six hours, starting at the interval between -6h to t 0 , and ending at -48h to -42h, sliding with a 50% overlap. For both analyses the AUROC was calculated when combining the predictions made for each patient.

To analyse the value added by the visibility graph indexes to the models, we performed a likelihood ratio test ( [START_REF] Harrell | Statistically efficient ways to quantify added predictive value of new measurements[END_REF], [START_REF] Chen | Discriminative information added by wearable sensors for early screening -a case study on diabetic peripheral neuropathy[END_REF]) on the best performing model, to compare its performance when the visibility graph indexes are included in the feature set to when they are not included. With this test, a p-value under 0.05 means that the information added to the model by the new features causes a statistically significant improvement on the model.

For the purpose of analysis and visualization of the predictions given by a particular MLA, trained on a given variant of the feature set, and for a specific patient, we compared the predicted probability of infection to a fixed threshold of 0.5 for all MLAs, although this might not be the optimal threshold for that MLA. We chose this method to simplify the comparison between the results from different MLAs, by comparing them all based on a set threshold. Thus we considered a false negative as a probability lower than 0.5 for an infected infant, and a false positive as a probability greater than 0.5 for a control patient.

III. RESULTS

In this section we first present the behavior of certain measurements of the HRV analysis in the whole population.

Afterwards we report the predictive performance of the MLAs with the different variants of the feature set. For simplicity, we only present the best results obtained. Next, we present the HRV measurements which showed statistically significant differences between LOS and control group, and then we analyze the effect of varying the calibration and learning windows on the predictive performance of the algorithm. Subsequently we consider the effect of including the visibility graph indexes. Finally we present the results for two patients of our population, one from the control group and one from the LOS group, as sample cases and to exemplify how our method could be used for monitoring in the NICU.

A. General Behaviour of Some HRV Parameters

In Figure 2 we present the comparison between the median values of the 24 infected infants (in red) and the 25 control infants (in blue) for some of the calibrated HRV parameters (∆features), for which a difference between both groups was easily observable. The green line represents the time t 0 . The features shown in the figure were calculated using a calibration period of 48 hours.

Figure 2a shows the median value of the ∆minRR for the control and sepsis groups, and it can be observed that the value is generally higher for the LOS group, which is consistent with an expected increase of the occurrence of bradycardia in the infected patients. In Figure 2b we observe that the ∆LFnu is generally lower in the infected population in the days around the t 0 , as compared to the control group; this could reflect alterations in baroreflex induced changes in HRV during sepsis. The ∆SD2 is also generally lower in the infected group, as observed in Figure 2c, which correlates to the decreased HRV associated with sepsis. Similarly, Figure 2d also shows evidence of a decrease of the ∆MD_VG in the infected infants, which also signals decreased HRV in this group.

B. Predictive Performance of the MLAs

We evaluated the predictive performance of all the MLAs using every variant of the feature set, created as explained in section II-E, for a time window comprising the six hours before t 0 . These results are presented in Table II, where we show the variant of the feature set that gave the best result, in terms of greatest AUROC, for each of the four types of MLAs we tested. The left side of the table shows the results when the visibility graph indexes were included in the feature set, while the right side displays the results obtained when these features were excluded. The Features columns specify which feature selection technique was used to construct the feature set (PCA, S_MW, or PCA_MW); the Calib. column indicates how many hours were used for the calibration period; the t i column shows how many hours before t 0 were used for the training window. Finally, in the AUROC column we present the AUROC, and its corresponding 95% confidence interval (CI), achieved by the algorithm using the given variant of the feature set on both the training and testing data.

In Table II we observe that the best performance for detecting whether a test patient is infected or not during the six hours before t 0 was obtained by the LogR algorithm, on the feature set that included the visibility graph indexes, with 87.7% AUROC on the testing data. All the MLAs performed better when the visibility graph indexes were included in the feature set, except the SVM which performed better without the visibility graph indexes. Similarly, all MLAs performed better when the calibration period used was of 48 hours, except the SVM which performed better with a calibration period of 72 hours. This might be due to a more robust calibration when using a 72h period, making it less sensitive to the heart rate variability changes normally associated with the first hours of life in neonates [START_REF] Oliveira | Early postnatal heart rate variability in healthy newborn infants[END_REF]; SVM might benefit from this more than the other MLAs given its sensitivity to any noise or outliers in the training data [START_REF] An | Fuzzy support vector machine based on within-class scatter for classification problems with outliers or noises[END_REF].

Regarding the feature selection, LogR and RandF performed better when trained with the PCA_MW features, while KNN performed better on the set of S_MW features, and SVM had a better performance on the PCA of all the features. Finally, the best training window for KNN was 30 hours before t 0 , for LogR it was 42 hours before t 0 , while for RandF and SVM the best training window was when t i = -6 hours.

We also observe that, as expected, the AUROCs obtained from the predictions for the training data are bigger than those obtained from the predictions for the testing data. For the Fig. 3: Progress of the AUROC for the best performing MLAs configurations evaluated on a sliding time window. KNN, RandF, and SVM the difference between training and testing AUROC ranges from 11.3% to 25.4%. Instead, for the LogR the difference between the training and testing AUROCs is smaller, at 0.7% when visibility graph features are included, and 1.3% when they are not. This suggests that the main reason why the LogR, despite being one of the simplest of the MLAs we tested, outperforms more powerful algorithms such as RandF and SVM, is because it is not over fitting on the training data, while the other algorithms are.

In Figure 3 we show how the AUROC from the testing data changes for each algorithm when evaluated on a sliding window of six hours, with a 3-hour overlap, between t 0 and t 0 = -48 hours. For this, each algorithm was evaluated using its optimal variant of the feature set, as presented in Table II. We observe that all algorithms have an AUROC above 60% for all time windows. Furthermore, LogR, RandF, and SVM have an AUROC above 70% since at least 42 hours before t 0 . In general, their AUROCs begin rising at 24 hours before t 0 and until t 0 , with LogR, RandF, and SVM ending with AUROCs of over 80%. We observe that the AUROC for all algorithms present some oscillations over time, with the LogR being the most stable of the four. This suggests that the oscillations might be due to the overfitting on the training data which, as observed in Table II, was more marked in the other three MLAs, and overfitting can cause small changes in the test data to translate into significant changes in the predictions made by the models.

Given that, when visibility graph indexes were excluded, the LogR model had a very similar performance to the SVM, which was the best performing MLA in this case, with a difference in AUROC of only 1.4%, and similar confidence intervals, compounded with the fact that LogR is a simpler algorithm, faster to train than the others, less prone to overfitting, and that it outperformed all other algorithms when the visibility graph features were included, from this point on we focus our results and analysis on the results obtained with the LogR algorithm, on the feature set variant as presented in Table II

C. Feature Selection

For every variant of the feature set, we also implemented the Mann-Whitney U test to compare the HRV calibrated features (∆features) of the control group to those of the infected group. We applied this to the training data in every iteration of the LOOC procedure. For simplicity, in this section we present the measurements that had p-value < 0.1, when the calibration period was set at 48h, and the and learning window at -42h, as presented in Table II for the best performing MLA, LogR. In Table III we present the HRV features which had statistically significant differences (p-value < 0.1) between LOS and control group in at least 50% of the LOOC iterations for said configuration. The column Occurrence shows the percentage of the LOOC iterations for which the given measurement showed statistically significant differences.

We observe that out of 28 HRV measurements considered, 24 were relevant (p-value < 0.1) in at least 50% of the cases, when using a calibration window of 48h and a learning window of -42h. Particularly, from the visibility graph features, four were relevant in 100% of the iterations: MD_VG, Tr_VG, r_VG, and MD_HVG.

D. Optimization of the Calibration Period and Learning Window

To analyze the effect of varying the calibration window and learning hours, we evaluated the AUROC for each possible combination when evaluated in the period of six hours before t 0 .

In Figure 4 we present the results for the LogR algorithm, using the PCA_MW method for feature selection for the training and testing data. We observe that the best results were mostly obtained when using the 48 hours calibration period, with the minimum AUROC for that case being of 84.8%. Using 72 hours of calibration also gave good results; the best AUROC for this case was obtained when using -42 hours in the learning window, at 86.7%, only 1% below the best AUROC for this algorithm, which was also obtained with t i = -42h, but calibration hours set to 48. Using the calibration period of 24 hours resulted in the lowest AUROCs for the LogR. The lowest AUROC for this MLA was of 76.4%, obtained when

E. Effect of Visibility Graph Indexes

As it was shown in Table II, the MLA with the best predictive performance for the six hours before the infection was the LogR, using a calibration period of 48 hours and the time for the onset of the infection set at 42 hours before the administration of antibiotics, using as input features the principal components of the features with p-value < 0.1 (PCA_MW), and including the visibility graph indexes in the feature set. This setting obtained an AUROC of 87.7%, which is presented as the blue solid line (Visibility) in Figure 5.

When excluding the visibility graph features, the LogR model, using the same calibration period, learning window, and feature selection method as before, obtained an AUROC of 80.9%, which is presented as the green dotted line (No Visibility) in Figure 5.

Thus introducing the visibility graph indexes in the feature set increased the performance of the MLA by 6.8%. Furthermore, we performed a likelihood ratio test to determine if the improvement obtained by the inclusion of the the visibility graph indexes to the feature set is statistically significant, obtaining a p-value of 3.5e-6 in the training set, and 2.9e-4 in the testing set. This indicates that the information added by the visibility graph leads to an statistically significant improvement in the fit of the model.

F. Sample Cases

The two cases presented in this section illustrate the results obtained for two patients: one from the LOS group and one from the control group. The patient from the control group was a very preterm male, born at 27 weeks gestation, with birth weight of 1220g and Apgar score of 2/8, and received 48 hours of antibiotics at birth for an unconfirmed suspicion of early onset sepsis. He was treated with caffeine and nasal continuous positive ventilation with 23% oxygen. He did not develop any infection during his stay in neonatology.

The patient from the LOS group developed LOS with positive blood culture which identified an Enterococcus Faecalis on the 14th day of life. This patient was also very preterm male, born at 25 weeks of gestation, with birth weight of 730g and Apgar score of 5/7. The patient received 48 hours of antibiotics at birth for an unconfirmed suspicion of early onset sepsis. He was treated with caffeine and nasal intermittent positive ventilation with 28% of oxygen, and fed through a venous central line. At the time of clinical suspicion of LOS an isolated increase in cardio-respiratory events was observed without other clinical signs. The results below show that the proposed method would have been able to diagnose the emerging infection at least 12 hours before the clinical suspicion.

In Figure 6a we observe the RR time series from the LOS patient, corresponding to a period of 30 minutes, three hours before the beginning of administration of antibiotics. In Figure 6b we present the RR time series of a 30-minute segment from the control patient. In this figure a difference we observe that the infected patient displays less variability in its heart rate in comparison with the patient from the control group.

In Figure 7 we present the visibility graph obtained from the RR time series presented in Figure 6. To facilitate the visual interpretation of the graph, for both patients we have zoomed into a window of only 300 beats of the time series, which is shown in the left side of the figures; we have also highlighted some interesting beats and their respective nodes in the visibility graph: in green, red, and magenta we highlight local maxima of the RR time series, and in grey local minima. We observe that the beats that are local minima in the time series, in the visibility graph convert into nodes that have very few connections and that are in the outer part of the clusters. On the other hand, the nodes that are local maxima convert into nodes that connect different clusters. In Figure 7a we observe that the low heart rate variability of the infected baby translates into a visibility graph with less connections within clusters. In comparison, in Figure 7b we observe that for the non-septic patient, the connections within each cluster are denser. The HRV measurements of these patients were calculated and calibrated. Thus, we obtained the features that would be used by the MLA, some of which are shown in Figure 8, where we compare the ∆features for the same non-septic (blue solid line) and septic (red dashed line) patients shown in Figure 6, during the 12 hours before (t 0 ). Differences in the HRV of the non-septic and LOS patient can be observed in the four different types of measurements we considered: time-domain (exemplified by minRR in Figure 8a), frequencydomain (LFnu in Figure 8b), non-linear measurements (SD2 in Figure 8c), and visibility graph indexes (MD_VG in Figure 8d). It's important to note that the ∆features shown in Figure 8 are the same as those shown in Figure 2, and that the features for these two sample patients follow the same tendency observed when comparing the entire LOS group to the control group.

Finally, the predictions of the probability of infection over time for both sample patients were calculated using the LogR model, with the configuration which yielded the best results in the six hours before t 0 , and including and excluding the visibility graph indexes, as presented in section III-B. In Figure 9 we present the predicted probability during the 24 hours before t 0 , and highlight in yellow the period corresponding to the six hours before t 0 .

The results for the infected infant are presented in Figure 9a, with the top row showing the predicted probability (in blue) when the visibility graph features were included, and the bottom row presenting the predicted probability (in blue) when these features were excluded from the feature set. The black dotted line represents the threshold probability of 0.5. In the case where the visibility graph indexes are included, observe that while the probability gets very close to the threshold in the period between -24h and -22h, it never actually crosses the line, so there are no false negatives. On the other hand, when these features are excluded there is a false negative in the period between -24h and -22h.

In the case of the patient from the control group, presented in Figure 9b, when visibility graph indexes were excluded there were false positives in predicted probability of infection (bottom figure), which was not the case when these features were included in the feature set (top figure).

IV. DISCUSSION This study proposes a method for estimating the probability of LOS in premature neonates using MLAs, with the aim to aid an earlier and more accurate diagnosis. Our proposed method is based on extracting HRV features from the continuous heart rate (HR) monitoring, and then using those features as input for the MLA. For this we use the traditional HRV features: time-domain, frequency-domain, and non-linear measurements. However, we also propose the inclusion of more novel measurements based on visibility graph indexes.

Previous studies have demonstrated that MLAs can detect sepsis in both adults and infants. Unlike the method we have proposed here, most of these studies rely not only on HR measurements, but may also include respiratory rate, blood pressure, motion, clinical signs, and laboratory tests ( [START_REF] Mithal | Vital signs analysis algorithm detects inflammatory response in premature infants with late onset sepsis and necrotizing enterocolitis[END_REF], [START_REF] Calvert | A computational approach to early sepsis detection[END_REF]- [START_REF] Joshi | Predicting neonatal sepsis using features of heart rate variability, respiratory characteristics, and ecg-derived estimates of infant motion[END_REF]). And among those studies that have focused exclusively on HR or HRV measurements ([11]- [START_REF] Ahmad | Clinical review: A review and analysis of heart rate variability and the diagnosis and prognosis of infection[END_REF]), we did not find any study that included visibility graph indexes or network-based analysis. In fact, we could find only one previous study that used visibility graph indexes of HR and blood pressure to diagnose sepsis in adults using MLA, which found a improvement of 7% in the AUROC when these measurements were included [START_REF] Shashikumar | Multiscale Network Representation of Physiological Time Series for Early Prediction of Sepsis[END_REF]. But to our knowledge, no previous study has used visibility graph indexes for diagnosis of sepsis on premature infants. Thus, one of our main findings was the contribution of visibility graph features to the performance of the MLAs. We found that the AUROC of the best performing MLA improved by 6.8% when the visibility graph indexes were included. The likelihood ratio suggests that the improvement introduced by these features is statistically significant, with p-value < 3e-4.

Another important aspect of our method is that we use the median value of HRV features for each individual patient during a calibration period as a baseline reference for that patient, and it is the difference between this reference value and the value measured for the following segments what is actually passed as input to the MLA. A similar approach has been proposed before, both in adults, where one study reports using the mean value of the HRV metrics over the first 24 hours of recording as reference value [START_REF] Ahmad | Continuous Multi-Parameter Heart Rate Variability Analysis Heralds Onset of Sepsis in Adults[END_REF], and in premature infants, where another study used a calibration period of 72 hours to predict sepsis based on HR, respiratory rate, and clinical signs [START_REF] Mithal | Vital signs analysis algorithm detects inflammatory response in premature infants with late onset sepsis and necrotizing enterocolitis[END_REF], however this study does not specify how the calibration was performed. But this type of method might be specially useful in the case of evaluating HRV in premature infants, for different studies have shown that differences in gestational age can imply significant differences in the HRV of the infant. In our study the best results were obtained when using the first 48 hours of recording as the calibration period.

In regards to the different MLAs evaluated, we found that LogR had a better performance than the others, even though it was also the simplest one used. This might be due to the fact that we did not have a very large population, and precisely due to its simplicity, LogR is less likely to overfit on the training set. This is supported by that fact that all the other MLAs had a significantly bigger AUROC on the training data than on the testing data, while this difference was very small with the LogR. However, with a larger population better results could possibly be obtained using more complex MLAs.

The fact that best performance was achieved when training the MLA using the 42 hours before t 0 , for both infected and control patients, might be due to this yielding a bigger dataset for the training of the MLA. Concerning the preprocessing for feature selection, we obtained the best results when choosing the measurements for which the comparison between LOS and control population had p-value under 0.1, and then performing a PCA on those, finally passing the components that represented 95% of the variance as input for the MLA. This might be explained by the fact that different relevant HRV metrics might reflect the same underlying information, and PCA helps to reduce this information into fewer features.

The method we propose could be deployed in real time in a NICU setting, updating the probability of late onset sepsis every half hour, based exclusively on the heart rate of the patient. Although this method would require a 48h observation period before the first prediction is made, in order to calibrate the system for the individual infant, this is not an impediment given that late onset sepsis is defined as sepsis occurring after the first 72h of life.

V. CONCLUSION AND FUTURE WORK

Based on our findings, we propose a method for LOS diagnosis in premature neonates using MLA based on HRV. Our first recommendation is to include visibility graph indexes, which are a novel method for HRV analysis, alongside the traditional metrics for HRV, to construct the feature set. Likewise, we recommend using a calibration period of 48 hours, proposing the median value over this time as the baseline for each individual infant, and then measuring the variation in regards to this value. For training the MLA we recommend using the period of 42 hours before the beginning antibiotic treatment in the case of the infected population, and continuous periods of equal duration in the control population. For feature selection we recommend performing PCA over the features with p-value under 0.1 when comparing the measurements for sepsis versus non-sepsis population. Finally, for studies with a small population we recommend using logistic regression for making the predictions.

Since this study was performed, more LOS and control infants have been included in the protocol. In the future we would like to carry out a study using some of the new patients to expand our training set, and the others patients exclusively as validation population. Finally, we would also like to evaluate the adaptability of our proposed approach to real-time monitoring in a NICU environment.

APPENDIX A CONSTRUCTION OF THE VISIBILITY GRAPHS AND CALCULATION OF THEIR INDEXES

To construct the visibility graph (VG) every data point of the RR time series is transformed into a node, and the connectivity between nodes is defined with the visibility criterion proposed by Lacasa et al. [START_REF] Lacasa | The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion[END_REF]. By this criterion, two data values of the time series (t a , y a ) and (t z , y z ) have visibility, and therefore are connected, if any other data point (t i , y i ) placed between them, so that t a < t i < t z fulfill the following condition:

y i < y z + (y a -y z ) t z -t i t z -t a
The horizontal visibility graph (HVG) is a subset of the VG, in which the connectivity between nodes is defined by the criterion proposed by Luque et al. [START_REF] Luque | Horizontal visibility graphs: Exact results for random time series[END_REF], by which (t a , y a ) and (t z , y z ) have visibility, and therefore are connected, if: ∀t i ∈ t a , t z : y a > y i and y z > y i

We then calculated four indices from the VG and HVG thus obtained, in order to give a numerical characterization of their properties.

a) Mean Degree (MD): The degree of a node if the number of connections (or edges) it has. The mean degree of the graph is then calculated as:

MD = 1 N N n=1 d n
Where N is the total number of nodes in the graph, and d n is the degree of node n [START_REF] Da | Characterization of complex networks: A survey of measurements[END_REF].

b) Cluster Coefficient (C): This index quantifies how connected the neighbours of a node are. The local cluster coefficient of node y n , c n , is given by: c n = number of triangles connected to y n number of connected triples centered on y n Where a triangle corresponds to three nodes that are connected to each other, and a connected triple is a set of three nodes which can be reached from each other. In other words, a connected triple is equivalent to a path formed by two edges, and in this with node y n as the central node.

Finally, the cluster coefficient C is calculated as the average of all the local cluster coefficients of all the nodes in the graph [START_REF] Da | Characterization of complex networks: A survey of measurements[END_REF] The factor three assures that assures that 0 ≤ C ≤ 1, given the fact that each triangle can be seen as three different connected triples, one with each of the data points as the central nodes [START_REF] Da | Characterization of complex networks: A survey of measurements[END_REF].

d) Assortativity (r): Assortativity is a correlation coefficient between the degrees of the nodes on opposite ends of an edge. It is calculated as:

r = 1 N N n=1 j n k n -[ 1 N N n=1 1 2 (j n + k n )] 2 1 N N n=1 1 2 (j 2 n + k 2 n ) -[ 1 N N n=1 1 2 (j n + k n )] 2
Where j n and k n are the degrees of the nodes at each end of the n th edge, and N is the total number of edges in the graph.

A network is assortative if the connected nodes have comparable degree (r > 0), otherwise the network is disassortative (r < 0) [START_REF] Newman | Assortative Mixing in Networks[END_REF].

ACKNOWLEDGMENT

This study received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 689260 (Digi-NewB project).

Fig. 1 :

 1 Fig. 1: Proposed approach.

Fig. 2 :

 2 Fig. 2: Median value of the ∆features over several days.

Fig. 4 :

 4 Fig. 4: Variations of the AUROC of the LogR as the calibration hours and learning windows change.

Fig. 5 :

 5 Fig. 5: Best predictive performance with and without visibility graph indexes six hours before administration of antibiotics.

  (a) Sepsis patient.(b) Non-sepsis patient.

Fig. 6 :

 6 Fig.6: RR time series for 30-minute segments observed three hours before t 0 .

Fig. 7 :

 7 Fig. 7: RR time series and its corresponding visibility graph

Fig. 8 :

 8 Fig. 8: Calibrated Features for Sample Cases.

  (a) Septic patient. (b) Non-septic patient.

Fig. 9 :

 9 Fig. 9: Predictions six hours before t 0 .

  (Tr): The transitivity index also measures the density triangles in the graph, and is a global version of the cluster coefficient. It is calculated as: Tr = 3 x number of triangles in the graph number of connected triples in the graph

TABLE I :

 I Characteristics of the study population.

	Patients in the LOS group were more premature than in the
	control group. LOS occurred at 8.4 (5.6-10.5) days after birth,
	with 17 cases of cocci gram positive bacteria on blood cultures.
	The LOS group consisted of 17 cases of central line-associated
	bloodstream infection, five cases of central line-associated
	infection without positive blood culture, and two clinically
	suspected infections in patients without central line. The
	bacteria involved were Staphylococcus Haemolyticus (n=5),
	Staphylococcus Epidermidis (n=4), Staphylococcus Warnerii
	(n=3), Staphylococcus Capitis (n=2), Staphylococcus Aureus
	(n=2) and Enterococcus faecalis (n=1).

TABLE II :

 II Best feature set variants for the six hours evaluation window and their respective AUROC. AUROC are presented as median value and 95% confidence interval.

  , both when including and excluding the visibility graph features.

	HRV Feature Occurrence	HRV Feature Occurrence
	meanRR	100%	HFnu	100%
	sdRR	96.4%	LF/HF	100%
	RMSSD	100%	SD1	100%
	maxRR	98.2%	SD2	96.4%
	minRR	100%	SampEn	100%
	Skewness	100%	ApEn	94.6%
	Kurtosis	100%	α1	73.2%
	AC	98.2%	α2	100%
	DC	98.2%	MD_VG	100%
	LF	100%	Tr_VG	100%
	HF	98.2%	r_VG	100%
	LFnu	100%	MD_HVG	100%

TABLE III

 III

	: HRV measurements with statistically significant
	differences (p-value < 0.1) between control and infected pop-
	ulation. The column Occurrence provides the percentage of
	LOOC iterations for which the feature is significant.

ACCEPTED MANUSCRIPT

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JBHI.2020.3021662, IEEE Journal of Biomedical and Health Informatics